FAOFAO FORESTRY PAPERPAPER 22/2

forestforest volume estimation and yield predictionrediction vol. 22-yiel1 -yield predictionprediction

by d. alder commonwealth forestry institute, u.k.u.k.

FOOD ANDAND AGRICULTURE ORGANIZATION OFOF THE UNITED NATIONSNATIONS Rome 19801980 The designations employed and the presentationpresentation of materialmaterial in this publication dodo not imply the expression of of any opinion whatsoever onon the part of thethe FoodFood andand AgricultureAgriculture Organization Organization of thethe UnitedUnited Nations concerningconcerning thethe legal status of any country, territoryterritory,, city or areaarea or of itsits authorities,authorities, oror concerningconcerning thethe delimitationdelimitation of itsits frontiersfrontiers oror boundaries. boundaries.

M-35 ISBN 92-5-100923-692-5-100923-6

The copyright in thisthis bookbook isis vestedvested inin thethe FoodFood andand AgricultureAgriculture Orga-Orga­ nization of the United Nations.Nations. TheThe bookbook maymay notnot be be reproduced,reproduced. inin wholewhole or inin part,part, byby anyany methodmethod oror process, process, withoutwithout writtenwritten permissionpermission fromfrom the copyrightcopyright holder.holder. Applications for such permission,permission, with aa statementstatement of the purpose and extent ofof thethe reproductionreproduction desired,desired, shouldshould bebe addressedaddressed to thethe Director,Director, PublicationsPublications Division,Division, FoodFood andand Agriculture Agriculture Organization Organization of thethe UnitedUnited Nations,Nations, ViaVia delledelle Terme Terme didi Caracalla, Caracalla ,00100 00100 Rome,Rome, Italy.Italy.

«: FADFAO 1980 - iiiiii -

Table of ContentsContent'

Page

1 • INTRODUCTION-INl'RoruCTION - THETHE PROBLEMPROBLEM OF GROWTHCROWTH AND YIELD PREDICTION ------"- -- - 1

1.1 REASONS FOR PREDICTING CROWTHGROWTH AND YIELD 1

1.1.1 Production Planning 1 1.1.2 Silvicultural Research and Planning 1 • 1 .2 Silvicultural Research and Planning 1 1.1.3 1 • 1 • 3 ECologicalEcological Research and EnvironmentalEnvironmental ManagementManagement 1

"1.21.2 'rHETHE METHODOLOGYMEI'HODOLOGY OF GROWTHCROWTH AND YIELD PREDICTION 2

1.2.1 The Estimation of Growth and YieldYield 2 1.2.2 The ConstructionCon, truction of a MathematIcaly",themat i cal ModelModel and ItsIts FittingFitting to GrowthGrowth 3 and Yield DataIBta 1.2.3 Testing of the Model for Validity 3 1.2.4 The Application of the ModelModel to the RequiredRequired End-UseEnd-Use 4

2. DESIGN OF YIELD PREDICTION STUDIESS'l'UDIES 5

2.1 SPECIAL FEATURES OF REGRESSIONREGRESSION PARAMETERPARAMETER ESTIMATIONESTIMATION 5 2.2 SAMPLING DESIGN FORFOR MODELMODEL CONSTRUCTION 5 2.2.12.2.1 Temporary Plots 6 2.2.1.1 Forest inventory 6 2.2.1.2 Growth estestimation imat ion from annual rirings Tl€'s 6 2.2.1.3 Sa,mplingSampling for allometric relationrelationshipss hips 6 2.2.1.4 Sampling toto define parameters of harvestingharvestin~ operationsoperations 6 2.2.1.5 Regeneration surveys 7? 2.2.2 Permanent SampleSample PlotsPlots 7?

2.2.2.1 Number of PSPs required 7? 2.2.2.2 Location of PSPs 8 2.2.2.3 Size of PSPs 8 2.2.2.4 Shape of PSPsPSPs 9 2.2.2.52.2. 2.5 Frequency and timing ofof remeasurementsremeasurements 9 2.2.2.6 Sampling with partial replacement 10

2.3 EXPERIMINTAL~PERIMENTAL DESIGNS 10

2.3.1 Randomized DesignsDesigns 11

2.3.1.1 RandRandr,mizedomized blockblock experiments 11 2.3.1.2 Factorial experiments 12

2.3.2 SystematicSyst emat ic DesignsDes igns 12

2.3.2.1 Single tree experiments 12 2.3.2.2 Clinal plots 13

2.4 EXAMPLES~LES OFOF GROWTHCROWTH AND YIELD EXPERIMENTS=ERIMENrS 14

2.4.1 Uniform Forests 14

2.4.1.1 Spacing experiments 14 2.4.1.2 Constant basal area thinning experiments 15 - iviv- -

2.4.1.3 Thinning experimentseXperiments usingusing gradedgraded thinningthinning treatmentstreatments 15 2.4.1.4 Factorial experiments with different components ofof thinning 16 treatment 2.4.2 Mixed Forests 16 2.4.2.1 Randomized bloc5.block design 17 2.4.2.2 Treatment definition 17 2.4.2.3 Measurements and plot design 17

3. PRCCElXIRESPRCCEDURES FOR DATADATA COLLECTIONCOLLEX:TION ANDAND PRIMARYPRIMARY ANALYSISANALYSIS 19 3.1 SAMPLE PLOT DEMARCATION 19 3.1.1 Location 19 3.1.3.1.2 2 PSP IdentificationIdentification on the Ground 19 3.1.3 Determination of Edge Trees 19 3.1.4 Marking Trees 20 3.1.5 Mapping Trees on the Plot 20 3.1.6 Identity Numbers for Ingrowth 20

3.23.~ SAMPLE PLOT MEASUREMENTMEASUREMMT FORMSFORMS ANDAND PRIMARYPRIMARY ANALYSISANALYSIS 21 3.2.1 Uniform Forests 21 3.2.2 Mixed Forest 24 3.2.3 Initial Assessment of Permanent PlotsPlots 25

3.3 STEI~STEM ANALYSIS PROCElXIREPROCEDURE 26 3.4 SPECIAL MEl'HODSMETHODS OF TREE INCREMENTINCRE)!ENT ESTIMATION 28 3.4.1 Simple Measurements 28 3.4.2 Remeasurement on ButtressedButtressed TreesTrees 28 3.4.3 Girth BandsBands 29 3.4.4 Growth Ring Measurements 30 3.5 INDIRECT TREETREE ANDAND DlINANTDOMINANT HEIGHTHEIGHT ESTIMATIONESTDIATION 31

DATA STORAGE SYSTEMS 33 4.1 ADVANTAGESADVANTAGE2 OF COMPUTER-BASED DATADATA STORAGE SYSTEMS 33 4.2 DATA VALIDATION 34 4.3 CONTRACTS FOR THE PREPARATION OF COMPUTER PROGRAMMES 35 4.4 STORAGE SYSTEMS FOR PLOT DATADATA 36 4.4.1 Introduction 36 4.4.2 File StructuresStructures 36 4.4.3 arrorError Checking and Editing Functions 37 4.4.4 Plot SummariesSummaries 38 4.4.5 otherOther Utilitiesutilities 38 4.4.6 Data Bas-,Bas o SecuritySecurity 38 4.5 DATA TRANSFER BEI'WEENBETWEEN COMPUTER SYSTEMSSYSTEMS 39 - v-

5. ANALYSISANALnIS OFOF GROWTHGROWTH ANDAND YIELDYIELD DATADATA PORFOR UN-LPORMUNIFORM FOREST 40 5.1 SITESITE CLASSIFICATIONCLASSIFICATION 40 5.5.1.1 1 • 1 Use of Dominant Height as an IndicatorIndicator of SiteSite 40 Construction of Site Index Curves 5.1.2 Construction of Site Index Curves 41

5.1.2.1 Graphical methods of construction 41 5.5.1.2.21.2.2 Mathematical methods of fitting site index curves 45 5.1.3 Site Assessment Models Based onon EhvironmentalEnvironmental FactorsFact ors 61

55.1.3.1.1.3.1 FUnctionalFUnctional models forfor si-tesite classclass predi~tionprediction 61 5.5.1.3.21.3.2 ConConstructionstruction and selection of environmental variables 62 55.1.3.1.1. 3. 3 Problems in the anplicationapplication ofof sitesite assessmentassessment functionsfunctions 62 55.7. 2 STATIC METHODS OF PREDICTING YIELD 63 55.2.1.2.1 Graphical ~!ethodsMethods BasedBas ed on ththee Diameter/Height Function 65 5.2.2 Direct statisticalStatistical EStimationThtimation of Mean Diameter Prediction FunctionFunctionss 65 5.2.35.2.1 Defining Treatment HiHistorystory in Terms of StockinEStockinf and Age 66 5.2.4 Static Yield Functions Predicting BasalBa sal Area or Volume 67 55.2.5.2. 5 LimitatioLimitationsns of Static Yield Models 67 5.3 DYNAMIC METHODS OF PREDICTING GROWTH AND YIELD 67 5.35.3.1.1 The Basal Area Increment Function 69 5.3.5.3.1.11.1 Basal arareaea incrementincrement asa s a function ofof dominantdominant height 69 5·3.5.3.1.21.2 otherOther methodmethodss of predicting basal area incrementincrement 70 55.3.1.3. 3.1 .3 Practical pproblemsroblems in analysisanalys i s of incrementincrement datadata 7070

5.3.25. 3.2 ConstructingCo nstructing a GrowthGr01-rth Model~!odel with Spacing Experimentaperiment Data:Data. Marsh'sMarsh ' s 7171 ReResponsesponBe HypothesisHypotheSis 5.3.3 ConversionConversi on of GrowthGrowth ModelsModel s to YieldYield ModelsModels byby IntegrationIntegration 73 55.3.3.1.3.3.1 Introduction 73 5.3.3.25·3.3.2 Ba.sicBasic theorytheory 7373 55.3.3.3.3.3.3 Application ofof an integralint egral yieldyield modelmodel toto differentdifferent thinningthinning 74 treatments 5.3.3.4 Example of use of an integral yield model 75 5.3.3.5Fitting compatible growthgrowth andand yieldyield modelsmodels toto incrementincrement datadata 77n 5.3.4 UseUs e of GrowthGrowth Models by Simulation 78 5.3.4.1 Requirements for a simulationsimulation modelmodel 7878 5.3.4.2 Method of construction ofof aa simulationsimulation modelmodel 7979 5.3.f.35.3 • .' .3 EXampleExample of a simple simulationsimulation modelmodel 79 5.4 THINNING 86 5.4.1 The Thinning Ratio 86 5.4.2 Estimating Thinning YieldsYields inin StaticStatic ModelsModels 8787 5.4.3 Estimating Thinning YieldsYields inin DynamicD,ynamic ModelsModels 88 5.5 MORTALITY 88 5.5.1 Establishment Mortality 88 5.5.2 DensityDependentDensity-Dependent MortalityMortality 89 5.5.3 Disease and Pest MortMortality ali ty 90 - vi --

5.5.4 Windthrow and Fire Damage 91 , 5.6 STAND VOLUME PREDICTION 91 5.6.1 Standstand Volume E4uationsEquations Based onon DominantDominant HeightHeight and BasalBasal AreaArea 91 5.6.2 Using Tree Volume EquationsEquations toto PredictPredict StandStand VolumeVolume 92 5.6.35.603 EstimationEBtimation ofof VolumeVolume toto aa TopTop DiameterDiameter LimitLimit 92 5.6.4 Volumes ofof Thinning 93 5.7 ADVANCED TECHNIQUESTEX:HNIQUES OF GROWTH AND YIELD PREDICTION 93 5.7.1 Size Class ModelsModels 9393 5.7.2 Tree PositionPosition ModelsModels 94

6. ANALYSIS OF GROWTH AND YIELD DATA FOR MIXEDMIXED FOREST 95 6.1 SITE CLASSIFICATION 96 6.2 STATIC YIELD FUNCTION FOR MIXED FOREST 98 6.2.1 General Principles 98 6.2.2 Data Analysis Procedures 99 6.2.3 Methods of Selecting a Yield Equation 99 6.2.4 Problems with\-lith StaticStatic YieldYield ModelsModels inin MixedMixed. ForestForest 100 6.2.5 Conclusions regarding Static YieldYield Modelslr10dels inin MixedMixed. Forest 100

6.3 TRANSITION MATRIX MODELSMODELS 101

6.3.1 Definition of a Transition Matrix ModelModel 101 6.3.2 Methods of Constructing Transition MatricesMatrices 102 6.3.2.1 SingleSingle treetree data 103 6.}.2.26.3.2.2 Size class data 103

6.3.3 Refinements to Transition Models 110505 6.3.4 Disadvantages ofof Transition Models 105 6.4 DISTANC~INDEPEND=DISTANCE-INDEPENDENT TREE MODELS BASED ON DIFFERENCE EZUATIONSEQUATIONS 106 6.4.1 DefiniDefinition tion 106 6.4.2 Allometric and DynamicD,ynamic VariablesVariables 106 6.4.3 Representation of Competition 106 6.4.4 Data Requirements and Approaches to Analysis 107 6.4.5 Basic Model Structure 107 6.4.6 Ingrowth, MortalityMortality andand HarvestingHarvesting 108 6.4.76.4.7 Conclusions regardingregarding Tree Models 108

7.7, VALIDATIONVALIDATION OFOF GROWTH ANDAND YIELDYIELD MODELSMODELS 109 7.1 THE ROLEROLE OFOF VALIDATIONVALIDATION 109 7.2 VALIDATION DATA 109 7.37.3 RESIWALRESIDUAL ERRORS 110 7.4 GRAPHICAL COMPARISONS 111 7.5 DEFININGDEFINING THETHE LIMITSLIMITS OFOF MODELMODEL UTILITYUTILITY 112 - viivii -

8. THETHE APPLICATION OFOF THETHE MODELMODEL TO THETHE REWIREDREJ:l,UffiED ETTA END USEUSE 113 8.8.11 INTRODUCTIONINI'ROOOCTION 113 8.28,2 EVEN-AGED STANDSSTANDS 113 8.3 NIXEDMIXED STANDSSTANDS 114 FOREWORD

There is probably little argument among forest managersmanagers that the abilityability toto estimate the volume of trees and standsstands and to predictpredict whatwhat thethe forestforest willwill produce,produce, onon differentdifferent sites, in response to particular types ofof silviculturalsilvicultural treatment,treatment, isis centralcentral toto allall rational planning processes connectedconnected withwith forestry.forestry. There is,is, however,however, a considerableconsiderable diversity of opinions over what constitutes "yield", andand howhow itit maymay bebe estimatedestimated andand projected intointo thethe future.future.

This manual is an attempt to codify current practicespractices inin thethe fieldfield ofof treetree andand standstand volume estimationestimation and forest yield prediction in a way that is practicable and useful to the person who isis chancedcharged with the resresponsibilityponsibility of producing volume estimations and yield forecasts,forecasts, butbut perhapsperhaps hashas notnot hadhad thethe benefitbenefit ofof extensiveextensive experienceexperience inin thisthis field.field.

It must be aopreciated,a?preciated, however, thatthat thisthis isis aa fieldfield ofof humanhuman endeavourendeavour thatthat isis currently ini n a statestate of rapid evolution, especially withwith regardregard toto forests growinggrowing inin tropical environments.environments. Consequently, all that is saidsaid in this manual mustmust bebe regardedregarded asas provisional and subject to future refinement for particular situasituationstions thatthat cancan arise,arise, oror new techniques that cancan bebe developed,developed, whilstwhilst otherother techniquestechniques maymay existexist whichwhich areare notnot referred to inin this texttext andand whichwhich maymay bebe superiorsuperior forfor particularparticular purposes.purposes.

Thus it is not a manual in the true sense; it is ratherrather a set of guidelines forfor thethe choice of procedure combined with more precise instructions concerning calculation tech­tech- nique for some specifiedspecified cases.cases.

The manual is done with special reference to the tropics and applies to natural as well as manman mademade forests.forests. Because ofof thethe greatgreat difficultiesdifficulties inin assessingassessing growth~rowth andand yieldyield of naturalnatural mixed and uneven agedaeed forests,forests, the methods giveneiven to construct growth models, however,however, mainlymainly applyapply toto eveneven agedaeed forests,forests. For mixed forestsforests nono specificspecific instructions are eivenEiven but rather some examples of possible ways of dealing with the problem.problem ..

The manual consists ofof twotwo volumes.volumes.. The first volume describes techniquestechniques ofof measurinemeasuring trees and the assessment of volume of trees and stands,stande, and the second volume deals with growth and yyieldield prediction. DesDescriptionscriptions of statistical and mathematical techniques, selected statistical tables, blank copies of calculation and data recordinrecordinge forms and an annotated bbibliographyiblioeraphy are includedincluded in a series of appendices.appendices.

Volume I of the manual has been written by Francis Cailliez, Centre Technique Forestier TropicalTropical (GTFT),(CTFT), NoNogentsurMarne,gent-sur-lolarne, France,France, andand VolumeVolume IIII byby DenisDenis Alder,Alder, Commonwealth Forestry Institute (CFI),(cm, Oxford, Great Britain, who also compiled the appendices. The work of thethe two authors has been coordinated by JuranJBran Fries, Swedish University of kTiculturalAgricultural Sciences,Sciences, Uppsala,Uppsala, Sweden.Sweden. The work was formulated and guided by Jean-PaulJeanPaul LanlyLanly andand KarnKarn DeoUeo SinghSingh ofof thethe ForestForest ResourcesResources DivisionDivision ofof FAO.FAD. Jean Clement (CTFT) was associated at the initial stagestage of the study.

The first draft of the manual was presented at the meeting ofof thethe TUFROIUFRO SubjectSubject Group S4,0134.01 (Mensuration,(Mensuration, Growth and Yield) heldheld inin Oxford inin SeptemberSeptember 1979, and was discussed for oneone fullfull dayday inin detail.detail. AmonAmongg the participants there were tropical forest mensurationists especially invited by FAD toto make aa thoroughthorough andand criticalcritical reviewreview ofof thethe cont~ntscontents of the manual. In addition,addition, the manual was also sentsent to a numbernumber of specialistsspecialists for comments. Based on these remarks, a revised versionversion ofof thethe manual waswas preparedprepared byby the authors concerned.ooncerned.

This manual, being the first ofof its kindkind inin thethe fieldfield ofof tropicaltropical forestry,forestry, hashas con-con­ siderable scope for further improvementsimprovements andand additions.additions. Particularly inin the casecase ofof mixedmixed uneven aged stands further complementary studiesstudies areare immediatelyimmediately needed.needed. All suggestions in this respect will be very much appreciated.

M.A.14.A. Flores Rodas Assistant Director-GeneralDirectorGeneral ForForestry eatry Department - 1 1 -

11. • INTRODUCTION THE PROBLE1>lPROBLEM OF GROWTH AND YIELD PREDICTIONPRIIDICTION

11.1 • 1 REASONSREASONn FOR PREDICTIM:PREDICTING GROWTllGROWTH ANDAND YIELDYIELD

In order to understand the reasons for the diversitydiversity ofof methods that areare being usedused for growth andand yieldyield prediction,prediction, itit isis usefuluseful toto examineexamine inin moremore detaildetail whywhy growthgrowth andand yieldyield prediction isis necessary.necessary.

11.1.1 • 1 • 1 ProductiProduction on Planning

Effective forest management involves thethe useuse of treatmenttreatment regimesregimes forfor controlcontrol ofof thethe growing stock in suchsuch aa wayway thatthat thethe increaseincrease inin thethe economiceconomic and/orand/or socialsocial valuevalue ofof thethe growinggruwing stock is more rapidrapid thanthan thethe interestinterest accumulatingaccumulating onon thethe costcost ofof treatment.treatment.

At the same time, all harvesting operations willwill depletedeplete thethe futurefuture growinggrowing stockstock toto a greater oror lesser degree.degree. Too heavy a rate ofof harvesting will ultimately liquidateliquidate the forest resource; too low a rate may both deprive a communitycommunity ofof immediateimmediate resourcesresources andand reduce potentialpotential growth in the forest for the future.future.

Clearly, rational decisions about treatment andand harvestingharvesting intensityintensity andand timingtiming cancan only bebe mademade if the response ofof the forest to these operationsoperations cancan bebe quantified.quantified. Growth and yield studies are thethe meansmeans toto thisthis end.end.

1.1.2 Silvicultural Research and Planning

Although the primary objective of growth and yieldyield studies isis probablyprobably thethe quantifi­quantifi- cation of forest production inin responseresponse toto treatmenttreatment andand harvesting,harvesting, therethere isis aa strongstrong twotwo · way relationship between growth and yield studies and more qualitative forms of silviculturalsilvicultural research. It is a two-way relationshiprelationship because:

(i) The silviculture of a species may determine the type of modelmodel thatthat cancan bebe usedused to predict itsits growth;growth; and may permit aa logicallogical basis forfor groupinggrouping speciesspecies inin complex forests.forests. Important silvicultural features of a species will suggest features and relationshipsrelationships thatthat mustmust bebe includedincluded inin aa quantitativequantitative modelmodel ifif itit is to be effective.effective.

(it) A quantitative model can be used, if it contains suitable relationships, toto testtest silvicultural hypotheses and to suggest experimental~xperimental designsdesigna andand treatmentstreatments thatthat are likely to provide usefulusefUl results.results.

11.1.3. 1.3 EcologicalEcolocal Research and Environmental ManagementMan ement

Quantitative models for growth and yield prediotionprediction may interact with thethe needsneeds of ecological research and environmentalenvironmental planningplanning inin severalseveral ways.ways. For example:exaJilple:

(i) A forest model may indicateindicate thethe amountamount ofof lightlight reachingreaching thethe forestforest floorfloor atat different stages ofof thethe growthgrowth cycle.cycle.

(it) A forest model may readily by adapted to show the biomass and rate of production of the tree crop.crop. - 2 -

(iii) The duration of the growth cyclecycle during which the forestforest cropcrop isis availableavailable toto large herbivores is important information for wildlife management.

On the whole, ecological modelling uses ratherrather different techniquestechniques fromfrom thosethose usedused in growth and yield studies inin forestry.forestry. This is because the latter has, ofof necessity, to focus upon a very precise prediction ofof the geometricgeometric propertiesproperties ofof thethe crop;crop; whilst inin ecology, it is possible to deal withwith populations andand levelslevels ofof anan ecosystemecosystem asas aa whole.whole. Furthermore, ecologicalecological modelsmodels tendtend toto concentra-teconcentrate onon describingdescribing oror explainingexplaining thethe mainmain qualitative featuresfeatures ofof anan ecosystem;ecosystem; a high level of precision is rarely eithereither possible or necessary. On the other hand, forest models must be reasonably precise ifif theythey areare toto justify their existenceexistence andand fulfilfulfil their aim.aim.

The techniques ofof ecologicalecological modelling dodo provideprovide aa numbernumber ofof usefuluseful pointspoints ofof contactcontact with forest modelling:

(i) In mixed forest, species compositioncomposition cancan be modelledmodelled by ecologicalecological populationpopulation dynamics techniques.

(11) Where mortality, defect, oror lossloss ofof growthgrowth isis attributableattributable toto specificspecific diseasesdiseases or pestspests of knownknown etiology, then this may be quantified as population dynamics model in which habitat information isis derivedderived fromfrom aa forestforest growthgrowth modelmodel andand tree growth is influenced by pest population levels.levels.

In thethe future, these points ofof contactcontact arear.e likelylikely toto enlarge.enlarge. In particular, the increasing interestinterest inin non-woodnonwood productsproducts fromfrom forestsforests andand the intractable problems of complexcomplex mixed species and age forests may be best accommodated by using modified forms of ecological energy flow/nutrient cyclecycle models.models.

1.21 .2 THE MMETHODOLOGYE'lHODOLOGY OF GROW'H!GROWTH AND YIELD PREDICTION

The methodology of growth and yield prediction may be thought of as containing four main phases, which are discussed in the following paragraphaparagraphs..

1.2.1 The Estimation of Growth and Yield

The estimation ofof growth oror yieldyield involvesinvolves twotwo kindskinds ofof problem.problem. One is the problem of definition ofof Whatwhat constitutesconstitutes yield.yield. This may be the timber volume ofof thethe crop,crop, oror itit may be the timber volumevolume ofof aa particularparticular groupgroup ofof species,species, oror it it may may be be some some non-timber nontimber product, such as bark, foliage, resins,reSins, etc..etc.. TheThe most commoncommon emphasisemphasis inin tropicaltropical countriescountries is on predicting the volume andand assortmentassortment ofof allall woodwood products,products, includingincluding timber,timber, pulpwood,pulpwood, poles and fuelwood.fuel wood. Because thethe speciesspecies compositioncomposition affectsaffects thethe utilityutility ofof thethe product, yield cannot be considered apart fromfrom speciesspecies compositioncomposition inin mixed stands.stands.

The measurement ofof yield and ofof growthgrowth isis relativelyrelatively easyeasy onceonce appropriate definitions have been made. The main difficulties areare practical onesones associatedassociated with access to the forest, demarcation and measurement of plots and the maintenance of permanent plots over long periods of time. These problems are dealt with in Sections 2 and 3.3. - 3 -

1.2.2 The Constructionconstruction ofof aTdathematicala Mathematical Model and itsits FjttingFitting toto GrowthGrowth andand Yieldyield DataData

Once data is available, a mathematical model cancan be constructedconstructed andand fittedfitted toto thisthis data. A mathematical model cansistsconsista of sets of equations oror graphsgraphs showingshowing thethe quantitativequantitative relationships betweenbetween the variables.

The process of fitting the modelmodel maymay bebe statistical, usingusing forfor exampleexample linearlinear regres­regres- sion, or it may be subjective, by drawing lines through datadata plottedplotted onon grapbs.graphs. Such hand­hand: drawn curves can be subsequentlysubsequently expressedexpressed asas equationsequations ifif required.required. Appendix A of this manual gives a number ofof methods ofof curvecurve fitting.fitting.

The types of curves drawn oror equationsequations fittedfitted may be basedbased uponupon someSOOle naturalnatural lawlaw ofof growth, or it may be empirical.empirical. In the latter case, the function oror equationequation is chosenchosen solely for its ability to represent a particular shape.shape.

At the present time, there is no genuine function forfor treetree growthgrowth basedbased uponupon aa naturalnatural law, although there are some, such as the Chapman-RichardsChapmanRichards functionfunction (described(described inin SectionSection 5) which are partially representativerepresentative ofof growthgrowth processes.processes.

1.2.3 Testing of the Model for Validity

Once a modelmodel has been constructed and fitted toto data,data, itit mustmust bebe testedtested toto determinedetermine its validity andand precision.precision. TbisThis is best done with a second set of data whichwhich waswas notnot usedused to fit any of the functions inin thethe model.model. The modelmodel is usedused to predictpredict thethe behaviourbehaviour ofof thethe stands which produced the testtest datadata andand thethe resultsresults areare comparedcOOlpared withwith thethe actualactual Observations.observations. It is often necessarynecessary toto repeat thisthis processprocess of validationvalidation aa numbernumber ofof times,times, withwith adjustmentsadjustments or corrections to the model as a resultresult ofof apparentapparent anomoliesanOOlolies showingshOwing upup atat eacheach stage.stage.

There are a number ofof reasonsreasons whywhy modelsmodels cancan performperform badlybadly whenwhen validated:validated:

(i) The original data set may representrepresent aa differentdifferent patternpattern ofof growthgrowth behaviourbehaviour toto the test set.

(ii) Inappropriate methodsmethode ofof fittingfitting thethe equationsequations maymay havehave beenbeen usedused inin modelmodel construction.construct ion.

(iii) SOOleSome of the functions may be extrapolated during the test with thethe validationvalidation data into a regionregion Wherewhere theythey areare inaccurate.inaccurate.

(iv) If the model involves a system of equations, it may becOOlebecome unstableunstable whenwhen treatedtreated as a whole, eveneven though eacheach functionfunction separatelyseparately maymay fitfit thethe datadata adequately.adequately.

(v) There may be various kinds of human error during transcription or application of thethe various equations oror graphs.

These points are speltspelt outout at someSOOle length in orderorder to emphasiseemphasise the importance of thoroughly testing anyany modelmodel beforebefore applyingapplying it it toto planning planning or or rese,arch. research. Section 6 deals with modelmodel validation in detail. - 44-

1.2.4 The Application of thethe ModelModel toto thethe RequiredRequired Ehd-UseEhdUse

Etsentiallytl'Bsentially, the growthgrowth andand yieldyield modelmodel maymay bebe appliedapplied inin oneone ofof threethree ways:ways:

(i) As a simple table oror graphgraph oror setset ofof tablestables oror graphs.graphs. TheseThese cancan be used by forest planners directly or can be fed inin inin tabular formfonn to a computercomputer forfor updatingupdating aa setset ofof inventoryinventory data.data.

(ii) AsAs aa programmeprogramme forfor aa computercomputer oror calculatorcalculator whichldrlch cancan produceproduce aa tabletable oror graphgraph of growth and yield forfor aa particularparticular setset ofof treatments.treatments. ThisThis isis appropriateappropriate when the model hashas sufficientsufficient inherentinherent flexibilityflexibility soso thatthat itit isis notnot possiblepossible to define all possible predictionsprediotions inin aneone setset ofof tables.tables.

(iii) As a computer programmeprogramme whichwhich formsfonns ae. sub-,model su1Hnodel withinwithin aa largerlarger computercomputer prog.. prog­ ramme for forest planning and whichldrlch will usually incorporateincorporate aa datadata basebase ofof inventory informationinfonnation and various economiceconanic oror technicaltechnical constraintsconstraints onon harvestingharvesting and traatmenttreatment operations.operati ons. - 5 -

2. DESIGN OF fiELDYIELD PREDICTION STUDIESS'lUDIES

2.1 SPECIAL FEATUlill3FEATURES OFOF RroRE5SIONREGRMSION PARAMETERPARAMEl'ER ESTDIATIONESTIMATION

Sampling or experimental designs for growth andand yieldyield studiesstudies shouldshould bebe conditionedconditioned by the type ofof model to be fittedfitted toto thethe datadata obtained.obtained. This is usually sanesome sort of regression model.

The following points shouldshould be borne inin mind:

(i)(0 When the modelmodel to bebe fitted is knownknown toto bebe linear in fonn,form, thenthen samplingsampling oror experimental treatments should be concentratedconcentrated atat thethe twotwo extremeextreme endsends ofof thethe line. For a surface relating three variables, the four extreme corners should be sampled.

(ii) More~lore usually, the precise shapeshape ofof the functionfunction to be fitted is unknown and likely to be somewhatsomewhat curved.curved. In this case, a good practice is to break thethe range ofof the predictor variable into 5 eaualequal sections and sample each section at the same intensityintensity (subject(subject toto (iii)(iii) below).below).

(iii) Sampling intensity in any part of the rangerangs should be proportional to the variance ofof the samplessamples aroundaround thethe model.model. This isis particularly relevant whenwhen predicting volume, asas discusseddiscussed inin PartPart II ofof thisthis manual.manual.

(kv)(iv) RandanRandom or systematic sampling byby area,area which is mostmoot appropriate for forestforest inventory, is inefficient as a basis for constructing growth and yield models, as it involves too high a sampling intensity in the central part of the range of response and -Lootoo Iowanlow an intensityintensity atat thethe extremes ofof response.

(v) In experimental designs for growth and yield prediction, extreme treatmentstreatments should always be incorporated, especiallyespecially withwith respectrespect toto standstand density.density. This will add greatlygreatly toto thethe accuracyaccuracy ofof thethe modelmodel whichwhich cancan bebe fitted_fitted toto thethe resul­resul- tant data.data.

2.2 SAMPLING DESIGN FOR MODEL CONSTRUCTIONCONSTRUCTION

Sampling is an alternative to experimentation in situations where thethe variablesvariables entering the model cannctcannot be controlled by the research worker. In gromthgrowth and yield studies,studies, this pl"ovisoproviso applies principally to sitesite variation. Forest type cancan be controlled by selec­selec- tion ofof the experimentalexperimental area, oror by establishmentestablishment ofof thethe desireddesired typetype ofof forest;forest; stand density can be controlled by silviculturalsilvicultural andand harvesting operations.operations.

Experiments are generally more efficientefficient and hence lessless expensive.forexpensive .for a givengiven accuracy and precision ofof prediction than sampling.sampling. However, both types ofof datadata are necessary if site variation is to be effectivelyeffectively includedincluded inin thethe model.

The real effects ofof harvesting operationsoperations areare alsoalso veryvery difficultdifficult to simulatesimulate experi-experi­ mentally andand mustmuet usuallyueually bebe determineddetennined byby aa samplingsampling programme,programme, carriedcarried outout shortlyshortly afterafter harvesting. - 66- --

2°2'12.2.1 2.21=ni=112tETemporary plots

Temporary plots areare primarilyprimarily usedused forfor estimationestimation ofof relationshipsrelationships whichwhich areare notnot time dependent. However, this distinctiondistinotion is blurred by the possibility of determiningdetermining tim~ependenttime-dependent relationshipsrelationships from annual ring information in situations where these are present.

2.2.1.1 Forest inventory

Forest inventory designs are primarily determined to give an accurate estimate of forest growinggrowing stockstock inin relationrelation toto landland area.area. However, much of the information gathered may be usefuluseful inin growthgrowth andand yieldyield studies.studies.

The general subject ofof forest inventory design and analysis is covered in the FAO Manual ofof ForestForest Inventory.Inventory.

It is generally inefficient to requirerequire thethe measurementmeasuranent ofof parametersparameters onon allall forestforest inventory plots thatthat areare onlyonly requiredrequired forfor growthgrowth andand yieldyield prediction.prediction. It is better to select a subset ofof plots for more detailed measurement.

22.2.1.2.2.1.2 Growth estimationestimation fromfrom annualannual ringsrings

Where clear annual rings are present, then studies on temporary plotsplots can bebe usedused in place ofof permanent samplesample plots.plots. In generalgeneral,, the use ofof annual ringsrings forfor incrementincrement estima-estima,­ tion is more difficult endand more expensiveexpensive than the use ofof permanentpennanent sample plots.plots. On the other hand, results areare Obtainedobtained muchmuch moremore quickly.quickly.

2.22.2.1.3. 1.3 smallagSa~pling for allcmetricallometric relati,onshipsrelationships

An allometric relationship isis oneone between oneone measurement onon a tree and another.another. For example, the relationship between crown diameter and bole diameter, oror betwebetusenen totaltctal height and bole length.length. Allometric relationships may be importantimportant inin somesome models.models. The necessary data is often not available in a suitable formfonn fromfrem a forestforest inventory, so80 it beCCXi1esbocones desirable to carrycarry outout a samplingsampling programmeprogramme toto determinedetennine thethe relationship.relationship.

The basic sampling unit is usually the singleSingle tree, although for convenienceconvenience plotsplots may be laid outout and allall treestrees onon thethe plotplot measured.measured. The number of samples will depend upon the relationship being studied.studied. A good general procedure is to analyse the data as sampling is beingbeing carried out and to terminatetenninate sampling onceonce thethe requiredrequired degree ofof accuracyaccuracy hashas been obtained.Obtained.

A tree volume tariff isis aa particularparticular exampleexample ofof anan allometricallometric relationship.relationship.

2.2.1.42. 2.1 4Sa_a_2_n Sampling tto oefinimetdefine parameters ers of ofharvesting_afrations harvesting operations

MostM.ost yieldyield predictionprediction modelsmodels acceptaccept asas inputsinputs thethe fonnalformal specificationsspecifications forfor inter­inter- mediate oror cycliccyclic harvestingharvesting operations.operations. It is possible to assume that the operation will be carried out as specified, or it is possible to carry out a sampling programmeprogramme toto examine the relationship between thethe theoreticaltheoretical specificationsspecifications andand thethe actualactual results.results. - 7-

Additionally, most models willwill requirerequire informationinformation aboutabout thethe harvestingharvesting treatmenttreatment that are not coveredcovered by thethe specifications.specifications. For example, it isis usually necessary to know the diameter distributiondistrib.ution ofof removedremoved stemsstems oror thethe relationshiprelationship betweenbetween thethe numbernumber ofof stemsstems removed and the basal areaarea removed.removed.

Sampling carried outout shortlyshortly afterafter harvestingharvesting willwill provideprovide informationinformation anon thesethese matters. Alternatively, concealed semi-permanentsemipermanent plotsplots cancan be set up which are measuredmeasured before and after thethe harvestingharvesting operationoperation toto givegive anan accurateaccurate determinationdetermination ofof treestrees removed.

A general feature ofof these studiesstudies isis thethe needneed toto useuse largerlarger sizedsized plotsplots thanthan otherother types ofof study.study. Typical figures would be:

Uniform forest 0.1 0.50. 5 ha Mixed14ixed tropicaltropical forest 5 10 ha

This is because real harvesting operations tend toto have veryvery heterogeneous effectseffects associatedassociated with extractionextraction tracks andand loadingloading areas.areas.

2.201.52.2.1.5 Regeneration surveyssurvey2

In mixed tropical forestsforests oror anyany otherother type ofof mixedmixed ageage forest,forest, oror uniformuniform forestsforests being regenerated by directdirect seeding,seeding, estimatesestimates ofof regenerationregeneration maymay bebe anan importantimportant partpart ofof a yield prediction model.

Plots for regeneration surveys needneed toto bebe small.small. They may be subplots within conven-conven­ tional forest inventory plotsplots oror theythey maymay bebe basedbased anon aa spearatespearate samplingsampling schemescheme carriedcarried outout 3-5 years after logging. Typically,Typically, thethe plotsplots areare subdividedsubdivided intointo quadrats;quadrats; on each quadrat, the presence oror absenceabsence ofof speciesspecies isis recorded.recorded. Actual counts ofof trees above a certaincertain diameter or height may also be made, but do not ueuallyusually add greatly toto thethe usefulnessusefulness of thethe information. Typical plotplot sizes are 0.01 ha (10 xx 10 m)m) or 0.04 haha (20 xx 2020 m),m), subdividedsubdivided in each casecase into 11 m2 oror 44 m2m2 quadrats.quadrats.

2.2.2 Permanent Sample PlotsPlots

Most foresters would considerconsider the data obtainedobtained from permanent sample plots (PSPs)(PSPs) toLo bebe the most important contributor toto aa growthgrowth andand yieldyield model.model. Although this remains true for many situations,Situations, experiments must be considered as a necessary adjunctadjunct toto introduceintroduce extremes of treatment that are not foundfound inin the forest;forest; whilst measurements onon annualannual ringsrings provide an alternativealternative toto PSPPSP measurements.measurements.

2.2.2.1 Number of PSPs required

It is not possible to define thethe numbernumber ofof PSPsPSPs requiredrequired fromfrom purelypurely statisticalstatistical criteria. The precision ofof aa modelmodel fittedfitted toto PSPPSP datadata willwill dependdepend uponupon thethe locationlocation ofof plots,plots, and the duration of remeasurement as we1lwell as anon the covariances of the various predictor variables andand coefficientscoefficients inin thethe fittedfitted model,model.

Experience suggests however that approximately 100 plots coveringcovering the range of site variation and stanstandd history may be sufficient in a given forest type or plantation species, unless there is evidenceevidence for distinctively different growth patterns onon part ofof the geographic range. -8 -

2.2.2.2 Location ofof PSPs

PermanentPennanent samplesarnple plotsplots shouldshould be placedplaced withwith equalequal frequencyfrequency in:in:

- poor sitessitea average sitsites es good sites and low density stands - stands of average density high density standsstands and youngyaang or recentlyrecently loggedlogged standsstands mid-rotation or midway through felling cycle at rotation age oror at the endend ofof thethe fellingfelling cycle.cycle.

This willwill probablyprobably resultresult inin anan areaarea distributiondistribution forfor plotsplots whichwhich isis quitequite uneven,uneven, and appear to be proportionately deficientdeficient inin thethe averageaverage stands.stands. However,However, thisthis isis thethe mostmost efficient method ofof samplingsampling toto estimateestimate regressionregression parameters,parameters, asas discussedindiscussed in sectionsection 2.1.2.1.

The type of stratification impliedimplied aboveabove maymay notnot bebe possible,possible, duedue toto aa lacklack ofof know-know­ ledge of forest growing conditions, in which case plots may bebe laid downdown systematicallysystematically oror using aa geographicalgeographical stratificationstratification toto givegive equalequal areaarea coverage.coverage. In this case, many more plots will be requiredrequired thanthan ifif thethe moremore effectiveeffective typetype ofof stratificationstratification describeddescribed aboveabove isis used.

2.2.2.3 Size of PSPs

In general, the sizesize ofof permanentpennanent samplesample plotsplots isis governedgoverned byby forestforest typetype andand thethe heterogeneity ofof stockingstocking andand speciesspecies distribution.distribution.

In mixed tropical forest,forest, aa sizesize ofof 11 haha isis usuallyusually appropriate.appropriate. This may conveniently be divided into 100100 1010 xx 1010 mm quadrats.quadrats.

In unifonnuniform forest,forest, sizessizes aroundaround 0.050.05 haha areare commonlycommonly used.used.

These figures may be varied aa greatgreat dealdeal forfor differentdifferent circumstances.circumstances. For experi-experi­ ments, larger sized plots are more nonnal.normal. The table below givesgives somesome details.details.

PERMANENTPERMANEVT PLOT SIZES

Forest type Mixed UniformUnifonn SarnpleSample plots 1-2 ha 0.04-0.08 ha Experiments (excluding surrounds) 1-5 ha 0.08-0.120.01>-0.12 ha StudiesStudies of real* harvesting operations 5-10 ha 0.1 - 0.5 ha

* As opposed to simulated operations,operations, whichwhich comecorne underunder thethe categorycategory ofof experiments.experiments.

These figures shouldshould notnot be treatedtreated tootoo rigidly.rigidly. - 9 -

2.2.2.4 Shape of PSPs

Generally speaking, PSPs may be eithereither rectangularrectangular oror circular.circular. In inventory, other shapes are used, e.g. crosses and clusters of circular pplots,l ots, whichwhich havehave specificspecific advantagesadvantages for area sampling butbut are not particularly useful forfor PSFS.PSRI.

Circular plots are faster to laylay outcut than rectangularrectangular plots forfor sizessizes belowbelow 0.10.1 haha in open stands, oror 0.05 haha inin densedense stands.stands. Their use is also recommended in plantations as effective area is nctnot relatedrelated toto thethe arrangementarrangement ofof plantingplanting rows.rowe,

Rectangular plots are more appropriate forfor plot sizessizes greatergreater thanthan 0.10.1 ha.ha.

The ratio of length to breadth forfor rectangularrectangular plotsplots cancan be alteredaltered asas required.required. On steep topography, a high ratio, up to 5 to 1, is better, with the length running upup andand downdown the slopes.slopes. On level ground, a square plot has a smallersmaller perimeter andand willwill thereforetherefore bebe easier to demarcate and measure.

2.2.2.5 Frequency andand timingtiming ofof remeasurementsremeasurements

The frequency with whichwhich PSPsPSPs shouldshould bebe remeasuredremeasured dependsdepends uponupon thethe growthgrowth raterate ofof the trees. It is also useful to remeasure a new plot after a shortershorter than normal intervalinterval in orderorder to benefit asas rapidlyrapidly asas possiblepossibl e fromfrom thethe growthgrowth datadata thethe plotplot provides.provides.

It should be noted thatthat inin general,general, thethe longerlonger thethe intervalinterval betweenbetween remeasurements,remeasurements, the more accurately the treetree incrementsincrements cancan bebe determined.determined.

In an organization responsibleresponsible forfor aa largelarge numbernumber ofof PSPs1PSPs, itit isis aa goodgood ideaidea tot o alternate remeasurements so that perhaps only one third of the plots are measuredmeasured inin ananyy one year.

RerneasurementRemeasurement intervals cancan be givengiven approximatelyapproximately asas follows:follows:

Forest type Measurement intervalinterval (yrs)(yrs)

Young plantations in

tropics 1 Older plantations or otherether uniformuniform forest in tropics 2-4 Mixed tropical forest 3-5 Temperate uniform forest 3-5

The timing of remeasurements shouldshould obviouslyobviously taketake intointo accountaccount seasonalseasonal effects.effects. If there isis a definite growinggrowing season,season, measurementmeasurement shouldshould be carriedcarried outout afterafter the growing season is finished, as timingtiming isis lessless criticial.criticial. In any case, a given plot should always be measured inin the samesame month Whenwhen annualannual measurements areare made,made, toto allowallow exactexact oneone year comparisons and incrementincrement estimation.estimation. With longer remeasurementremeasurement intervals and less seasonal climates, timing becomes less critical.critical. - 1010 --

In some cases, timingtiming ofof measurements maymay bebe restrictedrestricted byby accessaccess considerationsconsiderations oror seasonal availability ofof labour.labour.

2.2.2.6 EamangSampling with partial replacementreplacement

Strictly speaking,speaking, samplingsampling withwith partialpartial replacementreplacement isis anan inventoryinventory designdesign wherewhere semi-permanentsemi-pennanent plotsplots areare usedused toto supplementsupplement informationinfonnation fromfrom temporarytemporary plots.plots. However,However, thethe sanesame general conceptconcept cancan bebe appliedapplied toto permanentpennanent samplesample plots.plots.

The moremore timestimes aa PSPPSP isis measured,measured, thethe lessless informationinformation itit providesprovides comparedcompared withwith thethe previous measurement,measurement , unlessunless itit isis growinggrowing intointo anan age-site-standag&-sit&-stand densitydensity stratumstratum thatthat hashas not been wellwell sampled.sampled.

For uniform age forests,forests, twotwo basic samplingsampling strategiesstrategies arisearise withwith PSPs:PSPs:

(i) Plots are establishedestablished through allall ageage classes.classes. In this case, sampling is more efficient if a proportion of plots is replaced after thethe thirdthird oror fourthfourth remeasurements.

(ii) Plots are establishedestablished inin youngyoung plantationsplantations anlyonly (because(because nono olderolder ageage classesclasses exist). In this case, a proportion ofof the plots, saysay 30%,30%, mustmust be retainedretained tthroughouthroughOut thethe rotation.rotation. The remaining 70%70% are replacedreplaced after 33 oror 4 measurements.measurements.

InIn mixedmixed forest, an analogous situation exists,exists, exceptexcept thatthat insteadinstead ofof age,age, aneone isis concerned with the number ofof years sincesince thethe lastlast harvestingharvesting aperation.operation.

22.3. 3 EXPERDIENTALEXPERIMENTAL DESIGNSDESINE

ExperimentsEXperiments are the most efficient and useful sourceBource of data for constructing modelsmodels of growth and yield. However, the usefulness of many experiments that have beenbeen laid downdown in forestry research is limited by a failure to clearly envisage the mathematical modelmodel that the experiment isis designeddesigned. toto testtest oror parameterize.parameterize.

Growth and yield studies are notnet primarily concerned with determining significant differences between treatments,treatments, butbut withwith constructingconstructing responseresponse surfaces.surfaces. It is also a commoncammon error toto assumeassume that long term forestry experiments can provide useful solutionssolutions to problems that have their originsorigins in current and highly fluid economiceconomic conditions. It almost always happens that by the time the experiment starts to provide useful data, economic conditions have changed so that the results are irrelevant.

The solution to this problem is always to set up the experiments with the intention ofof definingdefining general prinCiples,principles, via a mathematical model,model, ratherrather thanthan toto selectselect thethe 'best''best' of a set of treatments.

Similarly, the parameters to be measured shouldshould nevernever bebe defineddefined inin economiceconomic terms,terms, but always in ecologicalecological oror silvicultural terms.tenns.

Short temterm experinl'entsexperiments that are defined in terms of economiceconanic parameters are necessarynecessary for the costing of silvicultural treatments and determinationdetennination of utilizable yields in rela.­rela- tion to specific harvesting methods, but these matters areare outsideoutside the scopescope ofof this present manualmanual.. - 11 -

The principal experimentalexperimental designsdesigns thatthat areare relevantrelevant toto growthgrowth andand yieldyield researchresearoh can be subdivided intointo randomizedrandomized oror systematicsystematic designs.designs. The former can bebe subject toto con­con- ventional analysis ofof variance, whilst the latter are usually satisfactory whenwhen regressionregression is the principalprinCipal method ofof analysisanalysis andand maymay bebe moremore economicaleconomical thanthan randomizedrandomized designsdesigns toto set outout anon thethe ground.ground. Randomized designs can however also be analysed by regression andand are thereforetherefore probablyprobably toto bebe preferredpreferred inin allall casescases exceptexcept plantationplantation spacingspacing experiments.expsriments.

2.3.1 Randomized Designs

A principal feature ofof randomizedrandomized designsdesigns isis thethe allocationallocation ofof treatmentstreatments toto thethe plots withinwi thin anan experimentexperiment byby somesome randomrandom process,process, usuallyusually aa tabletable ofof randomrandan numbers.numbers.

Another feature ofof randomizedrandomized designsdesigns isis thethe principleprinciple ofof replication.replication. Any treatment must be appliedapplied atat leastleast twice,twice, onon twotliO differentdifferent plots.plots.

These two features are the common characteristics of aa widewide diversitydiversity ofof experimentalexperimental designs, including fullyfully randomizedrandomized experiments,experiments, lattices,lattices, latinlatin squares,squareB, incompleteincomplete blocks,blocks, splitBplit plots, etc..etc •• Standard textbookB,textbooks, Buchsuch aBas Snedecor (Bee(see bibliography)bibliography) covercover thethe analysisanalysis and appropriateappropriate usageusage ofof suchsuch designs.deBigns. Dawkins, inin his well known book ofof Statforms,Statforms, givesgives calculation pro formesfonnas forfor many suchsuch designs.designs.

In growth and yield studies.studies, probably only twotwo randomizedrandomized designsdesigns areare widelywidely appr<>­appro- priate. These are the randomizedrandomized blockblock andand thethe factorialfactorial experiment.experiment.

2.3.1.1 Randomized block experimentsexperiments

The structure ofof a typical completecomplete randomizedrandomized block experimentexperiment isis illustratedillustrated below.below.

t2 b2 It4\t31t11t21t4t3tlIt21 t1 It1 t4 b1 t3 b3 It2it4ItlIt21t41t11t31t3

The treatments, of which>r (provided(provided therethere areare atat leastleast two),two), are designateddesignated byby t1,t1, t2,t2, etc..etc •• TheseThese areare grouped grouped into into blocks, blocks, labelled labelled b1, b1, 1)2, b2, etc..etc.. There may be anyany number ofof blocks. Each block containscontains aneone replicatereplicate ofof eacheach treatment.treatment.

The blocks are laid outaut so that the variations in site or forest condition are smallsmall within the blocks, compared to the variation between blocks.

The plots wiwithin thin a block do not necessarily have to be physically adjacent, as shown above, but usually the plots are relatively close compared with the distance between blocksblocks..

The treatments are assigneltoassigned to each plot withinawithin a block using a table of random numbers.

ßnAn incomplete randomized block arises Whenwhen aneane oror more treatments are not replicated in oneone oror more blocks. This may be a deliberate feature of design, especially whereWhere there are large numbersnumbere ofof treatments,treatments, oror itit maymay bebe thethe resultresult of loss of aneone or moremore plots through accidents. The analysis of variance ofof an incompleteincomplete randomizedrandanized block experiment is somewhat more complex than for a complete one,one, but asas farfar asas regressionregression studies.tudies are concerned, it does not make a greatgreat dealdeal ofof difference.difference. - 12 -

In growth and yield experimentation,experimentation, randomizedrandomized blockblock experimentsexperiments areare suitablesuitable forfor situations where thethe treatmentstreatments dodo notnot formform aa clearlyclearly defineddefined dimensionaldimensional continuum.continuum. For example, ifif thethe treatmentstreatments comprisecomprise initialinitial spacingspacing for a plantation crop, then this cancan bebe varied continuously andand factorialfactorial oror clinalclinal designdesign isis moremore appropriate.appropriate. But if the treat-treat­ ments are specifications for rainforest harvesting framed in termsterms of speciesspecies groupsgroups andand different cuttingcutting limitslimits forfor differentdifferent groups,groups, thenthen therethere isis nono clearclear continuumcontinuum betweenbetween treatments and aa factorialfactorial oror clinalclinal designdesign cannotcannot bebe used.used. A randomized blockblock design is therefore most appropriate.appropriate.

2.3.1.2 Factorial experimentsexperiments

Factorial experimentsexperiments are intended for situations where the treatment consistsconsists ofof two or more interacting factors.factors. For example,example, in a plantation thinningthinning experiment,experiment, oneone might designatedesignate thethe ageage ofof thinningthinning andand thethe intensityintensity ofof thinningthinning asas twotwo separateseparate factors.factors.

Each levellevel ofof aa factorfactor isis combinedcombined withwith eacheach levellevel ofof everyevery otherother factor.factor. Thus if there are 3 levels ofof oneone factorfactor andand twotwo levelslevels ofof aa secondsecond factor,factor, therethere willwill bebe aa totaltotal of six treatments.

All the treatmentstreatments shouldshould bebe replicatedreplicated atat leastleast twice.twice. It is efficient to group the replicatesreplicates intointo blocks,blocks, as thisthis allows the variation betweenblocksbetween blocks toto bebe accountedaccounted for.for.

Qualitative treatments such as for example pruning or no pruning, can bbee included inin a factorial experiment asas aa treatmenttreatment withwith twotwo levelslevels -- presentpresent oror absent.absent.

Factorial experimentsexperiments are well suited to studies in uniform forest involving timingtiming and intensity ofof thinning, initial spacing, pruning and the use of fertilizers and weed control. They are more difficult to apply meaningfully in mixedmixed forest becausebecause of thethe complex nature ofof the treatment definition andand effects.effects.

2.3.2 Systematic DesignsDesigns

Systematic designs are those in which treatment locations are not randomized, butbut are laidlaid outout inin somesome systematicsystematic patternpattern toto economizeeconomize anon thethe sizesize andand costcost ofof thethe experiment.experiment. Systematic experiments cannot bebe analysed by analysis of variance,variance, butbut they areare veryvery efficient as a means ofof providing datadata forfor regressionregression parameterparameter estimation.estimation.

The main application of systematic designs is towardsto'Wards spacing experiments in uniform forest. In any situation where there is the slightests l ightest doubt as to the likely outcome of the experiment, e.g. fertilizer experiments,experiments, aa randomizedrandanized designdesign shouldshould bebe used.used.

For spacing experiments,experimente, two basic design approaches are possible.

2.3.2.12.3. 2.1 Single tree experimentsexperiments

In the single tree systematic spacing design, the spacing varies between each tree in a continuous fashion.fashion. One well-known example is the NelderNeIder fan, in which trees are planted along radii going outout fromfran a central point, with the distancedistance between trees along radii increasing a the same rate as the distance between radii. The general appearance is illustrated below.below. - 13 -

• , , I , l I " , , ~ " , , , .-, ," , , ,°

0/e-

1 ---.--.----. e -o

NeIderNelder fans are a little difficult to plan and execute on thethe groundgraund andand somesome equallyequally effective rectangular designs are possible. The diagram belowbelow indicatesindicates oneone inin whichwhich spacingspacing between treestrees isis increasedincreased byby halfhalf aa metremetre for for each each treet, tree, inin bothboth thethe verticalvertical andand horizontalhorizontal direction. This plan has the advantage of testing all combinations of squaresquare andand rectangularrectangular spacing and replicating eacheach singlesingle combinationcombination twice.twice. By using a smaller incrementincrement thanthan -1-t metre, a larger experimentexperiment is is obtained,Obtainedt whichwhich isis lessless likelylikely toto bebe affectedaffected byby the loss of single trees.

, ... ,. ____ • ______e ______e: __ --- __ I .11' --1--I - QII Ie I ~ 'I ' , , , ~ - _,_ -- - -,_ - -- - - ~_ -- - - - e-I ______e:•

o I i o o, o I i I'i o I • ,I I I o• •I e,_ --e--1- __ _ _lee 1 __ - -- e-e - __ - --1--- ____ .0_I , o o" I, o o

o I I I o o I, I o I 'I , ' I ' ___ ..> ____ a _____ '. -- ---0------·"O''' ; , -e-- o ;, ' o i o o o o

o I, I I I I, I I I I I t I ; ___..-.41,-...-;--.....0;..._ ~ __ - .; ______e_ -- ______I. ____ - --~I! : ' , : I , ,

, 'I I I ,I, ___ • ___ .1.I ____ I I ___ - - -0-I ------•'

Single tree spacing experiments provide very useful data on the response of diameterdimmeter and crown diameter to spacing for uniform foresta,forests, but the data is not easilyeaSily compatible with that from conventional plots unless a singlesingle tree modelling strategy has been adopted. Single tree experiments are also very sensitive to the loss ofof trees, which upset their arrangement and design.design.

2.3.2.2 ClinalCurial plots

Clinal plots are those in which the treatments are arranged soso that successivesuccessive levelslevels are adjacent. The main advantage ofof clinal plots isis thethe possibility ofof eliminatingeliminating thethe plot surround except on the autsideoutside of the experiment.experiment. This is illustratedillustrat ed byby thethe diagramdiagram below.below. - 14 -

<.' _.. ...-,-,...;015....;:e.h.,k,,,-... ,....:- - .'" '-4. - ----.....4,,,,,.;.A:t.A.,!..w,,;.

;

, 1 1

1 t5 t4 t2 1 ti t5 !t3 1 i t1 1 I i I

' -. '. ... ,. ..,:.-,1-Z,..ie:ftg*.0,,Z,V.,i,t,i.'!. ftt'I',1,1rit*I:5:Ff2g;,;.?"-.4.)..,:.. ..ii

i!'¡61,tV'T.:;:t".f..;;:izt.r.iv.,:te....:,:l - ,i:.;..,,. ,,',W'''. ' i . s' - ..0.7!Ms-5.4..f.:¡;-: ...... ' '. 0.`.6.4

1 1

1 1 1

ti 1 1 t2 t3 1 t4 t5

1 1 1

AVAt., \ v'

Treatment plots Guard rowsrows 4-40m --T.-420m

For spacing experiments,experiment~, it is usually desirable to have from 5 to 8 treatments. The closest spacingspacing shouldshould bebe eithereither 22 xx 22 mm or or 2-i 2t x 22t m, depending onon whetherwhether a narrow crowned species (e.g.(e.g. Eucalypt oror Pine)Pine) oror widewide crownedcrowned speciesspecies (e.g.(e.g. GmelinaGmelina arborea)arborea) isis involved.involved. The widest spacing should be 7 x 77 m or 8 x 88 m. It is very important to always incorporate these two extremes,extremes, asas discusseddiscussed inin sectionsection 2.1.2.1 .

Data from clinal plots cancan be combinedcombined withwith otherother plotplot datadata withoutwithout difficulty.difficulty. The plots themselves are less sensitive to the loss of individual trees than single tree experiments.

2.42. 4 EXAMPLESElCAI~PLES OF GROimIGROWTH AND YIELD EXPERIMENTSEXPERDIENTS

There are literally an infinite variety of ways in which experiments may be designed and executedexecuted toto provide datadata forfor growthgrowth andand yieldyield prediction.prediction. The main purpose of this section is to present synthesized examples of thethe mostmost common types of experiment, withwith some discussion ofof their usefulness and the special problems ofof executionexecution and analysis. There is here a wide divergence inin technique between methods for uniform forests of a singleSingle age and mixed forests.forests.

2.4.1 Uniform Forests

Growth and yield studies in uniform forest have tended to concentrateconcentrate onon the effectseffects of stand denSity,density, fertilizationfertilization andand pruning.pruning. Fertilizer and pruning experimentsexperiments will not be considered directly in this manual as they are rarelyrarely relevantrelevant toto thethe direct problem efof yield prediction. Stand density is ofof primary importance, asas itit isis the major variable which the forester is able to control during the rotation ofof aa uniformunifonn forest.forest. There are four basic ways in which the interaction between standstand densitydensity andand growthgrowth maymay be:be studiedstudied experimentally.exp erim ent ally.

2.4.1.1 Spacing experimentsexperiments

Spacing experiments may be laidlaid outout eithereither asas singlesingle treetree experimentsexperiments oror asas clinalclinal plots. The latterlatter isis probably aa more usefuluseful typetype ofof experiment.experiment. Between 5 and 8 different spacings may be used.used. It is betterbetter to thin the plots to their final spacing in the second - 15 -

or third year of growth than to plant directly atat the finalfinal spacing;spacing; the latter approach is too sensitive to poorpoor survivalsurvival followingfollowing planting.planting. When the thinning to the final spacing is carried out, itit shouldshould bebe aa non-selectivenonselective thinningthinning to achieve thethe desireddesired spacing,spacing, notnot aa low or crown thinning.

A spacing experimentexperiment should be continuedcontinued overover the wholewhole ofof aa normalnormal rotationrotation ofof thethe crop. Once established, it should not be thinned.

It is very easy to analyse spacing experiment datadata toto provideprovide aa flexibleflexible growthgrowth modelmodel for the species inin question. The details are discussed inin sectionsection 5 of this manual.

2.4.1.2 Constant basal area thinning experimentsexperiments

In a constant basal area thinning experiment,experiment, plots areare laidlaid outout systematicallysystematically oror randanlyrandomly and allowed to grow until eacheach plotplot reachesreaches itsits designateddesignated basalbasal area.area. It is then thinned everyevery oneone oror two years to maintain the basal area atat thethe designated levellevel forfor eacheach plot.

Careful records mustmust bebe keptkept of thethe numbernumber ofof treestrees removedremoved andand thethe basalbasal areaarea removedremoved if this type ofof experimentexperiment isis toto be veryvery useful.useful.

AnalysisAnalyuis is more difficult than for a spacing experiment andand involvesinvolves thethe fittifittingng ofof a model predicting basal area increment as a function of basal area and other variables,variables, for example age andand stocking.stocking.

As a strategy for constructingCQlstructing a dynamicdynamic thinning experiment, this approach hashas thethe advantage ofof simplicity ofof design, although the execution may not be so easy, BSas careful and continuous record keeping isis requiredrequired ifif thethe resultsresults areare toto bebe fullyfully useful.useful.

2.4.1.3 Thinning experiments usingusing gradedgraded thinningthinning treatmentstreatments

In this type of experiment, a number of thinningsthinnings areare defined (typically(typically 4), whichwhich differ from eacheach otherother inin termstenns ofof bothboth timingtiming andand intensityint ensity ofof thinningthinning andand possiblypossibly inin terms ofof initial spacing also.also. These treatments cancan be roughlyroughly classifiedclassified fromfrom lightlight thinning toto heavyheavy thinning.thinning.

This type ofof experimentexperiment isis veryvery similarsimilar toto aa constantconstant basalbasal areaarea thinningthinning experimentexperiment in tennsterms of analysis and recordrecord keeping.keeping. Records must be kept of the numbers and basalbasal areaarea of stems removedremoved atat eacheach thinning.thinning. Analysis cancan be by severalseveral methods, thethe bestbest ofof whichwhich isis the fitting ofof a predictive model of diameter oror basal area increment as a function of basalbasal area, stem numbers, age,age, etc.etc.

The principal advantage over a constant basal area experiment is that the thinningsthinningu used are treatments which might be used in reality,reality, allowing costing of thinnings, thinning damage, windthrow risk and effects onon wood quality to be assessed on the experiment. Addi­Addi- tionally, thinning need not be performed so frequently as anon the constant basal area experi-experi­ ment, simplifying administration.administration. - 1616 -

2.4.1.4 Factorial experimentsexperiments with differentdifferent componentscomponents ofof thinningthinning treatmenttreatment

A thinning treatment inin aa uniformuniform forestforest cancan bebe brokenbroken downdown intointo aa numbernumber ofof compo-compo­ nents,nel'lts, viz:viz:

iniinitialtisl spacing age of first thinning proportion of stocking removed at each thinning time to elapse between eacheach thinning.thinning.

Otherother variations are possible,possible, asas forfor exampleexample thethe basalbasal areaarea atat firstfirst thinningthinning andand at subsequent thinnings, oror thethe useuse ofof heightheight insteadinstead ofof age.age.

When the experimentexperiment isis designeddesigned inin thisthis way,way, itit cancan bebe setset outout asas aa factorialfactorial experi-experi­ ment byby assigningassigning aa numbernumber ofof levelslevels toto eacheach thinningthinning component.component.

The result will be aa largelarge experiment,experiment, withwith aa considerableconsiderable numbernumber ofof treatmenttreatment plots,plots, but inin the analysisanalysis itit isis possible to separateseparate the different effectseffects of thinning age, thinning intensity, etc..etc.. Obviously the large number ofof treatmentstreatments meansmeans thatthat very carefulcareful administration is required, whilst the factorial nature of the e~erimentexperiment maymay bebe easilyeasily upset by accidental occurrencesoccurrences suchsuch asas firefire oror diseasedisease outbreak.outbreak.

2.4.2 Mixed Forests

The main function ofof experimentsexperiments inin mixedmixed forestforest isis toto provideprovide aa controlledcontrolled degreedegree ofof disturbance ofof the forestforest soso thatthat ultimatelyultimately aa modelmodel maymay bebe fittedfitted relatingrelating thethe incrementincrement ofof trees to the various parameters ofof thethe standstand followingfollOwing treatment.treatment.

A common problem with experimentsexperiments laidlaid downdown inin mixedmixed forestforest isis thatthat thethe treatmentstreatments are specified in terms which are irrelevant to the main parameters of the stand thatthat control growth. This isis discussed atat lengthlength andand withwith numerousnumerous examplesexamples byby SynottSynott (see(see bibliography).bibliography). However, aa typical casecase would involveinvolve the definition ofof fourfour treatments as for example:

1 -Log all merchantable speciesspeCies down to 30 cmem

2 - As for 1, but poison oror ring-barkring-bark allall non-merchantablenon-merchantable speciesspecies downdown toto 3030 cmem

3 - As for 2, but killing non-merchantable stems down to 10 cmem

4 - As for 393, but removingremOVing merchantablemerchantable speciesspecies downdown toto 2020 am.em.

The effect of treatments defined in this way upon the stand will depend entirely uponupon the conditioncondition ofof thethe standstand priorprior toto logging,lOgging, itsits sizesize classclass andand speciesspecies classclass distribution.distribution. It is possible for the most severesevere treatmenttreatment (e.g.(e.g. 44 above)above) toto havehave thethe leastleast effect.effect.

otherOther problansproblems have arisen by selecting only a subset of trees for increment measure­measure- ment, onon the basis ofof merchantability criteriacriteria which have anan unfortunate tendency to change during the intervalsintervals betweenbetween plotplot measurements.measurements. The method ofof measuringmeasuring incrementincrement hashas oftenoften been unable to cope with the developmentdevelopment ofof buttresses.

At the moment,moment, growth and yield research in mixed tropical forests is developing rapidly. The following recommendations are made therefore simply to help avoid the mistakes of the past and not in order to impose a strait-jacket of unnecessary regimentation on current work.work. -17-17

2.4.2.1 Randomized block design

It is suggested that a randomized design, withwith rePlication,replication, shouldshould alwaysalways bebe adoptedadopted in mixed forestforest experiments.experiments. The replicates should be blockedblocked withwith site andand species distri­distri- bution patterns, as well as past logging history, as uniform as possiblepossible withinwithin blocks.blocks. ThisThis implies a carefulcareful preliminary survey of the experimental area.area

2.4.2.2 Treatment definition

Treatments shauldshould be defined in terms of total basalbasal areaarea toto bebe leftleft afterafter logginglogging and/or poisoning oror girdling,girdling, ofof treestrees aboveabove aa minimumminimum sizesize ofof 1010 cm.em.

otherOther definitions of treatment are possible, butbut should alwaysalways bebe mademade inin termsterms ofof the remaining stand and not the material to bebe removed and should bebe independentindependent of economic criteria.

The treatments adopted should always include two extremes, oneane beingbeing an undisturbedundisturbed stand and the other being an extremely severe treatment,treatment, perhapsperhaps removingremoving allall materialmaterial overover 10 cmem diameter.

2.4.2.3 MeasurementsMeasurements and and 0..2-1.2[1221En plot design

Large plots are requiredrequired forfor experimentsexperiments involvinginvolving fellingfelling treatments.treatments. Typically a 200 x 200 m plot (4(4 ha)hal with aa 100100 m surroundsurround isis necessary.necessary. On the mainmain plot, all large trees (say(say overover 3030 cm)em 1 shouldshould bebe mapped.mapped. SubdivisionSubdivision of the plotplot into 20 x 20 m quadrats is desirable and allows locallocal competitivecompetitive effectseffects toto bebe includedincluded intointo modelsmodels ofof growth,growth, reregenerationgeneration andand mortality.mortality.

Detailed countscounts ofof seedlingsseedlings cancan bebe mademade onon aa systematicsystematic subsamplesubsample ofof quadrats.quadrats.

The methods ofof incrementincrement measurementmeasurement areare discusseddiscussed inin sectionsection 3.3. - 19 -

3. PROCEDURES FOR DATA COLLECTIONCOLLECTION AND PRIMARY ANALYSIS

3.1 SAMPLE PLOT DEMARCATION

3.1.1 Location

Permanent sample plots, inin particular, needneed toto bebe accuratelyaccurately locatedlocated onon forestforest mapsmaps and their precise Positionposition inin the forest determined through the use of survey tapetape andand compass. In addition to recording positions onon maps, itit isis helpfulhelpful toto placeplace stonestone oror con-con­ crete markers onon nearby forest roadsroads showing the bearing and distance toto thethe plotplot from thethe marker.

Temporary plots also need to be located on working maps, butbut thethe degreedegree of precisionprecision required isis notnot usuallyusually soso great.great.

Design prinCiplesprinciples relating to plot locationlocation areare discusseddiscussed inin sectionsection 2.2.2.2.

3.1.2 pspPSP Identification on the Ground

PSPs must be permanently markedmarked onon thethe ground.ground. Circular plots shouldshould bebe markedmarked atat the plot centre with a post ofof durable wood, concrete or metal bearing thethe plotplot identifi­identifi- cation number. This central point should also be indicated byby diggingdigging intersectingintersecting trenchestrenches 50 cm deep and approximately 2.5 metres long, with the point of intersection beingbeing thethe platplot centre. This provides a permanent markmark onon thethe groundground inin thethe eventevent ofof thethe plotplot centrecentre postpost being lostlost oror stolen.stolen.

Rectangular plots should similarlySimilarly be marked at the four corners with posts,posts, one of which carries the plotplot identificationidentification number.number. Trenches should be dug to intersect atat thethe corner post positions.positions.

The plot identity should also be prominently painted on a tree nearnear to the centre or a cornercorner postpost bearingbearing thethe plotplot identity.identity.

Rectangular plots subdivided into quadrats may also have smaller posts placedplaced atat the quadrat intersections.intersections. These should be of a different size to the corner postsposts toto avoid confusion. Alternatively, quadrats maymay bebe resurveyedresuzveyed atat eacheach measurement.measurement.

3.1.3 Determination ilf_Edgf_Treesof Eage Trees

Most treestrees willwill bebe eithereither clearlyclearly within,Jithin thethe plotplot oror clearlyclearly outsideoutside it.it. Some will intersect with the plot edge.edge. These should bebe included if the estimated centre of thethe treetree is inside the line demarcating the plot and excluded otherwise.otherwise.

The edgeedge ofof a circular plot isis determineddetermined by usingusing aa lineline oror rope stretched from the centre post. Care should bebe taken that the line does not have Significantsignificant elasticity (many light nylon lines are very elastic)elastic) oror become wrappedwrapped aroundaround thethe centrecentre post whilstwhilst working round the plot.

The following table showsshows radiiradii (i.e.(i.e. distancedistance fromfrom plot centrecentre post to edge)edge) forfor common plot areas ofof circular plots. - 2020- _

PlatPlot area (ha)(ha) Radius (m)(m)

0.04 11.28 0.05 12.62 0.08 15.96 0.10 17.84

With small rectangular plots,plats, thethe edge~e maymay bebe determineddetermined byby takingtaking aa lineline ofof sightsight between twotwo cornercorner posts.posts. With larger platsplots with intervening vegetation,vegetation, itit willwill bebe neces­neces- sary to insert extraextra edgeedge markersmarkers alongalong aa bearingbearing betweenbetween thethe cornercorner posts.posts. Small vegevege±aticntation may be judiciouslyjudiciously removedremoved alongalong the plat edge,edge, provided one is not destroying regeneration that formsforms partpart ofof thethe forestforest growinggrowing stock.stock.

3.1.4 Marking Trees

Trees on permanent plotsplats shouldshould ifif atat allall possiblepossible bebe permanentlypermanently identified.identified. There are two ways ofof doingdoing this:this:

1. By painting the identityidentity number onon the tree 2. By using embossedembossed aluminiumaluminium tagstags nailednailed toto thethe tree.tree.

If the tree number is painted onon, then the measurement point for diameter shouldshould also be markedmarked by paintingpainting aa ringring aroundaround thethe tree.tree. If anan aluminium tag is used it should be nailed to the tree a fixed distance above the diameter measurement point, usually 50 cmem above, soso that the latterlatter cancan bebe exactlyexactly relocated.relocated. The reference point for diameter measurement isis ofof coursecourse normallynormally atat 1.31.3 mm initiallyyinitially, exceptexcept forfor buttressed rainforestrainforest trees, but may changechange asas groundground levellevel altersalters overover time.time.

An aluminium tag should also be nailed to the stump of the tree, close to ground l""el.level. This helps to identifyidentify cutcut stumpsstumps andand givesgives addedadded securitysecurity againstagainst lossloss ofof tags.taga.

PbintPaint markingamarkings should not be used as the sole means of identification with species that shed their bark (e.g.(e. g. many Eucalyptusfucal;yptus species). ~larlcingaMarkings should be renewed at each remeasurement where they are becoming worn oror have been lost.

3.1.5 Mapping TreesTrees onon thethe PlotPlat

If possible, trees onon aa plotplat shouldshould bebe mappedmapped atat thethe initialinitial assessment.assessment. For circular plots, record the distance andand bearing ofof eacheach treetree fromfrom thethe plotplat centre.centre. For rectangular plots, subdivide the plotplat intointo quadrats,quadrats, eacheach ofof whichwhich isis notnat moremore thanthan 1010 m by 1010 m andand then measure the distance ofof thethe treetree fromfrom thethe twotwo quadratquadrat boundaries. Record these dis-dis­ tances as the coordinates ofof thethe tree withinwithin thethe quadrat.quadrat.

Mapping treestrees isis helpfulhelpful bothboth inin resolvingresolving thethe frequentfrequent confusionconfusion overover treetree identitiesidentities that occursoccurs and in the analysis ofof growthgrowth phenomenaphenomena anon thethe plot.plat.

3.1.6 Identity Numbers for Ingrowth

In naturalnatural forests, ingrowth present at each assessment will need to be given an identification number, a quadratquadrat number andand coordinatescoordinates onon thethe plotplat map.map. Great carecare isis necessary toto ensureensure thatthat thethe identityidentity numbernumber givengiven isis notnot oneone previouslypreviously assignedassigned onon thatthat plat,plot, includingincluding trees that havehave dieddied oror beenbeen removedremoved atat anan earlierearlier stage.stage. Otherwiseotherwise greatgreat confusion results when the datadata isis processed.processed. - 21 -

3.2 SAMPLE PWTPLOT MEASURU!ENTMEASUREMENT FORMSFOmlS AND PRIMARYPRIMARY ANALYSISANALYSIS

3.2.1 Uniform Forests

Sample plots in even-aged,even-aged, monospecific forestsforests (normally(normally plantations) requirerequire measurement of:of:

1. Diameter over barkbark at 1.31.3 m,m usingusing aa girthgirth tapetape calibratedcalibrated inTrominlTcm units, on each tree.

2. Tree heights onon a systematicsystematic samplesample ofof 88 trees,trees, togethertogether withwith heightsheights ofof dominantdominant trees not included inin thethe sample.sample.

Dominant height isis defineddefined asas thethe meanmean heightheight ofof thethe 100100 thickestthickest stemsstems perper ha.ha. Thu2Thus the number ofof treestrees requiredrequired forfor dominantdaninant heightheight estimationestimation isis thethe plotplot areaarea timestimes 100.100. For example, on a plotplot of 0.04 ha, 4 trees are required. Trees withwdth brokenbroken or significantly damaged tops areare notnot usedused inin heightheight sampling.sampling.

Additional characterscharacters cancan bebe notednoted onon eacheach tree.tree. Disease problems, dying trees, wind or insect damage and trees marked forfor thinning cancan bebe recorded.recorded. Such additional notes should be coded in a rigidlyrigidly standardizedstandardized wayway andand enteredentered anon thethe plotplot recordrecord form.form. The following coding suggestions maymay bebe adopted:adopted:

Code Description

no entry Tree healthy, undamaged, unmarked forfor thinning,thinning, singlesingle non-defectivenon-defective stem. A Animal damage.damage. B Tree broken by wind.wind. D Disease problem.problem. L Double oror multiplemultiple leaderlaader oror stems.stems.

M Tree marked forfor thinningthinn~ng (but(but stillstill standing).standing).

S Tree dying fromfrom suppression.suppressi on. T Tree has been felled (may(may bebe measuredmeasured onon thethe ground).ground). W Tree leaningleaning oror fallenfallen fromfrom windwind damage.damage. X Tree dead.dead.

Following the letters,letters, numeric codescodes cancan be placed indicatingindicating the degree of severity of the problem. The following scalescale isis suggested:suggested:

1 Da.mage/diseaseDamage/disease prepresents ent butbut very light.light.

2 MoreJ40re severesevere damagedamage - isis likelylikely toto significantlysignificantly reducereduce growthgro.rth oror impede utilization.utilization.

3 Very severe damage/disease. Likely to kill tree oror make itit unutilizable. - 22 -

otherOther codes cancan be introducedintroduced forfor specificspecific problems.problems. The important point is that forfor such a system toto be ofof anyany use,use, itit mustmust bebe rigidlyrigidly adheredadhered toto withoutwithout alterationalteration oror omissionomission over many years.years.

Form 3.1 is shown asas a recordingrecording formform forfor thisthis typetype ofof plot.plot. It is specifically designed to facilitatefacilitate automaticautomatic datadata processing.processing. Field entriesentries shouldshould be made inin softsoft pencil with arubbera rubber being used to ranoveremove errors.error8~

The primary variables calculated onon aa platplot are:are:

1. Stocking per ha (N).(N). Divide the total number ofof live trees onon the plot by plotplot area. From the example enteredentered inin formform 3.1:3.1: '

N = 9 / 0.040.04 = 225 stems/ha.

Diameter of the mean basal area tree (D g) 2. Diameter of the mean basal area tree (Dg ). Sum the diameters squared and divide by thethe numbernumber ofof treestrees onon thethe clot.plot. Take the square rootroot ofof thethe result.result. For the example:example:

2-,d2~d2 = 16237

DgD = ,/F6237-/(16237 / 9)9) = 42.5 cm.

3. Stand dominant heightheight Ho.Ho' This is the mean heightheight of thethe specified numbernumber of dominant trees onon the plot. For the example,example,

Ho = (32.1 + 29.6 + 30.8 + 33.1)/4

= 31.431.4m. mo

4. Stand mean height H. This is the mean ofof the systematic samplesample ofof heightheight treestrees or as in the example anon form 3.1,3.1, the meanmean ofof allall heights.heights. In the example,example,

H = 31.731.7m. m.

5. Stand volume V.V. This is usually calculatedcaloulated from an individual tree volumevolume tarifftariff entered by diameter andand height.height. There are two methods;methods:

(0(i) Calculate thethe volumevolume of thethe tree of diameter DgD and height H and then multiply by stockingstocking NN to givegive volume inin m3/ha.m3/ha.

(ii) CalculateCalcula-te individualindividual treetree volumesvolumes vv fromfrom diameterdiameter d and height h. Sum these volumes and divide by plot area to give volume per ha.

In thisthis second case, if h is not known for all trees, either estimate itit from a height/diameter regression (c.f.(c.f. section 3) or use H instead.

Both methods introduce an errorerror intointo volume estimation.estimation. The first aneone has an error that may resultresult fromfrom the distribution ofof diametersdiameters andand thethe secondsecond oneone fromfrom the estimationestimation of heights. The second method is generally preferable ' for accuracy, especiallyespecially if volume is beingbeing estimated to a merchantable top diameter limit. --23-23 -

Form 3.1 Sample P16~Plot Assessment Form - Plantations Sheet ...of...• •• of. ••

Forest District Compartment Plot number Species 3 Office 1110131/ 101012.1oo II 1a.IA2. 1 11.132_ Iq9 1+1I rn use Plot Slope Assessment 0o date 2 j area f 0 Is I NEI 1119iI9 I - month year no. ofof cardscards Tree no.no. Diameter bhob Height ------CodesCodec - - -- - I I 3 q r4- 2- g b It7 10 4- 3 I 3 2- I I¡3 3 3 b 4- B 3 I 6~ 4- 'g 5 Z 9 b g I/

2. I 4- I 2- I 1..4- :> 'il I 3" 4- "2. 4- 4 2- g '3 '8' 9 4- I.i- 'j "' "3 I

Notes .::~ ..•• I."!..... ~.~ •• ~.;J. • !"J.,,!~':'d> r. ...~'!>.K.~ .~ .":"::1. • ':t.~ .. .. .t.ead~-o 0000...... 00000000Un ...... 00470000...... 0.00...... 0000000000...... 0000000004.06000 ...... pn...... e ..000 ...... 000 .. .. 0000...... 0.. Assessed by ...•• ~>~~...... Date •• 3./ ..1- ./. ~1 USE BACK dFOF FORMFORM FORFOR HEIGHTHEIGHT CALCULATIONSCALCULA TI ONS ==_

- 2424-

Several different volumes maymay bebe calculated,calculated, usingusing differentdifferent volumevolume equations.equations. For example:

Over bark total volume (i.e. to the tip of the tree). Vob Over bark total volume (i.e. to the tip of the tree). Vub Under barkbane total volume. b Volume to7cm top diameter limit. VV77 Volume to 7 em top diameter limit.

V15 Volume toto 15 cm top diameter limit.limit.

The variousva.rious symbolssymbols andand unitsunits usedused forfor primaryprimary standstand variablesvariables areare listedlisted inin TableTable 3410

Table 3.13.1

SYMBOLSSYMBOl.'l FOR PRIMARYPRDlARY STANDSTAND VARIABLESVARIABLES

Symbol Description Units

A Stand age fromfrom planting years

Dg Diameter ofof the tree ofof mean basal area cmCM

G Stand basal area m2/ha

H Stand mean height m

Ho Stand dominant height,height9 defined as the mean height of the 100 largest diameter stems/hastEl!ls/ha m

N Stocking - number ofof live trees/ha trees/ha i V Stand volume. Type ofof volumev:.olume* isis m3/ha denoted by a subscript tog.e.g. Vub underbark volume V7 V7 - over banebark volume to 7 cmcm

Measurements mademade immediatelyimmediately afterafter thinningthinning maymay bebe indicatedindicated byby aa primeprime (').(,). E.g. N'N' stockingstocking afterafter thinnning.thinnning. Removals in thinningnthinnings should be denoted by a subscriptBubscript e.e. E.g. GeG would be basalbasal areaarea removedremoved. inin thinning.thinning. e

3.2.2 MixedMixed ForestForest

Primary analynisanalysis in mixed forestforest aimsaims mainly toto constructconstruct aa standstand tabletable givinggiving stemstem numbernnumbers groupedgrouped byby sizesize classesclasses forfor eacheach speciesspecies oror speciesspecies group.group.

Once the stand table has been constructed9constructed, thenthen variousvarious alternativealternative measuresmeasures ofof growinggrowing stock can be derived fromfrom it using differentdifferent criteria.criteria. For example9example, aneone may wish to derive total basal areaarea ofof treestrees inin aa givengiven combinationcombination ofof speciesspecies groupsgroups overover aa certaincertain size;size; on another occasionoccasion aneone maymay repeatrepeat thethe summarysununary usingusing differentdifferent speciesspecies groupsgroups oror sizesize limits.limits. - 25 -

A stand table cancan ofof coursecourse bebe constructedconstructed directlydirectly asas thethe plotplot isis beingbeing measured,measured, by recording onlyonly countscounts byby sizesize andand speciesspecies classes.classes. This procedure is not reconmendedrecommended with any type ofof inventoryinventory exceptexcept thethe mostmost preliminarypreliminary resourceresource assessments.assessments. Individual tree dimensions andand speciesspecies shouldshould always be recorded, even though the intention maymay bebe toto subsequently summarize thethe data.data. This is particularly important on permanentpermanent plots,plots, sincesince it allows individualindividual treetree incrementincrement estimates.estimates.

Measurement formsforms forfor plotsplots inin mixed forestforest may vary considerably,conSiderably, depending on the characteristics being recorded.recorded. The following types ofof situation are commonlycommonly found:

(a) Mixed tropical montane forest (A few light demanding species ofof mixed age).age). Form 3.1 may be used with the firstfirst 'code''code' columncolumn beingbeing reservedreserved forfor speciesspecies as a twotwo digitdigit oror twotwo letterletter code.code. A systematic sample of heights should always be meausred,meausred, toto allowallow aa height/diameterheight/diameter curvecurve toto bebe constructed.constructed.

(b) Tropical rainforestrainforest (Large numbers of mixed species, climbers, epiphytes, buttresses).buttresses ) . Large plots,plots, overover 11 ha, are normally used, subdivided intointo 10 x 10 m subplots or quadrats. Heights are not normally measurable, but trees maymay bebe classifiedclassified by crowncrown shapeshape andand crowncrown position.position. Two reference diameters should bebe measuredmeasured on stems forming buttresses.buttresses. There are normally severalseveral hundredhundred distinctdistinct species likely to occuroccur onon aa plot.

(c) SubhumidSub-humid woodlands woodlands (E.g.(E.g. MiomboMianbo forestforest inin easterneastern Africa).Africa).

Here treetree formform andand lengthlength ofof merchantablemerchantable bolebole areare importantimportant characteristics.characteristics. Height isis easilyeasily measurable because ofof thethe opennessopenness ofof the forest, butbut only to the nearest metremetre becausebecause ofof diffusediffuse crowncrown shape.shape. There are likely toto bebe over 100 possible speciesspecies present.present.

(d) AridzoneArid-zone woodlandwoodland Trees are likely to be multi-stemmedmultistemmed withwith heightheight andand speciesspecies beingbeing the onlyonly characters ofof importance.importance. Height shouldshould be measured toto thethe nearestnearest decimetre.decimetre.

Figure 3.13.1 showsshows alternativealternative recordrecord formatsformats forfor tree measurements for these four cases, together with the record format from Form 3.1 for plantations forfor comparison. A 'record' is assumedassumed toto bebe thethe amountamount ofof datadata thatthat cancan bebe enteredentered onon oneone 8O-column80column punched card, which represents thethe commonestcanmonest mediummedium forfor datadata inputinput toto aa computer.computer.

3.2.3 Initial Assessment of Permanent Plots

When a permanent plot is assessed for the first time9time, the following additional information is required:

(1) The exact area of the plot, as a plane projection i.e. corrected for slope.

(2) Basic sitesite informationinformation includingincluding latitude,latitude, longitude,longitude, aspect,aspect, altitude, slope,slope, slope position, forest history and past land use.

(3) Meteorological informationinformation fromfran thethe nearestnearest weatherweather stationstation givinggiving monthly precipitation andand mean minimum andand maximummaximum temperatures.temperatures. ,

-2626 -

(4) The positions ofof all the treestrees onon thethe plots. For circular plots, these shouldshould be recordedrecorded asas aa bearingbearing andand distancedistance fromfrom thethe plotplot centre.centre. ForFor rectangularrectangular plots, thesethese shouldshould bebe givengiven asas XX andand YY coordinatescoordinates inin decimetresdecimetres fromfrom thethe mostmost south-westsouth,west corner ofof the plot.

(5) Soil profile information including:including:

colour -texturetexture pHpH analysis for N, P, K,Kt Ca, MgMg depth bulk density

for each discernible soilBoil horizon.horizon. The soilsoil pits oror augerauger holesholes shouldshould bebe replicated onon eacheach plot.plot. Two samples should be sufficientsufficient exceptexcept onon veryvery largelarge and variable plots.plots. Analytical techniques may vary somewhatsomewhat accordingaccording toto methodsmethods already in use in a particular country and special conditions encountered, andand should be determined inin collaborationcollaboration withwith soilsoil scientists.scientists. -However, onceonce a system for soil analysis hashas been settledsettled upon,upon, itit shouldshould bebe modifiedmodified onlyonly forfor very strongstrong reasons and, at the time ofof modification, a series of samples (20(20 or 30) should be analysed by both the oldold andand newnew methodsmethods toto determinedetermine aa regression relation to allow results from the old methodsmethods toto bebe compared withwith the new ones.ones,

3.3 STEM ANALYSIS PROCEDURE

Stem analysis refersrefers toto the reconstructionreconstruction ofof thethe growthgrowth historyhistory ofof aa treetree by:by:

(a) FellingFelling the tree; (b) Cutting discs at intervals ofof around 22 m alongalong thethe stem:stem; (c) Careful counting and measurement ofof growthgrowth ringsrings onon thethe discs.discs.

A great deal ofof informationinformation aboutabout standstand dynamicsdynamics cancan bebe gainedgsined fromfrom stemstem analysis,analysis, but inin thisthis manual thethe main concernconcern isis thatthat ofof reconstructingreconstructing thethe dominantdominant heightheight growthgrowth history ofof aa stand.stand.

The procedure is only possible in seasonal climates and with species producing clearly defined rings.rings.

For height growth reconstruction, onlyonly the height ofof measurement and ring count need be recorded. With species and climates producing very unambiguous ringsringe this cancan be done in the field with little difficulty.

Where rings are not so clear, discs must be cut,cut, clearlyclearly marked inin the fieldfield as to their point ofof originorigin onon thethe treetree andand theirtheir orientationorientati on (i.e.(i. e. whichwhich wayway upup thethe discdisc waswas anon the tree), and returned to the laboratory forfor assessmentassessment by oneone ofof twotwo methods:

(1) PlaningPlaning and polishingpolishing the disc, followed by counting of ringsringe along two axes using a microscope anon a vernier rackrack and pinionpinion mounting.

(2) CuttingCutting of twotwo samples along a cross, withwith subsequentsubsequent analysis analysisbyX-ray by X-ray densitometry. Multi-stemmed numbered) Arid-zone format ( Rainforest buttressed number Sub-humid format (temporary Plantation species other Tem U1 0 hz:I porary NJ el- H e single - - d) (D° CD p cl-

forest ti trees • U) trees H- wo forest cl plot trees with or plot odl (17H)j0 got 0 1-1 trees. CD O not and. not cl- c+ Fi II " ~3 ¥' ~: r' ~"" 2 ~'X ",utlt.~t."T , ..,!2.~:t "0'" " \l , L I

SG1743,25 ... to. .,.. .; ., .e •• " .. , " I s RECORD I > I) ~~. , &0 1! cobE S"' lilT ~If , T . 0, T

tvg,qw/- T " t' ~.

.. bAMPre4 0- ~

. .0. Es -n e , i'" ~,:~~~~iJ~3 '" e", J~; 2 .; ~ I- % I

I C-0.08 unr ,. II. bD ~ , 2. ,i -rape' ~ ... " " U FO~IATS • " ~: tot 2 j!. H6lq Hl" .. , ~olles I, t; Ta.~6' _ 0 I, .. 1:1 ~

~ Pet 1es "" r , ,,,,, ;2~

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In either of cases (1)(1) or (2)9(2), false rings or very faint rings should bebe checkedchecked against climaticclimatic records.records. With X-ray densitometry,densitometry whichwhich producesproduces aa numericalnumerical estimateestimate of ring widthwidth andand density,density, directdirect correlationcorrelation withwith climaticclimatic variablesvariables isis possible.possible.

The age at which the treetree reachedreached aa givengiven heightheight isis gtvengiven byby thethe numbernumber ofof ringsrings atat the base ofof thethe treetree minusminus thethe numbernumber ofof ringsrings atat thethe givengiven height.height. From this,thiS, height cancan be plottedplotted directlydirectly onon ageage forfor thethe tree.tree.

If the sample tree is a well-fonnedwell-formed dominant, then thisthis height-ageheight-age curvecurve maymay bebe regarded as essentiallyessentially thethe samesame asas aa height-ageheight-age curvecurve derivedderived fromfrom aa permanentpennanent samplesample plot,plot, and may be analysedanalysed inin thethe samesame way.way.

3.4 SPECIALSPEX)IAL MEl'HOmMETHODS OF TREE INCREMENTINCR»IUlT ESTIMATION

3.4.1 Simple Measurements

·On pennanentpermanent plots, tree increment is estimated by taking the difference betweenbetween successive diameterdiameter measurements, divideddivided by the measurementmeasurement interval.interval. For this to be an accurataccurate e procedure:

(1) Individual trees must be clearlyclearly andand uniquelyuniquely identifiedidentified onon thethe plot.plot.

(2) The point ofof measurement anon thethe treetree mustmust bebe preciselyprecisely relocatable.relocatable. Two alter-alter­ native methods areare possible:possible:

(i) paint aa ringring atat thethe pointpoint ofof breastbreast heightheight measurement.measurement. This may be excessively conspicuous under somesome circumstances;circumstances;

(ii) insert a nail a precise distance above the point of measurement (50 emcm is suggested) and relocate the point ofof meaaurementmeasurement with reference to the nail. This nail may also bear the tree identification tag.

3.4.2 Remeasurement onon Buttressed TreesTrees

When trees areare developingdeveloping buttresses,buttresses, itit isis customary to measure diameter atat a refe-refe­ rence point about 11 metre aboveabove thethe buttresses. Since buttresses will extend between remeasurementsremeasurements, itit isis Obviouslyobviously necessary necessary toto movemove thethe referencereference diameterdiameter fromfrom timetime toto time.time.

This procedure makes anyany incrementincrement determinationdetermination fromfrom successivesuccessive remeasurementsremeasurements impossible.

Two approaches maymay bebe adoptedadopted toto countercounter thisthis problem.problem.

!l!:21First Use two reference diameters atat eacheach remeasurement.remeasurement. Then, if it is necessary to move the lower reference diameter, it replaces the upper one and a newnew reference diameterdiameter is formedfonned aboveabove thethe originaloriginal toptop referencereference diameter.diameter. In this way, direct increment estima-estima,­ tion is always possible.

It is recommendedrecamnended thatthat thethe twotwo diametersdiameters shouldshould bebe 1.51.5 mm apart,apart, withwith thethe lowerlower oneone

1 metre above thethe toptop ofof thethe buttresses.buttresses. - 29 -

The process is illustrated inin thethe diagramdiagram below,below.

-d~------~----- ~

firsfirstt measurement second measurement third measurement

L-1 ______ti I ~------~,I I~------r------~I

increment =0::= (d1 on 2nd meas. increment (d1 on 2nd meas. - increment = (d1(d1 on 3rd mmeas.eas. -.. d2 on 1st meas.) / timetime intervalinterval d2 onon 2nd meas.) / time interval

Note that these reference diametersdiameters shouldshould be markedmarked withwith paintpaint oror nails.nails. It is essentiaessentiall to climbcli,!,b the trees withwith laddersladders andand measuremeasure diametersdiameters withwith girthgirth tapes.tapes. OpticalOptical instruments suchsuch asas thethe RelaskopRelaskop areare notnot sufficientlysufficiently accurateaccurate forfor incrementincrement estimation.estimation.

Second The other approach isis toto use girthgirth bands, asas describeddescribed below,below, andand makemake increment estimatesestimates overover short periods (1-2(1-2 years)y ears) before buttresses can significantly influence thethe referencereference diameter.diameter.

3.4.3 Girth Bands

Girth bands may · be constructed locally suitequite cheaply and are particularly usefuluseful for intensive researchresearch in,in, forfor example,example, thinningthinning experiments.experiments. Without a vernier scale,scale, they areare accurateaccurate toto .± ± t mm, which isis quite adequate for increment estimation overover 1 year periods onon tropical species.species. With the additionadd~tion ofof aa verniervernier scale,scale, thethe accuracyaccuracy increasesincreases to ± 0.05 mm diameter.diameter. At this level, they can indicate physiological rresponsesesponses and seasonalseasonal growthgrowth fluctuations.fluctuations. (Conventional girth measurementsmeasurements by contrast cannotcannot give increment more accurately than .±± t anon diameter even onon quite smallsmall trees andand areare much worse onon very large trees.)trees.,

The basic requirements for construction are as follows:

(1) Basic materials requiredrequired areare aa rollroll ofof 11 cm wide stainless steel hand,band, a largelarge length~ength ofof 1/3 cmem coilcoil spring,spring, scalesscales templatestemplates forfor thethe mainmain andand verniervernier scales,scales, matt-blackmattblack ceramic ceramic paint,paint, aa shapedshaped holehole punchpunch forfor thethe springspring fixingsfixings andand a gas or electric oven.oven.

(2) Suitable lengths ofof handband are wrapped around emptyempty tins and painted anon one sidsidee with the black.paint.black ·paint. The tin, with the band, isis baked toto curecure thethe paint.paint. -3o-- 30 -

(3) A zero marl<:mark is scratched inin the paint atat oneone endend withwith aa metalmetal point.point. If a vernier scale isis toto be added, it is included at this point, with thethe positionposition of the graduations being defined by thethe vernier template.template.

(4) The approximate girth ofof the tree mustmuet be knownknown before addingadding thethe 'long''long' scale,scale, aboUtabout 1010 cmcm actualactual lengthlength (equivalent(equivalent toto c.c. 3 cmem of diameter), so that the band can be cut to the correct length and the scalescale putput onon at the correctcorrect point.point. TheThe scale is scratchedscratched ontoonto thethe band usingueing thethe appropriateappropriate scalescale template.template.

(5) Slots for thethe retainingretaining springspring areare punchedpunched inin thethe band,band, oneone outsideoutside thethe zerozero mark (&(& vernier scale),scale), thethe otherother insideinside thethe longlong scale.scale. The spring is fitted through oneone end,end, the band placedplaced aroundaround thethe tree,tree, takingtaking carecare toto removeremove looseloose bark and avoid sag at the backback of the tree, and then thethe free end of thethe springspring is slotted in.in. The spring should be cut to a length that gives a good positive tension to thethe band.band.

Increment is measured as the scales cale movement between the initialinitial andand finalfinal reading.reading. The scales are normallynonnally inTramin"em units,units, givinggivilfg directdirect readingreading ofof diameter.diameter. If the starting point of the long scalesoal e is placed an exact number of centimetres from thethe zero mark,mark, thethe handband alsoalso givesgives anan absoluteabsolute readingreading ofof diameter.diameter. If the long scale starts at an arbitrary point, then absolute diameter (as(as opposed to increment) should bebe measuredmeasured withwith aa separateseparate girth tape justjust aboveabove oror belowbelow thethe hand.band.

Templates for cuttingcutting thethe bands cancan bebe purchasedpurchased fromfrom forestryforestry instrumentinstrument suppliers,suppliers, or manufacturedmanufactured locally in any well equippedequipped workshop.workshop.

Various kinds of sophisticated girth bands,bands, some equipped for telemetry,telemetry, cancan bbee purchased directly. Because of their expense, these should only bebe used for the mostmost inten­inten- sive kinds ofof research andand underunder closeclose supervisionsupervision ofof thethe experimentalexperimental areaarea againstagainst animalanimal or human damage.

3.4.4 Growth Ring If,Measurements easurements

Where growthgrowth ringsrings areare present,present, theythey can can be be used used to to estima-te estimate increment.increment. The momostst reliable method is by thethe use ofof stemstem sectionssections onon completecomplete felledfelled trees,trees, takentaken atat d.b.h.d.b. h.

The width ofof the last three oror four annual rings should be measured on two diameters at right angles onon the section.section. These diameters shouldshould be alongalong thethe majormajor andand minorminor axes8JCes if the section is elliptical.elliptical. This gives the periodic underbark increment.increment. It is necessary to construct a regression relatingrelating underbark diameter measured directly, with overbark diameter measured with a girthgirth tape to convert the incrementsincrements underbark onto a common scale withwi th normal d.b.h.d. b.h. measurements.measurements.

Increment can also be estimatedestimated fromfran samplessamples boredbored fromfrom thethe tree.tree • . This is subjectsubj ect to numerOUsnumerous errors,errors, especiallyespecially inin speciesspecies with softsoft timber. The resultant core may be stretched oror spirallyspirally compressed.canpressed. It may not be accurately radial.radial.

Because the difficultiesdifficulties inin countingcounting growthgrowth ringsrings inin tropicaltropical areasareas oftenoften requirerequire resort to calibrated microscopes or X-ray densitometre, complete sections should be regarded as essential.essentia.l. The use ofof increment borers isis not usually aa possible option.option. - 31 -

3.5305 INDIrux:TINDIRECT 'rREETREE DOMINANTDOMINANT HEIGHTHEIGHT ESTIMATIONESTIMATION

Because tree height estimation is a relatively slow procedure,procedure, itit isis notnot usuallyusually desirabledesirable to measure more than 8-108-10 treestrees onon aa plot. If the heights of all trees onon the plot are required, then aa height-diameterheightdiameter curvecurve ofof thethe formform

h = bo4-bid b2d2

can bebe constructed. Some special points should be noted:noted:

(1) The calculation method forfor this regressionregression isis givengiven inin AppendixAppendix 2.2,2.2, togethertogether with an example.exampl e.

(2) With more than oneone plot inin ae. single stand,stand, thethe samplesample treestrees shouldshould bebe pooledpooled to fit the height/diameter regression.regression.

(3) Samples between stands ofof varying densitydenSity oror ageage shouldshould nevernever bebe pooledpooled forfor detenninationdetermination ofof the regressionregression unless comparisoncanparison ofof separateseparate regressionsregressions (Appendix A 2.8) shows therethere isis nono significantsignificant difference.difference.

(4) The fitted regression should notnot bebe usedused toto predictpredict heightsheights unlessunless threethree condi­condi- tions are satisfied:

(i)(0 The regression F value isis significantsignificant atat thethe 95% leVel.

(ii) The bb1 coefficient isis positive. 1 (iii) ThThee b coefficient isis negative.negative. b22

(5) Once the regression has been calculated and tested to see if it is uable, mean height H can be estimated as the height predicted when Dg is substituted forfor dd in the regression.regressiono

(6) Similarly, dominant height Ho is predicted whenWhen thethe meanmean diameter of thethe domi­domi- nants (i.e.(i.e. 100 largest diameter trees perper ha), symbolised byby Do.Do, isis entered inin the regression for d.

Referring back to point (4),(4), itit shouldshould bebe notednoted thatthat whenwhen thesethese conditionsconditions areare notnot satisfied, it is safer to assume that the individualindividual tree heights h used for volume estima-estima­ tion are equalequal toto thethe standstand meanmean heightheight H.H. This situation arisesarises inin manymany tropicaltropical speciesspecies because thethe variationvariation inin treetree heightsheights isis unrelatedunrelated toto diameterdiameter differencesdifferences andand thethe instru-instru­ mental errorerror involvedinvolved inin measurementmeasurement isis greatergreater thanthan thethe effectseffects ofof diameterdiameter on on height.height. Significant regressions and well-developedwelldeveloped relationshipsrelationships betweenbetween treetree diameterdiameter and height are more likely to occuroccur with more shadetolerantshade-tolerant species species and and at at higher higher stockings stockings per per ha. ha.

In mixed age stands, the regression will always be significant and can always bebe used forfor individualindividual tree heightheight determination.determination. However, care should still be taken not to pool regressionsregressions forfor different standsstands withoutwithout adeauateadequate statisticalstatistical teststests forfor thethe homogeneityhomogeneity of the data. - 32 -

If the height-diameterheightdiameter regressionregression isis fittedfitted toto anan ageage seriesseries ofof plantations (i.e.(i. e. aa series of stands of different ages) it willwill alsoalso bebe significantsignificant.. ItIt shouldshould bebe notednoted howeverhowever that this is a different kind ofof model to the one fitted within a unifonnuniform ageage standstand andand willwill nctnot reliable predict individual tree heights withinwithin one unifonnuniform standstand (although(althoughit it will predict mean height H asas aa functionfunction ofof meanmean diameterdiameter DgDg providedprovided thatthat standstand densitydensity isis constant).constant). Within a single-agesingleage stand,stand, thethe regressionregression reflectsreflects dominancedominance differences.differences. Between age classes, itit reflectsreflects aa time-dependenttimedependent growthgrowth relationship.relationship. - 3333 -

4. DATA STORAGESTORAGE SYSTEMSSYSTBIS

4.1 ADVAli'l'AGE!3ADVAVTAGES OF CCNPUTER-BASEDCONFUTER-BASED DATA STORAGESTORAGE SYSTEMSSYSTEMS

A computel'-basedcomputer-based data storage system will storestore allall permanentpennanent samplesample plot,plot, experi-experi­ mental plot andand temporarytemporary plotplot datadata anon magneticmagnetic tapestapes oror discs.discs. This data can be accessed quickly for summarization andand analysis.analysis. It can be subjected to automatic errorerror checking procedures toto Din-pointpin-point doubtfuldoubtful measurements.measurements. It can bebe updated or corrected relatively easily.

ComputerComDuter informationinfonnation storage (CIS)(CIS) stillstill requiresrequires maintenancemaintenance ofaofa conventionalconventional filing system on each permanentpennanent plot oror experiment,experiment, inin which>lhich originaloriginal fieldfield sheetssheets andand anyany queries,queries, notes, diagrams andand proceduralprocedural ordersorders areare kept.kept.

Un-tuUntill recently,recently, thethe capital costcost of aa computercomputer systemsystem andand thethe lacklack ofof skilledskilled person­person- nel inhibited the use ofof CIS. With the advent ofof the microcomputer, costscosts havehave fallenfallen drastically andand areare nownow ofof thethe samesame orderorder asas thosethose forfor aa motormotor· vehicle.vehicle. r,acrocomputersMicrocomputers usually operateoperate in BASIC, which is a language designeddesigned forfor easyeasy learning.learning. Anybody with a suitably facilefacile brain cancan teachteach themselvesthemselves BASIC,BASIC, givengiven accessaccess toto aa microcomputer.microcomputer. In addition, some forestry institutions offer specialized training in data processing forfor graduate or technical staff.1/

Lack of CIS is a significant inhibiting factorfactor inin thethe developmentd evelopment andand validationvalidation ofof yield models andand effectiveeffective yieldyield planningplanning andand control.control. With manual extractionextraction ofof data fromfrom files, onlyonly the simplestsimplest kindskinds ofof modelmodel cancan bebe constructed,constructed, whilstwhilst validationvalidation byby residualresidual analysis is nctnot possible because ofof thethe stupendousstupendous amountamount ofof workwork involved.involved. Nor is it pos­pos- sible to examineexamine alternativealternative modellingmodelling strategiesstrategies onon thethe sameBame datadata oror toto updateupdate thethe modelmodel functions as new datadata isis collected.collected.

Furthermore, the productivity of skilled staff using manual procedures is extremely low.10>1. Collection ofof adequate quantitiesquantities ofof datadata tendstends toto bebe inhibited,inhibited, becausebecause manualmanual pro-pro­ cedures are simplysimply unableunable toto copecope withwith it.it.

It is strongly recommended that allall forestryforestry organizationsorganizations shouldshould either:either:

(a) Have access to a large computer facility, with an absolute maximum job turn­turn- around of oneone day. This job turnaround should be consideredconsidered from the timetime thethe results are returned andand should take into consideration periodsperiods of thethe monthmonth when the main computer may be totallytotally unavailableunavailable (due(due toto prioritypriority allocationallocation of computer time to otherother users), transportationtransportation difficultiesdifficulties toto oror fromfrom thethe canputer,computer, etc..etc •• orOr (b) Purchase a microcqnputermicrocomputer systemsystem with:with: -32-64k bytes ofof memory -Twin drive diskette oror hard disc systemsystem -A printer

1/ E.g. The Commonwealth Forestry InstituteInstitute atat Oxford,Oxford, U.K.U.K. -34-- 34 -

A teletype that can be interfaced to the computer, with paperpaper tapetape reader/reader/ punch, for use in on or off-line data preparation A BASIC oror FORTRAN compiler.compiler. This type ofof system would cost around S10$10 000 in Europe or North AmericaAmerica atat current prices (mid-1979).(mid-1979).

Option (b)(b) gives the forestry organization an in-house, dedicated computer thatthat is likely to provide greater productivity andand moremore rapidrapid personnelpersonnel trainingtraining thanthan optionoption (a).(a).

4.2 DATA VALIDATIONVAlIDATION

Errors arise inin magnetically-stored datadata fromfrom thethe followingfollowing sources:sources:

(1) Field measurement errorserrOrs (2) Data entryentry oror keypunchingkeypunching errorserrors

(3) Programme errors.errors.

The third type ofof error must be assumed to be eliminated by thorough testing of any computer programmes used to store9store, correct, update, print oror select data. The testing of programmes is the responsibility ofof the programmer?programmer, Whowho is as directly responsible fOrfbr the errors of his programmesprogrammes as is the field worker whoWho fails to make correct measurements.meaeurements.

Data entry oror keypunching errors are largely eliminated by the process ofof verifics.­verifica- !iE!:!,tion, which should always be used with any largelarge mass ofof data. Verification involves entering every item of data twice, by two different operators or in two separate runs. The two data sets are then checked against eacheach other,other, automatically, and any inconsistencies reported to the operator, who can supply an appropriate correction. The actual details of the verification process will depend onon the data entryentr,y systemsystem used. Most data preparation organizations will offer verification as a normal service and this should always be expli-expli­ citly stipulated Whenwhen submittingsubmitting datadata forfor keypunching.keypunching.

Field measurement errorserrors cannotcannot be wholly eliminated,eliminat ed, but theythey cancan be reduced:

(a) By attention to the training andand morale ofof fieldfield workersworkers andand by provision ofof suitable instruments forfor theirtheir use.use.

(b) By running data~ checkingchecking programmesprogramnes anon thethe magnetically storedstored data.data. The data checking is based anon examinationexamination forfor logicallogical inconsistencesinconsistences suchsuch asas veryvery largelarge or very small measurements,measurements, negative increments inin diameterdiameter oror height, missing tree numbers oror tree numbers whichwhich havehave reappearedreappeared. sincesince aa previousprevious harvesting,harvesting, changes in species identificationidentification andand soso an.on.

Any such logical inconsistency isis reportedreported byby thethe datadata checkingchecking programmesprogrammes andand mustmust be examinedexamined to determine the likely source of the error. A correction must then be supplied,supplied, using a ~data editing programme, to amendamend thethe magnetically-storedmagnetically-stored data.data.

The en-tireentire processprocess isis anan ongoingangoing one,one, asas indicated diagrammatically below. --35-35 -

CoIlCollect ect data in the field

t Return toto field KeypunchingKeypunching to determine true value t ~ supply~ value 0fO t ° :~:!~tionscorrections VVerificationen ~ca l.on AnyAny errors?errors ? ---Yes./"! " No Data storagest~rage on "" magnetic media "" Is correction possible in the officeoffice Programme checks t Any logical errorserrol'S ? ----Yes-_ Yes ______--1

No1 Data ready for analysis

4.3 CONTRACTSCONl'RACTS FOR THE PREPARATIONPREPARATION OF COMPUTERCOMPUTER PROGRATIMESPROGRAMI>I:El3

Contracts for computer programme preparation should contain clausclauseses coveringcovering thethe following conditions:conditions:

(i) Full provision by thethe contractorcontractor ofof sourcesoullCe listinoslistings (in(in BASIC9BASIC , FORTRAN or other language used) of any programmes written.written.

(ii) Full documentation ofof allall programmes9programmes, including:including:

(a) A dictionary ofof meaningsmeanings forfor identifiersidentifiers oror variablesvariables usedused inin thethe programme.programme.

(b) Flow diagrams indicating the sequence ofof operation of any programmes or subprogrammes written byby the contractor andand the sequence andand naturenature of data transfers fromfrom externalexternal mediamedia andand magneticmagnetic storage.storage.

(c) aplicitExplicit definition ofof the recordrecord structure and usage of all magnetic files and all input andand outputoutput media.media.

(d) Explanation of theory behind the programme method, together with references to textstexts oror otherother backgroundbackgro.md material.material.

(iii) Liability by the contractor for any progrmmneprogramme errors or for any failure of the programmes to operateoperate as specifiedspecified whenwhen usedused inin accordance with the documentation.

(iv) Instruction by the contractor of some member of the forestry organization staff in the use ofof the programmes9programmes, up toto thethe point where the programmes can be demon-demon­ strated to operateoperate to the satisfacticnsatisfaction ofof thethe forest management withoutwithout any supervision by the contractor.contractor. - 36 --

(v) copyrightCopyright over all supplied documentation, programmes and reportsreports to be vestedvested in the forestry organization.organization.

It may be possible to reducereduce thethe costcost ofof aa contractcontract toto somesane degreedegree byby relaxingrelaxing pointspointe (iii) and (v),(v), but the other points shouldshould alwaysa lways be insistedinsisted anon ifif thethe programmesprogrammes writtenwritten are to be ofof any continuingcontinuing useuse toto thethe organization.organization.

4.4 STORAGE SYSTEMS FOR PlOTPLOT DATA

4.4.1 Introduction

It is not the intention to provide detailed descriptions of programmesprogrammes forfor etoringstoring and summarizing permanentpennanent plot, experimentalexperimental plot oror temporary plot data. These will proba-proba­ bly need to vary a goodgood deal according to the computingcanputing facilities available. Only the file structures forfor input and outputoutput and the types of functions the programmes can perfonn,perform, will be describeddescribed here.here.

4.4.2 File Structures

A file, in computingcanputing terminology,tenninology, consists of somesane body of machin~readablemachine-readable infonna-..informa, tion onon magnetic media (tape(tape oror disc) oror onon punched cards oror paper tape. Input filfileses provide data for a particular programme andand may themselves be outputoutput files from another programmeprogramme..

File structure defines, in the present context, the organization of the different types of informationinfonnation that must be grouped together to represent a single plot.

For pennanentpermanent plots, the basic informationinfonnation anon a single plot may be defined asfollows:as follows:

- Plot initial assessmentassessment

1st1 st plot measurement 2nd "n lt.. RepeatRepeated ed anon the 3rd ".. .. tape/disc for •0 each plot 0 0• •0 •0 •0 n'th plot measurement --Ehd-of-plotEnd-of-plot record

For temporary plots, the recordrecord structure is much simpler,simpler, being only:only:

Plot 1 assessment

Plot 22 11..

Plot 3 11.. • • • • • • Plot n assessment -

- 37 -

These record structuresstructures relaterelate toto thethe permanentpennanent datadata base,base, -whichwhich maymay bebe onan eithereither magnetic tapetape oror disc.disc. ForPor input of data for pennanentpermanent plots,plots, two typestypes of differentdifferent ininputput may be possible:

(i) Initial assessmentsassessments

(ii) Remeasurement data.data.

In cascasee (i),(i), a new plot would be created in the databasedatabase,, butbut thethe programmeprogramme wouwouldld needneed to checkcheck thatthat plotplot identificationidentification diddid notnot conflictconflict withwith anan existingexisting plot. plot.

In case (ii),(ii), an existingexisting plot would have to be present (otherwise anan error wwouldould bebe reported),reported), withwith thethe newnew measurementmeasurement beingbeing atat thethe appropriateappropriate point.point.

RxperimentalExperimental plots could normally bebe treatedtreated inin thethe samesame wayway asas permanentpennanent plots.plots. HO\;eHowever,v er , relatedr elated treatmenttreatment plots would be placed adjacentadj acent to eacheach other on ththee ttape,ape , for exampleexample asas :

Treatment 1 {Tre7ent Block 11 -

- TreatmentTreatment n

- Treatment 1 { Trea~ment Block 2 -

-TreatmentTreatment n •

{ Treatment 1 Block m TrealmentTreatment n

To retrieve all the data for the experiment from the tape,tape, n x m ploplotsts would be read ffromrom the ststartingarting position on the tape (or(or disc) or the first plot inin thethe experiment,experiment .

44.4.3. 4. 3 Error Checking and Editing Functions

The following error checks are required for pennanentpermanent plot information:information:

That the plot exists on tape (if(if a new plot is not being added to thethe syssystem)t em) or doesdoes notnot yetyet exist (if a new plot .!:.!!is beingbeingadded).aMed).

(2) That the remeasurement dates are in a consistent and ascending sequence.s equence.

( 3 ) That trees which have been thinned (i.e.(i. e. disappeared inin past measurementmeasurements)s) ddoo not reappear on later measurements.

(4 ) That trees do not change speciesspecies (on(on mixed forestforest plotsplots onlonly)y ) or plotplotss do not change species (on(on plantation plots). - 38 -

(5) That diameter increments areare positive andand notnot excesSivelyexcessively large.large.

(6) Checking of the size rangesranges ofof parameters suchsuch asas height,height, diameter,diameter, speciesspecies code, etc. for excessivelyexcessively largelarge oror smallsmall valuesvalues oror unrecognised codes.codes.

On temporary plots, onlyonly the checkschecks inin the 6th category areare possible.possible.

Data should be added to the database eveneven thoughthough itit containscontains errors;errors; an editing programme may then be usedused toto manipulatemanipulate andand amendamend thethe magnetically-storedmagnetically-stored data.data. With any significant amountamount ofof data,data, thisthis isis usuallyusually muchmuch moremore convenientconvenient thanthan attemptingattempting toto correctcorrect the original punched cards.cards.

4.4.4 Plot Summaries

The plot summary programme may incorporate the error checking proceduresprocedures (or theythey may be inin aa separateseparate programme).programme) . Its main function will be to produce summarizedsummarized informa-informa.­ tion relevantrelevant toto thethe particularparticular forestforest typetype inin question,question, forfor useuse inin furtherfu~her analyses.analyses. For plantations,plantations, the summariessummaries may includeinclude plot mean and dominant height, mean basalbasal area diameter, basal area,area, stockingstocking andand volume.volume. For mixed forest, the summaries will normallynormally also include stand tables ofof selectedselected characteristicscharacteristics byby speciesspecies andand sizesize classes.classes.

The summary programme shouldshould be ableable toto produceproduce itsits outputoutput bothboth inin aa printedprinted form,fom, with>lith titling text to make it comprehensiblecomprehensible and in machine readable form, without text, for direct inputinput toto datadata analysisanalysis programmes.programmes. The machine readable outputoutput may be put ontoonto magnetic tape oror discdisc oror punchedpunched anon paperpaper tapetape oror cards,cards, asas appropriate.appropriate. If the output isis

onto magnetic tape, this shouldshould normallynormally bebe donedone inin 'formatted'1 formatted' oror character-encodedcharacter-encoded form,form, rather than inin machine binarybinary code.code.

For mixed forests in particular, but also for plantations, it is useful toto have a facility so that only selected parts ofof thethe outputoutput areare printedprinted oror placedplaced onon tape.t ape.

4.4.5 Otherother Utilities

Two other programmes willwill probablyprobably bebe neededneeded withwith aa samplesample plotplot datadata base:base:

A sorting programme, able to reorder the sequence of plots on tape so that they are grouped by forest, district,district, species,species, compartment,comnartment, etc..etc.. Norma1ly,Normally,plots plots willwi1l be enteredentered ontoonto thethe datadata basebase inin anan arbitraryarbitrary order,.order,. whereasWhereas thethe summaries will probably be preferred withwith somesome logical sequence.

Archival and character-encodingcharacte~encoding programme,programme , to transfertransfer the main data base on disc or tape into a form suitable for archival (see(Bee below) or transfer to another computer site.site. This will involve character-encodingcharacte~encoding or formatting, the entire data base and writingwriting itit antoonto aa magneticmagnetiC tape.tape.

4.4.6 Data Base Security

A large data base stored onon disc or magnetic tape can readily be destroyed by acci-acci­ dents, programme oror computercomputer failures.failures. It is essential therefore that after each addition of a significant amOUJltamount of information, the entire data base isis copied ontoonto a spare tape or disc to give a completecomplete secondseccnd copy.copy. These spare copies can be rotated, so that at anyoneany one time there is an up tote date working version,vereion, an up - to date archived version and two or three previous archived versions ofof the datadata base. - 39 -

4.5 DATA TRANSFERTRANSFER BETWEEN COVPUTERC(),PUTER SYSTEMSSYSTEl~S

The necessity frequently arises for transferring datadata between computer systems,systems, either for cooperatcooperativeive research or toto permitpermit research workersworkers fromfrom oneone organizationorganization toto studystudy elsewhere using theirtheir ownown data.data.

Large amounts ofof datadata areare best transferredtransferred anon industry-standardindustry-standard magneticm"8'1etic computercomputer tapes. These may be 9-track9-track oror 7-track7-track tapes.tapes. The following information shouldshould bebe ascer­ascer- tained whenwhen thethe tapetape isis written:"Titten:

The number ofof tracks (7(7 or 9) The density (usually(usually 800800 oror 11 600 bits per inch)inch)

The parity (even(even oror odd)odd) - The inter-blockinteI'-block gapgap inin mmmrn (or(or failingfailing thisthis thethe exactexact markmark andand manufacturermanufacture; ofof the tape drive mechanism).

Tapes for transfer between computers should always bebe character-encoded or formatted and shouldshould preferably use fixed-lengthfixed-length recordsrecords ofof moderate size, probably less than 120 characters per record,record, toto facilitatefacilitate readingreading thethe tape.tape. The type of characte:r-encodingcharacter-encoding usedused (EGCDIC,(EICDIC, ASCII, BCD,BCD, etc.) should bebe determineddetermined ifif possiblepossible butbut isis notnot critical,critical, asas trans­trans- literation fromfrom oneone toto thethe otherother isis quitequite simple.simple. A listing of the first and lastlast fewfew hundred lineslines ofof the tapetape shouldshould be sentsent with it to helphelp check that when the tape is read?read, nOno records oror parts ofof recordsrecords have been lost.lost.

Header labelslabels and tapemarks are generally a nuisance when reading strange tapes, so it is better toto writewrite thethe tapetape asas aa singlesingle file,file, withoutwithout aa tapetape labellabel atat thethe start.start. The end of informationinfonnation isis normallynormally indicatedindicated byby aa doubledouble tapetape mark.mark.

Information concerning the mode of the tape (tracks,(tracks, density, parity,parity, charactercharacter code),code), the contents and the address ofof originorigin shouldshould be attached to the tape with a sticky label.label.

Tapes cancan normallynormally bebe sentsent easilyeasily throughthrough thethe post.post. However, high frequency metal detectors may eraseerase somesome oror allall the informationinformation anan a tape. The parcelparcel should therefore bebe clearly marked and easilyeasily openedopened forfor visualvisual inspection.inspection. 40- 40 -

5. ANALYSIS OFOF GROWTHGROWTH AND YIELDYIDLD DATADATA FORFOR UNIFORMUNIFORM FORESTFOREST

UnifomUniform forests areare thosethose inin whichwhich thethe mainmain cropcrop treestrees areare ofof aa known,known, uniformuniform age.age. They are usually composed ofof a single species or a few ecologically similar species.species. Uniform forests are, by definition, managed under a clear felling system, with or without intermediate thinnings.thinnings. Regeneration maymay bebe byby planting,planting, artificialartificial or naturalnatural seedingseeding oror from coppice.coppice.

In thisthis typetype of situation, thethe mainmain parameter"parameters of growthgrowth andand yieldyield predictionprediction areare wellVIell understood. There is a wide variety ofof possible models available.available. The main limiting factor on the effectiveness of a model is usually the availability of data for the forests in question coveringcovering aa widewide rangerange ofof sites,sites, agesages andand standstand densities.densities.

The gro,1thgrowth and yield ofof aa forestforest cancan bebe modelledmodelled atat threethree basicbasic levels.levels. These are the whole stand, the sizesize classclass andand thethe individualindividual tree.tree. For uniform forests,forests, stand models are usually adequateadequate forfor mostmost purposes.purposes. Stand models are also very much simpler to bothboth construct andand use.use. Consequently, this is the modelling strategy dealt with mostmost fully; the otheotherr two approaches are discussed briefly in section 5.7.

Even within this single levellevel ofof stand modelling, there is a wide variety of choice in the particular setset ofof functionsfunctions toto bebe incorporatedincorporated inin thethe model.model. Some alternatives are nresented inin thethe differentdifferent sectionssections whichWhich follow,follow, togethertogether withwith anan attemptattempt toto definedefine situa-situa­ titionsons in which a particular method is most appropriate. Other alternatives have been omitted simpi.simpl:; hecausebecause ofof thethe needneed toto keepkeep thisthis manualmanual reasonablyreasonably concise.concise.

On the Whole, techniquestechni.ques are incorporated which are characterized by simplicity, accuracy andand flexibility.flexibility.

Within a particular aspect ofof standstand modelling,modelling, suchsuch asas forfor exampleexample thethe productionproduction ofof sets ofof basal area/height/stand densitydensity curves,curves, anan attemptattempt hashas beenbeen mademade toto includeinclude bothboth graphicalgraphical methods and statisticalstatistical methods ofof differingdiffering degreesdegrees ofof complexity.complexity.

5.1 SITE CLASSIFICATIONCLASSIFICATION

The relatively high accuracy possible withwith growthgrowth andand yieldyield modelsmodels forfor uniformuniform standsstands resultsresults partly from the precision withwith whichwhich itit isis possiblepossible toto classifyclassify site.site. This is itself a result of the fact that ageage isis normallynormally knownknown fromfrom managementmanagement recordsrecords andand heightheight of the dominant trees can usually be measured byby hypsometershypsameters oror similarsimilar instruments.instruments.

5.1.1 Use ofof Dominant Height as an Indicator of Site

The height ofof aa uniform stand,stand, atat aa givengiven age,age, isis aa goodgood indicatorindicator ofof thethe potentialpotential productivitypr oductivity ofof thatthat typetype ofof forestforest onon thatthat particularparticular site.site. Hence the construction of height/age curvescurves correspondingcorresponding to differentdifferent sitesite classes is the first stepst·ep in growth and yieldyield model construction.construction.

However, the mean height of a stand is usually sensitive not only to age and site class, but alsoalso toto standstand density.density. Consequently, dominant height is normally used in defining the heightheight ofof aa stand.stand. Dominant height isis almostalmost entirelyentirely insensitiveinsensitive toto standstand densitydensity differences. - 41 -

DaninantDominant height can be defined in variousvarious ways, butbut thethe definitiondefinition with thethe widest currency is that the dominant height of a stand is the mean height ofof thethe 100100 thickestthickest stemsst ems per hectare.hectare. DominantDaninant height is also sometimessometimes termed'toptermed'top height'.height'.

Under some circumstances encountered in uniform forests in the tropics, dominant&eminent height ceases to be aa goodgood indicatorindicator ofof sitesite class.class. This occurs with youngyoung stands of VB1~·very fast growinggrot-ling crops and also with certain species which are notoriously variable in theirt heir height growth, suchsuch asas PinusPinus caribaea.caribaea. This situation can be detected by ranking permhlJepermanentr-.t plotolot datadata byby heightheight withinwithin eacheach ageage class.class. If the rank position of plots anon successivesucces.ive occasions is poorly correlated,correlated, then any sitesite classclass curvesCU~Jes constructed must be considered of doubtful value.value.

The problem arises simplysimply because of the great variability of height growth,gl"oNth, relativerelativ e to the effecteffect ofof site,site, onon thesethese typestypes ofof stands.stands. It could bebe partly overccmeovercome by a redefini-r-edefini­ tion of dominantdcminant heightheig..llt to requirer~quire a larger sample ofof height trees per plot, e.g.e.g. equivalentequivalent to 200 oror 4001\-00 stems/ha.stems/ha. linAn alternative idea is to correlate final productivity withl-rl:th envi-~nvi·­ ronmentalronrnental variables and use a sitesite classificaticnclassification based purely onon slope,slope, altitude,alt itude9 soilsoil typetype or other factors which appearappear toto be significant.significant.

5.1.2 Construction of Site IndexIndex CurvesCurves

The height-age-site indexindex relationshiprelationship isis basic toto uniformunifonn forestf vr ~' .:;~.; growthts."'l'owt h psedietioh.p ....~td1.c ti o.:L The relationship is usually referredreferred to simplysimply asas thethe sitesi-; e indelcinde::.c CUl"vescurves forfor a.a spspeciesecies inin aa given environment.environment.

Construction of sitesit e index curves may behe by graphicalgraphical methods oror byby regressionrGgressicri analysis.

5.1.2.1 Gra.hicalGraphical methods of coconstructionnstruction

Graphical methods ofof construction proceedproceed. asas follows:follows:

1. Plot all availablea'Tailable height-age data forfor standsstands ofof thethe speW.essp €G:.es in~.n question.qlH:sti on G DominantDaninant height should be used, not mean height,height, asas itit isis muchmuch moremol'~ ! indepuldentinder .';llde..."1t of variations inin standstand density.density. Both temporarytemporary andand permanentpeman"nt ploplots-, s mayma y be in-in­ cluded onon thethe graph.graph. With pennanentpermanent plots,plots, the pointspoints fromfrom succ6sssuccessivei"f e J:"Sremeasure-llea sure­ ments shouldshould be joined,joined withwith straightstraight lines.lines. This stage is illustrated in figurefigure 5.1, which shows data from Pinus patula standsstands inin Uganda.

2. Next draw curves by hand through thethe data.data. These should attempt toto follfollowoll tthehe trends of:of:

(i)(0 The plots onon thethe lowerlower edgeedge ofof thethe massmass ofof data;data;

(u) The median tendencytendency throughthrough thethe data;data;

(iii) The upper edgeedge ofof the data.

In each case the curves should follow as parallel as possiblepossible toto thethe tendenciestendencies of pemanentpermanent sample plots on that part ofof thethe graph.graph. The drawing ofof these three curves is shownshown inin figurefigure 5.2.5.2. Figure 5.15.1

DATA FROMFM PERMANENT PEmWlENT AND AND TEMPORARY TUlPORARY SAMPLESfJ-lPlE PLOTSPLCTS READYREADY FORFOR CONSTRUCTION OF OF SITESITE INDEXINDEX CURVESCURVES 35 -

+

30 + ~ + -5 1: +-sp + .... + :t ""CD 1l 25 t + +* + ~ + .;l + B A 20 1- + + .... + + f\)

15

10

s5

o __ .1 0o 2 4 6 86 10 12 14 16 18 20 22 24 26 2628 30

Age~ (years) Figure 5.25.2

HANDDRAWNHAND-DRAWN SITE SITE INDECINDEK CURVESCURVES SKEICSKETC HEM HED THROUGH'!'"dROUGR THETHE DATA TO REPRESENTREPRESENT THE NMINIMUN, ININUN 9 Jr.·l EDIANIIDI AN ANDA ND J.IAXTIHlt' NAXIMIIIi TRENDSTRENDS 35

+

30 ~ ~ ~+ E ~ + + ~ ~ t fo Qj 25 + .<: + § + .,..s:: + B 20 + I=l + .,. w

15

10

5

o 1 L __ o ;' 4 6 8 10 12 14 16 18 20 22 24 26 28 30

Age (years(years) ) FFigure'igu r e 5.3

TRACING OFOF THETHE THREETHREE HA MAINI r SSITEI TE INDEXniDE:< CURVESCURVES KWITHITH TWOTWO IINTERPOLATEDNTERPOLATED INTERMEDIATEINTEZIEDIATE CURVESCURVES ADDEDADDED AND FACHEACH CURVECURVE ASS ASSIGNEDI GNED A A SITE CLASSCLASS 35 - Site Class r 30

~ _1I -5 - ~1lI i... 25 ..... - Q) - .<: '" ~ -N' ....:: / 0 20 ~ /' - -- ..... - ~y ..,. I # I -' ~ 15 --

10 ,

5

o __ L _ -.t __ l _____ ._L __ o 2 4 6 86 10lo 12 14 16 1618 20 22I 24 26 26 30

Age (years) - 45 -

WhenWhen all of the data isis fromfrom temporary plots,theplots, the samesame methodcanmethod can bebe applied, but therethere is "a possibility of considerable error duedue toto thethe factfact thatthat thethe plotsplots atat differentdifferent ages may not be equallyequally representativerepresentative ofof differentdifferent sites.sites.

3. TwoTwo additional curves can now be interpolated between thethe upperupper and centralcentral curves and lower and central curves. The system can then be traced onto a separate piece of paper, giving the result shownshown inin figurefigure 5.3.5.3.

4. The curve system can be described as an equation usingusing thethe methodsmethods describeddescribed inin AppendixAppendix A.1. The curves themselves are usually numbered sequentially and referred to as site class curvescurves (or(or sometimessometimes yieldyield class,class, productionproduction class,class, etc.). ThusThus in figure 5.3 the site classes are numbered from I (most productive)productive) toto V (least(least productive).

The simplicity ofof thisthis techniquetechnique ofof constructingconstructing sitesite indexindex curvescurves byby graphicalgraphical methods isis ddvious.obvious. It has three significantsignificant disadvantages:disadvantages:

(i) The curves produced depend tc aa great extent upon the judgement of the person doing the work, especiallye"pecially if the data is sparse or largely from temporarytemporary plotsplots.. Different people will produceproduce differentdifferent setsaeta ofof curves,curves, whichwhich maymay bebe more oror lessleaa accurate and unbiased inin representingrepresenting thethe realreal trend.trend.

(ii) When there is a large amount of data and it is already stored inin aa formform acceptableacceptable to a computer (e.g.(e.g. onon 80 column Hollerith cardscarda or on magnetic tape), thenthen this isia a very slowslow method compared to statistical techniques which can bebe carried out by computer.computer.

(iii) When stage 4 above, ofof describing the curve system as an eauationequation is required (as whenwhen the curves are to be used within an inventory oror growthgrowth projectionprOjection progranune),programme), then thethe workwork involvedinvolved inin thisthis stepstep alonealone maymay bebe asas greatgreat asas the entire task of fitting the curvescurves directlydirectly byby oneone ofof thethe statisticalstatistical techniques.techniques.

5.1.2.2. Mathematical methods ofof fittinfitting site index curvescurves

Tathematical1·;athematical techniquestechniques ofof fittingfitting sitesite indexindex curvascurves havehave considerableconsiderable advantages over graphical methodsmethods when a computer is available and the amount of datadata is large.large. However,However, it should not be assumed that the resultsresults ofof these techniques are necessarilznecessarily moremore accurate than hand-drawnhanddrawn curves;curves; this will depend very much anon the correctness of the height growth model chosen and the validityvalidity ofof thethe statisticalstatistical assumptionsassumptions usedused inin thethe fitting ofof the parameters ofof thethe model.model. Mathematical techniquestechniques cancan bebe classifiedclassified intointo fourfour groupsgroups inin the following way:way: ,

temporary plot permanent platplot - data data I IE proportional minimumrninimurn- nested multiple curves maximum regression regressionregression method without withwith 1:a prioripri ori sitesite indexindex sitesite indexindex - 46 -

Of these four methods, nested regression is statistically thethe mostmost appropriateappropriate andand is also amenableamenable toto manualmanual calculation.calculation. Consequently we shall examineexamine this techniquetechnique inin most detail,detail, discussingdiscussing thethe otherother methodsmethods somewhatsomewhat moremore briefly.briefly.

AllAll fourfour methods can be related to a single model of height growth, Whichwhich is thethe SchumacherSchmnacher equation 1/:

Ho = Hmax.exp(b/Ak)

where Ho is dominantdominant height,height, Hm~ax is a parameter to be fitted andand representsrepresents thethe maximummaximum height the speciesspecies couldcould reach,reach, exp(exp( )) isis the mathematical notation to indicate that the expression in the bracket is a power of the constant e = 2.71828 (i.e.(i.e. exp(2) means e2)e2 ), b andand kk areare parametersparameters toto bebe fittedfitted andand AA isis thethe ageage ofof thethe stand.stand.

The shapes given by this equationequation are illustratedilluetrated in figure 5.4. By takingtaking logarithmslogarithms to the base ee (1n)(In) ofof bothboth sidessides ofof equationequation (1),(1), oneone gets:gets:

in H0 = ln Hm + b/Ak -(2)

If "ewe let a = Inin Hmax7~ax , then aa andand bb cancan bebe fittedfitted byby linearlinear regression,regression, providedprovided kk isis known.known. Appropriate values ofof k for mostm9st species lie between 0.2 and 22 and can be estimated by techniques described laterlater in this sectionsection oror by nonlinear estimationestimation as described in Appendix A.4.A.4. For many specieS,specie~, an assumedassumed value ofof kk =~ 11 willwill givegive aa satisfactorysatisfactory fit.fit. The bh parameterparameter inin equationequation (2)(2) shouldshould alwaysalways bebe negative;negative; if it is not, check calculations for errors.errors. The a parameter will normally be between 2 and 7; again, check for errors if ththereere is largelarge divergencedivergence fromfrom this.this.

FoForr proportional curves fit equation (2) to the set ofof temporarytemporary samplesample plotplot datadata asas a whole by linear regression, with the dependent variable Y as inIn Ho and the predictorPredictor variavariable'n le X asas 1/Ak.l/Ak. If k is not known, follow the suggestions in Appendix A.4 to determine it.

This gives the average height growth trend, assuming that in each age class, all sites have an equal likelihood of being represented. If it is known that, for example,example, older age c1assesclasses fall anon poorer sites and younger onesones anon the best sites, then do not use this method. Either construct hand-drawnhand,drawn curves or, if PSP data is available, use nested regression.

Once the mean height growth curvecurve has been fitted,fitted, curvescurves ofof thethe samesame shapeshape cancan bebe drawn to pass through different sitesite indexindex values.values. If the site index SS is defined as the dominant height ofof the standstand at anan indexindex ageage Ai,At, thenthen thethe aa parameterparameter forfor thethe curvecurve toto passpass through this site index, aia; isis givengiven by:

a;ai == Inln S - b/~ -(3)(3) where bb and k are from the average curve.curve.

111/ Schumacher,Schumacher, F.X., 19391939 A new growth curve and its application to timbertimber yieldyield studies.studies. J. ForestryForestry 37:819-820. - 474'1 -

Dfi.Figure ure_ 5.4

(,,)(a) GapGraph h ofof the Sc:huli1acher.ac1er equation with a=1a~log(Hmax), oon Hmax ) k=1,k~l HmaxFimax and differentdifferent valunsvaluns ofof bb b

5

10

20

o 101 0 20 30 50 Age (years(years) )

(b) Graph of thethe Schumacher Schumacher ecmion equation with wi. a=1 th a=log(Hmax),og (Hmax )2 b=5b=5 Hmax and differentdifferent valuesvalues ofof kk k

1

0.67

0.5

oo 10 20 30 11040 40o Age (years)(years) - 48 --

The muumum-maximumminimum-maximum method isis more flexibleflexible inin thethe typetype ofof curvecurve shapesshapes thatthat resultresult than the proportionalProportional curve method, · but it requires multiple observations inin each ageage classclass (at least 3),), andand hencehence cannot be used with limited amounts ofof data. The process proceeds as follows:

(1) In each age class, calculatecalculate the mean Ho forfor allall plotsplots andand thethe minimumminimum andand maximum values ofof Ho.Ho.

(2) Fit three separateseparate regressionsregressions ofof thethe typetype shownshown inin equationequation (2)(2) toto thethe maximum,maximum, minimum and mean setssets ofof observations.observations. The nonlinear k parameter can be assumedassumed as a constant forfor allall threethree setssets oror itit cancan bebe fittedfitted independently.independently.

(3) As a final step, the separate coefficients for each of thethe threethree lines can, if desired, be harmonized to givegive aa singlesingle equation,equation, usingusing thethe methodsmethods givengiven inin AppAppendix endix A.1.A. 1 •

A more complexcanplex variation on this method is justified with larglargee amounts of datadata in each age class, as might be obtainedobtained fromfran aa forestforest inventory.inventory. The height Observationsobservations inin each class are sorted into order, from maximum to minimum and each pointpoint is assigned aa site class SS from:

sS = (i - *)/t ) / n where i is the plot's position after sortingsorting and n isis the number ofof plots in the age class.class. Once the Plotsplots have been assigned a sitesite class, then the analysis can proceed usingusing multiple regression asas inin thethe lastlast methodmethod describeddescribed below.below.

It should bbee thoroughly appreciated that the above methods for use with temporary sample plot datadat a should be regarded as producing results that are anlyonly of provisional usefulness, asas theythey dependdepend criticallycritically uponupon thethe assumptionassumption thatthat allall sitessites havehave anan equalequal like-like­ lihood ofof beingbeing representedrepresented inin eacheach ageage class.class.

This isi s rarely the case in reality and hence the curves produced will"Till be inin somesome degree defective.defective. The only solution is to obtain recurrent height-age data fromfran permanent plots oror stem analysis trees,trees, whichwhich cancan be analysedanalysed byby aneone ofof thethe followingfollowing methods.methods.

Nested regression methods areare ofof twotwo types.types. There is first of all the use of condi-condi­ tional (or(or zero-one)zero-one) variables inin multiplemultiple regression,regression, asas describeddescribed inin thethe exampleexample inin Appendix 2.10.2.1 0. TbisThis method has not, to the author's knowledge, been used inin sitesite indexindex curve construction,c onstruction, probably because withwith anyany realisticrealistic numbernumber ofof plots,plots, thethe numbernumber ofof variables involvedinvolved inin thethe regressionregression wouldwould bebe enormous;enonnouB; but thethe approachapproach isis by nono means infeasible, givengiven aa speciallyspecially adaptedadapted programmeprogramme toto genera-tegenerate and handle the many zero-onezero-one variables. The second method, firstfirst described by Bailey & Clutter 111/ involves the use ofof the commoncommon slopeslope andand commonCQ'nffion interceptintercept estimatorsestimators fromfran covariancecovariance analysis.analysis. This method isis well suitedsuited toto sitesite indexindex curvecurve constructionconstruction andand isis sufficientlysufficiently simplesimple toto makemake manualmanual calculation ofof the parameters possible.

111/ Bailey, Rol,.R.L. andand ClClutter,utter, J.L., 1974 Base-Age InvariantInvariant PolymorphicPolymorphic SiteSite Curves.Ourves. Forest Science 20:155-59 - 4949-

The commonc omm on slopes lope regression model is depicted in figure 5.5(a) and is given by the equatequation:i on:

Y.=y = al.+b~+bX X (4)-(4) wherewhere ala i isis differentdifferent forfor eacheach plot,plot, butbut bb (the(the slope)slope) isis thethe samesame forfor allall plots.plots. The common interceptintercept model isis shownshown inin figurefigure 4.5(b)4.5(b) andand isis representedrepresented byby thethe equation:equation:

Y == a + bbii. X (5)-(5) whewherere thethe intercepti ntercept aa isis thethe samesame forfor allall plots,plots, butbut thethe slopesslopes bibi differ.differ. In terms of the SchumacherSchumach er equation,equation, eithereither model model can can be be used, used, with with Y Yas as in lnHo Ho and and X asX as1/,eJc. l/Ak• The common slopes lope mmodelod el corresponds in shape to sets of proportional curves, but there is an important ddistinctionistinction betweenb etween this approach and that for temporary plots, in that the distribution of sites in the differentdifferent ageage classesclasses hashas nono effecteffect anon thisthis method.method.

Figure 5.5

REGRESSIONSRIDRESSIOm WITHWI'll! commoNCOMMON SLOPES OROR CO!!J>!ONCOMON INTERCEPTSINTERCEPTS

(a) CanrnonCommon slope regressions for fourfour plots

Yy

x

(b) CanmonCommon intercept regressions for three plots

yY

x -50-- 50-

The statistical estimatorsestimators for the commonconunon slope and commonconunon intercept modelsmodels areare asas follows: .fli ni ni I(EX. ..Y..-E X,..E 12 12 . 12 . Y.3../n.)2 The comnonconunon slope b = --(6) (6 ) ni ikEkx.$)/ni)milli 2--Xi

ni m• nini Dotn4 ni 2 Ez Y. - I (I(l: x.j.f.x.Xij.I Xl. .r.' :'!Ii ./E ./I X. X . .'). ) i j j J J j 1J The commonconunon intercept a =~ _(7) MI((z1x. )v,..6xn714 .2) j jiJ

As these formulae appear rather complex (although(although in concept they araree quite simple) sanesome calculation pmpro formesfomas are provided with a worked example.example. These are formfom 55.1,. 1, parts 1 and 2.2. Part 11 carries out thethe within-plotwithinplot summations,summations, correspondingcorresponding to the E for the jj subscript fromfrom 11 to ni in the above formulae,fomulae, whilst part 2 of the formfom carries outout the between-plotbetweenplot summations,summations, correspondingcorresponding to:Eto2 for i fromfran 1 to m in the formulae.fonnulae.

The example uses the data shown in figure 5.65.6,, fromfran 66 permanent sample plots in Cupressus lusitanicalusitanica standsstands inin Kenya.Kenya. A k k parameterparameter valuevalue ofof 11 is assumed for illustration purposes. The height-ageheightage datadata areare transcribedtranscribed intointo the first two columnscolumns ofof partpart 11 of form 5.1. Two sheets ofof this formfom are necessarynecessa!';Y for thethe sixsix plots.plots. The transformedtransfomed X and Y values are enteredentered inin columnscolumns 3 and 4.4. X2 is enteredentered inin column 5 and X times Y in column 6.6. Calculations should be carried out to at least four significant digits. The totals for each plotplot (within-plot(withinplot totals)totals) for columns 3 to 66 are enteredentered inin thethe appropriateappropriate line.line. The number of points inin eacheach plotplot isis alsoalso entered.entered. One then turns to part 22 ofof formfom 5.15. 1 to continue the calculations. For each plot, the various within-plotwithinplot totalstotals (EX,(EX , ~YZY,, 2X2~X2 andand EXY) andand thethe numbernumber ofof pointspoints nn areare combinedcanbined according to the formulaefomulae shown atat thethe toptop of the columns ofof partpart 2.2. These figures areare thenthen totalledtotalled between-plotsbetweenplots toto givegive thethe itemsitems marked (1)(1) to (6)(6) at the bottom.bottan. Finally the common slope and commoncanmon intercept coefficients are calculated as shown inin the lastlast twotwo lines.lines.

The final reaultresult in this numerical exampleexample isis thatthat thethe commonconunon slopeslope coefficientcoefficient isis 9.222,-9.222, whilstwhilst thethe commoncommon interceptintercept coefficientcoefficient isis 3.583. If either of these models isis plotted asas aa setset ofof site"ite indexindex curves,curves, theythey willwill bebe foundfound toto bendbend overover muchmuch moremore sharplysharply than isis indicatedindicated by thethe datadata inin figurefigure 5.6.5 .6. This arises because the assumed value of k is much too large for this setset ofof data.data.

Bailey && Clutter, in the paper referred to earlier,earlier, showshow how itit isis possible to calculate the nonlinear k coefficientcoefficient directlydirectly usingusing aa regressionregression modelmodel containingcontaining thisthis coefficient in linear form,fom, provided that remeasurementremeasurement datadata (from(fran PSPsPSPs or stem analysis) is available soso that heightheight incrementincrement cancan bebe estimated.estimated. The method isis as follows:follows: - 51 -

1. Calculate a set ofof transformedtransfonned YY values forfor thethe 2nd,2nd, 1rd,3rd, etc.etc. Observationsobservations withinwi thin a plot, fromfrom thethe formula:formula:

) . inf rii-Hii-11 Al L A A I -

There is no Y value correspondingcorrespcnding toto thethe firstfirst heightheight Observation.observation.

2. Calculate a corresponding setset ofof transformedtransfonned XX valuesvalues fromfrom thethe formula:fonrrula:

X.. ln(2/(Aij+A1i_1)) 2.3

3. Fit a common slope estimatorestimator using formfonn 5.1 or equation (6) usingusing thesethese trans­trans- fonnedformed X and YY values. Note that if form 5.1 is used, anlyonly the first two columns and totals (1)(1) and (2)(2) are requiredrequired onon partpart 2.2.

4. Subtract 11 from the commoncommon slopeslope estimatorestimator obtained.obtained. The result is the estimate of k required.

When using manual calculation, form 5,25.2 can be used to carry out thethe transformationstransformations in steps 11 andand 22 above.above. It has been completed for the first plot in thethe exampleexample datadata toto illustrate thethe usage.usage.

The formulae for the common slope and common intercept estimators, togethertogether withwith thethe transformation techniquetechnique forfor estimatingestimating thethe k parameter, cancan easilyeasily be programmed for small computers oror programmable calculators.calculators. Any programmable calculatorcalculator withwith atat leastleast 15 data registers and 200 programme stepssteps shouldshould bebe adequate.adequate.

When k parameter is fitted in this >way... y to the data in figure 5.6, thethe following values areare obtained:obtained: k ~= 0.25 b = -6.6386.638 (common slope model) a == 6.1116.311 (camnon(common intercept model)

Let usUES examine the construction ofof aa setset ofof sitesite indexcurvesindex curves fromfrom thesethese results,results, using the commoncamnon interceptintercept modelamodel.

We have: 25 Inin HHo = 6.311 + b.!AO.bi/A°°25 -(8) o ~

The parameter bibi depends uponupon si-tesite indexindex SS. . For a selected site index, at indexindex age A.:Ai:

in S = 6.311 4-bi/Ai

0.25 ~. b. =( ( nn SS - 6.311).Ai0.256.311 ) .~ ~ - 52 -

If we wish to plot curvescurves forfor sitesite indicesindioes 16,169 189 18, 20920, 22922, 24924, 26 using an index egeage of 20 years, we have:

S bi 16 -7.4837.483 18 -7.2347.234 20 -7.0117.011 22 -6.8096.809 24 -6.66.62525 26 6.456-6.456

Then from equation (8),(8)9 substituting the bbi for each site index curve, values of Ho i can be calculated for selectedselected values ofof A.A. The curves that result from the above parameterparameter values are shown in figure 5.7,5.79 on the sarnesame scale as is used for the data in figure 5.6.

If aneone wishes toto calculatecalculate thethe sitesite indexindex ofof aa stand,stand, givengiven itsits ageege andand dominantdominant height, then use the formula:

in S = a + (ln Ho a).(A/A0k -(9)

For example, for the parameter values for a and k given above, supposeauppose wewe havehave aa stand of 14.5 m at 11.5 years. Then the estimatedestimated si-teBite indexindex is given by:

inIn S = 6.311 + (In(in 14.5 - 6.311).(11.5/20)°.256.311).(11.5/20)0.25

= 3.144 • :.•• sS = 23.2

Hence we can say that the sitsite e index of this stand is 23 m.

When the common slope model is used, instead of the common intercept model,model, thenthen thethe basic equationequation is:is:

Inln HoH .= a. + b/Akb/Ak -(10) o ai1 withwtth ,a.a. beingbeing dependent onon si-tesite index as:as: 1

/ k ai = In S -(11 ) with site index for aa selectedselected height-ageheightage observationobservation beingbeing givengiven by:by:

/ / lc% n S = ln +bk1/Ak 1/A ) -(12) Ho - 5353 -

Formo rm 5".1 rCommgnm n s]ove. Ose and commoncommon interceptsnterce _t reregressionession mOdelsmodels Part 1 Plot data summarization.smmarization. Use asas many partpart 1 sheetssheets as necessary for all plots. Data transformations used:used X = / / AqcA.... --~~~------Y = LocrELoc;-E /./E/gerII~";HT

- -...=44-SISIMI-V--=,..-L2,-._.-._.1 =i3.112111,---M- V L Pe 'Plot lot 5:3 Raw datadata Transformed data A H x2X2 X Y ifYXY __

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Form 5'.15.1 CQmmonommon slope -lo e andand common intercept intercept regressionrearession modelsmodels Part I1 Plot datadata summarization. summarization. Use as muymany part 1 sheetssheets as necessary for for allall plots. Data transformations used: X =. I, /Ave / Aft" Y == ------LoprL09. HFV/7"-He/flfl ----

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Plot Plot S'252 - Raw data TransfTransf oormedrmed data A IIH J('V -22 xy .-...... X...Y ...,... x XY 21.2-1-7 1 1'1/S.. /1 0.04-60gO .o~og .z.8'J2.2959 !>'9 0.002-12.0.002..12. 0.10. 1 3,-& ?,?4- L-1- 22.22.4 " 19.4-i?. 4- 0.00.0442-54-1J.2-S' 2.·"('<;;32.743 0-000.001'16 i 9 6 o.0. 13,], r12- 2. 21.·23-5 S' .20.02.0.0 0o.04-2.S5.o rr2.<;5 02.'1"15'12-9957 c'°0 . 00 t118 1 I 0.12.150 ·, 215 26.4:26· (, .2.2.22-3·3 O·0'37e;-q0.0375 3./04-4'g. to 4-4 0.00 lit. Ilj. It o·0. rII i Cl,-, 27.42. 7." 23.323· 3 (1).0360.036,2.23 3.3.(114.4.; I 1j..8 £ 0-000.00 III(3! o.i/14-1o. II 4- \ . 2.1l.~2g.S. 2.2 i-p-... 2.2. 0.03$'0'10. 035°5' 3.1864-3. (84,14- 0.00o.t.50 12..3i2.. o. I I 18 2.9·2?.4 If 1..1+·724-7 0.034-00.) 31t2 /I 3.l..o6l?3.2..o4 o0.00 . on Ifr/ 6 o· to?1091 1.._,I Totals nn-, p.l.0.2.7 7S'B SS 21.503/2.1. S03/ 0.01/000-0 f (00 o'~"""8

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Form 5.1 ammonCommon sloneslone and common j.ntercept interce t regresE!j.onre Te sion modellmodels Part 22 Totals betweenbetween plots plots and coefficientcoefficient calculationcalculation

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- o. 5.2.% Common slope slope b b= =(1)/(2) (1)/(2) == -O . 152.~bj;'S",t;0. of65(.5 = -'l.22.2. UI". S"bSC> - -ilo42,1)/f0.,2.1) Common intercept intercept aa == ( («6)-()/«5)-(4)) 6 )-( 3) )/( ( 5 )-( 4) ) = +(11-0-Opo - 38.2.,,"'S)38. 204-5)

=3.5'83 - 5656 -

Figure 5.6

Data from 6 permanentpennanent sample plots in Cupressus lusitanicalusitanica standsstands in Kenya.Kenya. Used inin text exampleexample for fitting site index curvescurves by nested regressionregression

.1,-93---e3 Measurement Measurement pointspoints && plot numbernumber forfor PSPPSP ,. "8 ,... - .. - ,, .' , ,. 30 , , ,.'o/ e 4.4o .' , , ~"'" ' ,0"e' .. a'e'­ , , .052, , , dr , , ,,-' ,0/ 0/ , "34.3l. 0' , , / ,·e I eo I , e 3 e.' , ,I/ ,,.",41( Ie.p e' ' .35 " / I I.'/ dp, e/I 15/5 I~" ..,,. ,~/I/ r/."e /' ,,; " , eV II//// ,t .' ..4, I ,e. ,.,- +# .,.-., i/I , 10 , I . .0/,, / •/

5

10 I~IS 2.020 2s25 30

Age (years)(years) Form £.2;-.2 Transformations to to ageage and and height height data data fromfrom PSP's PSP's to to fit fit k parameter inin SchumacherSchumacher equation byby COcommonRmon slope slope estimator estimator

( I da b c X d e f Yrjy. ! J A·I) X·IJ H··I) d 9 h IJ

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i 1 ;

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i . i : - -- L_ Transferm ___ ~"orL...,. to f formorm 5:15: - 5858 -

Figure 5.7

Site index curvescurves produced forfor the datadata inin figure 5.6 by Bailey & Clutter'sClutter's nested regres-regres­ sion method using the Schumacher equation

SSite ite indexIndex 35 26

24 -E - 30 22 .c C710'1 -00::; 20 :ccTi 1= 25 18 roru c: 16 E o0 0 20

15

10

5

o I-----.------r---.....---..,..---....,.-.-- --.- 5 10 15 20 2525 30 Age (yea(years! rs ) . , -- 5959 -

Multiple regressionregression methods ofof fitting site index curves can bebe usedused as an alterna­alterna- tive to nested regressionregression Whenwhen the site index or site class of the plotplot cancan bebe detennineddetermined prior to fittingfitting thethe model.model. There are perhaps five ways in which thisthis !!:a prioripriori sitesite classclass or site index determinaticndetemination cancan be made:

(1) For a long series ofof plot measurements whichwhich havehave passedpassed throughthrough thethe indexindex age,age, site index can nebe taken directly as the dominant height at the index age.age.

(2) ForPor series of measurements which do notnct passpass throughthrough thethe indexindex age,age, anan equationequation such as the Schumacher equation can be fitted independently to eacheach plctplot andand used to predict the dominant height atat thethe indexindex age,age, whichwhich isis thenthen takentaken asas the site indexindex forfor thethe plot.plot.

(3) For temporary plot data, sitesite class cancan be assignedassigned onon thethe basis describeddescribed above under the maximum-minimummaximumminimum methodmethod ofof fittingfitting site8i te curves.curves. This method should notnot bebe usedused with PSP data as it is wasteful ofof the information inherentinherent in such data.dat a.

(4) An existing set of site curvescurves can be usedused toto assignassign sitesite classclass toto thethe plots.plots. Alternatively, handhanddrawn- drawn curvescurves cancan bebe nademade speciallyspecially forfor aa particularparticular analysis,analysis, -Loto classifyclassify plots.plots. This technique is useful when existingexisting curves are almost satisfactory, butbut of not quite the correctcorrect curvature or oneaae wisheswishes toto approxi­approxi- mate hand-drawnhanddrawn curvescurves byby anan equation.equation.

(5) An environmental variable such as altitude, rainfall, etc.etc, can bebe usedused asas thethe site indicatorindicator variable.variable. This approach is rarely successful duedue toto thethe poorpoor correlation usually foundfound between aa singlesingle environmentalenvironmental variablevarianle andand heightheight growth.

Once the met;lodmethod of assigning site class to the plots has been determined,determined, one has a set of data in which threethree variablesvariables areare knownknown forfor eacheach observation:observation:

Dominant height H Hoo Age A Site class or index S

Multiple regressionregression cancan thenthen bebe usedused toto fitfit aa modelmodel whichwhich relatesrelates HoHo toto AA and and SS using various differentdifferent transformations.transformations. Two types ofof modelmodel havehave beenbeen used:

(1) Constrained models, for use with site index curves, where height is expressed relative to site index and age relative to index age A.• The regression fitted 1 is aneone withoutwithout anan intercept.intercept. An exampleexample wouldwould bebe thethe model:model:

(Ho s) = bi (A Ai) + b2(A Ai.)2

This kind ofof model isis forced to givegive aa dominantdominant heightheight Ho equalequal to site index S When the age is equalequal toto thethe indexindex ageage Ai.A;.

(2) Unconstrained models, with anan interceptintercept term.term. When used withwith sitesite index,index, rather than site class, the curvescurves must be conditionedconditioned after fitting to ensureensure that the dominant height correspondscorresponds toto thet;,e sitesite indexindex at the index age. -60- 60 -

SomeSerna examples of the types ofof unconstrained models that can be fitted are:

= b A + b1A + b2S + b3A° (13 ) Ho 0 + b4A2 -(13)

log Ho = 130+ bl/A + b2S + b3S/A -(l4)

Equation (13)(~3) is based onon thethe quadraticquadratic equation.equation. See figure A.2.1(e) in Appendix A for -the shapes that a single curve may take. Obviously forfor si-tesite indexindex curves,curve% only one case of the sseveraleveral possible shapes is desired. Equation (14)(14) is basedbased on the Schumacher equa-equ..... tion, oror figurefigure A.2.1(c).

An infinite variety of other modelsmodels is possible. In allall cases, it is essentialessential toto plot the curves after fitting and to overlay themthem on aa graphgraph of the data toto ensureensure thaeTthat thethe trends ~reve accurately followed. No reliance can be placed onon the statistical parametersparameters (e.g. R •9 residual standard deviation) associated with multiple regression in determing the suisuitabilitytebili ty ofof thethe curves.curves.

When sHesite index is used, as opposedopposed to site class,class, then it will be found that if the curves are plotted,plotted, the predicted dominantdominant height HoHo atat the indexindex ageage Al~ isis generallygenerally!!.2i not equal to the site index S.S. This is becausebecause there is no constraint in the fitting process to bring about this coincidence. o.uvesCurves for selected real site index values S*S* areare graphedgraphed bbyy calculatingcalculating thethe statisticalstatistical sitesite indexindex SS requiredrequired inin thethe modelmodel toto givegive aa lineline passingpass ing through S*S* atat A..A . • ~ Thus fromfrom modelmodel (13)(13) wewe have:have:

3* 2 S* ~ b + b A. + b S + b A . • S ...+ b A -(15) b0o + b1Ai1 ~ + b2S2 + h3-i.3 ~ b4Ai4 i

, • S = (s*(S* _ b b _ bbA A _ bA b A 2)/(b + b A ) •• 0o 11 4,4i b22 + b3Ai)3i (15 )

or frernfram momodeldel (14)(14) we have:have:

log S* = b +b /A+ b S + b.S/A. 0 1 i 2

S = (log S* 1)0 ) -(16) b,Ai )/(b2 + b3 /Ai (16)

The necessity for conditioning site indexindex curvescurves andand the mysterious andand confusingconfusing distinction between statisticalstatistical sitesite indexindex SS andand realreal sitesite indexindex S*S* hashas ledled manymany workersworkers toto use models ofof thethe constrainedconstrained typetype describeddescribed above.above. Unfortunately, itit isis oftenoften the casecase that many standardstandard multiplemultiple regressionregression programmesprogrammes dodo notnot havehave aa facilityfacility forfor fittingfitting equa-equa.­ tions withoutwithou:t an intercept term, soso thatthat thisthis qptionoption isis notnot availableavailable toto thethe researchresearch worker.worker.

With si-besite classclass curves,curves, thethe problemproblem doesdoes notnot arise, since there is no requirement for the curves to pass therough a particularparticular point.

Multiple regressicnregression techniques have the advlmtageadvantage of greatgreat flexibilityflexibility inin thethe typetypeof of model adoptedadopted .They They have have the the disadvantages disadvantages of of requiring requiring a !; p.tiori pritori estimationofestimation of sitesite index,index, ofof being inefficient statisticallyinstatistically in not making useofuse of the nested na-bureofRSPnature of PSP data.data andand ofof beingbeing based onon thethe invalidinvalid assumptionassumption thatthat thethe sitesite variablevariable isis knownknown toto aa highhigh degreedegree ofof precision.preCision. - 6161 -

5.1.3 Site Assessment Models Based onon EhvironmentalEhvironmental FactorsFactors

SiteSi-te index index oror sitesite classclass curvescurves areare obviouslyObviously onlyonly usefuluseful asas aa tooltool forfor predictingpredicting production potential whenwhen appliedapplied toto existingexisting stands.stands. A large class of forest managanentmanagement decisions require somesorne assesmnentassessment of potential production of a given speciesspecies beforebefore itit hashas been establishedestablished oror atat leastleast veryvery earlyearly inin the'the lifelife ofof youngyeung stands.stands.

This is onlyonly possiblepossible inin twotwo typestypes ofof situation:situation:

Very generalized predictions as to maximum productivity in aa givengiven regionregion byby comparison with optimaloptimal productivity ofof similarsimilar forestsforests growinggrowing under similarsimilar environmerrtalenvironmental conditionsconditions eIsel-mereeiseWhere inin thehe world.worldo

More detaileddetailed predictionprediction ofof sitesi'te class with reference to a particular set of height/age curves for a given species by construction of aa functionalfunctional relation­relation- ship for forests alreadyalready establishedestablished inin thethe regionregion inin questionsquestion, betweenbetween sitesite class and enviramnentalenvironmental factorsfactors suchsuch asas soilsoil nutrientssnutrients, depthdepth andand texturestexture, altitude,altitudes aspect andand rainfall.rainfall.

The first approach isis particularly relevantrel""""t forfor mixedmixed naturalnatural forests.forests. As far as man-made uniformunifonn forestsforests areare concernedsconcemedp itit corresponds more to the phase of species and provenonceprovencnce selectionselection thanthan toto thatthat ofof detaileddetailed yield prediction with which this manual is concerned.

5.1.3.1 Functional modelemodels for site claaaclass predictionrediotion

Si-teSite classclass predictionprediction modelsmodels areare constructedconstructed in the following stages:stages:

(1) Construction of site index or site class curves from permanentpermanent samplesample plotsplots inin existing forest,forest"

(2) Collection ofof enviramnentalenvironmental datadata frcmfrom permanentpennanent and/and/ oror temporarytemporary plots.plots. Each plot is also assignedassigned aa sitesite classclass onon the basis ofof dominant height and age from the site indexindex curves.curves, The environmental data collected should correspondcorrespond toto the suggestions givenglven inin sectionsection 3.2.3.

(3) The varicusvarious environmGntalenvironmental variables are tranefonnedtrensformed and selected as describeddescribed in the followingfoll~ding sectionssection, to produce a predictive model in the form:fonn:

S = b0bO + b1.e1b10e1 + b2.e2 + 0.,09...... •• • ++ bn.enbn.en

where bO,bOs blib1, etc.etc0 areare coefficientscoefficients fittedfitted byby multiplemultiple regressionregression andand e1el toto en are relevant transformationstraneformationt3 ofof environmental variables.variables.

(4) Confidence limits for predictionspredic'tions can be defined from the methods given for multiple regressionregression inin AppendixAppendix A.A. In generalsgeneral, it is also desirable to testtest the function by comparisoncomparison ofof actual andand predictedpredicted sitesite classesclasses forfor an inde-indE>­ pendent set ofof datadata toto thatthat usedused toto fitfit the"he function.function. This independent data should preferably be fromfran aa differentdifferent regionregion fromfrom thethe main data to test thet.he regional stability ofof thethe prediction function.:function. - 62 -

5.1.3.2 Construction and selection of environmentalenvironmental variables

The total number of environmental variables measured maymay bebe quite large, possiblypossibly aass many asas 100100 perper plot, whilst a usable model should contain the smallest possible numbnumberer of most easilyeasily measurable variables.variables.

Reduction inin thethe totaltotal numbernumber ofof variablesvariables cancan bebe achievedachieved inin severalseveral ways:ways:

By synthesissyntheSiS ofof somesome itemsitems ofof information.information. For example, climatic data maymay bebe synthesized to give the length of the wet ssPason~ason or growing saasonseason withwith tempera.­tempera- o tures overover 6°6 C,C, oror average temperatures during the wet Bessonseason or moremore canplexcomplex techniques based anon evapotranspirationevapotranspiration formulae.formulae.

Statistical selectionselection ofof thethe mostmost significantsignificant variables.variables. Principal components analysis can be used for this, but it is perhaps simpler to use stepwise multiplemultiple regressionr~eBBian analysis.analysis.

Graphical analysis ofof variables consideredconsidered likelylikely to be most important as limiting factfactors ore in growth, selecting anlyonly those with definite an~and obvious relationships with sitesite class.class.

The actual measured variables should ifif possible be transformed in order to give values that are likelylikely to be correlatedcorrelated withwith growth.growth. The most obvious exampleexample is aspect,aspect , which may be measured in degrees from 00 toto 360, where values around zero and 360 both corres-corres­ pond to northerly directions. Taking the sinesine ofof thethe angleangle divideddivided byby 22 givesgives anan aspectaspect code between zero for northerly directions andand oneone forfor southerlysoutherly slopes.slopes.

5.1.3.3 ProblemsPrOblems in the applicationaIlication ofof sitesite assessmentassessment functionsfunctions

To be effective,effective, aa site assessmentassessment model shouldshould be describeddescribed inin terms ofof predictor variables thatthat areare easilyeasily measurable.measurable. They should also have a relatively high corre1atirnlcorrelation coefficient, preferably overover 0.80.8 withwith 2020 oror moremore points.points. Otherwiseotherwise the relationship may be statistically significant,significant, but ofof nono practicalpractical useuse forfor predictionprediction purposespurposes because ofof lowlow precision. The predictive equationequation shouldshould alsoalso havehave asas fewfew predictorpredictor variablsvariab1s asas possible, preferably notnot moremore thanthan threethree oror four.four.

The predictor variablesvariables shouldshould be quantitiesquantities whichwhich areare readilyreadily determinabledeterminable usingusing conventionally availableavailable equipment.equipment.

An exampleexample isis shown inin figurefigure 5.85.8 ofof aa predictivepredictive modelmodel whichwhich fulfilsfulfils thesethese criteria.criteria. It is based anon aa singlesingle variable,variable, whichwhich isis thethe numbernumber ofof daysdays inin thethe growinggrowing seasonseason exceedingexceeding 6°0.6°C. Site index is expressedexpressed inin terms ofof meanmean annualannual volumevolume incrementincrement perper annumannum atat itsits maximum.

The fewer predictor variables are involvedinvolved inin aa model,model, thethe moremore likelylikely itit isis toto bebe usable overover aa widewide region.region. With many predictor variables, the model becomes very sensitivesensitive to the relationshipsrelationships between them, especiallyespecially ifif 'some ofof the predictor variables are highly correlated.correlated. Consequently, regionalregirn>a1 shiftsshifts in the balance between predictor varia- bles cancan easilyeasily invalidateinvalidate a complexcomplex model. -6363 -

Figure 5.8

SiteSi-te assessmentassessment modelmodel forfor ScotchScotch Pine (E.(P. sylvestris) forests in Sweden. Average productivityproductivity of regions is plotted against mean number of days exceeding 6~6°C perper year. (Reproduced from Fries, J.,J., 1978 "The assessment of growth and yield and the factors influencing it" Special paper toto 8th8th WorldWorld ForestryForestry Congress,Congress, Djakarta)Djakarta)

m'/ha013/ " / 8 1/ / 6 6,11 LYI w" /7 " 4 erg

! n I",00

, N /'V 2 V"

100 120 140 360160 180 200 ciayadaY' ~+6·C4-6'C

5.2 STATISTATIC C MnaTIODSMEI'HODS OF PREDICPREDICTING TING YIELD

StaticSta-tic yield yield predictionprediction methodsmethods areare thosethose inin whichwhich yieldyield isis predictedpredicted directlydirectly asas aa function ofof age,age, si-tesite classclass andand thethe history ofof stand density.density. The methods are staticstatie in the sense that the resultant yield functionsfunctions do not permitpenni t any variation in the history of stand treatment, except intcinto broad classes ofof alternative thinning treatments that are already present inin thethe data.data.

The components of yield that are ofof major interest to the forest manager are volume and mean diameter.diameter. To determine the volume of stands requires a knowledge of stem numbersnumbers and height, as well as mean diameter. Stem numbers are usually thethe basis forfor definingdefining thinning treatment, whilst height isis the most commoncommon means ofof sitesite classification.classification.

In this section, staticstatic methodsmethods ofof predictingpredicting meanmean diameterdiameter willwill bebe considered.considered. - 64 -

Figure 5.95.2 ShapeSha .e ofof thethe relationshiprelationshi between mean diameter and dominant_heightdominant height

(a) With temporary plot data, a scatter diagramdiagram isis 6btained.obtained. The true shape ofof the diameter/heightdiameter/height relationshiprelationship isis difficultdifficult toto define.define. Diameter of of treetree ofof mean basal. basal area Dg

• o

o • .o . a .to .. 0 . . '. ...00e ...... 0 0 * o o •0 0 0 o • • ...... '

DominantDoainant heigh'!;height Ho

(b) With thinning experimentexperiment data, the shapeshape ofof thethe relationshiprelationship andand its dependence onon stand density isis very clear. A represents the lightest thinning andand DD thethe heaviest.heaviest.

Dg Thinning treataentstreatments

,0 D / ..... (J " , / .' :Ii3...... , 0'.' ,,°- V ,.-' , _.. /,- .-- _-"". A / y ---6------...---," " / ,' .------41.-- ,- ,--- '2&'' ...A.....- .....' I ,.. e-

Ho --65-65 -

5.5.2.12• 1 GraphicalGra bical MethodsMethods based onon the Diameter/HeightDiametertHeight FUnctionFunction

When data franfrom aa massmass of temporarytemporary plotsplots isis graphedgraphed withwith meanmean diameterdiameter onon thethe verti­verti- cal axis and daninantdominant height onan thethe horizontalhorizontal axis,axis, aa diagramdiagram isis obtainedobtained similarsimilar toto figurefigure 5.9a. If the data is fromfrom aa thinning experiment,experiment, somethingsomething likelike figurefigure 5.9b is obtained. Treatment A represents thethe lightest thinning,thinning, i.i, e. e. the the heaviest heaviest stocking, stockingond and TreatmentTreatment DD isis the heaviest thinning, withwith thethe lowestlowest stockings.stockings.

ItIt can be seen that the curve is concave with respectrespect toto the horizontalhorizontal axis.axis. With temporary plotplot datadata or "ithwith permanentpermanent plots,plots, aa seriesseries ofof hand-drawnhanddrawn curves can bebe constructedconstructed in a mannermanner exactly analogous to that described inin sectionsection 5.1.2.15.1.2.1 forfor siteBite indexindex curves,curves, except that here the curvescurves demarcatedemarcate differencesdifferences ini.n standstand history history rather rather than than si-te site class.class.

With temporary plot data,data2 the shapeshape of thethe curvecurve maymay bebe defineddefined basicallybasically fromfrom thethe upper and lower envelopes of the scatter diagram, butbut obviously the curvecurve shapeshape cancan bebe easilyeasily misconstrued as a result of a few exceptional or anomolousanomalous pointspoints andand remainsremains largelylargely aa matter for subjective judgement.judgement. With permanent plot data on the other hand, the main curvecurve trends are readily visible from the slope of the lines joining remeasurements onon thethe samesame plot.

Height is used on the horizontal axisaxis asas aa site-dependentsitedependent indicatorindicator ofof age. Graphs can be plotted using age instead, but in this case a separate set of curves isis neededneeded forfor each site class.

Once a set of curves has been drawn, the stocking historyhistory represented byby eacheach denSitydensity class must be determined.detennined. This isis discusseddiscussed. inin liectionsection 5.2.3.5.2.3.

Hand-drawnHanddrawn curvescurves cancan bebe expressedexpressed asas equationsequations byby usingusing oneone ofof the various approxi­approxi- mation methods describeddescribed inin AppendixAppendix A.A.

5.2.25. 2• 2 Direct Statistical EstimationEstimation ofof Mean~!ean DiameterDiameter PredictionPrediction FUnctionsFunctions

Functions can be fitted directly to mean diameter data by methodsmethods analogous toto thosethose used forfor fittingfitting sitesite indexindex curves.curves. The predictor variablesvariables maymay bebe dominantdominant heightheight andand standstand treatment history (designated(designated by T)2T), or may include age and site index as separate variablesvariables instead of as the combined variablevariable dominantdominant haight.height. The methods that may be used toto fit the functions include:include:

(i) With temporary plot data:data: Sort data by diameterdiameter withinwithin heightheight classesclasses and assign an orderorder number after sorting. Adjust orderorder numbers withintlithin eacheach height class onto a commoncanmon scale by the transformation: T = iini/ n where i is the order numbernumber withinwithin aa heightheied classclass andand nn isis thethe numbernumber ofof points withinwi thin the class.class. T is an index ofof treatmenttreatment history.history. A multiple regression of one of the types discussed below can then be fitted using T as an independent predictorprediotor variable.variable.

Fit aamean mean trend line by simple regression ofof mean diametercmdiameter on daninantdominant height and then constructconstruct proportionalproportional oror parallelparallel (anamorrhic)(anamorphic) setssets ofof curves.curves. - 66 -

(11)(ii) WithIIi th permanentpennanent plotplot data:data: Simple nested regression, as described in section 5.1.2.2 for site index curves. This depends onon being able to find a transformationtransfonnation for the data that will result inin a figure like figure 5.5a5.5a oror 5.5b5.5b for a commoncanmon slope oror commoncanmon intercept model.model. The Schumacher equationequation may be suitablesuitable oror a simplerSimpler transfonnationtransformation ofof the type: k Dg a. a + b.Hokb.Ho Wherewhere kk isis aa powerpower betweenbetween O0 andand 1.1. Multivariate nested regression,regression, eithereither usingusing thethe conditionalconditional variable tech-tech­ nique describeddescribed inin AppendixAppendix A2.9A2.9 oror usingusing moremore advancedadvanced multistage regressionregression techniques. CbnventionalConventional multiplemultiple regression, after assigning a treatment history value T to eacheach plot.plot. This may be done in various ways, including a variation on the sorting method describeddescribed aboveabove forfor temporarytElDporary plotsplots inin which averageaverage values ofof TT calculatedcalculated forfor eacheach measurementmeasurenent areare takentaken forfor plotsplots managedmanaged according to a consistentccmsistent thinning schedule.schedule. With thinning experiments, TT values may be assigned forfor eacheach treatment simply as 1, 2, 3 etc. if the treatments cancan be clearlyclearly orderedordered according to their degree ofof intensity.intensity. A very commoncanmon approach to fitting multiple regressionregression models to permanentpennanent plots is toto useuse spacingspacing relativerelative toto hsightheight oror eveneven simplysimply stocking,stOCking, asas indices ofof treatment history.history. This method cancan appearappear toto workwork quitequite well,well, but introduces specialepecial conceptualccmceptual difficultiesdifficulties whichwhich areare discusseddiscussed belowbelow inin section 5.2.3.

The actual mathematical functionfunction thatthat isis usedused forfor thethe basisbasis ofof thethe fittingfitting methodmethod is not very importanteimportant, providedprovided itit isis capablecapable ofof assumingassuming thethe correctcorrect shape.shape. This will be a curve that is gently concaveconcave withwith respectrespeot to the horizontal axis,~s, i.e. bendingbending downwardsdownwards away from the straight line,line, as shownshown in figure 5.9b.5. 91>. Suitable models are:are:

2 Dg a b0bO ++ b1.111.bbl.Ho + + b2.T b2.T ++ b3.Ho.T + + b4.11402 b4.Ho k k Dg =- b0bO ++ b1.1iokbl.Ho ++ b2.Tb2.T + b3.Hob30Hok.T .T

The second model cancan be fittedfitted using trial values ofof k between 0 and 1 with linear regression methode,methods, selecting a value that gives a maximum coefficient of correlation oror itit can bebe fitted directly by nonlinear regression.

Whatever functionfunction isis usedused toto fitfit thethe data,data, thethe resultsresults shouldshould bebe examinedexamined graphically,graphically, overlaid anon the data, forfor anomolousanomolous behaviour.behaviour.

502.35.2.3 Defining TreatmentTreatmentI Historyto in TennsTenme of Stocking and Age

The claasesclasses ofof treatmenttreatment historyhistory maymay bebe defineddefined graphicallygraphically oror byby aa numericalnumerical method,method, as discussed in sectionssections 5.2.15.2.1 oror 5.2.25.2.2 respectively.respectively. Once this has been done, then the actual stockings involved may be determined by aa process ofof tabulation. If the function or graph relates mean diameter to dominant height, then forfor eacheach treatment class, constructconstruot a - 67 -

table of height classes and determine the averageave~e stocking per ha in each class. The height classes can subsequently be converted intointo classesclasses ofof ageage andand si-tesite indexindex byby referencereference toto thethe relevant site indexindex curves.curves.

When the meanmean diameter :functionfunction hashas been fitted using stocking or relative spacing as an index ofof treatment historyshistory, then it is not nscessarynecessary to detenninedetermine thethe stockingstocking corres­corres- ponding to a particular treatmenttreatment hietory.history. HowevergHowever, the coefficientsooefficients detennineddetermined for suchauch a function are dependent upon the partparticulari cul ar relationship betweenbetween ageage and stocking observed in the data. FailureFailurs to realise this can mean that that the function maymay bebe usedused forfor age/stockingage/stocking combinationsccmbinations not represented in the datesdata, givinggiving moremore oror lessless erroneouserroneous results.results.

5.2.4502.4 Static Yieldneld Functions PredictingPredictin, Basal Area------or Volume

As well as constructing static yield functionsfunctiane toto predictpredict meanmean diameter,diametergit it isis possi­ncesie ble to use identicalidentical techniquestechniques toto predictpredict basalbasal areaarea oror volumevolume perper hectare.heotare. The actualaotual shapes ofof the the functionsfunctions willwillbesunewhat be somewbat differentdifferent andand willwill bebe asas showninshown in figuresfigures 5010a5.10a oror b.b.

In general9general, use ofof mean diameterdiameter involvesinvolves certaincertain advantagesadvantages inin termstenns ofof overallOITerall model simplicity.Simplicity. If basalbasal area or volume are predicted,predicted9 then itit is usually necessarynecessa.ry toto have some function whichwhich willwill allowallow meanmean diameterdiameter toto bebe determineddetennined subsequently.subsequently.

Also,Alsos withwith basalbasal areaarea oror volume per hectare9hectare, a large component of thethe response toto different stand densities is siMplysimply the multiplicative effecteffect ofof differentdifferent stockings.stockings. With the mean diameter :function,functiong thisthis influenceinfluence isis removed: remOlTed p soso that resultant relationehipsrelationships focus upon the realreal effectseffects ofof competition.

5.2.5 Limitations ofof Static Yield

Static yield models havehave threethree significantsignificant disadvantages:disadvantages,

(i) ItIt is difficult to combine together data fromfrom standsstands withwith radicallyradically differentdifferent or variablevariable treatmenttreatment histories and obtain aa consistentconsistent andand effectiveeffectivy yieldyield :function.function.

(ii) Once the model has been constructed9constructed, it cannot bebe usedused to predictpredict yieldsyields for alternative treatments apart from those represented byby the treatmenttreatment historieshistories incorporated in the model.

(iii) DetenninationDetermination of thinning yields is difficult,difficult9 unlessunless accurate records of thethe diameters of removedremOlTed stemsstems areare kept.kept. Very oftenoften with PSPaPSPS or even with experi-experi­ ments such datadata. are unavailable.unavailable.

On the other handshand, static models are undeniably easier to construct and 11236use than dynamic models and,andgin in appropriate situations9situations, at least as accurate.

5.3 DYNAMICDYNAKIC MEI'IIODSMETHODS OF PREDICTING GROWTHGROlfl'll AND YIELD

A dynamic model is one which models Tatesrates ofof change within a system.system. As farfar asas forest yield studies are concernedconcerned9, this means tthmthat the basic prediction is of increment in diameter9diameter, basal areaarea oror volume.volume. - 6868 --

av_Kt_5112Figure 5.10 ShapeShae of thethe volumevolume and basalbasal area functionsfuncti ons over daninantdominant heie;htheight

(a) Volume per hectarehectare anon daninantdaninant height.height. Note that the solid line represents a limit notnat exceeded at at very high standstand densities. If logarithmic scales are used onan bothboth axes,axest the relation appearsappears as a straiglotstraight line.line.

vV decreasing j stand densitydensity

Ho

(b) Basal area per hectarehectare onon dominant height. The solid line is a limit not exceededexceeded at very high stand densities.

G decreasing standstand Idensity

Ho -69-- 69-

Dynamic models have the advantage of being more realistically representative of theths true causecause andand effecteffect dependencedependence between stand densitydensity and stand yieldyield than are static yield models. They are free from the limitation that the data shouldshould representrepresent aa consistentconsistent seriesseries of stand histories and consequently can be used as the basisbasis for !Illalysinganalysing andand synthesiSingsynthesising data fromfrom veryvery diversediverse typestypes ofof experimentsexperiments andand permanentps:nnanent samplesample plots.plots.

Growth models are based around functions that predict incl'ementincrement (of meanmean diameter,diameter, basal areaarea oror volume)volume) overover shortshort timetime intervals,intervals, asas aa functionfunction ofof stand density, expressed in termstenns ofof basal areaarea and/orand/or ageage andand standstand ageage andand sitesite class.class. The latter two variables may be combinedcombined asas dominantdominant heightheight asas waswas donedone forfor the staticstatic yield models.

To produce yield predictions, the growth function must eithereither be integrated mathe..·mathe- matically oror iterativelyiteratively summedsummed overover aa successionsuccession ofof years.years. This latter process is usually achieved by writingwriting aa smallsmall computercomputer programme, whichwhich becomes by definition a computer simulation model ofof forestforest growth.growth.

In the sections following, these techniquestechniques areare discusseddiscussed inin moremore detail.,detail, withwith partiparti-­ cular reference toto functionsfunctions ofof standetand basalbasal areaarea increment.increment. The techniquestechniaues are not greatly affected by thethe use ofof mean diameterdiameter oror volume as alternativealternlltive parameters, but basal area increment isis perhaps the simplestsimplest toto model.

5.35.3.1. 1 The Basal Area Increment FunctiFunction on

5.3.1.1 Basal area increment asas a function ofof dominant heiheightt

Stand basal area increment for most plantation species, when plottedplotted over daninant height, showseho'1B the followingfollOwing type ofof trend:trend.

maximum increment increment "

" N1

Ho Ha Hb Hc where I Ho is dominant height and N1 where I is current annual basal area increment per ha, Ho is dominant height and N1, N2, etc,etc. argar~ different levelslevele ofof constant stocking, with N4 greatergreater than N3,N , which isis greatergreater 3 than N2'N , and soso on.on. The curve has three regions which can be distinguisheddistingttlshed for the purposes 2 of practical datadata analysis:analysis: - 70 -

1. The range HaHa te to Hb, '\ showsshows increment rising sharply to a maximum and then falling again. There are strong differences between stockingstocking classes.classes. This part of the curve is difficultdifficult toto fitfit withwith simplesimple regressionregression models.models. With somesane species, it occurs at tootoo earlyearly anan ageage toto bebe representedrepresented inin samplesample plotplot data.data.

2. The range Hb1ft, to E.,He showsshows aa strongstrong declinedecline inin basalbasal areaarea incrementincrement withwith timetime (increasing(incraasing Ho)He) and marked differences betweenbetween stockingst ocking classes.classes. This part ofof the curve can easily be modelled by eeveralseversl of the regression functionsfmlCtions of the type shown in figure A.2.1A. 2. 1 inin Appendix A. The following model includes the effect of stocking:

In 1g b0 + b1 H b2 N Ho-k

The coefficient k can be given assumed values (1,(1. 22 etc.)etc.) oror fitted by nonlinear regression. NaturelNatural logarithms (base(base e) are used to transfonatransform I as this sim-sim­ plifies the use ofof an integral formfom ofof this model (see(see section 5~3.3).5f3.3).

3. Beyond He. the differences between stocking classes becomesbecanes negligiblenegligible andand the basal area incrementincrement willwill be almoetalmost constant.constant. This part ofof the curve is beyond the culmination ofof mean annualannual volume incrementincrsnent inin most epecies9species, butbut willwill bebe relevant to yield prediction in long rotation sawlog stands. Model (1)(1) above produces a satisfactory type ofof responseresponse inin thisthis region.region.

It is difficult, as hashas been noted,noted, toto findfind aa wellwell behavedbehaved linearlinear functionfunction thatthat fitsfits the entire rangerange ofof thisthis curvecurve withoutwithout biasbias overover "any any portionportion ofof it.it. Because thethe pointpoint Hi° Ht, usually occurs early in the life of the stand, it is betterbetter toto fitfit aa function suchsuch asas (1) above only to datadata beyondbeyond thethe maximummaximum IgtIgi andand toto predictpredict standingstanding basalbasal areaarea G G at at HbHb fromfrom a yield function in termstel'l1ls ofof stocking.stocking. For example:

Go eel:31N+ble -(2)

where Go isis thethe initialinitial basalbasal area,area, atat a.a defined dominant heightheight Hj"14109 and and depends depends onlyonly anon the stocking at thatthat point inin time,time, regardlessregardless ofof previous standstand history.history.

5.3.1.2 otherOther methods ofof predictingredictin basal areaarea incrementincrement

BasalBaaal areaarea. incrementincrAnent can be predicted as a function ofof agaage oror ofof standing basalbezel area. When age isis need,used, differentdifferent setssets ofof incrementincrement curvescurves areare requiredrequired forfor eacheach si-te site index class. With standing basal areasarea, eatisfaetorysatisfactory models cannot be Obtainedobtained for stands with very varia­varia- ble histories.histories. Consequently, the prediction ofof basal area incrementincrsnent fromfran daninant height and stocking,stOCking, as describeddescribed above,above, isis recommended.recanmended.

5.3.1.3 PracticalL.ctinHreman_valisofincrementdata. problsns in analysis of increment data

Increment data isis alwayealways varyvery variable.variable. This inis the combined effecteffect of:of:

1.1, Year to yearyear variationvariation inin incrementincrEment duedue toto climaticclimatic fluctuations.fluctuations.

2. Instrumental error.error. Simple measurEmentmeasurement systems such as diameter ta.pestapes maymay havehave thathe same orderorder ofof errorerror asas the incrementincrEment being measured,meamll'ed. - 71 -

A consequence ofof this is that large amounts of data are required to obtain satis­satis- factory estimatesestimates ofof thethe regressionregression coefficients.coefficients. Furthermore9Furthexmore, because thethe effectseffects ofof different standstand densities show. out only weakly through the general variation,variation9 extreneerlreme differences inin standstand densitydensity mustmust be present inin thethe data if a density-depeadentdensity-dependent model is to be constructed.constructed.

The best incrementincrment fnnctionsfunctions may malyonly have coefficients of detexminstiondetermination of between 0.7 and 0.8.0.8. To test the suitability of sa particular model underunder thesethese circumstances,circumstancee9 it must be usedused inin itsito integralintegral formfoxm (or(or asas partpart ofof aa simulationsimulation model) to predict final yields. Small biases in the increment functionfunction which do notnct show up graphically or by direct residual analysis can produce largelarge errors inin yield prediction because error in the increment functicnfunction is always cumulative.

5.3.2 ConstructinConstructing a Growth Model with sualag22±12mLa121_jantILLE2a2-1222Spacing Experiment Det... 14arsh's Response .1n2ihatiaI\ypotheais

It is possible to develop growth modelsmodels based based on on spacing spacing experiments experiments of of thethe type described in section 292, using simplified graphical techniques?techniques, which can give accurate pre­pre- dictions for the grouthgrowth ofof stands subject to a variety ofof differentdifferen"t thinning treatments. The basis ofof this method isis March'sMarsh's hypothesis:hypothesis:

"The incrementincrenent of thinned standsetands is equivalent to that of unthinned stands of the same stocking (measured(measured in aternsstems per unit area) and densitydenSity (neanured(measured in basal area or0:1' volumevolume per unit area)?area), but ofof youngeryounger ageage (i(i. .e. e. thethe ageage atat whichwhieh theythey hadhad thethe samesarne basal area or volume per unit area)". 111/

This hypothesis has been vulidatedvalidated mithwith several sub-tropical species and appears to give accurate and nnbiased\U1biaeed estimatesestimates ofof growth follomdngfollowing thinning9thinning, provided that the growthgrCl>ri:h is measured overover not less than 33 years.years. The method proceede as follows:

(1) First graph the spacing experimentelq)eriment reaultsresults using smoothed means for each stocking level?level, using baaalbasal areaaroo averover ageage (figure(figure 5.11).5.11).

(2) Define the thinning treatment to be used iniii terms of both the basal area at which-which thinning thinning takes takes plaoe place andand thethe basal areaarea andand stockingstocking to to be be removed. renoved.

Suppose for example we have a treatment schedule 0.8as foll"",,:follows:

Plant atat 11 300 trees/baotrees/ha. G after N after Thin.Thin,'' noono. Thin at Basal Area ThinningThinxthg Thinning

1 28 22 900

2 35 28 500

3 35 30 300 CF 35 clsarf'ellclearfell

ReBll_ onse of Even-aged Pine StandsStands to Thinning. 111/ .Marsh? Marsh, E.K.E.K. andand Burgers?:Burgers, T.F. 1973. The Reeponse of Forestry in S.S. Africa?Africa, 14:141 103-111.103-111. - 7272 -

Figure 5.115.11

Analysis ofof spacingspacing experiment ""Periment using Marsh'sMarsh's hypothesish.ypothesis

2 Basal areaarea (m2/ha)(m /ha) stems/ha.Stems/ha. 60 '' 1300

900 50 -

500 40

300

30 - 200

20

10 -

A2 ar /4r.. A3--I.r 41+-A4-*I

o0 1 56 1016 151'5 202I0 25 Age (years)(years) - 7373 -

(3) Draw the treatment schedule on the graph (broken lines on figurefigure 5.11) and from Marsh's hypothesishypothesis determinedetennine thethe timetime intervalinterval betweenbetween thinningp.thinnings.

Age Thin nownO... Apparent Interval Real !£eAmat at Time ofof Thinuas~in"g

1 0-8 8 8 2 8.5-13.5 5 13 3 14.5-18.514.5-18.5 4 17 CFGP 20-23 3 20

(4)( 4 ) Reconstruct the basalbasal a:rea/agearea/age rela:tionshiprelationship usingusing thethe correctcorrect ageage axisaxis forfor thethe thinned stand.stand. It is then possible toto construct thethe variousvarious other yieldyield stati­stati- stics for ·thethe stand in relationrelation to age.age.

A variation anon this method allows a single diagram ·toto bebe usedused for construcing yield curves for different sitesite classes.cla:sses. This involves usingueing dominant heightvheight, insteadinstead ofof agepage, as the x-axis ofof thethe graph.graph. HeightHeigirt isis thenthen usedused a:sas aa sit&-dependentsite-dendent transformatipntranBfonna~ioll of of a41(2 ~ .nthwith the age intervals being determineddetennined viavia thethe sitesite indexindex curves.ClU"res. ~'hl.sThis method cancan alsoalso bebe use3.used where the "Pacing, pacing experimentexperiment datadata do not adilquatelyadequately covercaver a range of S1.sitesp tes, to provide hype­hypo- thetical yieldyield tablestables forfor suchsuch sites.sites. NoteNot e howeverhowever thatthat thethe stibstitutionsubstitution of dominantdominant height for separate site-agesite-a.ge combinationscOOlbinations may not always give a sufficiently accurateaocurate representationrepresentati.on of site-dependentsit&-dependent responses.responses.

Increment fUnctionsfunctions Obtainedobtained from pennanentpermanent samp/esample plots of uncontruncontrolledolled spacing or fromfrOOl tree increment cores can be used to constructconstruct curvesourves for standetand ba:sa.lbanal area of unthinned stands at different stockingsvstockings, which"hich maymay thenthen bebe analyseda.ns.lysed graphicallygraphically as as above.abave. This is convenient when the users ofof a particular growth model do not have aceessaccess tot.o computingcOOlputillg equipment.

5.3.3 Conversion of GromhGrowth ModelsModels to Yield r~odelsModelz by IntefnmtionIntegration

5.3.3.1 Introduction

The mathematical integration of aa banalbasal areaarea incrementincrement functionfUnction for a givengiven series of thinning operationsoperations resultsresults inin aabasal basal area yield functionforfUnction for thatthat thinning series. Math~Mathe- matical integrationintegration has advantagesadvnntages over simulation as a way ofof using aa growthgrowth modelmodel inil1 thatthat no special computingcomputing equipmentequipment isis needed.needed. On the otherother handhand, t manymany deceptiveldeceptivelyy Simplesimple gl'owthgrowth models (e.g.(e.g. equationequation (1)(1), section 5.3.1.1) may be very difficultdifficult toto integrate.l.ntegrate.

This method assumes anan elementaryelementary knowledge of differential and integral calculusoalculus anon the part ofof thethe resresearch earch worker.

5.3.3.2 Basic theory

If we have an increment function ofof the type shownsho"" below2beloWI --74-74-

q

where q is any growth raterate (basal(basal area, diameter oror volumevollJlJle growth) and t is the time axis, then the growthgrowth ''aterate equationequation isis represented very generally by:

q f(t) )

whilstWhilst the area under this increnElltincrement curvecurve is is the total accumulatedaCCUlllUlated yield.yield.. Thus the accu­accu- mulated yieldyield upup toto timetime t,t, is given by:

1 qedt (2) Q tt Qo + '1 q.dt Wherewhere Q0Qo is ·thethe accumulated yield at the start of the period, to.

For an unthinned stands,stand, thethe basalbasal area/agearea/age curvecurve representsrepresente thethe integral function equivalent to equationequation (2) aboveab ove with with Q Q =- 0 and ttaO.= 0e libnce&nee anyany function fittedfitted to data franfrom un·thinnedunthinned stands provides an alrea8.alreagyy integrat~integrat2d fo,,"form of growth model.model. Consequently, if any any data data from from aa spacing spacing exp_eriment experiment isis fitted withIdth a yield function dependent onan stand density, then that yieldyield ftinctionfunction cancan bebe usedused to accuratelyaccurately predict thethe growth of thinned s·tands.stands. This is a generalization ofof Marsh's hypothesish;ypothesis to anyany typetype ofof growthgrowth functionfunction using anyany variables.1/variables. 1 /

5.3.3.3 Alicatio_pyrEApplication of an intesml1 yieldeld modelniosto to ditya.r.ten different thinningthinning_keatments treatments

If an integral yield model has been fittedfitted to data franfrom unthinned stands which predicts basal area from dominantdaninant height and stocking, thus:

G = f(B.f(H , N) o

Maialma504,1131.alca-14M.M.VAImmo

1/.1/ Marsh's)larsh' B h;ypothesishypothesic as as discussed discussed earlier earlier isis equivalentequivalent toto integration of a function ofof the form:form: = f(G N) - 75 -

Then if the standstand is thinned at a dominantdaninant height HHt to a stocking NNtt , thethe yield up to the' time ofof thinningthinning willwill be:be: t t

is(Ht5No) ( 2 ) where NoN isis thethe initialinitisl stockingstocking (after(after earlyearly mortalitymortality isis accountedaccounted for).for). IfIf the thinning ratioratio (c.f.(g.f. sectionsection 5.4)5.4) is:is:

Tr m :. %

• .. G' ' T /0 then for this particularpai'ticular thinning the standing croporop basal area after thinning will be:be:

G' ~ t ( 3 ) ------Tr/Gt

Whilstwhilst the yield at an interval t + i after the thinning will be:

G. = Gt f ( .) - f(ili 9 Nt ) (4)

5.3.3.4 2...b21214a1PExample of°f useuse ofot_miattEal.21.211-_Et2A21 an intepy.l :yield model

Figure 5.125012 shows shows data data fran from a spacing a spacing experiment experiment in Pinus in Lin.u.patula stands at Kwira,Kwirag TanzaniagTanzania, designated EXperimentExperiment 345. In all there were 8 treatmentsgtreatments, laidlaid outout systematicallyvsystematically, with 2 replicates forfor eacheach treatmenttreatment 'usingusing rectangular plotsplots of 0.08 ho.ha. Figure 5.12 shows only thethe data franfrom 4 plots at stockingsstocld.ngs ofof 173,173g 3471347, 694 and 11 388 etems/ha.etEms/ho. A model to predict standing basal area was""s fitted to the Wholewhole set ofof data (192(192 Observations)observations) byby weighted multiple linear regressiontregression, USingusing the model:

InGIn G a= b +b H*+bE+bEH*H* ( 5 ) boo 1,11 b2E 2 b3EH* 3 whereWhere H*H* m 1/(Ho- 1.3)1.3) l/(H o - and E - 100100/,(-7/ IN

The model was weighted by G to avoid the excessiveexcessive bi~bial to~towards lower values thotthat often results when aa logarithmiclogarithmic trsnsformationtransformation isis used.used. An R of 0.950.95 waswas obtained with coefficient values:

b 4.0865 bo0 b 1.5991 b11 1.5991 b 0.047e38 b22 0.647838 b -4.3063 b33 The model is shown on figure 5.12 overlaidoverlaid onon data fromfran 4 treatments. - 7676 -

Figure 5.12 spacingSpacing experimentuperiment 345 at Kwirat~. , Tanzania, in PilausPinus patulapatula stamlsstands

60 - !l: Ey11;::1 TOTo bAIeDATA + 1388t'liB sre-msh4A "T~..s/MA 0O 694.6 '14- 0 $TEMS/WI...... 60 a-a 341l4' " '":z: • X 1'73(73 --l: " '-' -- P'ITTE'bF I TTED MOto"Ot)E1 .. /fr

i 30 <7c.t 4~ !'" \I)V) 2..02,0 -

10 -

o o 5 /0 /5 :1..020 252.5 Dorn/NA/4T1-1riqHT

It was desired to construct a yield table for stands planted at 1 100 stems/ha, thinned onceonce at 9 years using systematic thinning to 700 stems/ha and clear felled at 16 years.

For averageaverage sites,sites, 9 years correspondedo01'1'eeponded to a dominantdaninant height ofof 17 mym, and the rotation age ofof 1616 yearsyears toto 2424 m.m. TheThe basalbasal area at 9 years, prior to thinning, can be calculated directly fromfran equation (5):(5): -77-

H* =- 1/(17 - 1.3) =- 0.0637 E ~ 100/100//7170 j1'"'1"OO

= 3.02

ln -D 4.0865 4.+ 1,59911.5991 xx 0.06370.0637 + 0.047838 x 3.02 - 4.3063 xx 3.02 xx 0.06370.0637 G9 -=3.5043 2 :. 0G = e3.5043 =• 33.2633~26 mm2/he /ha 9

Since thethe thinningthinning fromfran 11 100 to 700 stems/hestems/ha waewaa toto bebe systematic,eystematice thethe proportionproportion of basal area removed wouldwould be thethe samesame asas thethe proportionproportion ofof stockingstocldng removed.removed. FromFran equa-equa,­ tion (3)(3) above,abovee with TTr -= 1.0 by definitionldefinition: r at Gi - 222Zi_122700/1 100 9 1/33.26 2 - 21.17 mm2/ha /he

Therefore the basal areaarea extractedextracted in thinning was:wae I

G - 33.26 -- 21.1721.17 e 2 -= 12.1 m2/ham /ha

The yield at clearfellingclee.rfelling is obtainedobtained from equationequation (4)e(4), Wherewhere f(H i'.2 N)Nt) corres-corres­ t t ponds to equationequation (5) entered withuith HH =- 2424 mm, and N =- 700 stems/halstems/ha, giving giving: :t+19 +

f(Ht+ief(H + , Nt) ~= 35.98 t i

and f(Ht9f(H , lit)Nt) is eqaationequation (5)(5) entered using H =- 1717 m: t

f(Ht9f(H , Nt) = 26.97 t = so that basal area yield at clearfellingclearfelling isis givengiven by:byl

a= 21.17 ++ (35.98 - 26.97) CF 2 _ 30.18 mm2/ha/he

5.3.3.5 aWtbalm.JELLIalxmletkj:Ajan2j1:22.toFitting canpatible growth and yield models to increment data

When no no suitable suitable spacingspacing experimente:q>eriment data is available forfor direct fitting ofof the integrated formform of a yieldyield model,modele then a growth model can be fitted to increment data fromfrom permanent plots oror stem analyses whichWhich can subsequentlyeasequently bebe integratedintegrated to give a yield model. Mutterl/Clutter!! proposed proposed aa gromthgrowth model of the type:typel .

111/ ClutteryClutter, JO,.J .L. 1963. . CcrnpatibleCompatible growthgrowth andand yieldyield modelsmodels for LabiallyLoblolly Pine,Pine. Forest Sci. 99 (3):(3): 354-371 354-371 - 78-78 -

1 I =• G(aa(a + bSbS -- lJlinG)A-1 a)A- 'gg (6) whereWhere I is basal area increment/ba/year,increment/ha/year, S is sitesite index andand aa andand bb areare coefficientscoefficients whichwhich can be fstimatedgstimated by linear regression by fittingfitting thethe function:function.

(A I + lnIn a)G) =• a + bS (7) -1!.1E a where the expressionexpression onon the leftleft hand side is the dependent Y variablevariable and site index S is the predictor variable.variable. This model ,integrates to give the yield equation:equation.

lnG = a+bS- Ao (a ++bS- b S - InIn a ) (8) Go)o A

where Gtat is the yield at the endend ofof the period andend Ga 0 is the yield at the start of the period. No thinnings may occur duing the predictionointerval,prediction interval, butbut thethe basalbasal areaarea afterafter thinning becomes Goa forfor predictingpredicting thethe basalbasal areaarea immediatelyimmediately priorprior toto thethe nextnext thinning.thinning. o 5.3.4 Use ofof Growtharowth Models bby Simulation

5.3.4.1 aguirementsRequirements forfor aa simulationsimulation modelmodel

Simulation modelling is a much easiereasier method ofof using a growth model than mathematical integration in the majority of cases, but it does normally require access to a small computer. To use a growth model by simulation, the following points mustmunt be defined from permanent sample plots oror experimentalexperimental data:data.

(1) The height/age/siteheight/áge/site index function:function.

H -• f (A,S) Hoo fi(AS)1 (2) The basal area increment function, in terms of height basal area or age and and / or and site index stocking/hastocldng/ba

I E= f OH(H ' . N)N) g 2 oo'

or I • fOH(H '. a)G) g 2 o'o

or I E= f (A,N,S) g f2(A'Nr,S)2 or I =• f2(A1f (A, a,G1 S)s) c.f. Clutter's model, g 2 section 4.1.1.5 for an example

or other combinations ofof AIGOV,H0A,a, N,S,H on the right hand side.side. o - 7979- -

(3) The relationshiprelationsl!ip between stocking ranovedremoved and basal area ranovedremoved for each thinning type:type:

N'iN = f3(GVG) (4) The volume functionfunctiQ1 fromfran height and basal area:

Vv •= f4(HooG)f (H 'O) 4 o

(5) The initial basal area of the stand at Ba certain reference age or heightheight at WhichlIhich the simulation commencesvcamnences, as a functionfunctiQl ofof the planted stocking:etocking:

G =E ff5(N)(N) Goo 5 0 These functions need not be continuouacontinuous mathematicalmathanatical relationships.relationships. Each function may be aa set ofof equations,equations, oneone ofof which1ibich isis selectedselected by aa conditionalconditional process;process; or it may be a set of tabulated values, as will be seenseen in the exampleBDIIIple below.

5.3.4.2 Method ofof construction of a simulation model model

Simulation modelsmodele are constructed by coding the functional relationships andand thethe logical interconnectionsinterconnections betweenbetween them,them, asas statemientsstatanents in a computer language, usually FORTRAN9roR'l'RAN, BASIC, oror ALGOL. ThisThis' 'source source language' model is thenthen' 'compiled' compiled' by the computer into its owown internal machine codecode andand runrun inin thethe computer.computer. AnAs it runs,runs , it will require data defining:

1. The site index and initialin!tial spacing of the stand 2. The timing and intensity ofof thinnings.thinnings.

The programme will then generate output,output, in the formfom ofof a stand table or graphtgraph, as designed by the computercanputer programmer. The simulation can be divided into two logically distinct phases:

(1) The initializatianinitialization phase, where>ohere the initial valuesvalu!'s ofof standstand basal area, height and stocking are defined and table headingeheadings areare printedprinted out.out.

(2) The d.ynamicdynamic phase in whichlIhioh thethe growth ofof the standstand overover A is computedcomputed and addedadded to existingexisting growinggrowing stock;stock; any required harvesting operations are carried out;out, and the age of the stand is incrementedincranented by BB andand thethe process repeated.repeated. During the dynamic phase, summariessunmaries ofof growinggrowing stockstock areare printedprinted out.out. The dynamic phase is terminated When>ohen thethe standstand isis clearfelledclearfelled oror aa previouslypreviously setset timetime limit is reached. (A isis aa peri'odperiod" of time (usually(usually oneone to fivefive years)years) andand BB isis the length of the period.)

5.3.4.35.3.4:3 &!!mple&le ofof aa simplesimple simulationsimulation model

InIn this example, wewe construct a simple simulation model model in BASIC for even-aged""",,-aged Pinusf!l:!2! patula stands.stands. The functions in the model areare asas follows:follows: -80-80 -

(1) The height-age function is taken from Alder 111/ and is:

in H = bo + bi/A+((In S-0042)(b2 + b3/A)

where bb9 0' b , b , b are obtainedobtained from regression analysis fromfran PSPPSI' data and 2 3 for PoP. patuIc3atulai a wDichch3have have the values:

b = 3.6068 boo b =~ -17.513 1

b m= 0.008057 b22 b =~ 0.3308 1)33 whilst cc , c are defined as:as: 1 c22

= b + b /15 cl boo + b11151

C2c B b + b/15 2 - b22 + 1)3/15 The index ageage forfor sitesite index is 15 years.

(2) The basal area incrementincrElllent function is derivedderived franfrcm Experimentfteriment 3451345, KwiralKldra, Tanzania, by tabulating mean basal area incrementincrElllent by classes of stocking and dominant height,height, givinggiving thethe repultresult shownsho"" inin TableTable 5.1.5.1. This table isis used directly inin thethe programmeprogramme withoutldthout furtherfurther analysis.analysis.

(3) TheThe thinning ratios are defined byby the folloldngfollowing simple assumptions: (a) Systematic thinning: T -= 1.0 Trr

(b) Selective low thinning: First thinning T ~= 0.7 Trr Later thinningethinnings TrTEO.= 0.9 9 r (4) The volume function is the PinunPinus patula standstand volume equationequation for Kenya: 2 V B= N(-0.0072N(-O.0072 ++ 0.0000~8870.0000887 Dg2Dg +.0.00002077 HH D D t +4. 0.000032765 Dg:a)Deli) g

This gives tctaltotal volume. The merchantable volume to 20 cmem diameter top is calculated by:

Vm = VVt(0.97352(0.97352 - 21.973721.9737 axpexp (-0.15407(-0415407 Dg))Dg» VM t (5) Initialization ofof basalbasal areaarea isis donedone by by -Lining using aa standstand basalbasal areaarea ofof zerozero atat age 3 years.

1/11 Alder, D. 1977. A Growth and ManagementManagement ModelModel for Coniferous Plantations in East];hat Africa,Africa. D.Do Phil. thesis, Oxford University. - 8181 -

L1212_541Table 5.1

VAUlESVALUES OFIlF BASAL:BASAL AREAAREA IATREMLIVIINCmMENT TABULATEDTABULATED BY STOCKIAUSroCKIKl AAMAND HEIGHT FOR EXPKRIMENTEKPERDIENl' 345, 345, KWIRA1KWIRA, TAA2ANIATANZANIA

Spacing class,class, m.m. 2-3 3-4 4-5 5-6 6-7 7-8 8-9 DominántDominant Height, m. 3-5 3.89 2.87 5-7 6.59 3.69 77-9-9 7.22 4.49 9-11 7.337033 6.10 3.83 11-13 5.90 50395.39 4.55 3.56 3.17 13-15 4.59 4.59 4.41 3.50 2.84 1.91 15-17 2.82 3.30 4.35 2.93 2.34 2.182. 18 1,512.51 17-19 2.24 2.22 3.34 2.50 2.00 2.11 1.54 · 19-21 2.09 2.032. 03 2.40 1.73 1.42 1.72 1.43 21-23 2.01 11.57.57 1.99 1.32 1.28 1.37 1.271. 27 23-25 1.86 1.93 1.59

Values wiwithin thin thethe table areare basal basal area area increment incranent per per ha ha per per annum. annum. E.g. at a spacing oi' m and a daninent height of 12 m, incremE

The programmeprogramme listing listing isis givengiven below:below:

5 SELECT PRINTPRINT 005(6)4)005(6~) 10 REM EXAMPLE OF STAND GROWTH SIMULATION MODEL 20 REM INITIALIZATION PHASE 30 PRINT "SIMULATION MODEL FOR PINUS PATULA PLANTATIONS" 40 PRINT "GIVE INITIAL STOCKING AND SITE INDEX, PLEASE" 50 INPUT N,S 60 REM READ TABULATED INCREMENT VALUES 70 DIM T(7,11) 80 MAT READREAD TT 90 DATA 3.89,6.59,7.22,7.33,5.90,4.59,2.82,2.24,2.09,2.01,1.863.89,6.59,7.22,7.33,5.90,4.59,2.82,2.2~,2.09,2.01,1.86 100 DATA 2.87,3.69,4.49,6.10,5.39,4.59,3.30,2.22,2.03,1.57,1.932.87,3. 69,4.49,6.10,5.39,4.59,3.30,2.22,2.03,1.57,1.93 110 DATA 0,0,0,3.83,4.55,4.41,4.35,3.34,2.40,1.99,1.59 . 120 DATADATA 0,0,0,0,3.56,0,0,0,0,3.56,3.2.93,2.50,1.73,1.32,0 3. ~ : c, 2.93,2.50,1.73,1.32,0 130 DATA 0,0,0,0,3.17,2.84,2.34,2.00,1.42,1.28,0 140 DATADATA 0,0,0,0,0,1.91,2.18,2.11,1.72,1.37,0 150 DATA 0,0,0,0,0,0,1.51,1.54,1.43,1.27,0 152 REM READ SITE INDEX CURVE COEFFICIENTS 153 READ BO,B1,B2,133BO,B1,B2,B3 1541511 DATADATA 3.6068,-17.513,0.008057,0.3308 - 8282 -

155 LET C1=B0+B1/15Cl=BO.B1/15 156 LET C2=B2+B3/15C2=B2.B3/15 160 REM SET STAND BASAL AREA TO ZERO,AGE TO 3 170 LET G=OG=0 180 LET A=5 190 REM READ DETAILS OF STAND TREATMENT 200 PRINT "HOW LONG IS THE ROTATION, PLEASE" 210 INPUT R 220 DIM A1(10),N1(10),T$(10)Al(10),N1(10),T$(10) 230 PRINT "HOW MANY THINNINGS, PLEASE" 240 INPUT M 250 IF M == 0 THENTHEN 300300 260 FOR 1=1I=1 TO M 270 PRINT "GIVE"GIVE AGE,STEMS/HAAGE,STEMS/HA LEFT,LEFT, ANDAND TYPE(S/L)TYPE(S/L) FORFOR THINNINGTHINNING ";I";1 280 INPUT A1(I),N1(I),T$(1)Al(I),Nl(I),T$(I) 290 NEXT II 300 LET M=1M=1 305 SELECT PRINTPRINT 215(120)215(120) 310 REM PRINT TABLE HEADINGS 320 PRINT 330 PRINT TAB(20);"GROWING STOCK";TAB(80);"THINNINGS" 340 PRINT 350 PRINTUSING 360360 360 % AGE AGE HDOM STEMS/HA DIAM(G) BA/HA VOL/HA VOL/HA(20CM) VOL/HA VOL/HA(20CM) 370 REM DYNAMIC PHASEPHASE OF SIMULATION 380 REM CALCULATE DOMINANT HEIGHTHEIGHT 390 LET H=EXP(B0+B1/A+((LOG(S)-C1)/C2)*(B2+B3/A))H=EXP(BO+B1/A.«LOG(S)-C1)/C2)"(B2.B3/A)) 400 REM SELECT HEIGHTHEIGHT ANDAND STOCKING CLASSCLASS FORFOR GROWTHGROWTH INCRMENTINCRMENT 410 LET H1=(H-1)/2Hl=(H-l)/2 411 IF H1(12 THENTHEN 420420 412 LET H1=11H1=11 420 LET E1=SQR(1/N)*100-1E1=SQR(1/N)"100-1 430 IF T(E1,H1)=0 THEN 920920 440 REM ADD INCREMENT TO CURRENT BASAL AREA 450 LET G=G+T(E1,H1)G=G+T(El,Hl) 460 REM TESTTEST IFIF THINNINGTHINNING REQUIREDREQUIRED ININ CURRENTCURRENT YEARYEAR 470 IFIF A=A1(M)A=Al(M) THENTHEN 670670 480 REM CALCULATE MEAN BA DIAMETER, & VOLUMES 490 LET DD =SQR(G/(N*0.00007854))=SQR(G/(N*0.00007854)) 500 LET V=N*(-0.0072+0.00002887*D!2+0.00002077*H*D+0.00003276*D!2*H)V=N*(-0.0072.0 .00002887*D!2.0.00002077*H*D.0.00003276*D!2*H) 510 LET V1=V*(0.97352-21.9737*EXP(-0.15407*D))Vl=V*(0 .97352-21 .9737*EXP(-0 . 15407*D))

515 IFIF V1V1 ) 0 THEN 520 516 LETLET V1V1 = 0 520 REM PRINT GROWING STOCK DETAILS 530 PRINTUSING 540,A,H,N,D,G,V,V1;540,A,H,N,D,G,V,V1; 540 % ###uon ##.#no.n UUU####### ####.#Huun.' ###.#HI,.n ',#U.#####.# ####.#I'HH.n 550 REM PRINT THINNING DETAILS IF A THINNING WAS PERFORMED 560 IF T2=0 THENTHEN 600600 570 PRINT TAB(64);TAB(64); 580 PRINTUSING 590,N2,D1,G1,V2,V3;590,N2,D1,G1,V2,V3; 590 %% #####UUD ##0.####.# #eu###.#.n #nno.#####.# #Rnl####.#.n 600 REM INCREASEINCREASE AGEAGE BYBY 1 YEAR, & REPEAT DYNAMIC PHASEPHASE 605 PRINT 610 LET AA == A+1A+1 620 LET T2=0T2=0 630 IFIF A(= RR THENTHEN 370370 640 PRINT 650 PRINT TAB(20);TAB(20);"END "END OF SIMULATION" 660 STOP - 83 -

670670 RE~REM SECTION TO SIMULATE LOW OR SYSTEMATIC THINNING 680 REMREM DETERflINEDETERMINE THINNING RATIO FROM THINNING TYPE 690690 IF T$(M)="L"T$(M)="L" THEN 720 700700 LET T3=1T3= 1 710710 GOCOTO TO 760760 720720 IF M)lM)1 THEN 750 730 LET T3=0.7 740740 GOTOCOTO 760 750 LET T3=0.9 760760 REMREM CALCULATECALCULATE STOCKING REMOVED N2 & BA REMOVED G1Cl 770770 LET N2=N-NlN2=N-N1(M)(M) 780 LET G2=N1(M)/N/(T3/G) 790 LET GG1=G-G21 =G-G2 800 REM ADJUST STOCKING & BA OF RESIDUALRESIDUAL STAND 810 LET G=G2 820 LET N=N1(M) 830 REM COMPUTE MEAN DIAMETER ANDAND VOLUMESVOLUMES OFOF THINNINGSTHINNINGS 8408h0 LET Dl=SQR(Gl/(N2*0.00007854))D1=S0R(G1/(N2*0.00007854)) 850 LET V2=N2*(-0.00724-0.00002887*D1!2s.0.00002077*H*D1!24-0.00003276*D1!2*H)V2=N2*(-0.0072+0.00002887*D1!2+0.00002077*H*Dl!2+0.00003276"D1!2"H) 850860 LET V3=V2*(0.97352-21.9737*EXP(-0.15407*D1!2))V3=V2*(0.97352-21.9737*EXP(-0.15407*Dl!2)) 861 IF V3 )) 0 THEN 870 862 LET V3 == 0C 870 REM INCREMENT THINNING NUMBER ANDAND RETURNRETURN TOTO MAINMAIN PARTPART OF 880 REMREN SIMULATION 890 LET M=M+1M=M+l 900 LET T2=1 910 GOTOCOTO 480 920 REM SECTION TO ABANDON PROGRAM WHEN SIMULATIONS GOESGOES 930 REM OUTSIDE LIMITS OF AVAILABLE GROWTH DATADATA 940 PRINT TAB(20);"SIMULATIONTAB(20); "SIMULATION EXCEEDSEXCEEDS RANGERANGE OFOF GROWTHGROII'TH FUNCTION"Fm!CTION" 950 PRINT TAB(20);"RUNTAB( 20); "RUN ABANDONED"ABANDONED" 960 END

The outputoutput produced by by oneone run run is is shown in in figurefigure 5.13.5.13. Table 5.2 lists thethe definitions ofof eacheach ofof thethe variablesvariables usedused inin thethe programme.programme. This simulationAimulatioa programme is intended to be an example of of the the flexi­flexi- bility of of the the technique technique andand is not intendedintended toto suggestsuggest thatthat thethe specific functions used are inin anyany sense the best oror most most preferable methods.methode. PigureFigure 5.13

CanputerCuter rizit printout producedro_p_ou by standand growth simulation model

SIMULATION MODEL FOR PINUS PATULA PLANTATIONS GIVE INITIALINITIAL STOCKING AND SITE INDEX,INDEX, PLEASE 1700 23 HOW LONG IS THE ROTATION, PLEASE 16 HOW MANY THINNINGS, PLEASEPLEASE 1 GIVE AGE,STEMS/HA LEFT, AND TYPE(S/L) FOR THINNING 1 8 700 S

....coCX> GROWING STOCK THINNINGS I

AGE HDOM STEMS/HA DIAM(G) BA/HA VOL/HA VOL/HA(20CM) STEMS/HA DIAM(G) BA/HA VOL/HA VOLIHAVOL/HA(20CM) (20CM) 5 6.1 1700 7.0 66.5.5 8.6 0.0 6 8.5 1700 10.10.1 1 13.813.8 45.3 0.0 7 10.8 1700 12.5 2'21.1 .1 96.0 0.0 8 12.9 700 14.2 111.1 1 . 1 61.8 0.0 1000 1414.2.2 15.915 .9 138.8 0.0 9 14.8 700 16.916.9 15.7 101101.6.6 0.0 10 16.5 700 18.6 19.0 137.7 0.0 111 1 18.1 700 19.6 21.2 168168.4. 4 0.0 12 19.5 700 20.5 23.2 198.7 10.0 13 20.7 700 21.4 25.3 230.1230.1 38.5 1414 21.9 700 22.1 26.8 257.8 63.1 15 22.9 700700 22.7 28.4 285.9285.9 89.4 16 23.9 700 23.5 30.3 317.9 122.6122.6

END OF SIMULATION -85-- 85-

Table 5.2

DEFINITIONOFDEFINITION OF VARIABLES VARIABLES USED USED IN IN STAND STAND GROWTH GROll'J1I SIMULATION PROGRAKME PROGRAMME

Variable Size DescrintionDes cripti on ofof useus e

A Stand agevage, yearsyears

Al 10 List of ages for thinningsthinnings

BO Bl Coefficients for height growth function,fUnction, representing b 0' b , b , b in texttext B2 representing boy bll b292 b33 B3

Cl Coefficients for height growth functionvfUnction, representing 019c ' cc2 inin text ] l 2 C2 1

D Mean basal basal area diameterdiameter ofof growinggrowing stockstock

DIDl Mean basal basal areaarea diameter of trees removedremO'/ed in thinningthinning

ElE1 The standstand densitydensity class,class9 going fromfrom 1 i for spacingsspacin",'" ofof 2-3 m toto 77 for spacingsepacings of 8 m oror more

G The basalbasal area/are/ha he. ofof the the growing growing stockst ock

GlG1 The basalbasal arare/haea/he. ofof trees renovedremoved in thinnings

G2 The basal area of growing stock immediately after thinning H The dominant height ofof the standstand

HlH1 The dominantdominant height classtclass, goingguing fromfrom 11 for heightsheights 3-5 Invm, to to 11 for heightsheights overover 23 23 m m

I A counter for thinning number during thethe T-eaaingl'eruling inin of thinning details

M Initially,Initially9 the number ofof thinnings.thinninga. During the dynamic phase, representerepresents the next thinning numbernumber duedue

N The stockindhe.stocking/ha ofof the standstand

NlN1 1010 TheThe lietlist of residual stockings toto be left afterafter each each thinning

1(2N2 TheThe numbernumber of trees perper he.ha removed in a thinningthinning R The rotation age for the stand, in years

S The site index of the stand,stand, inin m m dominant dominant heightheight atat ageage 15 T 7911 The tabletable of diameter increnentsincrements corresponding to to thethe spacing class (1st dimension)dimension) and height class (2nd(2nd dimen-dimen­ sion). ValuesValues as per table 5.15.1 in text. - 86 -

Variable Size Description of use

T2 Set to 11 if the stand has been thinned inin thethe currentcurrent year of simulated time, otherwiseotherwise setset toto O.O. Controls the formatfo:nnat of printing.

T3 Gives the thimlingthinning ratio Tl'Tr asas defineddefined inin thethe texttext forfor each type of thinning. Set toto 11 for systElnaticsystematic thinning, 0.7 for 11st st low thinning, 0.9 for otherother lowlow thinnings.

T$ 10 A list of thinning types, for eacheach thinning. "5"S"" represents systElnaUcsystematic thinning and "L" thinning fromfran belowbelow..

\ V The total volume overover bark per hahe ofof thethe growinggrowing stock.stock.

V1vi The volumevolume o.b./heo.b./ha to a top diameter limit ofof 2020 cm,an, forfor tthe he growing stock.st ock. V2 Total volume/havolume/6 ofof thinnings.thinnings.

V3113 Volume/heVolume/ha to 20 cmem limit ofof thinnings.thinnings.

5.4 'mINNIIDTHINNING

The economiceCCOlanic componentcanponent of growth in unifonnuniform stands managed for timber will be partly removed in commericalcanmerical thinnings. In addition, non-canmercialnon-commercial thinningsthinnings maymay bebe carriedcarried outout to reduce stand density and givegive aa correspondingcorresponding greatergreater standstand mean diameterdiameter inin subsequentsubsequent thinnings and the final crop.crop.

Obviously, the modelling ofof thinningthinning is centralcentral toto anyany exerciseexercise inin yield prediction. A thinning operationoperation isis very largelylargely characterizedcharacterized by thethe numbernumber ofof stemsstems per hahe andand thethe basal area per hahe removed.removed. In single tree models,models, it is necessary to describe thethe distri­distri- bution ofof removedremoved trees;trees; but this is not necessary for stand models, which are the main subject ofof thisthis section.section.

5.4.1 The Thinning Ratio

A useful way ofof characterizingcharacterizing thinningsthinnings isis inin -ternaterms of the thinning ratio:ratio'

No.Nó, of stems leftleft/Vb. No. stEInSstems beforebefore thinninthinning Thinning ratio Thinning ratio a Basal areaarea left/Basalleft asal areaarea beforeb e f ore thinningthi·nnl.ng

liithWith a little algebra, the thinning ratioratio cancan be seenseen asas being equivalentequivalent alsoalso to:to:

(MeanMean BA diameter before thinning)2thinnin )2 Thinning ratio = (Mean(~Iean BABA diameter after thinning)2thinning 2

Typical values for the thinning ratioratio forfor differentdifferent typestypes ofof thinningthinning are:are:

Thinning typet:ype Value ofof thinning...rags.thinning ratio Early low thinnings 0.6-0.8 laterLater low thinnings 0.8-1.0 Non-selective thinning 1.0 Crown thinningthinning 1.1-1.3 -87- 87 -

The thinning ratio can be detemineddetermined empiricallyenpirically using tenporarytemporary plotsplots anon which thinningsthinningp are carredca.rred outout by contractorscontractors oror Whichwhich areare marked for thinning without actually felling the trees. The first technique gives a more realistic assessment, but maymay not be appropriate for experimentalexperiJnental types ofof thinning which\lhich one simply wishes to evaluate on a dynamic model.

Estimates of thinning ratios dbtainedobtained from pennanentpermanent sample plots maymay not be relia­relia- ble, as PSPs areare normallynonnelly clearlyclearly marked inin thethe forestforest andand hence are unlikely to receive typical treatment.treatment.

5.4.25. 4. 2 EstimatingEst Thinning Yielda in StStatic ati c Modelsod els

In order to estimate the thinning yield for a particular static yield function, then the actualactual thinningthinning intensity,intensity, defineddefined inin termstems ofof stockingstocking removedrenoved oror basal areaarea removed, and the thinning ratio, as defined above, mustmunt be known. It is also necessary toto know whether thethe yieldyield functionfunction isis basedbased anon measurements made immediatelyimmediately before thinning, immediately after thinning or, as is mostmoat normal,nonnel, an uncontrolled mixture of measurementsmeaaurements before, after andand between thinnings.

Assuming that the yield function predicts mean basal area diameter and itH is based on an uncontrolled mixture of data,datao then the diameter predictedpredioted at the time of thinning may be assumed to be a mean ofof before andand afterafter thinningthinning diameters,diameters, viz:

DpDP E (Db + De.)/2Da)/2 (1) where Dp is diameter predicted fromfrom thethe staticstatic yieldyield functionfunction andand DbDb andand DaDa areare thethe beforebefore and after thinning diametersdiameters at thatthat pointpoint inin time,time, whichwhich havehave toto bebe determined.detemined.

The thinning ratio alsoalso relatesrelates thethe unknownunknown diametersdiameters DeDa andand Db,Db givinggiving twotwo simul-simul­ taneous equations:equations: / Tr = Db2/Da2 (2)(2 )

These twotwo equations.equations cancan bebe solved to givegive DaDe andand Db:Db:

Da 2Dp/(11m/i7)

Db = 2Dp -- DeDa (4)

Knowing the before and after thinning mean basalbasal area diameters,diameters, thenthen itit isis possiblepossible to work out the basal area renovedremoved knowing the stocking removedremoved or thethe stockingstocking removedremoved knowing the basal area removed.removed.

In the case where thinning intensity is specified in ternetems ofof -atocking,stocking, then the mean basal area diameter of the trees removed in thinningethinnings is givengiven by:by:

DaifTr.Nb- Dt = (5 ) ~r~Nb_-NaNa}L vb Na. f or Dt(5)(6) Dt nJ{Nb ~ ~~Tr} Dt(6)Na - 888!i -

Note that equations Note that equations (5)(5) andami (6) are independent of the aseumptionassumption (1) that predictedpredicted diameters are averages of values immediately before diameters are averages of values immediately before and after thinningthinning.'•. Thus either equation (5)or (6) can be used to determine mean diameter of thinnings (5) or (6) can be used to detennine mean diameter of thinnings if the functions are defined in terms of predictions of after thinning diameters in tenns of predictions of after thinning diameters or beforebefore thinning diameters respectively.respectively.

AllAll thethe algebraalgebra leading to thethe aboveabove fonnulaeformulae is basedbased anon the simple relation:relation' 2 G -= N.k.DgN. k.Dg2

where G is basal area, where G is basal area, ~ is stocking, k is thethe constantconstant 0.000078540.00007854 for diameters in centi­centi- metres, basal area in m /ha and stocking in numbers/ha, metres, basal area in m /ha and stocking in numbers/ha, and Dg is the mean basal area diameter.diameter. Once the meanmean diameter of thinnings is known,known, then total and merchantable volumes can be readily oalculatedcalculated fromfran a suitablesuitable volume equationequation.

5.4.3 Fstimati¥Estimating Thinning YieldsYielda inin DynamicD;yn..,ic Modele

As with static yield models,models, both the thinning intensityintenSity and the thinning ratio must be known.known. From these two statistics, the basal area removed for a given stockinstockingg or alternatively the stocking removedremoved forfor a givengiven basal area cancan be readilyreadily determineddetennined and both parametersparameters ofof thethe growinggrowing stockstock adjustedadjusted forfor thethe thinning,thinning, withwith thinningthinning removalsremovale being calculated directly. ExamplesEamples havehave alreadyalready beenbeen discusseddiscussed inin sectionssections 5.3.3 and 5.3.4..5.3.4.

An essential feature ofof thinning in dynamic models ieis that the subsequent growth isi s influenced by the intensity ofof thinning, asas stand density willwil1 be modified followingfol1owing treat-treat­ ment. In a static modelmodel this does not happen and, hence, the danger of obtaining incorrect results with static modelsmodela by applying thinning treatmentstreatmerlts that dodo not correspondcorrespond to the fitted functions.

5.5 MORTALITY

Mortality can oftenoften be regardedregarded asas negligiblenegligible inin manymany typestypes intensivelyintensively managed-managed­ uniformunifonn forest,forest, onceonce aa particularparticular cropcrop hashas becomebecome established.established. In other cases, however,however, there isis aa markedmarked reductionreduction ofof stemstem numbersnumbers overover thethe passagepassage ofof timetime whichwhich mustmust inin somesome wayway be accountedaccounted forfor byby thethe growthgrowth andand yieldyield model.model. There are severalseveral types ofof mortality whichwhich must be considered.considered.

5.5.1 EStablishmentEstablishment MortalityMortality

Establishment mortality definesdefines thethe percentagepercentage ofof viableviable seedlingsseedlings whichwhich failfail toto survive thethe firgtfirst year.year. In the case of plantations, it is Obviously quitequite easyeasy toto definedefine in the field, but inin thethe casecase ofof standsstande establishedestablished byby artificialartificial oror naturalnatural seeding,seeding, itit isis probably better to considerconsider thethe absoluteabsolute number ofof seedlingsseedlings that become established,established, rather than to considerconsider aa ratio definingdefining percentagepercentage eurvivalsurvival oror percentagep.ercentage stocking.

Survival can also be defineddefined inin termstenns ofof percentagepercentage ofof thethe areaarea thatthat isis fullyfUl1y stocked,stocked, as discusseddiscussed inin sectionsection 6.6.

The factors that most strongly influence survival and must be considered in any model constructedconstructed to predict itit are:are'

- The method and intensityintensity ofof sitesite preparation.preparation. The weather atat thethe timetime ofof establishment811t&bHsba ...t andand forfor thethe monthsmontha following.following. - 89 --

The aspect ofof thethe site.sit e. The degree ofof exposureexposure ofof thethe site.site. The nature ofof the top 1010 amem ofof the soil, in particular, and ofof the soil generally.generally.

For plantations, the following mustmuzt also be considered:

- The age ofof the seedlings.seedlings. - Procedures for handling the seeldings from nursery to the forest. The season of the planting in relation to dormancy.

For seeding by artificialar-tificial oror naturalnatural methods,methods, oneone mustOlust consider:consider:

PesticidalPesticiaPlcoatings coatings oror treatmentstreatments toto reducereduce seedseed predation.predation. Seed pr~treatmentpro-treatment to stimulate or improve germination.gennination.

Obviously,Obviously, determinationdetennl.nation ofof allall these parametersparRlllaters andand theirtheir inclusioninclusion inin aa usefuluseful quantitative modelmoel isis virtuallyvirtually impossible.impoesible. It inis better to construct simpleSimple and robust models that cancan easilyeasily be related to practices followed in aa particularparticular areaarea oror pointpoini ofof time.time.

The model will usually be ini n the formfonn ofof aa multiple regressionregression betweenbeiween survivalsurvival (relative or absolute) and t",twoo or three of the above parameters, coded in appronriateappropI'"-'.ate ways-yo. .

5.5.25.5.2Derl.:1z7.1)2pendent Density-Dependent MortalityMc.)rtality;

Density-dependent mortality may bebe aa directdirect result ofof suppressionsuppression,, butbut Inoremore UJ.!uallyusually is indirect, in that the less vigorousvigorous trees,trees, withwith crowscrowns lowerlower iinn thethe canopy,canopy, becomebecome sus­sus- ceptible to disease oror animal damagedemage toto aa much greatergreater degreedegree thanthan moremore vigorousvigorous trees.trees.

This type of mortality can usually bebe anittedomitted fromfrom modelsmodels ofof standsstands plantedplanted atat widewide spacings andand grown grown on short rotations or subject to adequate thinning. HoHowever,wever, many growth and yield predictions relate to unthinned stands or to stands in Whichwhich thinning has been delayed and hencehence someseme efforteffort mustmust bebe mademade toto includeinclude thisthis typetype ofof mortality.mortality.

A useful model is the ReinékeReineke line, ini n 1Ihichwhich stocking is plottedplatted against dominantdomInant height, usingUSing logarithmiclogarithmic scalesscales onon bothbcr',h axes.axes. This givesgives a diagram of the type shown in figure 5.15. The Reineke line defines the maximum stockingstocking thatthat cancan be sustained at anyany given dominant height.

If stands are normally well thinned,thinned, itit maymay bebe difficultdifficult toto obtainobtain thethe necessarynecessary data to constructconstruct thisthis line.line. There are advantages in establishing in a given plantation a series ofof aboutabout 1010 plots ofof forestforest establishedestablished atat 11 x x 1 1 oror 22 xx 22 mm anon widelywidely differentdifferent sites to provide thisthis data.data. This obviously does not apply in cases whereWhere therethere areare extensive unthinned stands.stands.

In static yield models, density-dependent mortality is implicit in thethe meanmean diameter/diameter/ dominant height functionsfunctions and need not be explicitlyexplicitly defined.

In dynamicdynamic growth models, the reductionreluption in stocking caused byby mortalitymortality willwill bebe effectedaffected byby a a simulatedsimulated lowlow thinning,thinning, withwith aa thinningthinning ratioratio aramndaround 00690.6, whenever thethe stockingstocking/height ft.eight relationshiprelationship movesmoves toto aa pointpoint toto the the right right of of the the Rein-eke Reineke line.line. -90-90

Figure 5.15

RelationshipRelationshi betweenbetwesn stockingstocking aridand dominantdominant heightheight averover time in an unthinned.unthinned heavilyheavil stocked unifonnuniform fprestforest

stocking/ha,Stocking/ha. (log(log scale Reineke linelino initial.initial onset ofof den.sity.-dependent density-dependent density mortality

The arrowarrow indicatesindicates thethe direction ofof progressionprogression over timetime

Dominant heightheight (log scale)Bcale)

Note that although the Reineke line will be very well defined for a single stand, its position and slope may bebe dependentdependent onon site.site. The lowest part ofof the line may also tend to bend downwards, givinggiving a slight curve.curve.

5.5.3 Disease and Pest Mortality

As mentioned in section 5.5.2,5.5.29 some aspects ofof disease and peatpest related mortality can be implicitly included inin a simplesimple density-dependent model. However,However, therethere areare many cases where the diseasedisease oror pestpest incidenceincidence isis epidemicepidemiC inin naturenature andand tendstends toto occuroccur asas out-out­ breaks followingfollowing particularparticular patternspatterns ofof weatherweather andand movingmoving outwardsoutwards fromfrom epicentrosepicentres ofof infection. Techniques for modelling suchsuch problems areare somewhatsomewhat beyond thethe scopescope ofof this manual. Such models are oftenoften stochastic, inin that onlyonly the probability ofof anan outbreakoutbraek cancan be predicted.

The probability ofof anan outbrealcoutbraek may be a function ofof weather pattern, condition ofof the forestforest growinggrowing stockstock oror ofof distancedistance fromfran anotheranother outbreakoutbreak area.area. FromFrom aa managerialmanagerial pointpoint of view, the probability ofof an outbreakoutbraek cancan be used toto assessassess thethe erpectedexpected valuevalue ofof thethe crop at some future point inin timetime andand hencehence thethe opportunityopportunity costcost ofof pre-emptivepre-emptive fellingfelling or ofof taking oror failingfailing to taketake controlcontrol measures.measures. - 91 -

5.5.4 NindthrowMlndthrow andand FireFire Dama-'eDamage

As with pest and disease problemsvproblens, both windthrow and fire damage effects have density-dependent andand density,-independentdensity-independent effectsveffects, with the latterlatter tending to be catastrophic only be pre- inin nature, i.e.i.e. destroyingthedestroying the whole"hole stand. Again, the catastrophiccatastrophic componentcanponent cancan only be pre­ the dicted asaas a probabilityprobability of of occurrences, occurrence, dependentcmdependent CIt climaticclimatic andand sitesite factors,asfactors, as wellwell asas the conditioncondition ofof' thethe growinggrowing stock.stock. WithWith fire,fire, additional factorsvfactors, such as the time since the last controlled burning and the use ofof lowlow pruningvpruning, may be important variablesvariables inin determiningdetermining the probability ofof catastrophiccatastrophic damage.damage.

The successfulsuccessful predictionpredicti on ofof thethe probabilityprcbability ofof aa catastrophiccatastrophic eventev.."t isis extremelyextremely difficult where suchsuch eventsevents areare rare.rarea WhereWhe?e firevfire, wind damage or certain peetspests or diseases are normal, suchsuch modelsmodels maybemay be constructed; but it is stillstill likely to require a careful process ofof datadate. gatheringgathering andand evaluationevaluation overover aboutabout 2020 yearsyears beforebefore anythinganything usefuluseful maymay emerge. It should also be noted that the parameters forfor suchsuch models must be continually revisedvrevised, as theythey are likely to be veryvery sensitivesensitive toto climaticclimatic fluctuations.fluctuations.

5.6 STAND VOLUMEVOWME PREDICTION'PREDICTION

In the the foregoing foregoing parts of sectionparts 59 of various section methods 5, of predictingvarioua dominant methods heightv of predicting danina."t height, basal area, area, stocking andand mean basal area diameter havehave been discussed.discussed. The final stage in a yield prediction systemsysten for a single stand is to use these variable.variables totO predictpredict to'taltotal andand merchantable volume.volume.

Part I ofof this manual hashas dealtdealt exhaustivelyexhaustively withwith thethe techniquestechniqu.ea of0::: measuringmeasuring andand modelling thethe volumevolume ofof individualindividual trees.trees. This section otilvC!'Ily concernsconcerns iicelfiteelf with with a aresum6 resurn~ of methodsmethods particularly appropriate to modelsmodels of stand volume franfrom thethe parametersparameters normally calculated in a yield model.

5.6.1 StandSancIVolquatjons Volume Equations Based Based on Dominant on Dominant Heht Height tuqc1 and 33asal Basal AreaArea

TctalTotal stand volume, eithereither toto thethe tiptip oror toto somesome smallsmall diameterdiameter limitlimit suchsuch ase.s 1010 cm,ern. can bebe accurately predic'tedpredicted by anan equationequation usingusing basalbasal areaarea andand dominantdominant height.height. Typical equatequations ions are:

(1) V -~ b0bO +-> bl.G.Hb1.G.H (linear)

(2 ) viaV/G ~= bOb0 ++ b1.Hb1.H (linear(linear, weighted by G)

(3 ) log VV ~= b0bO + b1.10gblolog G + b2.1og H (logarithmic) 2 (4) V/GV/G ~ bOb0 ++ b1.Hbl.H ++ b2b20H2. H (quadratic,(quadraticv weightedweigtted byby G)

A wide variety of otherether models isis possible. A£As has been discussed in PartPart I.Iv sanesome form of weighting is normally desirable for fitting volumavolume equations, as thethe residual error tendstends to be approximately proportional to the vollllle.volume. JiquationsEquations (2),(2) (3) and (4) all achieve this,this. Model (2)(2) is very simple to fit graphically.

The quantity V/G is widely known as formfo:rm height and has the dimension metres.metres.

The degree of fit obtained whenwhen stand tetaltotal volumevolume is regressed withwith basalbasal areaarea andand dominant heightheigtt with models such as the above is usually very high9high, often with correlation coefficients exceeding 0.990.99.. - 9292 -

5.6.2 UsinUsing Tree Volume Equationseations to Predict Stand VolumeVolume

It is a fairly commonplace practice to use a tree volumevolume equation to predict stand volume by enteringentering thethe equationequation with mean basal area diameter and Lorey'sLorey's mean height. The, latter is the mean of a systematic sample of tree heights weighted by tree basalbanal area. Alternatively, aritbneticarithmetic mean height or dominant height are used.

This method gives a biased result in most cases because the equation coefficients have not been influenced by the changing diameter distribution and form of trees with stand parameters.

In manymany cases the bias, which is normally towards underestimation of stand volume, is small enough to be regarded as negligible. The methodmethod willwill tendtend towardstowards serious undel'­under- estimation of volume however if the equation is predicting volume to a cut-off diameter and the mean diameter isis closeclose to thisthis limit.limit.

5.6.3 EatimationEBtirnation of Volume to a TopTgp Diameter Limit

It is often the case in uniform forests that volume yields are required to be esti-esti­ mated for two oror three top diameterdiameter limits.limits. This may bebe done as follows:

(1) Fit a standstand volume equationequation asas describeddescribed above(s.abOlTe (s. 5.6.1.),5.6.1 . ), eithereither overbarkoverbark oror underbark, asas required.required.

(2) For each sample plot, calculatecalculate the ratioratio ofof volume toto thethe selectedselected top diameterdiameter limit to total volume.volume.

(3(3) ) A graph of these ratios against stand mean diameter will produce a diagram ofof the type shownshown below.below• •E =- VV /1T mIv ror 1.01 0 ------

" , "

, ., " " , " •

o TailToP diameterdiameter

(4) This shape can be fitted by thethe function:function:

R =c 1 _- a.exp(b.Dg)

which can be transformed into:into:

log(1log(l - R) =c a* + b.Dg

and fitted by linear regression.regression. Not thatthat a is exp(a*). - 93 --

With severalsevsral merchantable ratios, an attempt can be made to harmonize the coeffi-coeffi­ cients so that they are themselvesthemselves functionsfunctions ofof thethe toptop diameterdiameter limit.limit. Alternatively, oneone could attempt to construct aa multivariable extensionextension ofof the above model incorporating mer-mer­ chantallechantable diameter.diameter.

When fitting the modelemodel, itit is important that data points with zero ratios are exclexcluded.uded. When the function is actually used, if a negative ratio is predicted, the mer­mer- chantable volume should bebe takentaken asas zero.zero,

5.6.4 Volumes of ThinningsThinningE

Various methods ofof calculatingcalculating thinning volumes cancan be adopted.adopted. The stand volume equation may bebe entered using the basal area of thinnings. AlternativelyeAlternatively, a tree volume table may be enteredentered withwith thinningthinning mean diameterdiameter andand estimatedestimated meanmean heightheight (perhaps(perhaps fromfrom aa height/diameter regression).regression). The most unbiasedunbiased method is to construct a stand volume equation using only thinnings, i. e.e. a a regressionregression ofof thinnedthinned volumevolume onan thinnedthinned basalbasal areaarea and stand dominant height.height. This regression can bebe testedtested toto seesee ifif itit differsdiffers Significantlysignificantly from the general stand volume equationequation ands,and, ifif not,note thethe latterlatter maymaybe be used.

Because thinning volumes areare oftenoften notnot measuredmeasured. it may not be possiblepOSSible to constractconstruct such a model. It is suggested that inin such cases,oases, thinned volumevolume isis calculatedcalculated asas thethe dif-dif­ ference between stand volume immediately before and afterafter thinning.thinning. This techniques maymay bebe applied to both total and merchantable volumes.

5.7 ADVANCED TECHNIQUEE TECHNIQPIS OFOF GROWTHGIlOWTI! AND AND YIELD YIELD PREDIC'l'IONPREDICTION

There are two types ofof model Whichwhich cancan be appliedapplied toto uniformuniform standsstands whichwhich havell3ve notnot been referred to directly in the foregoingforegOing sectiOnssections,, in spite of thethe fact tll11tthat theythey areare inin useuSe in a number ofof forestryforestry organizationsorganizations. These are size class models and tree position models.

5.7.1 Size Class ModelsModels

In a size class model ofof aa uniformunifo"," stand,stand, thethe growinggrowing stock stock at at any any point point in in timetime isis described by a frequency distribution ofof treetree sizes.sizes. The increment on each size class is calculated separately, usuallyusually as a function of site,sitee age and stand densitydensity andand thethe relatie!',relaticn between the size classclass diameterdiameter andand thethe meanmean oror dominantdominant diameter.diameter.

Thinnings and mortality must be describeddescribed inin termsterms ofof frequencyfrequency distributionsdistributions alsovalso, with the growing stock distribution beingbeing modifiedmodified forfo:, eacheach thinningthinning or0:' mortalitymortality event.event.

The advantage of this type of model for unifonnuniform stands is that it gives moremore detaileddetailed information about the size assortment of yield, especially if a stem tapertaper function is alsoalso used for volume calculation. The disadvantage is that thethe analysisanalysis ofof thethe growthgrowth datadata isis more complex than for whole stand models and requires consideration of additional parametersparameters for diameter distributions; whilst the use of the model necessitates access toto an electronic canputer.computer.

Models ofof this type areare notnat necessarilynecessarily moremore accurateaccurate inin nredictingpredicting whole stand parameters such asas basal areaarea oror meanmean diameter;diameter; but they dodo provide more detailed informa-inform..... tion about the stand.stand. - 94-94 -

5.7.2 Tree Position Models

Tree position models are those in which the actual spatialspatial relationship betweenbetween all the trees inin aa simulatedsimulated plotplot areare representedrepresented byby thethe model.model. Inter-treeIntel'-tree competitioncompetition isis dependent anon the relative sizes and positions of neighbouring trees. This information isis used to provide aa competitivecompetitive index for eacheach tree Whichwhich is used to reduce the actual growth of the tree, relative to that which it would have if it were growing free of any competitioncompetition.. The latter is dependent anon thethe ageage andand site.site.

Tree position models require a formidable number of calculations and stretch the resources of eveneven quitequite largelarge computers.computers. As a technique for modelling the growth of uni-uni­ form stands, they are unnecessarily complex, sincesince they do not provide significantly more useful information aboutabout stand growthgrowth than aa sizesize classclass model?model, Whilstwhilst requiring probably two ordersorders ofof magnitude more calculations.calculations. The principleprinciple advantagesadvantages of treetree positionposition modelsmodels are in relation to physiological researchreeearch and to growth and yield prediction in mixed stands.

For readers interested in pursuing the literature anon these more advanced types of model, the bibliography published by the CommonwealthCommonwealth AgriculturalAgricultural Bureau entitledentitled "Compu"Compu­ terized Methods inin Forest PlanningPlanning and Forecasting" isis recannemied.recommended. Details are in Appendix D.D. - 9595-

6. ANALYSIS OFOF GROW'lHGROWTH ANDAM YIELDYIELD DATADATA FORFUR MIXED FORESTFURE3T

Mixed forest,forest, inin thethe contextcontext ofof this manual, implies any forest in wchwhich thethe indi­indi- vidual standsstands thatthat composeCOO1pose thethe basicbasic mensurationalmensuraticnal unitunit containcontain aa mixturemixture ofof ageage classes.classes. TYPicallyTypically, mixedmixed forest forest willwill containcontain alsoalso aa varietyvariety ofof apecies,species, Whichwch may bebe ecologically quite similar oror composedcOO1posed ofof severalseveral ecologicalecological groups,groups, withwith eacheach groupgroup tendingtending toto bebe dominant in a particular stratum of the canopy or anon particularparticular microsimicrosites tes oror inin differentdifferent successional phases followingfollowing gapgap formationformation oror harvestingharvesting operations.operations.

The distinction between mixed and uniform foresteforests isis not absolute.absolute. There are cases in which a mixed standstand consistsconeists essentiallyessentially ofof anlyonly twotwo ageage classesclasses - very small understoreyunderstorey regeneration and a dominantdOO1inant overstoreyoverstorey - and where harvesting is essentially aa clearclear fellingfelling operation. In these cases, the techniques of unifolmunifoim stand yieldyield predictionprediction maymay bebe largelylargely applicable. Similarly, standsstands of uniform age, but of mixedmixed species, maymay needneed toto bebe analysedanalysed using methods appropriate forfor mixed stands, in orderorder to allow for different proportimsproportions of species mixtures withinwithin aa singlesingle model.model.

Mixed forestsforests generallygenerally involveinvolve much more severes""ere mensurational and sampling problems than uniform forests.forests •. .The The value per unit area ofof a mixed forest will normally bebe lower than that for a uniform forestforest atat maturity andeand, hence, onceonce cannot easily justify an equivalent sampling intensity.intensity. At the same time, thethe dispersiondispersion of valuablevaluable epecieespecies andand thethe variabilityvariability of the forest, implies that the aamplingeampling intensity required to obtain thethe sernesame predictivepredictive power as for a uniformunifonn standstand model must be much greater.

The mensurationslmensurational problems imposed by poor accees,access, deneedense understorey,understorey, buttreesesbuttresses and the near impossibility ofof treetree total height measurement in many cases, means that measurement costs are much higherhigher forfor a given level of precision. This tends to :f'u.rtherfurther reduce the amount ofof samplingsampling thatthat cancan be undertaken within a fixed cost budget.

Consequently, most attempts at yield prediction in mixed forest have beenbeen basedbased uponupon relatively small quantitiesquantities ofof data;data; the resultant models not unnaturally haTehave muchmuch lower precision than wouldwauld be consideredconsidered adequate in uniform forestforeet management. There are eanesome exceptions to this, as forfor exampleexample the yield prediction work in the PhilippinesPhilippinee reportedrepetr!;ed by Revilla !i,1/, wchwhich cites resultsresults fromfran Dipterocarp forestforeet involving over 240 permanent sample plots.

Because ofof the variety ofof apeciesspeciee composition,composition, floristicflorlstic structure,etructure, ecological situations and silvicultural practices that are possibleposeible in mixed forest, there are an eveneven greater diversity ofof modelling strategiesstrategies available than for uniform stands. Many ofof thethe published methods areare essentiallyessentially untesteduntested exceptexcept onon thethe basisbasis ofof veryvery smallBmall numbers ofof plots.plots. The techniques selected forfor discussiondiscussion in this section areare perhaps the most robust methods and fall intinto 0 three groups:groups,

Yield functionsfUnctions applicableapplicable toto simplerSimpler typestypes ofof mixed forest.forest. TransTransition i ti on matrixmat rix models.models. Distance-independentDistanceindependent treetree modelsmodels basedbased onon differencedifference equations.equations.

!i1/ Revilla, A.R. 19791979 Yield prediction inin cut-overcutover DipterocarpDipterocarp standsstands inin thethe Philippines.Philippines. Paper to SEminarSeminar onon ManagEmentManagement ofof Dipterocarp Forests, Metro Manila. 20 pp. - 96 --

ItIt is probable that in the long term,tenn, a fourthfourth groupgroup ofof models, the treetree positionposition models, will be the mostmost effectiveeffective basiobasis forfor yieldyield predictionprediction inin mixedmixed stands.stands. However, such models requirerequire moremore powerfulpowerful computerscomputers thanthan areare normallynormally availableavailable todaytoday andand considerableconsiderable investment into research into appropriateappropriate sampling techniques before they become a viable nroposition.oropoeition.

In this chapter, we do not attempt to givegive any specificspecific instructionsinstructions concerningcCllcerning thethe -onstruction of growth models and yield forecasting for mixed forestsforests but ratherrather somesome examplesexamples of possiblepossible wayswaye ofof dealing with these problems.problems. ElrtensiveEktensive work is however going on in the field of modelling forfor mixed forests andand itit isis hopedhoped thatthat viableviable methodsmethode willwill shortlyshortly bebe rleveloped.developed.

",' . ' SITE CLASSIFICATION

Because mixed forestsforeste normally comprisecomprise manymany species,species, thethe conceptconcept ofof sitesite indexindex mustmust neoe related either to aa speciesspecies withinwithin anan individualindividual treetree modelmodel oror toto aa speciesspecies associationassociation inI n a yield model. There are many specific techniques for determining eitesite index that havehave een proposed. ~'heyThey are all equivalent however in their origins.origins. Site indicators areare either2either.

Mensurational parametersparameters inin variousvarious combinations.oombinations. (E.g. dominant height and age in the conventional uniform stand site indexindex curve).curve).

EnvironmentalFrivironmental factors,factors, suchsuch asas altitude,altitude, Aoilsoil type,ype, rainfall,rainfall, etc.;etc.; or indirect environmental indicators such as indicatorindicator speciesspecies oror goegraphicgoegraphic groupings.groupings.

The first classclass ofof site indicatorindicator repreaentsrepresents inin factfact thethe observedobserved residualresidual fromfrom aa 'articular;larticular model, inin whichwhich sitesite isis notnot included.incluhed.. ThunThus if oneone devises a model, such asss forfor example,',xample, a transitiontransition matrixmatrix model,model, forfor a mb>dmizmi stand, and applies it toto aa numbernumber ofof plots,plots, oneDne will findfind a.a differentdifferent residual errorerror fozfOI eacheach plot.plot. That error can b.be coded on a scale such aeas 1,1 t 2, 3,3, 4,4, 5 etc., with a medianmedian v£:tuevejue representing a zero error midand oneone then has a mensurational sitesite classclass ay-tensytsm applicable toto those plots.plots.

To'1'0 use such a site class system in ieactice,I mctice, the erroreerrors (i.e(i.e. site indices)indices) mustmust be.be -tetenninedietermined at oneone pointpoint inin timetime andand thanthen usilus"i asas aa compensatingcompensating inputinput toto thethe modelmodel forfor aa prediction ata-h aa secondsecond pointpoint inin time.time. '!'hisThis is precisely what is done with a conventionalconventional set of site indexindex curvescurves forfor aa plantation.plantation. From a known height-ageheight-age input,input, sitesite classclass isis determined,detennined, whichwhich isis thenthen fedfed intointo thethe modE.mod<_ at a second point in time in order to obtain a much more accurate predictionprediction ofof aubsequentsubsequent height than wouldwould be possible from the mean height-age curve.curve.

Once a setBet ofof sitesite classclass values haehal, been assigned by determining residuals from a Rite-independentsite-independent model, then the residualsieanresiduals ;oan be correlated with owironmentalenvironmental factors,factors, to determinedetennine whichwhich areare mostmost effectiveeffective inin aeplaininga'.plaining thethe residualresidual variationvariation ofof thethe model.model. Indicator species are includedincluded bybY' uningUSing ajzero-onea.Jzer~one variablevariable forfor eacheach species:speoiest zero for absence fromfrom thethe plotplot andand cineone for presentpresent.on' on the plot.plot. otherOther qualitativequalita:U,ve factors, such as soil type, can be codedooded inin the sanesame~. way. ~'Appendix~pendix AA discussesdiscusses thethe useuse of ze~onezero-one variables in multiple regression. 111- -1-

-97 -

The use ofof mensurational variables as sitesite indicatorsindicators isis lesslese easye • .sy with a mixed stand than with uniformunifonn stands and can onlyonly effectivelyeffectively be done with a static yield medel.model. However, environmentalenvironmental factorsfactors cancan be usedused asas sitesite indicatorsindicators withwith anyany type ofof model;medel; with mixed forests, environmentalenvironmental sitesite indicatorsindicators willwill thus bebe more importantimportant than with unifonnuniform stands. In mixed foresta,forests, environmentalenvironmental parameters are a necessary part of an effectiveeffective model, whereas withwith uniformuniform standsstands usingusing heightheight asa,s aa basis forfor sitesite determination,detennination, environ-environ­ mental parameters areare asan optionaloptional extra.extra.

It should be appreciated fromfran the above discussion that the convenitonal tennsterms site index, site class,class, etc.etc0 are to somesane degree misnaners.miBnaners. What one is really dealing with are self-calioratingself-calibrating models,medels, using an error estimate at one point in time toto reducereduce thethe predic­predic- tion error for aa laterlater estimate.estimate. There is nothing inherently one-dimensionalon&-dimensional about the effect of the environment onon a" forestforest growthgrowth model.medel. In addition, somesane part of the error that self-calibrationself-calibration can effectively compensate for may be unrelated to the environment and may be due to historical factors (past(past stand treatment),treat.ment), stand density,denSity, if this is not adequately incorporated in the medelomodeltend and genetic effectseffects (provenance(provenance oror epeciesspecies variation).

These points cancan be summarisedsummarised symbolicallysymbolically esas follows:follows:

TimTime T Growth/yield model.medel. e 1 Solve:Solve •

= f(X I) Y1 • f(X191,I) ~

for I such that Y -Y*~Y* 1 1

Fit regression between II, and E,E J J

I, • G(BoBi)g(B,E,)

/ J

Time Use I q= g(B,E ) T2 g(BE2)2

Compute:impute,1 Y •= f(X°f(X ,I)I) Y22 2 to determine predicted yield

5.32:bolsSymbols B Coefficients in the regression between sitesite indexindex andand environmentalenvironmental parameters. E Set of environmental parameters onon a series ofof plots, usedused to fitfit regression.regression. j E2 EhvironmentalEnvironmental parameters onon plot atat timetime T2T2 E2 rOf() Any mathematical modelmedel usedused asas thethe growthgrowth model.medel. -98-

gOg0 Any mathematicalmathematical function (linear or nonlinear) usedused asas thethe sitesite index­index- environment regression.r~ession. I ERtimatedEstimated site index value forfor aa plot.plot. I Set ofof site indexindex values anon thethe samesame plotsplots asas E.E werewere determined.determined. Ijj j X)x1) Independent variables used in the growthgrowth oror yieldyield modelmodel atat timestimes 1, andand 2.2. ,1)) X ) X2)2 Y,Y )) Yield estimated fromfrom the model atat timestimes' 1 and 2.2. 1\) Y ) Y2)2 Y~Y* Actual yield onon plot onon whichtdlich X1X, werewere determined.determined. 1

6.2 STATIC YIELD FUNCTIONS FOR MIXEDMIXEM FOREST

6.2.'6.2.1 General PrinCiplesPrinciples

Static yield functions have beenbeen discusseddiscussed in section 5 with reference to uniform forest. 'l'heThe principal feature of a static function is that time is included in thethe modelmodel as a total elapsedelapsed timetime fromfrom somesome referencereference point.point. In a uniformuniform forest,forest, thethe reference pointpoint is usually the date of planting oror establishmentestablislrnent and time is the age of the forest. In a mixed forestforest model,model, thethe timetime basebase maymay be the lastlast harvestingharvesting qperationoperation oror thethe lastlast forest inventory oror it may bebe somesome otherother point.pOint.

The model must take thethe generalgSReral form:form:

Y f(x1,x2?0000Xn,t) where Y isis the measure ofof yieldyield ofof interest.interest. This may be timber volumevolume of merchantable species, basal area, fuelwood tonnage/ha,tonnage/ha oror non-timbernon-timber yieldyield suchsuch asas fruit,fruit, seedseed oror bark productiproduction. on.

The variables x.x, are any variables which fulfilfulfil two criteria:criteria: ~

(,) They are useful inin predictingpredicting Y.Y. That is, they add Significantlysignificantly in a statis­statis- tical sense to the goodness ofof fit ofof the funtion f()fO to the yield.

(2) They can be determined without requiring remeasurementrsmeasurement ofof the forest at time t. They may be inventory statistics at the base time tt0 , or they may be regional site indices derivedderived fromfrom soil,soil, topographictopographic or or climgtic clim~tic data. TheyThey may be qualitative variables denoting the particularparticular typetype ofof harvestingharvesting treatmenttreatment the forest has received.received.

The time elapsed from the time base is t. This must be present ifif thethe functionfunction isis to behe useful as a planning tool.

The form of the function will vary tremendouslytremendously dependingdepending uponupon thethe yieldyield being pre-pre­ dicted, the data involved and,and, to somesome extent,extent, thethe sophisticationsophistication ofof thethe analyticalanalytical toolstools available. -9999- -

6.2.2 Data Analysis ProceduresProcedures

The data may be from temporary oror permanent plots or from experiments.experiments. The simplest type of data analysis that oneone isis likelylikely to undertake will be multiple linear regression,regression, coupled with graphicalgraphical analysis ofof data and ofof residuals. An electronic computer isis there­there- fore essential, although itit couldcould be quitequite aa small one.one.

The data from the plots must be summarized priorprior to determining the yield equation. The summary willwill derivederive thethe followingfollowing typestypes ofof informationinformation forfor eacheach plot!, plot, anon eacheach occasionoccasion of measurement.

(1) Plot identity; for experimentsexperiments this will include block number, treatment number and replicatereplicate number.number.

(2) YieldslYields; there may be onlyonly oneone type ofof yield to considerconsider (e.g.(e.g. totaltotal merchan-merchan­ table volume) but more usually yield will be expressedexpressed by speciesspecies groupsgroups andand in size/quality classes, so there may be eight or more types of yield.

(3) Possible predictor variables.variables. These include:

- Basal area/has:rea/hA by speciesspecies groupsgroups at timetime to.t • o - StockingSt ocking % by speciessp eci es groupsgroups atat time to.t • o - Logging intensityint ensity atat timetime to.t • o - Silvicultura].Silvicultural class. - Forest type code.code. Environmental indicatorsindicators (soil(soil type,type, altitude,altitude, rainfall,rainfall, etc.).etc.).

Once the plotplot summaries have beenbeen derived,derived, theythey s:reare maintainedmaintained inin computercomputer accacces-es­ sible formform forfor the various subsequentsubsequent analyses.analyses.

For analysis purposes, variousvarious proceduresprocedures suchsuch asas principalprincipal componentscomponents analysis,analysis, stepwise regression, combinatorial regression can bebe used to give the automatic seleselectionction of the predictor variables whichwhich' 'best' best' predictpredict thethe yields.yields. However,However,! a priori selection ofof predictor variablesvs:riables in a relatively simple equation is generally preferable, combined with a carefulcs:reful graphical analysis of residuals and preferably some canmonsensecommonsense relationship between the formform ofof thethe functionfunction andand thethe realityreality ofof thethe biologicalbiological situationsituation predicted.predicted.

6.2.3 Methods of SelectingSelecting a Yield Equation

Section 5 has considered the various forms of equation that can be used to model asymptotic growthgrowth processes.processes. In Appendix A, figure A.2.1. shows a varietyvariety of basicbasic functionsfunctions which cancan representrepresent otherother shapesshapes besidesbesides thethe asymptotic.asymptotic. In developing a curve or function to model yield development in natural forest, it is best first to graph the data, plotting yield againstagainst timetime fromfrom thethe timetime basebase (Whether(whether aa fellingfelling oror inventory).inventory). The data should bebe classified by basal areas:rea ofof thethe standsstands atat thethe basebase time;time; separate graphegraphs maymay bebe drawn for different sites oror forests ofof different speciesspecies composition.composition. - 100 -

On the basis ofof suchsuch graphs,graphs, aa suitablesuitable equationequation cancan bebe selectedselected fromfran thethe variousvarious forms described elsewhere inin thisthis manual.manusl. The data maymaybe be fittedfitted directly,directly, usingusing multiplemultiple regression oror nestednested regression;regression; or aa hand-drawnhand-drawn line may be sketched through the data andand subsequently approximatedapproximated byby anan equationequation ifif required.required.

In many instances wherewhere anlyonly lightlight selectionselection fellingfelling isis carriedcarried outout andand thethe forestforest is a complexcanplex mixture ofof many species and ecologicalecological types, there may be no direct relation­relation- ship discernible overOller timetime betweenbetween yield,yield, basalbasal area area and and si-te site class.class. In such cases, predic­predic- tion ofof growth withwith aa simplesimple yieldyield modelmodel isis notnot anan effectiveeffective strategy.strategy.

6.2.4 Problems with Static YieldYield Models inin MixedMixed ForestForest

Because static yieldyield models are relativelyrelatively straightforward to constructconstruct and use, it might seemseem that they offeroffer somethingsc:mething ofof aa panacea forfor thethe problemsproblems ofof yieldyield prediction in mixed forests.forests. Unfortunately, thisthis isis farfar fromfran thethe case.case. The problems are essentiallyessentially ofof the same kind as those discussed inin sectionsection 5.2.5 regarding yield models forfor uniformuniform forests,forests, but are ratherrather moremore severesevere inin degree.degree.

There are two basic kinds ofof problem:

(1) A particular model of yield has implicit in it a historical sequence ofof events corresponding to those occurring in the data set used to constructconstruct the model. The model cannot be applied reliably to data which has experiencedexperienced a different history.

(2) There is a problem ofof compatibility.compatibility. Suppose for example aneone predicts three types of volume; VmVM for merchantablemerchantable species; Vp for partially merchantable species; and Vu for unused species.epeciee. The total of these three should logically represent the total volume of the forest.forest. However, if aneone comparescc:mpares (Vm+Vp+Vu)(Vm+Vp+Vu) with total volume, either from actual data oror fromfran aa fourthfourth functionfunction fittedfitted directly to total volume, oneone willwill findfind aa considerableconsiderable bias.bias.

Neither of these problems is insurmountable inin principle.principle. With adequate quantities of data franfrom permanent plots and long term experiments, historical factors representingrepresenting different sequencessequences ofof treatmenttreatment cancan bebe incorporatedincorporated asas additionaladditional qualitativequalitative variables.variables. Compatibility problems may be overcomeovercane byby fitting constrained regressions whichwhich areare forcedforced to satisfy particular requirements.requirements.

Static yield modelsmodels aboveabove allall suffersuffer framfrom thethe basicbasic limitationlimitation thatthat theythey cannotcannot utilize data fromfran varied historical sources (e.g.(e.g. different types of experiments, permanent sample plots and short term tree increment plots); nor can they be adaptedadapted toto predict yieldyield for historicalhistorical regimes (i.e. sequences of treatment) other than those implicit inin thethe datadata used to construct them.than.

6.2.5 Conclusions regardingregarding Sta-ticStatic YieldYield ModelsModels in Mixed Forest

Static yield models areare relativelyrelativelv simplesimple toto construct,construct, givengiven adequateadequate data.data.

In manymany types of mixed forest,forest , cLearc" ear relationships between time, treatment intensity and si-tesite areare notnot obBel"'V1'\.~le.cibservie. In such cases, static models cannotcannot be used. - 101 -

Care must bebe taken with a static model to recognize that it is tied to predefined historical treatment sequencessequences implicitimplicit inin the data used to constructconstruct it.

lfuenWhen predictingpredicting severalseveral typestypes ofof yield,yield, eithereither constrainedconstrained regressions should bebe fitted or it mus'tmust bebe accepted thatthat the differentdifferent predictionspredictions cannot bebe regarded as additive to anan unbiasedIlIibiased total.total.

6.3 TRANSITIONTRAlllITION MATRIXMATR17 MODELSMODFLS

'l'ransitionTransition matrix models provide a method for making short term predictions of forest growing stockst ock anon the basis ofof largelarge quantitiesquantities ofof poor qualityquality datadata fromfrom recurrentrecurrent measure-measure­ ments. They can be developed from, for example, continuous forest inventory data based on permanent plots oror temporary plotsplots nearnear thethe samesame locations,locations, inin which.mich trees havehave been measured only by diameter andand speciesspecies class.class. The method canCEll'! be used with data that contains signifi­signifi- cant numbers of erroneouserroneous measurements. Short temterm predictions, in thisthis context, implies periods up "coto fortyforty oror fiftyfifty years or oneOIle cutting qycle.cYcle.

The method is objective,objective. butbut notnot necessarilynecessarily veryvery accurate.accurate. Its principal advantage is that the data base can be analysedanalysed automaticallyautomatically and may be strictlystrictly conventionalconventional inven-inven­ tory data. Its princpal disadvantage is that it is not usually possible to construct adequatelyade«uately refined transition matrices unlessunless largelarge quantitiesquantities ofof datadata areare auailable.~lable.

6.3.1 Definition ofof a a TransitionTransition 1-Tatrix Matrix Wod.elKodel

A transition matrix model requiresrequires thatthat . aa systemsystem be representedrepresented by aa rowrow ofof variables .mtchwhich may be calledcalled a state vector.vector... One can imagine this as, forfor example:example,

Xm =~ (:xrn1,(xm1, Dn2,xm2,.....xmn) ••••• :xmn)

where eacheach xmi"",i representsrepresents thethe numbernumber ofof treestrees inin thethe ii thth diameterdiameter class.class. The first sub-sub­ script (m)(m) denotes the time period. The x'sx' s do not need to representrepresent sizesize classes.classes. They could, for example,example, be total biomass inin differentdifferent trophictrophic levelslevels ofof anan ecosystem;ecosyst ... ; or development stages in an animal population. But in growthgrowth models for mixed forests,forests, the state vector elements will normallynomally bebe sizesize classes.classes.

The transition matrix itself, denoteddenoted by TvT, consistsconsists ofof anan nn x x n n tabletable ofof elements,elements, like this:this,

T = t11 t12 t13 .• • • • • t1n t21 t22 • • • t2n • .O •n •0 •O0 • •Ou oo •e0000 • • • • .0 •0 •0 •0 0• O• •0 0 •0 •0 •O •0 0 •0 • tn1 tn2 • ...... tnn• • • • • • tnn

fuchEach element tij represents the proportion ofof elementelement xmixmi thatthat becomeshecanes xm+11j"",+l,j overover the intervalinterval mm toto m+1.m+1. The total value ofof xm+10xm+1,j willwill be the sum ofof allal1 the tij times the xmi. That is:is,

xm+1j (1) - 102 -

By repeating equationequation (1)(1) for each value of j fromfl'Olll 1 to n9n, one calculates an entire new set ofof values forfor XM+1Xm+1 fromfrom Xm0Xm .

The following exampleexample may clarifyclarify matters.

The table below assumes a state vector (X)(X) ofof diameter classes9classes, as indicatedindicated::

DiameterDiam et er class at end of period (204.20 20-40 40-604~60 60-806~80 80+ Total

Diameter <<20 20 0.6 0.39 00.01. 01 0o 0 11.0. 0 I at start 2~4020-40 0 0.68 0.30 0.02 0 1.0 of 4~6040-60 0 0 0.75 0.21 0.04 11.0.0 period 6~8060-80 0 0 0 00.9. 9 0.1 1.0100 80+ 0 0 C0 0 1.0 1.0100 1

The transition matrix probabilities are given within the table°table. Note that the zero values ofof the lowerlower diagonaldiagonal areare indicativeindicative ofof thethe factfact thatthat treestrees dodo notnot movemwe intointo smallersmaller size classes. The addition of the transition probabilities to 1.0 along the rowsrows is neces-neces­ sary if all the states ofof the systemsystem areare truly defineddefined by thethe statestate vector.

Now assume that the diameter class distribution at the startstart ofof the period is:is:

diameterdiamet er <<2020 20-402~40 40-604~60 6o-8o6~80 80+ frequency 1037 219 88 22 7 trees/hatreesilla

Construct a table as shown below,below9 withwith thathe classclass totaltotal inin thethe rightright handhand columncolumn andand the otherother figures obtainedobtained by multiplying the total by the corresponding transitiontransition prob~proba- bility. Then:

Diameter class at endend ofof period

(20<20 2~4020-40 40-60 60-80~80 80+ Total 4~60 ----- Diameter <20 623 404 10 1037 at start 2~4020-40 149 66 4 219 of 40-604~60 66 19 3 88 period 6~8060-80 20 2 22 80+ 7 7

Total 623 553 142 43 12

add the columnscolumns verticallyvertically toto getget thethe newnew frequencyfrequency distributiondistribution forfor thethe stand.stand.

This example ignores ingrowth9ingrowth, mortalitymortality andand harvesting.harvesting. These are consideradconsidered in the following sections°sections.

6.3.2 Methodsmethods ofof ConstructinConstructing Transition MatricesMatrices

The methods for constructing transition matrices may be differentiated accordingaccording toto whether single tree, permanentpennanent plot remeasurementsremeasurements are available or whether aneone is dealing only with diameter classclass measurements. - 103 -

6.3603.2,1. 2.1 Single angiLimuak tree data

Second measurement

IIABCD. A B C D • •e • • M H Total First I measurement A B C D • • • M H

The framework of the transition matrixmatrix isis shownshown above.above. The letters A,B,C,D,A,B,C,D, ••• denote successive diameter classes. M denotes mortality and H harvested stems.stems. I denotes ingrowth.

The transition matrix is constructedconstructed as follows:follows:

(1) Define the initial class and final class, overover 1 timeUme period, ofof aa. tree. Tally that tree inin the appropriateappropriate row/columnrow/column intersectionintersection andand inin thethe total columncolumn anon the right. The latterlatter isis basedbased anon thethe initialinitial classclass value.value. Ingrowth trees are defined as those which appear in a class at the second measurement,msasurement, but were not present inin anyany classclass atat thethe firstfirst measurement. measurement. MortalityMmiality treestrees are thosethose whichwhich are dead at the secondsecond measurement;measurement; and harvested trees those whicht.hlch have been removed.rQ1\oved.

(2) Once all the trees in the entireentire data setset have been tallied, divide the tran-tran­ sition tallies by the total in the right handhand column to givegive aa propurlion,proportion, rounded to two decimal places.

(3) Check thatthat the transitions alongalong eacheach rowI'<>wadd add toto 1.0.1.0. IfIf there isis aa smallsmall discrepancy (±(:1: 0.01 oror 0.02), it will be aa roundingrounding error;error; adjust somesane figuresfigures at randomrandan until they total 1.0.1. O. If there is a large discrepancy, therethere hashas been an ariarithmetictbnetic errorerror inin that row.row.

If one ofof the totals columnscolUl)ms isis zero,zero, thatthat classclass mustmust bebe amalgamatedamalgamated withwith anan adjacentadjacent size class, as there are no data definingdefining transitionstransitions relatingrelating toto it.it.

6.3.2.2 Size class data

The diagram below illustrates aa transitiontransition matrixmatrix asas itit mustmust bebe constructedconstructed ifif onlyonly size class data is available. It must be assumedassumed thatthat I

- AllAl]. ingrowthingrowth occursoccurs intointo thethe lowest size class°class. - 104 -

Outgrowth fromfran oneOne classclass cancan anlyonly occuroccur intoint 0 thetbe nextnext higherhigh.,.. class.class.

The number of stemssterns harvested in each class is known from an independentindppendent source or dE>iuceddeduced from the treatment prescription.

The shaded blocks in the table are not usedlused; they are zeroszeros within the main tran-tran­ sisitiontion matri:x.matrix. I HHA A , I ? "'~~(~, ~ Po, ~ , v ~p. ~ '"(~ r, 4 .'" ~/ ~4 ~,//4 H 1 ~4 ~4 %~, '"~ fA0 B3 I/'/"~ ~

Diameter classesclasses inin the standstand atat thethe firstfirst measurementmeasuremmt areare enteredentered inin column A.A. Diameter classes at the secondsecend mePsurementmeasurement go in rawrow B. Harvested stems are entered in column H. The diagonals Po and Pr are calculated from the following formulae:fonnulae:

Po ~m B - A + Po* + H (2 )

Pr am A - Po (3 )

In equationequation (2),(2), A isis the rowrow totaltotal and BB the columncolur.m totaltotal forfor a particular Po:Po; Po* is the value ofof PoPo inin thethe nextnext rowrow down.down. Po* is zero by definitiondef"inition for the largest size class. H is the harvested stemsstems forfor thatthat row.row. Po is calculated recursively9recursively, stlU"tingstarting at the bottom right cornercorner with.lith Po*Po* ma 0,0, andand thenthen workingworking upup toto thethe left.left.

After the values ofof PoPo andand PrPr havehave been calculatedcalculated inin stems/ha,stems/ha, theythey areare convertedconverted to proportions by dividing by thethe correspondingcorreeponding rowrow totaltotal A.A.

The statistics Po andand PrPr representrepresent realreal processesprocesses asas follows:follows:

(1) In the ingrowth raw9row, 19I, PinPr isis thethe proportionproportion ofof seedlingsseedlingp whichwhich doesdoes notnot enter the lowestlowest chame-terdiameter class.class. Po is the proportion which does and is accounted as ingrowth.ingrolorth.

(2) In the other rows, representing successivesuccessive sizesize classes,classes, PrPr isis thethe proportion of sternsstems in that class which remainrsnain thsrethere during any oneone time period. - 105105 -

(3) Po is the proportion of stemssteme growing out of oneane class and into the nextnext higher classclass during any time period.

TheThe figure shows thethe valuevalue of AA corresponding toto thethe ingrowthingrowth row asas aa? ?. Any arbi-arbi- trarilytrarily large figure maymay bebe used,used as thisthis row rspresentsrepresenta the total number ofof seedlings forming the potential ingrowthingromh pool.pool. However, ifif regenerationregeneration surveyssurveys areare conducted,conducted, then thisthis figure may be estimatedestimated absolutely oror modified in proportion to a percentage stocking.

Note also that the transition H/H isis shownshown asas 1.01.0 becausebecause 100%100% ofof thethe stemsstens harvested in period A (the(the earlier period) will remain harvested at period B~BL

MortalityNortality cancan be added to the matrix inin aa wayway exactlyexactly similar to harvesting. Like harvesting,harvestin~, it must be measured,moosured, whichwhich means thatthat thethe numbernumber ofof dead trees in eacheach size class must be assessed at period B. It is also added to equationequation (2)(2) inin thethe samesame may~ asas R.H.

6.3.3 Refinements to transitiont~dnsition models

Two basic refinements must be added to a transition matrix model for mixed forest beforp,before itit becomes aa workable~orkable tool:tool:

(a) Species must be groupedgrouped andand separa-teseparate matricesmatrices constructedConstIUcted forfor eacheach group.group. The number of groups should not be too large,largs, otherwise one may have too little data for many transitions.

(b) Data should behe grouped into basal area classes for different forests, to allow a different transition matrix to be used forfor different standstand densities.

6.3.4 DieadVlUltagesDisadvantages ofof Transition'Prs...,sition ModelsYodels

Transition matrix models have seve4alseveral disadvantages.disadvantages.

It is difficult andfllld tedioustedious toto representrepresent dynamicdynamic interactions.interactions. For example,example, the interaction between stand density andand growthgrowth rate.rate. It cancan onlyonly be done byby having a separateseparate transitiont.ransition matrtxmat:t

The precision ofof a transition model isis limitedlimited by thethe needneed toto workwork withwith broadlybroadly defined classes;classes; otherwise too many undefined transitions occurOCC1lI" oror the matrices become too largnlarge for easyeasy computation.computation.

Transition models are inefficient inin terms ofof the number ofof parametersparameters requiredrequired to define a growthgrowth process.process. A 9 x 9 9 matrix (81(81 parameters) mightmight onlyonly correspondcorrespond to a single treetree parameterparameter growthgrowth equation.equation.

- Because of their inefficiencyinefficiency andand lacklack ofof dynamic interaction, transition matrixmatrix models offerofhr little scopeBcope for improved understanding of forest growth processes. - 106106 -

6.4 DlSTMICE-INDEPENDENl'IMTANCE-INDEPMIDENT TREE TREE )I0DEU3 MODELS MSED HASID ON ON DIFFEREN'CE DIFFERENCE mD'ATIONS atuvrian

6.4.1 Definition

A tree model isis oneone inin whichwhich eacheach tree in a stand is individually represented by a set ofof variables, describing, forfor example,example, treetree species,species, tree diameter,diameter, heightheight and crowncrown condition. In a distance-dependent model, the tree position in the stand is also represented, typically as x-yx-y coordinates.coordinates. In a distance-independent model, tree position is not repre­repre- sented.se.'lted. Distance-independentDist,."ce-independen"~ models are generallygenerally much more economicaleconanica1 in terms of computercanputer resources than tree positionposition models;models; but give a less realistic and detailed representation of inter--tI'eeinter-tree competitivecanpetitive processes.processes. In this manual, onlyonly the distance-independent modelmodel isis considered, as experiencee~erience has suggestedsuggested thatthat for gross projections inin yield, there is no specificepecific advantageadVlUltage inin goinggoing forfor thethe moremore complexcanplllX and computationallycanputationally demandingd

6.4.2 .4,k11.ometricAl1anetric and _and D,ynamic Rynanlic Variables Variables

As has been saidsaid, each tree in the stand is representedrepresented byby aa setset ofof variables.variables. In the simplest case,csse, aa treetree willwill bebe representedrepresented by onlyonly oneone variable,variable, itsits diameter. Usually, species willwill alsoalso be identifiedidentified andand perhapsperhaps heightheight oror crowncrown class.class. Otherother tree variables, such as crowncrown diameter,diameter, volumevolume oror biomass,bianass, maymay bebe derivedderived by allometricallanetric relationshipsrelationships withwith tree diameter andand height.height.

Dynamic variables are those Whichweh are predicted fromfran the state of the tree at a previous time period. AllanetrieAllanetric variables areare thosethose Whichweh representrepresent eta-ticstatic relationships between differentdifferen"~ dimensionsdimensions ofof thethe treetree atat thethe samesame pointpoint inin time.time.

6.4.3 Representa.tion"Representation'of of CanpetitionCon etition

The individual tree dynamic variables, suchsuch as diameter, will be predicted fromfran an equation incorporating tree diameter atat thethe lastlast time period andand alsoalso some measure ofof standstand densitydensity.. Stand density may be represented in a number ofof ways, as an absolute meaSUI'emeasure such as number ofof trees over a certain size limit or the basal area ofof trees inin the standstand oror asas a relative measure suchsuch as basal areaarea divideddivided by thethe maximumma.:rimum basal areaarea possible onon that site; or itit could bebe measuredmeasured in sanesome novel way by considering, for example;example, total leaf biOOIassbiomass onon aa givengiven area.area.

HO>lsvHoweverar stand densitydE!1lBity isis measured, itit will retainretain the propertyproperty thatthat it isis inin somesome sense "thethe commationSUl!mIation ofof the individual treetree variables inin thethe stand.stand. ThusTIma basal area is the sum of the individualindivid.ual tree sectionalsectional areas.areas.

Stand density obviously varies fromfran place to place in a stand. If the model repre-repre­ sents a large plot,plot, ofof saysay 11 or 10 ha, then the overallaverall stand density will not necessarily reflect very accurately importantimportant variationsvariations suchsuch asas thethe occurrenceoccurrence ofof geps~gaps. This difficultydifficn1ty can be avoided by dividing thethe simulated plotplot intointo quadrats,quadrats, ofof saysay 1010 xx 1010 m,m andand calcu-calcu­ lating the stand density as it affects anyanyone one quadrat as the average density ofof that quadrat and its eight neighbouringneighbouri"g quadrats.quadrats. For edge quadrats, the neighbours can include the quadrats on the oppositeopposite sideside ofof thethe piaplot in order to avoidavoid edgeedge effects.effects. - 107107 -

6.4.4 Data Requirements and Approaches to Anal:ysisAnalysis

Data must be available fromfran pennanentpermanent or temporary sample or experimental plots in order to define all the dynamicdynamic andand allometricallanetric relationshipsrelatiCllships inin aa model.model. Any variables included in the model must havehave been actuallyactually measuredmeasured onon samplesample trees.trees. If competition is toto bebe defineddefined in tennsterms of quadratquadrat standstand denSities,densities, asas discusseddiscussed aboveabove (seotion(section 6.4.3),6.4.3), thenthen thethe sample plots or experiments must also have been measured so that trees cwldcould bebe placed in quadrats, and aa quadratquadrat by quadrat competitioncompetition index calculated.calculated.

With mixed speciesspecies forests, it may well be that rare species will not be represented by many points. In these cases it is better to group the rarer species together and provide them with a commoncommon setset ofof growthgrowth and allometricallometric functions.functions.

Unlike a transition matrix model,model, thethe processprocess ofof fittingfitting thethe variousvariaus equationsequations inin thethe model to the fieldfield datadata requiresrequires considerableca,siderable skill.skill. The various problems are discussed elsewhere in thisthis manual,manual, notably in sectiCllsection 55 and Appendix AA,, but the research workerworker has to consider: the problem of ••selecting1eoUng a Punctiont'\moUon whiohwhich producesprod"" .. the appropriate shape fitting the data using a suitable and adequate statistical and/orand/Or graphical techniquetechnique,, together with an examination of residuals for bias what will happen if in applying the model, an extrapolation will be demanded from a particular function and if the rresultesult it gives under such ciroumstanoescircumstances is reasonable, if not necessarilyneoessarily accurate.acourate. 66.4.5. 4.5 Basic Model Structure

The averalloverall model structurestructure isis veryvery simile.:similar toto thatthat for dynamic stand modelsmodels, dis-dis­ cussed in section cussed in section 5.3.4,5. 3.4, and is illustrated inin thethe diagramdia,;mm below.below. There is first ofor' all an initialization phase, where the statestate ofof the standstand at the start of the simulation is given. This could be derived from actual inventory data or it could be generated randomly or from a set ofof functions.functions.

Initialize tree variation t r ______Compute competition indices for each

quadrat! .

Compute growth for a given period of time, r------quadrate.g. 1 year, tree by treetree Increment time period by 11 Remove 1harvested trees, compute yield

t¡- LRemove dead trees 1 . OeneratGenerate e new ingrowthingrowth treestrees 1 ~ ______Print summary ofof plot statisticsstatistics for this period - 108 -

Then the modelmcdel enters a cyclic or iterative phase.phases whereWhere each cyclecyole represents aneone unit ofof thethe basicbasic growthgrowth periodsperiod, typicallytypically 11 or 5 years. In each iterationsiteration, a series ofof calculations areare repeatedrepeated followingfollowing thethe mainmain stagesstages indicated.indicated. GenerallysGenerally, the orderorder in which these stages areare carried out is notnot important;important; so that one couldscould, for examplesexample, removeranove dead or harvested trees before,before' rather than aftersafter, calculating growth onan the residual stand.stand.

6.4.6 Ingrowth.Ingrowth., MortalityMortalit and HarvestingHarvestiqg

FranFrom the point of view ofof tree model conetructionsCalstruction, ingrowthsingrowth, mortality and harvesting have in commoncommon the following:

- they invo/veinvolve thethe creationcreation oror romar.alranoval of tretrees.... fromfran thethe listlist of treMtraen in the model - they are inherentlyinharErltly randomrand.. processesprooees88 and cannotoannot easilyeesily be treated as simple functional relationships .

adequate descriptive datadata isis oftenoften lackinglacking anon thesethese parameters,parameters.

Ingrowth andand mortalitymortality are often bothboth partiallypartially density depenaent;dependent; so that the number ofof ingrowthingrowth trees oror the numbernumber ofof treestrees ayingdying maymay both bebe functionsfunctions ofof standstand density.density. Once this number has been determined'detennined, thethe actualactual selectionselection ofof individualsindividuals may involveinvolve aa variety ofof ad hoc rules.rules. With ingrowth,ingrowths one typicaltypicallyly creates trees of randanrandom eizessizes and species with limits determineddetermined perhapsperhaps byby aa probabilityprobability distributiondistribution based based on on data, &A% aror per-pel'­ haps onon an arbitrary maximum sizesize thatthat appearsappears reasonable.reasonable. With mortality, the likelihood of any individual tree dying may be a function of its size wadand species.

Stochastic oror probabilistic processesprocesses ofof this kindkind cancan be modelled veryvary easilyeasily inin aa canputercomputer simulation by using aa uniform randomrandan number generatorgenerator thatthat producesproduces randomlyrandanly distri-distri­ buted numbers betweenbetween zerozero andand one.one. Such functions are often standard library functions in FURTRANFORTRAN programming and are part ofof thethe standardstanda:rd languagelanguage inin BASIC.BASIC.

SuppoeeSuppose for example that a functionfunction indicatesindicates thatthat aa particular treetree inin thethe modelmodel has a 0.7 probability ofof dying inin thethe currentcurrent year.year. A randoml'IIlldan numbernumber betweenbetween 00 andand 11 is generated; if it is.!!!!!is less than 0.7,0.7s then the tree isis presumed to havehave dieddied andand aa mortality routine is entered which records thethe detailsdetails ofof thethe treetree forfor thethe plotplot summarysummary andand removesremoves it fromfran the table of trees alive inin thethe model. If the randomrandan number isis greatergreater thanthan 0.790.7. then the tree survivessurvives andand growsgrows untiluntil thethe nextnext period.period.

Harvesting processes are similar to mortality in that they involve thethe rmlovalremoval ofof trees, but usually inin accordanceaccordance withwith definitedefinite rulesrules relatingrelating toto thethe speciesspecies andand size.size.

6, 4 7 Conclusions RegardingRegrág_____arcli Tree Models

Individual treetree modelsmodels are highly flerlble,flexibles butbut requirereauire considerableconsiderable skillskill inin bothboth canputercomputer programming andand inin datadata analysisanalysis anon thethe partpart ofof thethe researchresearch worker.work?r. They also require the use ofof a largelarge computer.computer. There is no single stereotypedstereotyped method ofof constructingconstructing such modelsmodels.. The interested reader should studystudy referencesreferences suchsuch asas thosethose givengiven inin thethe CABCAll (1977) Bibliography listed in Appendix B.

Individual tree models areare notnot aa panacea.panacea. Like all otherother forecastingforecasting methods they require an extensiveerlensive data base of remeasurements anon carefully maintained and assessed pennanentpermanent samplesample and experimentalexperimental plots. Also, like other methods discusseddiscussed earlier,earlier, their sophistication is no guarantee ofof theirtheir accuracy.accuracy. - 109 -

7. VALIDATION OF GPOWTHGJ10WTH AND YIELD MODELS

7.1 THETETE ROLEROLE OFOF VALIDATIONVALIDATION

Validation is,is, literally,literally, thethe process ofof determiningdetennining whetherwhether oror not a model trulytruly represents reality.reality. However, thethe followingfollowing pointspoints mustmust bebe borneborne inin mind:mind:

Model predictionspredictions approachapproach realityreality as aa setset ofof successivesuccessive approximationsapproxir.1ations asas more and more efforteffort isis put intointo datadata collectioncollection andand modelmodel construction.construction.

At somesane point, the effort involved in obtainingobtaining more data or producing a more statistically sound model becanesbecomes more expensive""Pensive than the marginal improvementimprovEment in model predictions warrant.warrant.

Forest models do not have a unifonnuniform degree ofof errorerror over their wholeWhole range ofof predictive behaviour, but willwill bebe more aror lessless accurateaccurate overover differentdifferent rangesranges ofof predictor variables.variabl~~.

The process of validation may be as time consuming and expensiveexp""sive Paas that ofof model constconstruction.ruction. Special experiments may need to be constructedconetructed to test the model;model; data pr<>­pro- cessing systems may need to be set up to allow models to be validated by comparisoncanparison with large numbers ofof plots;plots; considerable statistical analysis maymay bebe requiredrequired toto estima-teestimate thethe covariances ofof residualresimlal errors.errors.

This input of effort into the validation process isis inin no senseBense wasted. Science differsdiffers fromfran philosophyphilosophy and.and religionreligion in beingbeing subject always to the criterioncrl.terion ofof empiricalEmpirical validation ofof theoriestheories andand models.models. A model which is no~no validatedvalidated isis simplysimply speculation and guesswork.

Furthermore,Furthennore, thethe naturenature ofof modelmodel constructicnconstruction impliesimplies thatthat itit mustmUet interactinteract withwith validation as aa cycliccyclic process.procesa.

Data collection A

Model Validation construction

Validation will show weaknessesweaknes7es in modelmodel behavi""rbehaviour >obichwhich willwill leadlead tota improvAdimproved modelmodel structure or to the necessity ofof collectingcollecting moremore fieldfield data.data. EXperimentsExperiments onoil models, eveneven those 'whichHhich are quite invalid may leadlead toto alternativealternative conceptsconcepts ofof experimentalexperimental designdeflign andand data capture.capture.

7.2 VALIDATION DATA

In order to validate aa model, itsits behavioUrbehavioUr mustmust bebe comparedcompared withwith observationsobsel"'rations frcmfrom real situations whose history andand treatmenttreatment areare preciselyprecisely known.known. This data.data cancan bebe called validation data and may be obtained from experiments,""Pariments, pennanentpermanent sample plotsplots or temporarytemporary --110110 -

sample plots.plots. Two distinct situationssituations uslinllyusually existexist withwith reepectrespect to validation data:data:

- The data used totO validate the modelmodel is the same asas thatthat usedused toto construct thethe various functionsfunctians formingforming partpart ofof thethe model.model. We can call thisthis self~ validation.validation.

The data usedused to validate the model has notnot beenbeen usedused toto estimate any of thethe function parameters in the model. This situation may be called InimmisLindependent vaHdativalidation. on.

There is nothing extraordinaryextraordinary aboutabout thethe conceptconcept ofof selfself validation.validation. It is the normal procedure forfor regressionregression analysis,analysis, forfor example,example, wherewhere all statisticalstatistical estimatorsestimators areare based onon the errorserrors bet"eenbetween model predictionspredictionsandthe and the observedobserved valuesvalues usedused toto fitfit thethe model.model.

Self validation cancan however,however, bebe dangerouslydangerously misleading.misleading. RegressionRegression analysisanalysis provides a good example:example: if the assumptions of the statistical model used (uniform(uniform error variance, normally distributed uncorrelated errors,errors, predictorpredictor variables knownknown withoutwithout error)error) areare incor-incoI'­ rect then the statisticalstatistical estimatorsestimators andand parameterparameter valuesvalues obtainedobtained will will al-so also be incorrect.

Self validation is particularly dangerous withwith smallsmall numbersmunbers ofof observationsobservations andand with very complex models. With simple models (perhaps(perhaps only one or two f'unctionB)functions) and with large amounts ofof comparison datadata itit cancan be quitequite acceptable.acceptable.

Independent validation is a much more satisfactorysatisfactory procedure fromfran everyevery point ofof view. It gives an absolute reflection ofof how effectiveeffective a model is as a predictive tool,tool. The main difficulty with independent validation isis thatthat aa considerableconsiderable bodybody ofof datadata maymay need to be ignored Whenwhen constructing thethe model. Probably the best procedure, and one widely adoptedadatad byby the the systemssystems modellingmodelling community, canmunity, isis thatthat ofof halvinghalving thethe datadata setset byby aa randomrandom oror systematic process and using half the datadata forfor model constructionconstruction andand halfhalf forfor validation.validation.

Unfortunately, there isis aa certaincertain greygrey areaarea betweenbetween selfself validationvalidation andand indspendentindependent validation. It is possible to construct a model using oneone type ofof informationinfonnation fromfrom aa setset ofof plot or experimental data and validate it using another, independent statisticstatistic fromfrom thethe samesame plots. Such partially independent validation may be regardedregarded asas lesslees satisfactorysatisfactory thanthan fully independent validation, but more indicative of a modd'smodel's truetrue validityvalidity thanthan selfself validation.

7.7.3 3 RMIDUALRESIDUAL ERROREmlORS

Validation ofof models isis usually based uponupon anSll analysisanalysis ofof residualresidual errors.errors. These are defined as:

Residual error •..-- Observed value - predicted value

This is exactly analogous toto thethe residualresidual errorerror usedused inin regressionregression analysis.analysiS. It is suggested that the reader refer to the notes in Appendix A.2.6; identical techniques can be applied to errorerror analysisSllalysis from models.

If the residual errorerror isis toto givegive aa truetrue indicationindication ofof aa model'smodel's performance,performance, itit mustmust be assumed that thethe model cancan be setset upup inin suchsuch aa wayway thatthat thethe independentindependent variables giving rise to the observed value are identical to those for the predicted value. If this is not the case,case, then thethe errorerror ofof thethe modelmodel willwill tendtend toto bebe exaggerated.exaggerated. --111111 -

ForFor example, a dynamic stand model requires inf'ormationinformation aboutabout thethe plantingplanting density and timingtiming andand intensityintensity ofof thinningsthinnings ifif itit isis toto makemake anan accurateaccurate prediction. Such a model requires""quires longlong termtenn permanentpennanent samplesample plots or thinning experimentexperiment data for accurate validation.

A growth and yield model willmill generally predict several statistics, for example mean diameter,diameter, dominantdcminant height, total volumevO'lume and merchantable volume. Residual errors generally increase (relative(relative to the statistic predicted) franfrom height to diameterdiameter,, to total volume to merchamerchantablentable volume. Consequently, itit isis usuallyusually sufficientsufficient toto carrycarry outout validationvalidation studiesstudies on tototaltal vovolumel um e oror merchantable volume;volume; unless there is a specialspecial needneed toto lcrtowknow thethe precisionprecision of diameter and heighheightt estimates,estimates, these need not be the subject of separate validation studies.

Residual errors can be summarized in several ways:

- GraphicallyGraphically,, as plots ofof residualresidual errorserrors against predicted values oror predictor vvariables.ariab l es. The commentscamnents in AppendixAppendix A.2.6A.2.6 applyapply inin thisthis case.case. 2 - As a ccoefficientoefficient of detennination,determination, analogous to RR2 in regression studies.studies. This iiss calcalculatedculated as:as :

Sumsura_2LETIams of squares of resresilualidual errors. , 1 - . Sum ofof squares ofof dbservedobserved values

-As a residualr esidual standard deviation,deviation, calculatedcalculated as:as:

Sum of ssquaresares of residualresi dual errorserrors No,No. of vvalidationalidation samples - No,No. ofof predictorpr edictor variablesvariables

- As a pel'Centagepercentage residualr esidual standardstandard deviation (equivalent(equivalent toto aa coefficientcoeffi cient ofof variation)variati~) calculatedcalculated as:as :

Residual standard deviation

lMean~ ean predicted value

In quoting or using thesethese statistics,statistics, thethe readerreader shouldshould appreciateappreciate thatthat theythey areare approximations which cannot necessarily bebe relatedrelated toto specific confidenceconfidence intervalsintervals oorr llevelsevels of significance.

7.4 GRAPHICAL COMPARISONSca.!PARISON3

Analysis ofof residualresidual errorserrors isis aa somewhatsomewhat abstractabstract technique.technique. An alternativealternat ive approachapp.-oach to model validation isis thatthat inin which>ihich aa gisaphgraph isis plottedplotted of thethe statisticstatistic ofof interestinterest against somescme predictor variable forfor bothboth realreal standsstands andand thethe yieldyield model.model. For example,example, aneone couldcould plot the development ofof volume againstagainst timetime forfor actualactual standsstands andand for a modelmodel..

This type of approach gives a more concrete appreciation of the strengthastrengths and wweak-eak­ nesses of a model, than residual analysis. HbweverHbwever,, it cannotcannot effectively bebe usedusedto to summarize the behaviour ofof aa model involvinginvolving manymany predictor variables: nor can it easily be used to represent results with large amounts ofof validation data. - 112 -

It is suggested that both types ofof validationvalidation areare usuallyusually necessary.necessary. Residual analysis can be used with large sets of data and to summarize results into aa smallsmall numbernumber of statistics,statistics9 overover the wholewhole rangerange ofof predictorpredictor variablevariable values.values. Graphical comparisons are effective for presenting the key aspects of model behaviourbehaviour for publicationpublication or canrnunicationcommunication..

7.5 DEFINING THE LIMITS OF MODEL UTILITY

The residual errors from a modelmodel will generally tendtend toto increase asas oneane movesmoves toWBl'dstowards more extremeextreme values ofof predictor variables. This is likely to be especially true if one'sone's validation datadata coverscovers aa broader rangerange ofof sites,Sites, agesages and growing conditions than the data used tot o constructconstruct the model.

However, there will be limitslimits to the validation data, as there will have been to the data used in model construction.construction. It is important, as part ofof the validation process,process, thatthat one examines how ththee modelmodel behavesbehaves outside these limits andand estimates, forfor each typetype ofof pr~pre- dictor variablesvariables,, a range of values outside which the model predictions become obviouslyObviously inaccurate and unusable.unusableo

This is important because many applications of growth and yield modelsmodels maymay bebe in situations where absurdabsurd valuesvalues areare notnot immediatelyimmediately apparent.apparent. For example,example, as part ofof an economic forest sector model or as a component in a programme forfor calculating cutting planaplans..

The limiting range of values within which a model may be reasonably precise (to quoted specifications) should therefore be explicitlyexplicitly defined during the validation process.process. - 113 -

8. THE APPLICATION OF THETHE MODELMODEL TOTO THETHE Ill!XPIllEOREQUIRYD EN>Em USEUSE

88.1. 1 I Nl'RODUCTIONNTRODUCTION

!Bsentially,Essentially, the growthgrowth andand yieldyield modelmodel maymay bebe appliedapplied inin oneone ofof threethree ways:wy":

(i) As a simple table oror graphgraph oror set ofof tablestables oror graphs.gl'Sphs. These canCM be used bbyy forest planners directly or can bebe fed in tabular form toto aa computer forfor updatingupdating a set of inventoryinventory data.data.

(ii) As a programme for a computer or calculator which can produceproduce aa tabletable or graphgraph of growth and yield for a particular set of treatments. ThisThis is appropriatappropriatee when the model has sufficient inherent flexibility so thatthat it is notnot possiblepossible to define allall possiblepossible predictionspredictions inin oneone setset ofof tables.tables.

(iii) As a computer programme whichwhich formsforms aa sub.,modelsub-

Alternatives i and ii have been sufficiently dealt with in 5.3.4.5. 3.4. Remains ttoo comment on alternative iii where the model is used in connection with inventoryi nventory data in forest planning.

A necessary prerequisite in this case is that the variables included in the model as parameters (predictor(predictor variables) also are included in the inventory data.

8.28. 2 EVEII-AGEDETTN-AGED STANDS

In the casecase when thethe forestforest consistsconsists of even-agedeven-aged standsstands thethe follOwinfollowingg threethree functions might beb e includedincluded inin thethe model:model L

Ho = f (Sp,(Sp, s,S, A)

Ig = f (Sp, s,S, A, G) or Ig ~= f (Sp,(Sp, Ho,Ho, N)

Hf =~ f (Sp, Ho, N)N) or Hf = f (Sp,(Sp, Ho,Ho, ~/i G)

Wherewhere Sp denotes species, SS sitesite qualityquality class,class, HoHo dominantdominant height,height, A age, NN numbernumber ofof trees per hectare,hectare, GG basal areaarea perper hectare,hectare, IgIg basalbasal areaarea incrementincrement perper hectareh~ctars andand yearyear andand HfHf form height, defineddefined byby thethe functionfunction VV = G.HfG.Hf whereWhere V is volume per hectare.hectare. In this case, the variables Sp, A, Ho, lJN sndand C;G have to be known forfor thethe inventoryinventory datadata whilewhile sitesite class,class, SS,, may beb e calculated by meansmesns ofof thethe firstfirst function.function.

We maymay assume that thp.the model is calculated by mesnsmeans ofof observationsobservations on growth and yield plots.plots. We may further assume that the inventory-data,inventory 'data, to WhichMUch 'the model will be applied, is derived from observationsobservations onon samplesample plots.plots. If the plot sbesize is the sameBame in bothboth cases the model may directly be applicable to the inventoryinventory data.data. If, however, there is a difference inin size inin thattfuit the inventoryinventory plotspl ots forfor exampleexample areare smallersmaller thanthen thethe growthgrowth andand yield plots, the model may givegive biased results.results. There are two reasons forfor this. One is that the site classclass will be somewhatsomewhat underestimatedund

compared to largerlarger plotsplots (0.05-0.1(0.05-0.1 hectares).hectares). ThisTbis bias isis, however, usually only a few percent. The otherother reason forfor biasbias isis thatthat thethe effecteffect ofof competitioncompetition fromfrom treestrees outsideoutside thethe plot will increase with decreasing plot size. This in turn meansmeans that aa growth function containing competition or density variables like N or 0 is correctcorI'ect onlyonly forfor plots ofof thethe size for which it was calculated. The size of this biasbias depends on thethe differencedifference inin sizesize between the growth plots and the inventory plots, the type of growth function in the modelmotel and finally on the variation within the inventoried stands. Nothing can therefore be said in general about the sizesize ofof thisthis error.error.

Provided that the growth plots and the inventory plots are of more or less equal size,size, the application of the model to thethe inventory datadata should bebe donedone byby forecasting thethe development forfor everyevery singlesingle inrentoryinventory plot.plot. At the end of the forecasting period,peried, the plots may be sortedBorted into strata according to speciesspecies, Bitesite class andand agp.age a.'ldand thethe oumssums and means forfor eacheach stratumstratum calculated.calculated.

If, however, there is a great difference in plot size in that the growth plot.splots are much bigger than thethe inventoryinventory plots itit might bel,e advisableadvisable firstly to sort the inventoryinventor"J data according toto species,species, si-tesite classclass andand ageage andand thenthen toto apply the modelmotel to t.hesethese strataRtrata instead ofof toto eacheach separateseparate inventoryinventol"J plot.plot.

8.3 MIXED STANDSSTA?IDS

InIn thethe case of a foreetforest consisting of mixed (uneven aged) st""dsstands the following functions might be included in the forecasting model:

IvIr = f (S,(S, F9F, G(1),0(1), 0(2), ••• O{n),G(n), N(1), N(2)N(2), ••• N(n)ll{nl, L(1), 1,(2)L(2), ••• 1,(n))L(n) )

yV = f (Sy(S , F,F9 0(1),G(1), G(2)90(2), 00.••• 0(n)9O{n), N(1)9N(1), N(2),11(2), ••• N(n))N(n»

where TvIv is growth inin merchantable volume perper hectarehectare andand year,year, S is site class defined by environmental indicators (soil(soil type, altitude, rainfall,rainfall, etc.),etc.), F is forest type, 0(1), 0(2), ••• G(n)O(n) is basal areaarea afterafter logginglogging by speciesspecies groupgroup perper hectare,hectare, N(1), N(2) ...••• N(n)II(n) isis the number ofof treestrees afterafter logginglogging perper speciesspecies groupgroup andand hectarevhectare, L(1),L(1), L(2)L(2) ...••• L(n)L(n) isis logging intensityintenSity perper speciesspecies groupgroup andand finallyfinally VV isis merchantablemerchantable volumevolume afterafter logginglogging perper hectare.

To enable forecastingforecaeting thethe samesarne variablesvariables (predictor(predictor variables)variables) havehave toto bebe recordedrecorded in the inventoryinventory asas thosethoBe includedincluded inin thethe model.model. The discussion anon plot sizesize inin relationrelation to forecasting foreven-agedfor even-a,ged standsstands appliesapplies inin principleprinciple alsoalso toto mixedmixed. stands.stands. It must be admitted, however, that very little so far has beanbeen done in the field of growth modelsmodels forfor tropical mixed foresteforests andand thatthat thisthis fieldfield stillstill isis inin aa stagestage ofof research.research. It is thus not possible toto givegive anyany preciseprecise instructionsinstructions anon howhow toto make forecastsforecasts concerning the development of tropical, mixed forestsforests afterafter logging.logging. For tenperate,temperate, mi:7::edmixed forests, modelsmod.els have been developed by whichwhich reliablereliable forecastsforecasts may be donedone (see(see e.g.e.g. Monserud 1980). It might be worthwhileworthwh:tle trying thesethese models alsoalso forfor tropicaltropical mixed forests,forests , but this has so far not been done.

Monserud, A,A, Ek,Ek, A,A, 198011980, CanparisonComparison of two stand growthgrowth models forfor northernnorthern hardwoodshardwoode -- Wright H9H, (editor)(editor) 1980. Planning, performance and evaluation of growth and yield studies. Meeting of IUFRO 54.01.54.01. Commonwealth Forestry Institute, Oxford, GreatOreat Britain, pp.8.pp.8. - 115 -

Appendix A

STATISTICAL AND MATHEMATICALMAmEMATICAL TECHNIQUESTECHNIQum PORFOR YIELDYIELD STUDIESSTUDm3

EtatPage 1.lo EQUATIONS AND GRAPHS

1.1.11 The StraightStraight LineLine 117 1.2 Transformations for Cln'VeCurve Fitting 118 1.3 Example of Approximation of a Curve Using Transformation to 119 a Straight Line 11g.4 PolynomiaJ,sPolynomials forfor Curve Approximation 121 11.5.5 Graphs Involving Three Variables 125 1.6 Example: Fitting anan equationequation toto aa systemsystem ofof curvesourves byby 126 harmonizationharmonization..

2. CURVE FITTINGFITTING BYBY LINEARLINEAR LEASTLAST SQUARESSQUARES ANALYSIS 129 2.1 Simple Linear Regression 129 2.2 RegressRegressioni on with Two PredictorPrediotor Variables 132 2.3 Data Transformations and Curve Fitting 133 2.4 Multiple Regression Analysis 140 2.5 Calculations for Estimating the Parameters of a Multiple 2.5 Calculations for Estimating the fBrameters of a Multiple 141 Regression Model 22.6. 6 Residual Analysis 150 22.7. 7 Weighted Regression 152 2.82. 8 Comparison of RegressionRegrossion LinesLines 153 2.9 Nested Regression UsingUsing ConditionalConditional Variables 160

3. SOLUl'IONSOLUTION OF EQUATIO~EQUATIONS 163 3.1 Solution of the QuadraticQuadratio EquationEquation 163 33.2. 2 GraphicalGraláhioal SolutionSolution of Equations 163 3.3 Numerical Solution of Equations 165

4. FITTING NON-LINEAR MODELSMODIug 166 - 117 -

1.1 • EQUATIONS ANDAD GRAPHSGRAPHS

This sectionseotion concernsconoerns methodsmethods ofof ez.plessingexpressing graphicalgraphical relationshipsrelationships asas equations.

1.1 The StraightStraiOat Line

The simplestsimplest typetype ofof grail'gTaph is thethe straightstraight line:line:

y

b ·= Il Y / .6 x

a

o +------"x

This can be represented asas anan equationequation by:by:

yyea = a + b.x a is the value of y at the point Wherewhere thethe lineline crossescrosses thethe yy axisaxis (i.e.(i.e. at x.0),,.,0). b is the slope of the line, and can be conveniently workndworked out from a graph as follows:

(i) TaknTake any convenient intervalinterval LixA x (say(say forfor example,example, 1010 imiteunits ofof x), andand drawdraw aa line AB of that length parallel to the x axis (see(see diagram above).

(ii) Measure the length AyLl.y ofof thethe lineline parallelparallel toto the yy axis from BB to thethe graphgraph line. If BC is measuredmeasured upwards,upwards, Ll.yAy isis positive.positive. If C is belowbelow BB thenthen Ll.yAy is negative.

(iii) The slope b is given by fly/ay/4x. AX. Note thatthat,6y Ay and CixAx are measured in the units of the y and x axes, andand not!!2! inin actualactual distancedistanoe onon thethe paperpaper inin centimetres etc.eta.

An alternative algebraic method of calon]Rtingoalculating the aa andand bb coefficientsooefficients forfor aa straight lineline isis asas follows:fo llows:

(i) TakeTakn any two oonvconvenientenient points on the line,line, designated byby (x1(xl' Yj)) and (x2,(x , Y2)'asy2),as shown on the diagram below. 2 -118-- 118 _

y

I I I I I I I I Y2y2 -----1------I I I I

X2 x1

(ii) Calm/IPA()Caloulate thethe slope bb as:as:

b (72-7.1)/( xrx1

(iii) CaloulateCalculate the interoeptintaroept as:as:

a y

1.2 °transformations'I'ransfonmtions for Curve FittiPittillg

JIe.nyMany relationshipsrelationehipe between variables in forest yield prediotion are in the form of o=vescurves whenWhen represented graPhioally.grarhioally. The model,ormodeloor particularpartioular equation that represents the OUSVE4ourve, must be known if an ~exact representation of the curveourve is to be worked out.out In practice,practioe, the correotoorreot model is not usuallyusually known and only an approximation is possible.

PerhapePerhaps the most easily fitted and used approximatingapprorlmat1Dg equationsequations areare thosethose whiohwhioh imrolveinvolve transformations of the x or yY units, but leave thethe generalgeneral equationequation inin linearlinear form.form. Common transforystionstransformations are:are:

log10 logarithm to the base 1010 10g'0 x it 11 tt logelog or inIn " It e (2.71828....)(2.71828 .... ) e- " " 1/x1/X reoiprooalreoiprocal JX square root ~x2 square

The transformationstraneforuations canDan be applied tew both the Xx and y variables to give variousva.r10tlS combinations andresultant curves.ourves. --119-119 -

Data that that is obviously not linear in form oancan be plotted using variousvarious transfo~transforma- tionations until a linear shape is obtained. The work is reduoedreduced if one hashas somesome idea in advanoeadvance of the transformations likely toto bebe successful.sucoessful. In thethe mainmain part of this manual examplesexamples for particularpartioular types of data areare suggested.suggested.

1.31.3 ExampleExamele of of Approximation Approximation of of a aCurve Curve.Akg_ltauformation Using Transformation tota.ajam121121422 a Straight Line

Figure A.1.1(a)A01.1(a) shows a ourvecurve representing a relationship between between tree volume (v)(V) and diameter (D).(D). We assume"",sume that the curveourve has beenbee" drawn by hand and it is desired to express itit as anen equation.equation.

Points at convenient intervals onon thethe x-axis>:-axis areare tabulatedtabulated fromfrom thethe graph:graph.

PointsPoints from Original Function FUnction way. coass.ncuutt..M.....arcl Transformations_ansformations D V D2 log DD log VV 1010 0.05 100 1.00010000 -1.301 20 0.20 400 1.301 -0.699 30 0.45 900 1.477 -0.347 40 0.85 1 600 1.6021. 602 -0.071 50 1.50 22500 500 1.699 0.176 55 '.901.90 3 025025 1.740 0.279

The first transformationtransformation triedtried ieis shownshown inin figurefigure A.1.1(b),A.1.1 (b), where D2 isis plottedplotted against V.V. The points from the original curveourve almost followfollow a straightstraight lineeline, but therethere isis still a8. slight curvatureourvature apparent.

Another transformationtransformation isis shownshown inin figure figure A.1.1 A.1.1 (c (0 ).). HereHere wewe have taksntaken logarithms to base 1010 ofof bothboth axes.axes. The transformed values values are are lishd listed forfor the 6 selectedseleoted pointspoints inin the tabletable above.above. It canoan bebe seenseen thatthat thisthis traneformation transformation givesgives anan almostalmost perfectperfeot fit, exceptaxoept for the smalleetamallest pointpoint whiohwhioh isis slightly aboveabove the lineeline. From the the figure wewe canoan oaloulata!calculate the slope and interceptinteroept forfor thisthis line.line. Taking the two pointspoints (1.301,-0.699)(1.301,-0.699) and ((1.740,1.740, 0.279)0 .~79 ) shownshown by by the the doubledouble oiroles oiroles on the figurefigure andand asas DD atat 20 20 andand 5555 cmom inin the table above,above, wewe have:have.

b D= (0.279 -- (-0.699))/(10740(-0.699»/(1.740 - 10301)1.301)

e 0.978/0.439

~u 2.228

a =D -0.699-0.699 -2.228-2.228 xx 1.3011.301

'c -3.597

So our equationequation approximatingapproxima-ting the curveourve inin figure figure A,1 A.1.1 01(a) (a) is:

logioV = -30597 + 2.228 log10p - 120-120 -

FigureFigure A.1.1A. 1 .1

(a) 2 V

1

10 20 30 4.040 50 60

(b) 2 V

1

~• ____ ~ ____ ~ ____ ~D2 1000 2000 300~

(c) (c) 0,5 log VV

0,0 log D 1,1 -0,5 -.

-1,0-1.0

-1,5

Transformation of a hand...u-awnhand-drawn curveourve to give a straight lineline approximation.approrlmatioll. (a) Original free-handfree-he.nd curveourve with selectedseleoted points for oaloulationcalculation of transformed.transformed values. (b) x-axis transformed by taking square of selectedseleoted points. (0)(o) xx andand y axesaxes trauformedtranaformed by taking logarithmslogarithms toto base 1010 of the selectedseleoted points.points. - 121 -

1.4 E9),,non_lia,18Pol:ynomials 1br fbr Curve Curve Approximation Approximation

Equations involving only two ooeffioientscoeffioients are oonvenientconvenient for thethe approximationapproximation ofof funotionsfunctions because they cancan bebe drawndrawn asas straightstraight lineslines withwith suitablesuitable transformations.transformations. Ho_How- ever, some shapes of curveourve cannotcannot bebe treatedtreated inin thisthis way.way. In this oasecase the teohaiqueteohnique of fitting a quadratic or cubicoubio polynomialpolynomial toto thethe functionfunotion maymay yieldyield goodgood results.results.

A polynomialpolynomisl is anan equationequation of thethe form:form:

2 y ~ b + b x + b x 2 + b x 3 + •••• +bxnn b0O + blx1 b2x2 + b3x33 bnx

where nn isis thethe order2!:!!!!: andand thethe bibi areare ooefficients.ooeffioients. The higher the order the greatergreater thethe flexibility of the funotion. On the other handhand, y highhigh orderorder polynomialspolynomials areare diffioultdifficult to fit by manual techniques.

For quadratioquadratic (2nd order) and oubiccubic (3rd order) polynomials, manualmanual caloulationcalculation ofof the ooeffioientsooefficients is possible in about 30 minutes with the help of aa 4-funotion4-function (+9(+, -,-9 X9x, 4.-)t) caloulator.calculator. A form is providedprovided toto assistassist thethe useruser (form(form .1..1)AA) whichwhich shows in this case the fitting of a cubic equationequation toto thethe lineline inin figurefigure A.1.2.A.l.2. FburFour points are taken fromfrom thethe function to be fittedfitted and enteredentered inin tabletable (1)(1) ofof thethe form.form. The points need not be in any order, but it isis desirable that:that:

(i) One point each should representrepresent thethe twotwo extremesextremes ofof thethe function.(Infunotion. (In thisthis case, Xl and X .) case, X1 and X3.)3

x4should not be an extreme point if the quadratic equation coefficients are (ii) X4 should ~ be an extreme point !! the quadratic equation coeffioients are to be calculated.caloulated.

(iii) The other points should bebe moremore oror lessless evenlyevenly spaced.spaced.

(iv) lIhenWhen allyonly the quadratioquadratic equation is to be caloulated,calculated,three three points are sufficient.

Form A.1A.l explainsexplains thethe neoessaryneoessary caloulations.oaloulations.

QuadraticQuaBratic and oubiocubic curvesourves provide a useful means of approximating hand-drawn ourvescurves by an equation. WhenlIhen usingusing themthem thethe followingfollowing detailsdetails shouldshould bebe noted:noted:

(i) wAlayEAlways checkcheok thethe calculationscalculations andand thethe suitabilitysuitability ofof thesethese curvescurves byby drawing the functionfunotion on top of the original curve.ourve. It is usuallyusually preferablepreferable toto startstart by fitting the quadratioquadratic equation. Then, if this isis not sufficientlysuffioiently accurate,accurate, compute the cubicoubic equationequation coefficients.ooeffioients.

(ii) !!m!:Never use a polynomial equationequation to extrapolateextrapolate beyondbe;yond the two extremeextreme points used in fitting it.it. If extrapolationext~polation is neoessary,necessary, extend the hand-drawn curve first and then fitfit a new equationequation with a new extreme point. - 122-122 -

Figure A01.2A.1. 2

Hand-drawnHand-

yY 25 y3 ______-- 20 yk ______J2 ______15

10 _~ ______

5 , , I ,X2 :X1 IX!. :X3 X 5 10 15 20 ~ 25 - 123 -

FOrmForm A.1 CoeCoefficientt_112Ef fioients for anan approximatingapp tinguedraticor quadratic or cubic oubio curve ourve alt2-li_xe:Objective: To calculateceloula.te the coeffioients bibi inin eithereither ofof thethe equations:equations: 2 y. b + b .X + bb2.X2·x ++ b3'~b.X3 (cubic,(oubio curve) ' bo0 + b1X1 2 3 orOr y. b + b .X .. b .x? (quadratic curve)curve) IC= boo + b1.X1 b2.X2 2

Items markedmarked <> .0 are are notnot requiredrequired forfor the quadraticquadratic curvecurve oalculations.calculations.

(1)(i) Tabulate 3 3 oror 44 datadata points for the functionfunotion and computeoompute squaresBquares andand cubes.cubes.

Ii 1 2 3 4 <> 9.2 16.0 21.0 Yj 21.0 19.4 <> 5 10 20 15 XI <> 25 100 400 225 X~ 11111111111111 <> <> <> <> 125 1000 8000 3375 X?I

(2) From table (1)(1) calculatecalculate differencesdifferenoes as.... sbown:sholm: 3 I )( X _X3 't, - ~ Xi1+,- Xi X~+, -X~ i+1i+' i <> 1 6.8 5 75 875 <> 7000 2 5 10 300 7000 <> <> <> <> 3 -1.6 -5 -175 -4625--4625

(3) From table table (2) constructconstruot thethe termsterms shownshown in thethe tabletable below below accordingaocording to thethe following definitioBS'definitions: I U· W. Vi u.u G (Y1+'-Yi)/(X + -X ) I I i i4,1 i 1+1i 1 i <> fx2...x2Nitx._a N 1 1.36 viv .• 15 175 i (X~+,-~)/(Xi+1-Xi)i.0-1 ill 1+1ii <> 2 0.5 30 700 Wi - (\+1-\)/(X + -X ) <> <> <> wi ''(X3i+1-X3i)/(Xi+1-Xi)i 1 i 3 0.32 35 925 / ... --124-124 -

FbrmForm A.1 (continued...)(oontinued ••• )

(4)(4) From table (3)(3) calculateoaloulat8 thethe termsterms inin thethe following table from these defini-· tions:tiODDI

Ii Vi+1-Vj p- qj Vi+--V Pt Pi· (ui+1-ui)/(vi+1-vi)(ui+1 -u )/(vi+1 -v) <> 1 15 ..0.05133-13.05733 35 (w )/(v -v ) <> <> <> qi - (w1+1-;,)/(v1+1-i)i+1 i i+1 2 5 .c1.0.036-0.036 45

(5) CalculateCaloulate the eoeffioientscoeffioients accordingaccording toto thethe following forroulae:fomulae.

CoeffioientCoefficient CubicCubio ean,m. quadraticQaadra,tio ecm.azaz

None b3 (p2-131)/(q2-011) - 0.002133

b p1P -b30q1•Q P b22 1 3 1 P11 - -0.1320-0.1)20 -

b u -b ·v -b ·"1 u -b ,v b11 u11 -b2v2 t -b3ow13 u1-b2..v11 2 1

• 2.9612.9677 ' -

b .X3 Y -b bo0 Y1Y1-b1·X1-b2·~-b3·~ -b1 .X1 -b2.X2-b31 1 Y1-b1'X1-b2'~1 1 1 2 1

x -2.5995

* * * ...... * * * --125-'''5 -

,1.5 .5 GraphsGras InvolvinInvo Iv! ng Three Variables

All the graphs consideredconsidered in the above paragraphs relate a y variablevariable withwith an x variable. l(anyMany relationships involve a third variable .miohWhich wewe maymay calloall aa z variable.variable. The graph may appear somethingsomething likelike this:this:

y

+-______x

Here, eacheaoh levellevel ofof z0 (z(0'1' z2,"2' z3. eto.) forms a distinotdistinct relationship be~between.... en x i and y.y. Such casesoases are common with with sito site iodexindex ourvesourvss Where.mere height (y)(y) dependedepends on age (x)(x) for different levels of site index (z),(z) oror volumevolume oror basalbasal areaarea. curvesourves dependingdepending onon height (x) and stand density (z).(z).

Such syntemnsystems of ourvesourves having been drawn by hand throughthrough data maymay need to be expressed as equations for caloulation or tabulation purposes, forfor 11238use in a computer programme, or simply forfor convenienceconvenienoe of communication and and analysis. analysis.

The system of curvesourves canoan be expressed either:either:

(i) as a seriesseries of separateseparate equationmequations with aa differentdifferent relationshiprelationship betweenbetween xx and yy forfor eaoheach levellevel ofof z;"; .2!:or

(ii) as a single equationequation inin which.mioh z" entersenters asas aa distinotdistinot variable.variable.

The firstfirst methodmethod canoan leadlead toto thethe seoondseoond byby thethe prooessprooess calledcalled harmonization.harmonization. SetaSets of equationnequations of the first type have twotwo disadvantagesdisadvantagess I

(i) There is no method of determining a value of y given aa •z valuevalue intermediateintermediate between the levelslevels chosen.ohosen.

(ii) A large number of coefficientscoeffioients areare requiredrequired toto describedesoribe thethe curveourve system.system.

Equations of the second type are therefore to be preferred; they areare IOOremore conoiseconcise and canoan be used to calculateoaloulate y forfor any given valusevalues of x and z (within thethe rangerange ofof thethe ourvecurve system).

The example belowbolow showsshows howhow aa systemsystem ofof harmonizedha.rIOOnized curvesourves canoan bebo constructed.constructed. The general prinoipleprincipio involves graPhinggraphing the ooeffioientcoefficient values againstagainst thethe •z variablevariable levelslevels and forfor each coefficient,ooeffioient, deriving a new expressionexpression to predictprediot the ooeffioientcoefficient valuevalue given a particularpartioular z.z. - 126 -

1.6. 6 Example:Exam le: Fitting enan equationequationto to aámstem system of of curves ourves by by harmonization harmonization

Figure A.1.3 shows a system of height (H) over diameter (D) ourvescurves for individual trees in four different age classesolasses (A).(A). The ourvescurves were originally drawndrawn byby handhand throughthrough data.data.

TheThe first etagestage is to fit a separateseparate equationequation toto eacheach line.line. ItIt is deoideddecided to use a quadratiCqlmaratic function as the approximating equation.

Figure A.1.3A.1. 3

Hand-drawnHanddrawn curves ourves ofof height height onon diameter diameter forfor differentdifferent ageage classes,classes, toto bebe approximatedas. .ximated byb harmonizedharmonized quadratiouadratio eequations uations (seesee examexBlDple le inin texttext). .

30 H A 25 ~-- ,20

----15 20

10 15

10

5

o o 4 8 12 16 20 --127-127 -

The calculations,oalculations, whichwhioh are not shown, involveinvolve selectingseleoting 3 points from each line and calculatingcaloulating the coefficientsooeffioients according to the method of form A.1. The selected points are shown on eacheach curvecurve inin thethe figurefigure byby e'.a The tabl"table below gives the ooeffioientcoefficient valuesvalues forfor each age class:olass:

Coefficients Age Age b b b ------Class b0O b11 2 5 2.2 1.0625 -0.0219 10 .66,6 1.0500 -0.0250 15 9.8 1.1375 -0.0281 20 12.5 1.2750 -0.0313

In order toto reducereduce thethe systemsystem ofof four four quadraticsquadratio equations to a single system, the ooefficientscoefficients bi mustmust bebe mademade toto dependdepend onon age.age. The form of this dependenoedependence can best be seen by graphing the coefficientsooeffioients against age classolass (figure(figure A.1.4).A0104)0

From thatthat figurefigure itit canoan bebe seenseen thatthat thethe relationrelation betweenbetween 1)0 bO endand age follows a.. gentle curve, whilst bibl versusversus ageage isis aa steeper,steeper, moremore asymmetricasymmetric curvecurve; b2 versus age appears almost exactlyexactly linear.linear. TheT'ne equations were calculatedcaloulated (again(again using form A.1) for these three curvescurves asas follows:follows:

2 b =- -0.0187 4.1.153+1.153 AA -3.100-3.100 A2A -(1) O

,-55 3 b , = ~ 1.2251.225 -0.0508 -0.0508 A -',-0.00400A +0.00400 A2,f,.2 -6.67 xx 1U10- A

b2 =-0.0188 -0.000627 A (3 )

The general equationequation forfor thethe linealines inin figurefigure A.1.4Aala4 isis the crivdratioquadratic equation, asas selected earlier, withwith thethe form:form:

4-1)D4bD2 -(4) H.b0 1 2 The coefficientooeffioient valvesvalues in equation (4) canoan now be deriveddarived for any age between thethe limits 55 endand 20 using equationsequations (1)( 1) toto (3).(3).

This is a fairly complex example of harmonization intended toto illustrate thethe fullfull soopescope of the principlesprinoiples involved.imTolved. In practicepraotioe some simplifioationsimplifiostion isis often acceptable.acoeptable. ForPor example, in the above ossecase a quitequite oloseclose approxima.tionapproximation toto thethe original hand-drawn hsnd-drawn functionfunotion canoan be obtained ifif averageaverage valuesvalues ofof b/b, andend b2b2 areare usedused andend onlyonly bobe dependsdepeni13 upon age as a linear function:funotion:

b =~ a +4- a , A O0 a0O 1 the overall equation, substituting for blac)O will be:

H s aao + alAa,A + biDb D + b2D2 b D2 O , 2 - 135128 -

Coefficient values plotted againstagainnt age 01a28olass A forfor the quadraticquadratio equationsequationn fitted fitted toto the lineslines ofofflgure figure A.1.1A.1.3 (a)( a) b0bO 14

~. 12 ~ , ~ 10 , ~ , , .. ,/ 8 , , / , ,...457.- 6 , , , / 4 , , , , 2 A 5 10 15 20

(b)( b) ~bl ,• 1.250 , , , I I I I 1.200 I , I I 1.150 I 1.150 I Olt I • I 1.100 I 1.100 , I , .- , , , ~ - 1.050 N A 5 - ,---- 10 15, 20

( c) b (c) -0.020 b22 -0.022 , , , , , , -0.024 et, -0.026 -0.028 , -0.030 , , .. -0.032+------~------~----~-0.032 A 5 10 15 20 - 129 -

This model is not a perfeot fit (as is thethe oneone workedworked outout above)above) butbut thethe errorerror overover the range of the hand-drawnhand~awn functionfunotion is within .:!;+ t m ofof predicted predioted heightheight whichwhioh maymay well bebe acoeptableacceptable if the original data was well scattered and consequently the hand~wnhand-drawn lines subject to some uncertainty.uncertainty.

As with polynomials, harmonized functionefunctions should nev_21,...1_3ee.Lar....e_p_asztracaationAnever be used for extrapolation' If extrapolationextra.polation is necessary, draw the extended curve set by bendhand to cover the range ofof intended use and recalculate all the coefficients of the approximating function.

2. CURVE FITTING BYBY LINEARLINEAR LEASTLEAST SQUARESSQUAHEl'! ANALYSISANALYSIS

22.1. 1 SimpleSiplç Linear Regressionession

Linear regression analysis is dealtdealt withwith in manymany texttext booksbooks inin greatgreat detaildetail andand withwith numerous examplese=mples (c.f. Snedecor,Snedecor, DraperDraper &&Smith Smith andand SeberSeber inin thethe bibliograPhy).bibliography). It isis proposed here only to give the barest outline of the fundamental ideasideas, f basiobasic caloulationscalculations and most essential statistical parameters.parameters.

Linear least squares narmlysisanalysis isis a.a slightlyslightly moremore preciseprecise namename forfor linearlinear regressionregression analysis. It refers to a techniqueteohnique forfor fittingfitting predictivepredictive equationsequations toto rawraw datadata (i.e.(i. e. unamoothedunsmoothed observations) based on the prinoiple of minimizing the squares of the deviationsdeviations from a straight lineline throughthrough thethe points.points. When the assumptions, whichwhiCh areare listed below, areare fully satisfied the parameters (coefficients)(ooefficients) for the fitted equation willwill be those whichwhich are most likely to be correct in a statisticalstatistioal sense.sense, I-When/hen the assumptions are !!21not satis­satis- fied, thenthen thethe method will still give parameter estimates,estimates, but they willwill no longer be the bestnest ones obtainable; better parameter estimates couldoould be obtained by deriving from first principlesprinoiples correotcorrect "ma.zimum"maximumlikelihoodequations" likelihood equations" forfor thethe partice1pepa.-ticular classclass ofof problemproblem involved. This latter subject isis outeideoutside thethe scopesoope ofof thiethis appendix.appendix.

The definition of the simplest typetype of regression problemproblem canoan be shown with the help of figure A.2.1. It involves a relationshiprelationship betweenbetween twotwo variables,variables, xx andand y.y. TheThe x variable is assumedassumed to be the one that is to be used to predictprediot the y variable. The statisticalstatistioal model is:is:

y a + b.x + e -CO- (1) where the ei are the random deviations of eachaaoh point fromfrom a line.line. InIn figurefigure A.2.1A.2. 1 the solid line represents the partpart

y a +bex usually called the regressionregression equation,equation. a and b are the coefficientsooeffioients of this equation whose estimation from the data is the primary purposepurpose of the analysis. 'lb.eThe eiei areare calledcalled 'residuals, 'residuals' and represent the vertical distancesdistanoes betweenbetween the pointspoints andand the line.line. -130 -

Form A.2A,2 Calculation of statisticsstatistics forfor simplesimple linearlinear regressionregression usingusing StatformStat form 1111 from Dawkins,Dawkins? 19681968 (reproduced(reproduced uithwith pgrmission).permission). The datadata. isis thatthat shownshown AllirAy:einfi.-rore A21A.2.1.• • •

LINEAR REGRESSION and CORRELATIONCORRELATION C-- LiHeLine olweinctr-on-s iI N'tlIfCNature ofof thethe observations:observations: HsiH'1I::-r'"·.£.J:;':,-fe-t--,-tL ....e· ~ ~~~.541.1,. - -~~~~.'" ct:4mil:or . O~con t.".i pct: I:it~I- taimai Tr&E 1.. 12 D,,:pl:ndclHDependent or 4.ft-handleft-hand variable,variable, y: {+fl..< y/-.t., t\t. 3 independentIndqlL'lldcllt or right-handrighe-hand variable,variable. x: 4 1,--xxlej-42,r,~~, 00<. c". ;s

6 SumsSUlllS ofof squares s'l u;'Jrcs and :md prOduCES. products. In ... '3 ...... pairs of observations x 7 (Ey)2(Ey)' - (,0'>/6 0S. 1 e, EEy yx xEx Ex = 6,bl'l'l2.. (992_ (Ex)2(Ex)' =:= C3f.3 5<;O,+- 04- y x . - - H Iy'! = 71 1L ...5?'g .... Eyx . . 1113.4 ExrX22 _= 53~J~'+ I 4- Iv' = ...... 71. . " :JI:?"L ... III t g8- 9 (~'y)(Ey)2/n'/ n = 672-1-F672..4- EyEx/n ,-_-- 6bUg V F V (Ex)'/n(Ex)2/11 = 70~b70 5 L 9 -- = = 2..2-2- 2- n11 10io SSy -=-- i'i'q-'i9~ SPyx =- '1'£1/ 5 1 SSx -= 10-1'1/01? - tI (0(" 2--2-1 I IIii LinearLincar JJ.-Coefficient;Cocfiiciellt; SPyx/S5x Sl'yx/SSx =- 0-l::),git2-.J gt 2- .4't.. . ~--, b 2..L 2., 111 2 regressionrl'grcsslon I, Constant;COllStant; y-bxy--bil. =.--= .... .26 .., 431/11.... , . ~1 ..1 If - a 2:i2..'1 3S3 s 5 3 9 .9 6 13 Rt..'grcssiollRegression 5S SS- = bxbx SSPyxl'yx = ...... \l>!l . '1G . :',\-+ 32..3 7._ 14q Total SS is theth e SSy. Total d.f.d.f. =-= 11-1ni == •• H ••• 8'K ...... 2.102_ to 2S2..~ I;1 3 ResidualRt.'sidual SS5S and d.f.d,f. obtainedobrained byby subtraction.subtraction. 14-4-2- 2- Lt:> r, 6 ANOVAR Variance Table Lrr::- 17 SourccSource SS d.Ld.f. MS ratio F 4-Lt >. > 18I H Regression .glg38.8,jI~ Y 1I .. B?"],,,"ES 7 .o 31'17-' :z. ,,-:-9---: Is :U. . I4- @ fze. II 19 Residual L?'?.:'?1 5 5- 0 ... 7 '=0, 2010 Total qq'l-9- 8 21 Coefficient of determination,dc[crmillJ.tion, r2,r2,

1222 RegressionRegressioll SS/T SS/Total otal SS SS -= lr}~g 3 1 I9

252S Residual standardstandard deviation deviation --= VResid.VResid. MS -= .. /t4-. :.-1] .IL . --=-- SD.SO. rcsid.mid. 26 Coefficient of residualresidual variation%variation ' }~ -=- = toox IOOX SI)resid./F SI) resid .------./y = 1,'11%=i.i...i.X= CV%CV% EyIy ExIx 27 Standard error for random samples == SDSO resid./ mid.1 %/77 Vii = --= /,>7 ¿$7 7 - SE. mid. resid. 2..+'-2-+ k)2--S--'2-Q. 1828 Student's t t(or for 11-2 nz d.f. d.f. andand PosP.OS isis ...... 2.~., .. .~16 . b. .. y x 2?.27.1 i 2-2jl g 29 Sampling errorerror °,;) % =- Iootx!ootX SE. SE. resid.h7resid)y -= ("qJ,C?o~ (P 434 = E,E~ ~ , 30 StalldardStandard error ofof the the coefficientcoefficient bb isis -IRcsid.N/Resid. MS/SSxMS/SSx = 0 1::>.f. ,.4-('+ -3 ..3 . ..$ ...... =- SEbSE() 313 ! Its t-r

Stntj(lrmStatform 11 CommonwealthCOllmlOlllllt'olth ForestryForestry Instilllte.Institute, OxfordOxford 1968 --131131 -

azureFigure A.2.IA.2.1 A simplesimple linearlinear regression model fitted fitted to datadata

y 50

40

/ / • , / / / / / / 30 / /0/ / / / . / ,. / / • / / / / / / /

/ / / • . / / ,. 20 / " ,. / /

/ " • ,. / , / , ,. • ,. , 10 / / , ,. , , / o x 10 20 30 40 50

Regression lineline y = a + b.x

Confidence intervalinterval atat 95%95% levellevel

o Point formed by mean x and y values in data - 132132-

The assumptions mademade inin simplesimple linearlinear regressionregression areare that:that:

(i) The model isis trulytruly linear,linear, asas representedrepresented byby equationequation (1)(1) above,above, andand notnot curved inin anyany sense.sense.

(ii) The residuals are normally distributeddistributed withwith aa constantconstant variancevariance overover thethe range of x values underunder consideration.oonsideration.

(iii) The residuals are indepenaentindependent ofof eacheach other;other; they are not correlstedoorrelated or grouped inin anyany way.way.

(iv) The values of the sample pointspoints onon thethe xx axisaxis cancan bebe determineddetermined exactly.exaotly.

The caloulationcalculation of the various statistiosstatistics required for linear regressionregression analysisanalysis areare shown in form A.2,A.29 whiohwhich reproduces Statform 11 from Da.wkinsDawkins (see bibliography for details)details) with the working of thethe datadata shownshown inin figurefigure A.201.A.2.1. The most important statisticsstatistios are the coefficients (lines 11 and 12), the variance ratio (line(lille 18), the coefficient ofof determina,­determina- tion r2 (line 22), the residual standard deviation (line 25) and the confidence limits forfor predictions (lines(lines 32-35).32-35).

The r2 values and thethe residual standard deviation are useful for the comparison of regressions. The higher thethe r2r2 thethe moremore preciseprecise thethe relation.relation. When r2 isis oneone, the residuals are zerozero andand thethe datadata fallfall exactlyexactly onon thethe line.line. Similarly, the closer the residual standardstandard deviation approaches to zero,zero, the more precisely the regression predicts the y values. The confidenceoonfidence intervals of the regression line show the limits wiwithin thin which the true mean of a selected number of yy values shouldshould lie.lie.

It will be noted that the regression lineline will always pass through the point formed by the mean of thethe xx andand yy values.values. The confidenoe bands are also curved in form andand becomebecome rapidly wider as one moves outside the range of the data contributingoontributing to the regression.

The x variable in a regression analysis isis aalledcalled the independent or predictor variable; it provides the basis on which..n.ich thethe predictionsprediotions willwill bebe made.made. The y variable is called the dependent oror responseresponse variable;variable; it is assumed to be controlledoontro lled in some degree by the level of the predictorprediotor variables.

2.2 Regression with Two Predictor Variables

Regression with two predictorprediotor variables assumes the model

Y = b0 b1X b2Z where the X and Z are knownknown predictorprediotor variables andand YY isis thethe dependentdependent variable; the bi are the coefficientscoeffioients toto bebe determined. The assumptionsassIDDptions and general principlesprinoiples are the sanesame as forfor simple linearlinear regression.regression. The ooeffioientscoefficients and statistics for this model canoan be calculated by hand although the method is rather moremere involved than forfor simple linearlinear regression. It is a useful model for fitting a variety ofof curves,curves, as will becomebeoome apparent in section 2.3.2.3. - 133 -

Form A.3 showsshows how thethe calculationscaloulations proceed.proceed. The example uses the data from figure A,2.1A.2.1 and and form form A.2 A.2 forfor thethe height-diameterheight-diameter relationshiprelationship ofof 9 sample trees, but this time with the addition of an extra variable (diameter)2,(diameter)2, soso thatthat thethe equationequation beingbeing fittedfitted becomes:becomes:

H = b0 + blD + b2D2

After working through thethe formform thethe coefficientcoefficient valuesvalues calculatedoalculated are:are:

bb0== 8.31177 O b = 0.371583 b11 = bb2=2 = 0.00953469

all to 66 significantsignificant digits.digits. It is important whenWhen oarryingcarrying out thethe calculationscalculations toto workwork to 6 or more signifioantsignificant digits and not to round.round small numbers (such asas gl'g1, g2'g2, g3g3 inin thethe example)emample) to a few decimal places;plaoes; otherwise considerable lossloss of accuracyacouracy maymay result.result.

The lower variance ratio and t-values for the ooeffioients obtained with this regression as compared withmdth the earlier linear one (figure AA.201,. 2.1, form A.2) refleots the fact that the additio;'addition of the extraextra variable increasesinoreases the uncertaintyunoertainty of the parameter estimates with respeot toto thethe populationpopulation from£'rom whichwhich thisthis samplesample ofof pointspoints waswas drawn.drawn. In this case neither of the t-values is significant for the two variable coefficients b1 andand bb2,2;°this this indicates indicates that that thethe modelmodel wouldwould be juetjust asas effioientefficient forfor prediotionprictioned purposespurposes if one oror otherother variablevariable werewere ommitted.omitted.

2.3 Data Transformations and Curve Fitting

Various types of curvilinearourvilinear functionsfunotions may be fittedfitted using linear regression tech-tech­ niques by making appropriate transformationstransformations ofof thethe dependent and predictorpredictor variables. It is desirable when making transformationstransformations ofof data to proceedprooeed through the followingfollowing stages:

(i) Plot the untraneformeduntransformed data on normal graphgraph paper and examineexamine itit to Beesee whether any curvature in the trend lineline isis apparent.apparent.

(ii) If a curvature is apparent, select an appropriate transformation for the dependent and/or predictorprediotor variables andand calculatecalculate transformedtransformed valuesvalues forfor eacheaoh data point.

Plot the transformed valuesvalues on normalnormal graphgraph paperpaper andand examineexamine thethe pointspoints to see whether thethe trendtrend isis nownow linear.linear. If it is, fit a regression using the transformed data values.

The!!he selection of a suitable transformationtransformation canoa.n bebe aidedaided byby thethe diagramsdiagrams inin figurefigure A.2.1, which illustrateillustrate some commonlyoommonly usedused curveourve shapes.shapes. FtnctionsFunctions (a)(a) to (d)(d) canoa.n be fitted by simple regression. Functions (e)(8) andand (f)(f) requirerequire twotwo predictorprediotor variables;variables; consequently the traneformedtransformed values cannotoannot bebe examinedexamined graPhicallygraphically asas inin (iii)(iii) above.above. 1

- 134 -

FbrmForm A.3 Coefficients andand statistiosstatistics of of a aregression regression mith with two two predictor variablesvariables (part 1)

Description ofof problem:problem: "/~tJJr-,p.·(),,,,~e.-1-1-u?ki, -amid- ee- cd.carakfra,v,ClII.~f(;;QA e,..~\;' ~ AA.2.1 '2- I • (,..4.-,,;..r... "' - 1- ~ (1) RegressionRegression data ( 4) CorrectedCorrected products .._ IyC.Ey = LYxyc._(LY)c./ -(BY) /n = 'i~ oj-.OOO Y X ZZ n ci`1 +.000 IXYzxy == zXY-EC.I.Y/n.IXY-IX.XY/n = "/51. 000 ¡I g 1/ g 42'-If.It.. EzyIzy = IZY-IZ,IY/nZZY-EZ.EY./n = 5'2.>2. 33 H21 . (,. 6--, H2.2. 11 LJ~24 9 2 Ix2Ex2 = XXEX2-(EX)2/n_( IX) 2/n = ,(78(o7B·ooO.000 i 6 1.12-1 C+'l-I44.1 l:XzExz == EXZ-EX.IXZ-IX. EZ/n.IZ/n == ;l1/b.o511i . ooo 0 2-1 8<+1841 22-2'" /. 2-1 32.1S.>o/.,S'b Iz2Ez2 -= l:Z2_UZ)2jn - 32.3 3S0/SS. C., 11. ... S IZ2-"122-61= 1.129 H35 t 2_2-5 (5)( 5) Determinant && GaussGauss multipliers multipliers 3'1-3 Li 3:1-32.. 10so?...q. l..'/- D == zxIXc..f---T7T---7o l:zc._(IZ -- Exzil:xz) c. = /.5.15'1."1. 4,743 ',02.0. 9,020.? 'f bH(.2.-S Jj, z.~z5 2- b g1=gl= Zz2/Dl:Z2/D = o o.. o 0;/,0 zo 2.4. 2.4- 4.4-1 --/ Ij.}.... If-SurS 1.01.Lois- <; -Exz/D 0 /4-2- g2= -l:xz/D = -0.-0.000360003(,07'\1 rt1 to) II.Iqt era0-0 g3= Zxl:X2/DD 'l-lI-Li-4 'fO g3= = 6'.138~ 13a2 ::/t)-')~ 10-" ((6) 6) Regression coefficientscoefficients ! bi=~= g1.Exy-t-g20Zzygl,l:xy+g2' l:zy = = o·.nl0.3.7t5S3 S83 b2=b2= g20g2,l:xy+g3,l:zy Exy+g30zzy == C.o. caoco 95~5 3q-Cci 34-''1 bo=bO'" 'l-bT-b1Z-b2.Zl ,I-b2·Z' == gIf. 3 ,,,,1311 77 77

( 7) Regression sum ofof s'Luaress uares -772 zy b b 4... l:~2 ='" b10Exy+b2.Ezyl , Ixy+ 2, Izy == 8,52_,852..14- (8)(8 ) ResidualResidual sumSUDl ofof squaressquares --- Id~",Ea= Eyr.y~-IY~ = /4-1. toft, - - (9(9) ) Residual degrees ofof freedomfreedom ( 2)2) TotaasTota1.s andand -productsproducts Idf=df_ n-3 == lc,10

Iy2Y ' 111'67 -ng EyIY ~4-"(9 EXYIXY 19~.,783, zzYI'4Y 274-",2.741 n '1 zxIX :z..~2- zZIZ ?13'11:3 +

IX2 '11 1'1> i:kzIXZ ~~~.,&~ (3) ~!eans 4- l::,I2 /" i'89.j 3,< Y = XY/n = 27. :a LELI__g2fit , X = IX/n '" LS B Z' '" IZ/n '" 90 3.i - 135-135

Form A.3 ((...Part ••• Part 2)2)

(10)10) Correlations betweenbetween variables 7 rryxyx = Exy/l(ix2.zy2)r.x:y I {( r.x • ry7) = (9.1(87O·"H87 r = r.yz/{(ZyzAlkI-. rz2.Zz2 .Ey2r.y2) = ryzYz = = O.'p0.92-24.. 2\f- r Zxz/y rtk Ix22 Ez2%2 rxzxz = r.xz/{(r.x .r.z ) = o.0.9-16g 'llb&

1(11) MultipleYultiple correlationcorrelation coefficientcoefficient RR andand RR~ R2 = r9''Iry=

( 12) Analysis of variance sum of mean variance squaresSQua.res d. f. square ratio

2 4-2.~.ll 18.13.o~0 Re,q:ressionRegression r.S'2 8!)2852.34.. 1~ 2 2 Residual IÇ12r.d /1j.1.!.b(4.1Gb df ben :n.(.,12-4.(01 = s2 2 Total rzy2 y 991}..Cro"I'Ilf. .0"0 n-1n-l11 gS

(13) Residual stst .dard deviation

s =. {s22 = 4·86(43 6

(14) Standard error and t-valuet-value of coefficients . Coefficient bl b 2 ¡Standard error Standard error sb5b s.{fh=s.igi= O·"'1IS0,0tS s.{g3=oig3= 0.00.012 1:1 (12- (,2- Itt= = b/abblsb 0o .-31{?,'" 0O,,~b .--1%-6

(15) Confidence intervalsintervals forfor predictionspredictions For a series ofof m estimatesestimates ofof YY ·atat aa givengiven levellevel ofof XX andand Z,Z, the standard errorerrcr of thethe mean predictionprediction isis givengiven by:by:

y =S.V.(1/111+1/n + g1.x2 + 2.g2.x.z + g3.z2)

where x = (X - Y)X) and z = (Z - 7).Z). For a large sample,swnple, l/m11m will be zero; forfor aa singlesingle pointpoint itit willwill bebe unity.unity. - 136136 -

FigurePigure A.2.1 Curve shapes resultingresulUng fromfro.. different functionsfunotions that can be_fittedbe fitted byb;y linearlinear regression.regression.

(c = lOa10a or eae& depending on whether logarithnslogarithms to base 10 or base e are used.)used.)

((a) a) log yy = a + b loglog xx (b) y = a ++ bb loglog xx y Y »=1 , , / a- , / b

b

(e)( c) log y = aa ++ b/xb/x ( d.)d) log y == a ++ b.xb.x y y e b>O e __ b

______b>O e b

azureFigure A.2.1A.2.1 (oontinuedcontinued. ••••e) )

e) 3T= a+ bx+ b2x2

x

(f) log y ; a + ~log x + b x log y = a + bilog x + b2x2 y

b1<0' b2>0

x - 138 -

The effeoteffeot ofof datadata transformationstransformations uponupon thethe basicbasio assumptionsassumptions ofof linearlinear regressionregression involves three important points:points:

(i) Regressions inin whichwhioh thethe samesame dependentdependent variablevariable hashas beenbeen subjectsubject toto differentdifferent transformations cannotcannot bebe comparedcompared directlydireotly forfor goodnessgoodness ofof fitfit usingusing thethe correlation coefficientcoefficient RR (or(or R2).R2).

(ii) 'ilieThe distribution of the residuals will be influencedinfluenoed by anyany transformationtransformation ofof the dependent variable.variable.

(iii) The regression may be biased by a transformation ofof thethe dependentdependent variable.variable.

For comparison ofof regressionsregressions forfor goodnessgoodness ofof fit fit whenwhen severa],vtransformations severalytransformations ofof the dependent variable areare involved,involved, thethe FurnivalFurnival IndexIndex mustmust bebe used-7.used • This isis calculatedcalculated as:

FIFI= = s.s. ((nf'nf (y)( Y) -1)-1) 1/n1/n

or more simply:simply:

FI . s. antilog (( Elogf'(y)-1)/n)

Here ss isis thethe residualresidual standardstandard deviationdeviation fromfrom thethe fittedfitted regression;regression; n isis the number of data points;points; and f'(y)-1f'(y)-1 isis thethe reciprocalreoiprooal of the derivative of the transformation applied to the yyvariable variable withwith respectrespect toto y.y.

Since the non-mathematicalnon,mathematical reader will not be familiar with derivatives, appropriate forms for the commonestoommonest transformationstransformations areare givengiven below:below:

-11 Transformation f'(yr

None 1 logioylog1OY 2.3026 y log y Yy logeye y/wY/w w 1/y -I-"1.2 :I-yk 1/(qk-111(kYk-4) )

In the above y is the original data variable, w isis any weight to be used in fitting the y to normalise residuals (see(see laterlater sectionseotion on weighted regression forfor more details), k is any constant that is used to transform y by raising itit toto aa constantconstant power.power. E.g. if .;y2 were the dependentdependent variable,variable, kkwouad would bebe 2.2. Note that Whenwhen no transformation isis applied the Furnival Index is identical with the residual standard deviation.

111/ See Furnival, G.14.GoM., 19611961 An index for comparingcanparing equations used in constructing volume tables. Forest Science 7(4)337-341. - 139 -

Given a set of regressionsregressions for Whichwhich FUrnivalFurnival Indices have beenbeen computed,computed, all involving the same basiobasic dependent va.ria.ble,variable, the equation whichWhich fitsfits bestbest willwill bebe thatthat withwith the smallest index.index.

The effect of transformations upon the residua.ls'residuals' distributiondistribution andand uponupon thethe biasbias ofof the regression willwill be consideredconsidered inin detaildeta.il in in the the sectionsection uponupon residualresidua.l analysisa.na.lysis and weighting. FbrFor simple usage of the tra.nsforma.tiontransformation principle in associationassociation withwith ma.nua.lmanual caloula.tion,calculation, the correctioncorreotion factor for bias resulting from a logarithmic transformationtransformation ofof the dependent variableva.riable shouldshould bebe mentioned.mentioned.

This correction factorfaotor for bias, due to MeyerY,Meyer], assumesassumes thatthat thethe modelmodel beingbeing calcu,. calcu­ lated hashas the form:form:

logalog Y = f(x) + eie. a ~ wheretheerrors.where the errors eiel are normally distributed anda.nd homogeneous withwith aa standardstandard deviationdeviation s,s, which is the residualresidua.l standard deviation calculatedcalcula.ted in the regression analysis. The model that is to bebe actuallyactua.lly appliedapplied isis however:however:

(r(x)(f(x)) ) y=y = aa

wherewhepe a isis eithereither 1010 oror ee dependingdepending onon whether commoncommon oror naturalnatural logarithmslogarithms are to be used. If the assumptions about error distribution are correct, thenthan a systematic error will occur whichwhich canca.n bebe compensatedcompensated forfor byby usingusing aa correctioncorrection factorfactor CC soso thatthat the above formulaformula. becomes:

(f(x)) y=y = C.aC.a (f(x))

The correction factorsfactors forfor commoncommon anda.nd naturalnatural logslogs are:are:

(i) Common (base(base 10)10) logs:logs: s2 C ~= 10.1.1513 s2 (ii) Natural (base(base e)e) logs:logs: s2 2 C = ees2/2/ where ss isis thethe residualresidua.l standardstandard deviationdeviation obtainedobtained fromfrom thethe originaloriginal regressionregression withwith loglog y.

The validity of the correction factorfactor isis dependent uponupon assumptions about errorerror distribution whichwhich needneed notnot bebe correct.correct. Hence,Henoe, itit is not possible to state in any fixedfixed or dogmatic fashionfashion that thisthis correctioncorreotion Shouldshould oror shouldshould notnot bebe usedused inin aa pprticularparticula.r case;oase; preferably the distribution ofof the residualsresidu:;.ls shouldshould bebe examinedexamined graphicallygraphically asas describeddescribed inin the section on residual analysisanalysis beforebefore arrivingarriving at anya.ny decision.deoision.

In practice, when the degree ofof fitfit obtainedobtained isis high (i.e.(i.e. R2R2 over about 0.9) then the variousva.rious arguments about alternative fittingfitting methods,methods, errorerror distributions,distributions, etc.,etc., areare essentially academic;academio; the fitted function may be safely manipulatedmanipula.ted and transformed as it it were a deterministic algebraioalgebraic function.funotion.

JVY See Meyer,Meysr, H.A., 1944. A oorreotioncorrection for systematicsystematio error occurringooourring in the application of the logarithmiclogarithmio volume equation.equation. PennsylvaniaPennsylvania. StateState UniversityUniversity FbrestForest ResearchResearoh PaperPaper 7. -140140 -

2.4 Multiple Regression Analysis

Simple linear regression and regression with two prediotorpredictor variablesvariables representrepresent models,models, respectively, ofof thethe type:type:

Y =a a + b.X -(1) Y a a + b1 .X + b2.Z -(2) where a,a, byb, b1b1 andand b2b2 areare thethe coefficientscoeffioients to be estimatedestimated and X, Z and Y are variables. Both of these are examplesexamples ofof thethe generalgeneral multivariatemultivariate linearlinear regressionregression modelmodel whichwhich hashas the form:form:

Y. b0 + b1.X1 + b2.X2 + b3.X3 + + b .X m m -(3) where thethe bibi are the coefficientscoeffioients to bebe estimatedestimated by regression and the Xi are different predictor variables. This can be written inin short formform as:as:

m Y == b ++ ¡t b..X.b .• X. -(4) 0O 1 1

The various predictor variables canOan bebe ofof thethe followingfollowing differentdifferent sorts:sorts:

(i) Each Xi may bebe aa separateseparate independentindependent variable.variable. ForFbr example,example, inin a particularparticular applicationapplioation X1X1 maymay bebe heightheight andand X2X2 diameter.diameter. The independent variables may be more or lessless correlatedcorrelated amongamong themselves,themselves, providedprovided thethe correlationcorrelation isis notnot perfect (i.e. R = 1). In the latter caseoase one of the variables involvedinvolved mustmust be ommitted.omitted.

(ii) Some variables maymay bebe combinationscombinations and/orand/or transformationstransformations ofof otherother variables.variables. For amample, X3 might be log(X2) and X4 might be X1.X2. However, additive Fbr example, X3 might be log(X2) and X4 might be X1.~. However, additive combinations (e.g.(e.g. X3X3 = X1X1 ++ X2X2 oror X/X, - X2)~) are notnot permitted;permitted; they result in perfect correlation among variiples.vari \~les.

(iii) Some variables maymay bebe conditionalconditional variablesvariables havinghaving valuesvalues ofof onlyonly zerozero oror one.one. ForFbr example, in a particularpartioular regression, X/X1 might be zero for data from one seed sourcesouroe and one forfor data fromfrom a different source. This is discussed further in the section on comparisoncomparison ofof regressionsregressions andand covariancecovarianoe analysis.analysis.

In most applications in forest yield predictionprediotion only a few basiobasic independent variablesvariables are involved,involved, often only 22 or 3, but with perhaps 2 or 3 other variables constructed as transformations inin order to provideprovide particularpartioular curveourve shapes.shapes. Fbr example, the site index model: log H·H = bb0 ++ b,lAb1/Á ++ bb2.5.S + b .S/A O 2 b3.SA3 involves three basic variables: height (II),(H), ageage (A)(A) andand site index (S).(S). TheseTheee are trans-trans­ formed and combinedoombined to give thethe regressionregression model:model:

Y=Y = b0b ++ b1.X1b ·X ++ b2.X2b .X + b .X O 1 1 2 2 b3.X33 3 where Y isis log(H),log(H), X1X isis 1/A, X2X isis S andand X3~ is S/A.S/A. 1 2 - 141 -

2.5 Calculations for EstimatingliBtimating the ParametersParameters ofof aa Multiple RegressionRegression ModelModel

As in the previous cases ofof regression with one and two predictor variables,variables, in mulmul multiple regression one is prim~rilyprimvily concernedconcerned toto calculatecalculate thethe coefficientcoefficient values, the coefficient of determinationdetemination (R(R ),)t the residual standard deviation,deviation, the variance ratioratio ofof the residuals to the regressionregression mean square, the standard errors of the coefficients and the statistics necessary for constructing confidence intervals for predicted values.values.

'fueseThese parameters areare usuallyusually estimatedestimated usingusing aa computeroomputer programme.programme. Manual calcula-oalcula,­ tions are very tedioustedious andand itit isis veryvery easyeasy forfor errorserrors toto slipslip in.in. However, the manual calculation procedures are given here both for the benefit of those whowho wishwish to use themthem as they standstand andand asas anan algorithmalgorithm thatthat will provideprovide thethe basisbasis forfor a computer programme if it is necessary toto writewrite oneone fromfrom scratch.soratch.

It is best to considerconsider thethe calculationscalculations asas followingfollowing aa numbernumber ofof stages:stages:

Stage 11 Define the model to be fittedfitted andand tabulatetabulate thethe transformedtransformed variablevariable values.values.

'fueThe general model is:is: m Y.Yj~ = bbO+fbi'\j+ Eb. .X + e.e. 0 i3 3J where are the dependent variable observations (j = 1 to n); are the predictor variable where YjYj are the dependent variable obeervations (j ~ 1 to n); XijXij are the predictor variable observations with mm variablesvariables perper observationobservation (i(i =~ 1 to m) and n observations.obeervations. 'fueThe bibi are the coefficients to be estimated, the ej are the residuals between the oobservedbserved Y8Ys and the YaYs which will bebe estimatedestimated fromfrom thethe linearlinear model.model.

In the calculation exampleexample wewe wishwish toto fitfit thethe model:model:

Y =~ b + b ,X + b ,X + b ' X b00 b1.X11 1 b2X22 2 b3.x33 3 given thethe data:data:

Y X X1Xl X22 x3X3

1 4 9 4 3 3 9 9 7 3 6 6 9 5 8 3 14 4 7 1 18 6 4 5 19 5 3 7 21 7 1 9 24 6 3 4 26 7 2 5 whereWhere n is 1010 (number(number of observations)observations) andand mm isis 3 (nutber(nUlllber of predictorprediotor variables). - 142-14 2

Stage 2 FbrmForm the totals and cross-productscrossproducts ofof thethe datadata defineddefined algebraicallyalgebraically as:as:

(i) Totals n T.T = EX.[X.. for ii = 1 to m 1 j l.J13

T = [Y TyY = EYij

(ii) Cross-productsCrossproducts

n PX = [\joXkjEX .X for ii andand k=k= 11 toto m ik .13 kj J n PY. = EYH.oX .X . . for ii= = 1 to m l.1 j jJ l.Jij

FbrFor the example we have thethe followingfollowing totalstotals andand products:products:

k y 1 2 .23 Tk 142 50 52 53

k 1 2 3 P\kPXik i 11 270 227 267 22 350 256 3 339

k 1 2 3

811 522 752 PYk

The total sum of squaressquares ofof thethe YsYs mustmust alsoalso bebe calculated:calculated:

EY2r.l = 22 714

ayeStage 33 Correct the sums of productsproducts forfor thethe means.meanso From the values PX.PX , PY.FY. and ikl. k l. y2Y2 the following corrected values are formed:formed:

QX(gikr.= PXPXik - TioT,jnTi.Tk/n ik ik

QY = PY. T .T/.T /n QY1 = PYi - TioT/n El2 T;Al. rlEY2 = ty2 _ Tin - 143-143

The calculated values forfor thethe exampleexample are:are:

QX i~k~ ______~1 1 ______~2 2______~3_ ik 3 1 20 -3333 2 2 79.6 19.6 3 58.1

k 1 2 3 101 -0.6 QYk 216.4-216.4 0.6

2 r.yZY2 = 697.6

stageStage 4 Solve for the coefficient valuesvalues and at the same timetime invertinvert thethe cross­cross- product matrix usingusing JordanJordan elimination.elimination.

QX and QY are arranged inin aa tabletable (which(which wewe willwill callcall C)C) inin thethe followingfollowing way:way:

1 2 3 •0 • • k •0 .4> 0• m 1 2 2

• QQX X • • 1kik QY • i i • • • •o • •o m m

At the same time, the values QX91,QX l' QX QX , etc. are filled in as having the 31 32'32 same values as QX12, etc. Fbr te example we have therefore: same values as QX12' QXn'139 eto. For t~e example we have therefore:

C .. j 1 2 3 4 Cij~J i

1 20 33-33 2 101 2 33-33 79.6 -19.619.6 -216.4216.4 3 2 -19.619.6 58.1 -0.60.6

Having drawn up the table or matrix C we are nownow readyready to proceed with the steps of the Jordan elimination process. --144144 -

Jordan eliminatelimination ion k'th columnoolumn

oC. .. c c l.J C.cikik cijij

oc c k'th row ckj kk ckjkj k'th row

0..c .. c c .. ijl.J cikik cijl.J

Perform m reductionsreduotione of the table.table. On the klthk'th reduction replace eacheach elementelement cijCij of the tabletable byby aa newnew valuevalue cii*cit accordingaccording toto thethe followingfollowing sequence:sequence:

let c * = 1/0 (0 is called the 'pivot.) kk kk kk

(ii) let each element of the k'th column,column, excludingexcluding thethe pivot,pivot, bebe replacedreplaced by:by:

C.-*" _c. .0 * ik kk

(iii) let all the elements c0. .. , ~but excludingexcluding thethe k'thk'th rowrow oror thethe k'thk'th column,column, bebe ij --- replaced by:by: l.J

c..*..cc *' == c + cC *.0*.o ij ij ik kj

(iv) let the elements of the k'th row, excludingexcluding thethe pivot,pivot, bebe replacedreplaced by:by:

* = c .0* kj kj kk

After the last reduction then the elements c to c mm compriseoomprise thethe 'inverse'inverse matrix'matrix' c11ll mm of the regression problem.problem. These elements,elemento, as we shall see, are important in calculating the variances and covariances of the coefficientsooeffioients and henoehence the confidenceoonfidenoe limits of the regression estimates.estimates. The elements in thethe m+l'thm+rth columncolumn areare thethe regressionregression coefficientscoefficients b to bm. For the numerical emamoleexample thethe threethree reductionsreduotione areare shownshown below:below: b1l to bm.

REDUCTION 1 5.05000000 0.05000000 -1.65000000 0.10000000 5.05000000

1.650000001.65000000 25.15000000 -16.30000000-16.30000000 -~9.75000000-49.75000000

-0.-0.10000000 10000000 -16.30000000 57.90000000 -10.70000000 - 145 -

REDUCTION 2

0.15825049 0.06560636 -0-0.96938369.9693836J 1.78608349

0.06560636 0.03976143 -0.64811133 -1.97813121

0.96938369 0.64811133 4747.33578528. 33578528 -42.94353876-42.94353876

REDUCTION 3

0.17810238 0.07887895 0.020478810.02047887 0.90664807

0.07887895 0.04863523 0.01369178 -2.56610485

0.02047887 0.01369178 0.02112566 -0.90721086

Although Jordan eliminationelimination soundssounds fairlyfairly complexcomplex whenwhen explainedexplained inin full,full, andand isis certainly very tedious to carry out manually, it can be programmedprogrlU!llled very easily for aa com­com- puter. Below isis a section of oode in BASIC which inverts and solves the regression problem in the matrix C.C. The variable lettersletters usedused oorrespondcorrespond exactlyex'$ctly toto thethe explanationexplanation above.above.

Code in BASIC for Jordan Eliminations

100 FOR IC=K= 1 TOMTO M 110 LETLET C(K,K)C(KK) == 1/C(K,K)l/C(K,K) 120 FORFOR I I== 11 to M 130 IF I ~= K THEN 190 140 LETLET C(I,K)C(I,K) == -C(I,K)*C(K,K) 150 FOR J = 1 TO MilM+l 160 IF J=KJ=K THEN 180180 170 LETLET C(I,J).c(I,J)+C(I,J)=C(I,J)+C(I,K)*C(K,J) C(I,K)*C(K,J) 180 NEXT J 190 NEXT I 200 FORFOR J=J = 11 toto MilM+l 210 IFIF J=ICJ= K THEN 230 220 LET C(K,J)C(C,J) == C(K,J)*C(K,K)C(K,J)*C(KK) 230 NEXT J 240 NEXTNEXT K

TheThe coefficients b to b have now been calculatedoaloulated and are stored inin 0 m+l to o1,1114,11 c 1.. Thus, forfor thethe nurlericaT example we have: ' m,m+ n~ericaf have;

b = 0.9066 b11 = b =-2.566=-2.566 b22 b =-0.90720.9072 b33 roundingrounding to 4 significantsignificant figures. The intercept 1)0b isis calculatedcalculated from:from: 4 O m 130 = (T - ib.T)/n bO = (TyY - Ibi .Ti)/n - 146 -

For the numericalnlDllerical exaxnpleexample wewe have:have: b = (142 -- (50(50 xx 0.90660.9066 + 52 x (-2.566)(-2.566) + 53 x (-0.9072))/10 b0O = 53 x (-0.9072)00 = 27.8227.82 (to 44 sig. figures)

The elements 0 1 to c of the final table after the lastlast Jordan reduction wewe willwill . c11 1 IJIIlIMm refer to as thethe inverse1nverse matrixmatr4X c.c.

Stage 5 Caloulate total, regression and residual SIDIlsum ofof squares andand hencehence coefficientcoefficient of determinationdetermination (R2)9(R2), variance ratioratio (F)(F) andand residualresidual standardstandard deviation.deviation.

The total sum of squares (TSS) is gy2,ri, as calculated at stagestage 3.3.

The regression sum of squares (RSS) is given by: mrn RSS = !b.lb .•.QY. QY. 1D. 3. 1

The residual sum of squares (DSS)(DSS) isis obtainedobtained asas thethe differencedifference betweenbetween thesethese two:two: DSS == TSS - RSS

The degrees of freedom for the total sum of squares isis n-1.n-1. For the regression itit is m. For the residuals itit isis n-m-1.n~1.

The mean square of the regression (RMS)(mIS) is:is:

RNSmIS = RSS/m

The mean square ofof thethe residualsresiduals (DMS)(DMS) is:is:

DMSDNB == DSS/(n-m-1)reS/(n-m-1)

The variance ratioratio is:is:

F = mIS/moERTC/DMS with n-m-1n-m,1 and m degrees of freedom.freedom.

This F value can be used to test the statistical significancesignifioance of thethe regression.regression.

FbrFor the numericalnumerioal exampleexample wewe havehave therefore:therefore:

TSS == 697.6 (see(see'i.iEy2 in stage 3)

RSS = 0.9066 xx 101 ++ (-2.566)(-2.566) x (-216.4)(-216.4) .4+ (-0.9072)(-0.9072) xx (-0.6) =. 647.4647.4 resDSS = 697.6 -647.4 DNS= 50.2/(10-3-/) = 50.250.2 E= 8.3678.367 RMS = 647.4/3 F = 215.8/8.367 = 215.8215.8 = 25.7925.79 with 6 and 33 d.f. - 147 -

The F values fromfrom tablestables with these d.f. and forfor P=0.1%P-oO.1% isis 23.70. Hence we may say that therethere isis lessless thanthan 11 chancechanoe inin 11 000000 ofof thethe relation between the dependent and predictorprediotor variables beingbeing duedue purelypurely toto chancechanoe factorsfactors outsideoutside thethe regressionregression model. model.

The coefficientcoefficient ofof determinationdetermination R2R2 isis givengiven by:by:

R2= RSS/TSS r2 The multiple correlationoorrelation coefficientcoefficient RR ==A/R .,fR2.

For the exampleexample wewe have:have:

2 RR2 = 647.4/697.6647.4/691.6 = 0.92800.9280

R = 0.96330.9633

The residual standard deviationdeviation ss isis thethe squaresquare rootroot ofof thethe residualresidual meanmean square:square:

s = .,fmlS which forfor thethe exampleexample is:is:

s = 2.8932.893

Stage 6 Calculate standard errors and t-statisticststatistics forfor thethe regressionregression coefficients.coeffioients.

The standard errors of the coefficients b1b to bmb arearo given by: 1 m

s(b.)s(b) = s • ..,fc .. 1 11

TheThetvaluesoftheb.are t-values of the \ are given given by:by:

t(b.)t(bi) = b./S(b.)bi/s(bi) 1 1 1

These t-valUBstvalues canoan bebe testedtested forfor significance withwith n-m-1nm-1 d.f.d.f. For the numerical example we have:

Standard Coefficient t SignificanceSignifioance Error

b 0.90660.9066 1.2211.221 0.742 b11 b -2.566 0.6380 -4.022 -M-31- b22 2.566 40022 ** b -0.90720.9072 0.4205 -2.1572.157 b33

** indicates signifioanoesignifioance at the 1% level level (t-3.71(t=3.71 with with 6 6 d.f.). d.f.). CoeffioientsCoefficients b1 and b3 are " regression. One or not significant.signifioant. This doesdoes !!2!not mean that ~both can be removedremoved from thethe regression. One or the other (preferably the least significant, herehere b1) could be removed and the results recal-recal­ culated by the procedure explainedexplained later.later. The t-valUBBtvalues ofof thethe remainingremaining coefficientscoefficients would bebe found to havehave inoreasedincreased and both would probablyorobably be significant inin thethe newnew regression.regression. - 148148 -

Stage 77 Standard errors of predictionspredictions fromfrom thethe modelmodel areare derivedderived fromfrom thethe formula:formula: mm Ss= = ss,/-(1/k fi.l/k ++ l/n1/n + Er E'i cc ..x.x.) x) y . . 1J 1 J 1i jJ

, where xixi= - (X;. - Xi), Xi is a given value of predictor variable i, X is the mean value for the regression model data and k is the number of repeated estimates of Y to be made. SyS. is the standard error of the mean estimate of Y. For a single prediction,prediction, kk == 1;1; for the population standard error,error, l/k1/k= c 0.O.

For the example, suppose we wish to calculatecaloulate the standard error for a single estimate of YY atat XlX1= C 8,8, X2 X2== 4 and X3X3 =C 6.6.

First calculate thethe xi:~:

x = 8 - 5.0 .= 3 1 xX22 = = 4- 4 - 5.2,=5.2 = -1.2

x .= 6 6 -- 5.35.3 =C 0.7 3

Then caloulatecalculate the sumstmt ~ yc.I: c ...x.x.x.x. 1Jij 1 J 3(3 x 0.17810.1781 + (-1.2) xx 0.078880.07888 ++ 0.70.7 xx 0.02048)0.02048) + (-1.2) x (( 3 x 0.078880.07888 ++ (4.2)(-1.2) xx 0.048640.04864 ++ 0.70.7 xx 0.01369)0.01369) + 0.7(3 xx 0.020480.02048 ++ (-1.2) xx 0.013690.01369 ++ 0.70.7 xx 0.02113)0.02113)

= 1.1781.178 ThenThenS S-= 2.893-/2.893-/(1 (1 ++ 1/101/10+ 1.178) y = 4.366

Thus, the standard errorerror forfor a singlesingle estimateestimate ofof YY fromfrom thisthis model at X1Xl C= 8, X = 44, X3 = 6 is +Z 4.366.4.366. X22 X3 =

This formulaformula willwill alsoalso givegive thethe standardstandard errorerror ofof thethe interceptinteroept 1)0,b ' if required, O using l/k1/k= = 0 and xiXi == Xi'

This completes thethe mainmain calculationscalculations forfor multiplemultiple regression.regression. Two other types of calculation are commonly performed. These are the computation of the correlation matrix between variables and the shortcut calculation for removal of variables from a regression.

The correlation matrix This is a table showing the correlationoorrelation between any pair of variables. It is constructed byby arrangingarranging aa matrixmatrix VV asas follows:follows: - 149149- -

1 2 3 4 •...... m+ • • • • • m +1 1

2 1 yY QY

2 3

• QY QXQlt • • • • m+ 1

and then noting thatthat thethe correlationoorrelation betweenbetween variablevariable ii andand variablevariable jj is:is:

r = Vi/ .V ) 11 jj

This can be done at any time after stage 3 in the regression calculation.calculation.

Since the matrix V is symmetrical about the diagonal VV1111 to VVmm,mm , it is not necessary to calculate the lower half, as, forfor example,example, r23 ~= r , and soso on.on. r32,32 Removing a variablevariable_from from the_regressionthe regression When a variable is found toto bebe non-signifi­nonsignifi- cant it may be desirable to recalculate the regression statisticsstatistios with that variable omitted.

'lbTo do this the values of the inverse matrix c and thvJth" coefficients are adjusted according to thethe followingfollowing formulafonnula wherewhere uu denotesdenotes thethe subscriptsubscript ofof thethe unwantedunwanted variable:

b*=b* = b ib -ci.bu/c .b,lo i i iuu uu

c,C. .** =... Cc .. - C .•oC 0.,10 u/C 1Jlj 1J 1U J UUuu For the numerioalnumerical example suppose we wish to omit the effect of variable X1.Xl. The original inverse matrix and coefficientscoefficients werewere (to(to 6 significantsignifioant figures):figures):

1 2 3 b

1 0.178102 0.0788790 0.0204789 0.906648 2 0.0788790 0.0486352 0.0136918 -2.56610 3 0.0204789 0.0136918 0.0211257 -0.907211

Eliminating variablevariable 11 wewe have:have:

bb* * == -2.56610 - 0.07887900.0788790 xx 0.9066480.906648 // 0.1781020.178102 2 2 .~ -2.96764 b3*b3*= = -0.907211 - 0.02047890.0204789 xx 0.9066480.906648 / 0.1781020.178102 = -1.01146-1.01146 -150- 150 _-

The revised inverseinverse matrixmatrix is:is:

2 3

2 0.01310010.0137007 0.00462197 3 0.00462197 0001877100.0181110

The various other statisticsstatistios from stage 5 on can then be recalculated for the newnew regression. A second, third, etc.etc. variable may be successivelysucoessively eliminated in this way.

This completes thethe summarysummary ofof multiplemultiple regressionregression calculations.calculations. ThisThie work is usually performed by computercomputer butbut cancan bebe donedone byby hand forfor smallsmall problems.problems. Great oarecare should be taken not to loselose significantsignifioant digits byby roundingrounding toto a fixedfixed numbernumber ofof decimal places during the matrix inversioninversion process.prooess.

2.62. 6 Residual Analysis

Residual analysis is a tool that should be associated with the intelligent use of multiple regression.regression. Its purpose isis threefold:threefold:

(i) ToTb determine whether thethe residuals fromfrom thethe regressionregression conformoonform to the assumptions of the mciel,model, i.e. are uncorrelated,uncorrelated, normally distributed and of uniformlmiform variance.variance.

(ii) PoTb assess 'lack'lack of fit'fit' inin the model fromfrom systematicsystematic trends in the residuals.

PoTb examine visually thethe shape of relationshipsrelationships betweenbetween residuals and possible predictorprediotor variables not yetyet introduced intointo thethe regressionregression model.

Residual analysis (like(like most other aspects of regressionregression calculations!)calculationsl) isis very tedious to perform by hand. It should however be part of every good computer programme for multiple regression analysis.

The residual e is defined as the difference betweenbetween thethe actual value of the dependent variable Y and the valueva~ue predictedpredicted fromfrom thethe regression modelmodel Y.Y. That is:is:

e = Y, Y. i Yi... ~

When thethe eiei areare plottedplotted againstagainst thethe YiYi thenthen severalseveral possiblepossible kindskinds ofof graphsgraphs maymay bebe obtainedobtained::

( i) e.

• • ...• ••.. .. •...... o ------• ••• . . • - 151151 -

~ ThisIhisistheidealsituationwiththevariationinthee.uniform is the ideal situation with the variation in the ei uniform with with respect respect to to Y.Y. The model appears toto have nono undesirableundesirable characteristics.oharacteristics. ( ii) ...... ' . .... ' .... . " .' .' .. .. : . " . . ' . . ' . . . " ---- ~~'.:.- ~---- ~ '~'.:.a. Z. - -- o · . '. . '. .'., .-- · '.' . :'.":':- . .. . . ·" . . . ' ......

This indicates systematic behaviour of the e. with respect to Y and shows lack of fit. In other words, there is a better model than the current one, using the same basiobasic prediotorpredictor variables but with different or additional transformations that could be fitted and would remove the systematicsystematic trendtrand inin thethe ei.ei'

( iii)

. : " - . . k. o0 - - - - -< .... .;.;•. .:. .• ~~:.;:~ ~... ."'"7 •.-:-.: __ .. , '0 • .• -.:. • ...

y.

Here thethe variances of thethe eiei areare notnot uniformuniform withwith respectrespect toto Y.Y. In this case, better estimates of the coeffioients can bebe obtained eithereither byby usingusing a transformation of the Y variable or through the useuse ofof weightedweighted regressionregression (see(see nextnext sectionsection 2.7).2.7).

When the ei are plottedplotted againstagainst XX variablesvariables notnot yetyet includedinoluded inin thethe modelmodel thenthen oneone may have eithereither a random scatterscatter oror somesome systematicsystematic patternpattern similarsimilar toto (ii)(U) above.above. In the latter case the X variable could be includedincluded inin the regression.regreesion. The shape of the general trend in the pattern may not bebe linear,linear, butbut followingfollowing oneone ofof thethe trendstrends shownshown inin figurefigure A.A.2.1. 2 .1. In this case the appropriate transformationtransformation ofof thethe XX variablevariable cancan bebe used.used.

For example, suppose we have thethe model:model:

Y b0 + b1X1 + b2X2 where thethe residuals plotted against an additional variable X3 give the followingfollowing pattern:pattern: X3 -152152 -

e. .." ...... ;::~ :~.~:.~.~ . "."'-:"'d: . . __ __ ._ ·_·~.l:.::'. ;; .' ______o 17=MS M., .. ;:,,,::..t .. ' .". " ~ .",

X3 The dotteddotted lineline (-( )- appears-) appears to tofollow follow a slighta slight curve curve which which could could bebe allmdedallowed forfor by adding X3 and X2 to thethe regressionregression (c.f.(c.f. figurefigure A.201(e))A.2.1(e)) toto givegive thethe model:model: ~ and X~3

Y= b + b X + bl + bl + b0O + b1X11 1 + b2X22 + b3X33 + b4~b4

After fitting this secondsecond model the trend of the residuals from the new model against X3 would appear similar toto thethe firstfirst residual patternpattern shownshown inin (i)(i) above.

2.7 Weighted Regression

Weighted regression isis used when the residuals do not have uniform variance with respect to the Y values. This situation commonlyoommonly arises when complex transformations ofof the Y variable are used toto obtainobtain particularparticular curvecurve shapes. In yield prediction it is also commonly found with volume data where the variance tends to be proportional to the volume.volume.

Associated with eacheach observation is a particular weight wi.Wi' The value of this can be determined empiricallyempirically oror itit may be derived fromfrom somesome theoreticaltheoretical reasoning. ForFbr the best fit the weights wiWi should bebe proportionalproportional toto 1/si1/s? where sis. isis the residual standard l. l. deviation atat Yi.Yi •

Empirical weights cancan be calculated as follows:follows:

(1) Fit the unweighted regression model usingusing the proceduresprooedures described in the preceding section.seotion.

(2) Calculate the squaredsquared residuals:residuals:

2 2 e(Ye2 = (y(Y - )\) i i (3) Then either:either: 2 Tabulate mean valuesvalues ofof w."l.' (=( = 1/e.) for classes of Y.;Y ; (i)2,l. i or 2 (ii)(n) Fitaregressionrelatinge.toFit a regression relating e to Y (possibly a simple regression: i Yii 2 ee2 = a + b.Y will bebe adequateadequate forfor mostmost cases).cases). Use this regression to predictpredict aa , 2 weightw.(.1/gi)weight w.l. (=1/11l.?) forfor each each Y..y l..• - 153 -

In the case of theoreticaltheoretical weightsweights thenthen oneone usesuses whateverwhatever formulation is involved totocalculatethew.for calculate the wi for eacheach YY.• i For weighted linear regression with one predictor variable, the caloulationscalculations areare asas follows:

(1) Compute weighted totals,totals, sisessums ofof squaressquares andand products:

2 A= lw.Iw. B = Zw,oX.~w . • x. C= 1.w .• x. ~J. ~1 ~1 ~ ~ 2 Dn= = I,.,.Ew..y. .•y. E == ~wZw..y. .• y. F=iwF= iw..x..y. .• x .• y. 1~ 1~ 1~ ~1 ~1 1~ 1~

(2) Calculate weighted meansmeans andand correctedcorrected productsproducts andand squnres:squares:

x = B/A yy= = D/An/A 2 SS = C - B2/AB /A SP =. F - B.n/ABONA SSxx = xy

(3) All other calculationscalculations thenthen proceedprooeed asas fromfrom lineline 1111 ofof formform A.2A.2 (Statforn(Statform 1111 of Dawkins, 1968).1968).

For multiple weighted regression the gamesame general principles apply. Most advanced regression programmes have a facility forfor weighting.

Weighted regression should not be used unless there is utrongstrong !a priori theoretical justification for its_use,its use; or alternatively, a clear indioatedindicated trend in the squared residualsresiduals when plottedplotted againstagainst YY thatthat maymay provideprovide thethe basisbasis forfor empiricalempirical weighting.

FOrFor well defined functions with high R2R2, weighting isis unlikely to result in much oracticalpractical improvement in the degree of fit. It is a techniqueteohnique that may be regardedregarded asas a refinement to analysis to be used onceonoe preliminary analyses of a set of data have been completedcompleted..

2.8 Comparison of Regressionession Lines

It is frequently necessary to try and decide wiether or not two or more regression lines areare so similarsimilar thatthat they can effectively be replaced by a single line. This case may arise, for example, when it is desired to poolpool data fromfrom different speciesspeCies with similar growth habits or from different forests,forests, regions,regions , sitesite types or provenances.provenances.

The regressions to be compared may be simple regressionsregressions with a single predictor variable or multivariate regressions. Two basicbasio techniquesteohniques existexist forfor thethe comparison:comparison:

(i) Covariap2Itánálm12.Covariance analysis. This is perhaps more suited to manual computation and simple regression, although itit cancan equallyequally wellwell bebe extendedextended toto thethe multivariate case.case.

(ii) SignificanceSi- ificance teststests onon conditionalconditional variables.variables. This is quite suitable for complex models and interactions, but leadsleads to Largelarge multivariate problems. It is a useful method ofof alaptingadapting standardstandard computercomputer multiplemultiple regressionregression packages to regression comparison problems.problems. -154-- 154 -

Covarianoe analysisanal:ysis isis a largelarge subject.subjeot. Here,Hereo only the techniques neoessary for 00oom-.... paring the slopes and interceptsintercepts betweenbetween simplesimple regressionsregressions areare considered.considered. The aalculationscalculations are explained in relation to an example representing hypotheticalhypothetical datadata for thethe volum...-Jleightvolumeheight line from two geographically distinctdistinct plantationsplantations (called(called II andand II)1II). usingusing 1010 plotsplots onon each.each. The data is illustrated inin figurefigure A.2.2A.2.2 andand setset outout inin tabletable A.2.1.A.2.1. The problem is to determine whether itit isis reasonablereasonable toto combinecombine thethe datadata toto fitfit aa commonoOlllll1On regressionregression lineline basedbased on all 2020 plots.plots. The model to be fittedfitted isis thethe logarithmiclogarithmio volumevolume lineline withwith thethe form:form:

log VV == a + b. log H )

This is the linear form of the model with the parameters a and b being estimated byby Dimplesimple regression. By taking antilogs to both sides thethe equationequation becomes:becomes: b V == A.H (2)-(2) where AA == 1010aa • InIn this form points from the fitted line cancan bebe calculatedcalculated forfor plottingplotting onon normal graph paper.paper.

Figure A.2.2 Hypothetical volume-heightvolumeheight datadata usedused forfor comparisoncomparison of regression lines example. section 2.9. (an

Volume (m(m3/ha)3/ha) 250 ( ;n + Plantation I •o Plantation II 200 +

150

.. r / / / ./ 100 / . /. ,./ / /' , • ;I­ , .y 50 , /' , , ,

o 5 10 15 20 25 30 Standstand height (m.)(m.)

---- individual regressionregression lines;lines; ------combined regression --155-155 -

Table A.2.1. Original and transformed data fromfrom twotwo plantations used in the regression oomparisoncomparison example.example.

10=. Volume Height log. VV log. H Plot No.No. V~lume Height m3/ham/ha. m. y x

Plantation I

1 30.0 12.9 1.4771 1.1106 2 23.3 11.6 1.3674 1.0645 3 212.6 24.9 2.3276 1.3962 4 15.1 10.8 1.1790 1.0334 5 50.4 15.8 1.7024 1.1987 6 189.4 25.8 2.2774 1.4116 7 68.7 18.2 1.8370 1.2601 8 221.2 25.4 2.3448 1.4048 9 123.9 22.1 2.0931 1.3444 10 60.8 16.7 1.7839 1.2227

Plantation II

11 70.6 15.4 1.8488 1.18751.1875 12 131.3 18.5 2.1183 1.2672 13 201.7 24.6 2.3047 1.3909 14 156.6 22.922.9 2.1948 1.3598 15 136.2 20.8 2.1342 1.3181 16 87.0 16.7 1.9395 1.2227 17 95.0 16.5 1.9777 1.2175 18 108.4108.4 17.7 2.0350 1.2480 19 93.6 18.0 1.9713 1.2553 20 57.7 15.3 1.7612 1.1847 - 156 --

The sums, sunssums of squAressquares andand productsproducts ofof thethe datadata (table(table A.2.1)A.2.1) are calculated both for the separate sets of data and forfor thethe twotwo setssets combinedcombined asas follows:follows:

StatisticStatistio Source of Data

I II Combined (C)(C) IY 18.38954 20.28547 38.67501 2 SY!Y 35.37950 41.39075 76.77025 IxIX 12.44696 12.65166 25.09862 Ix2sx2 15.67703 16.05102 31.72805 :EXYYxY 23.43337 25.76286 49.18623 n 10 10 20

2 2 SSy=SSy ~ yy2-(yy)2/nIY -(iY) /n 1.56198 0.24072 1.98243 2 / .2/2/ SSx == I:XEx -(£x)--Ex) /nn 0.18435 0.04457 0.23101 SPxy= sxyl-rx.yy/nIxy-rx.:Ey/n 0.53398 0.09837 0.65176

coefficients: b == SPxy/SSxSPxy/SSx 2.89657 2.20716 2.82135 a == (!y-b.:Ex)/n(y-b.lx)/n -1.76640 -0.76388 -1.60685-1.60685 regression sums of squares:squares: RSS == b.SPxy 1.54671 0.21712 1.83884

From these basicbasio calculationsoalculations thethe analysisanalysis ofof variancevariance cancan bebe performed.performed. The following additional quantities areare calcInated:calculated:

(i) Sums of squares betweenbetween bb coefficients:coefficients: SSb = !SESS RSS -_ (ISPxy)2/ISSx(!Spxy)2 /ISSx For the example thisthis is:is:

SSb = (1.54671 ++ 0.21712) -- (0.53398(0.53398 + 0.09837)2/0.09837)2/ (0.18435 + 0.04457)0.04457) = 0.017080.01708

(ii)ii) The sums of squares betweenbetween aa coefficient:coefficient:

SSa = ~SSY1SSy - !RSSzESS = (1.56198(1.56198 ++ 0.24072)0.24072) - (1.54671(1.54671 ++ 0.21712)

.,_= 0.03887

(iii) The residual sumsum ofof squares:squares:

DSS = SSycSSYc - (SSa(SSa + SSbSSb ++ RSSc)RSS ) res C where the G0 subscript -denotesdenotes quantitiesquantities fromfroMthe- the -combinedcombined -regressiohregression statistics on thethe tabletable above.above. - 157 -

ForFbr the example, the deviation sumsum of squaressquares is:is:

DSS = 1.98243 - (0.03887 + 0.017080.01708 ++ 1.83884)1.83884) = 0.087640.08764

(iv)(iv) The degrees of freedom forfor thethe above quantities are asas followsfollows wherewhere nTn andand nIl are the numbernumber of points in thethe separateseparate regressions,regressions, rr isis thethe numternumter ofof nII regressions being compared and nnC is the total number ofof points:points:

statistic i!.!:.d.f.

SSyc nCn - 1

resDSS 2r ninr+nn-nII SSb )) rr- 1 SSa ))

RSS 1 C

The analysis of variancevaria.nc e tabletable forfor thethe comparisoncomparison of0 f regressionsregressions maymay bebe setset outout asas follows: Sum of Mean Variance Source of Variation Squares .!hL.d.f. Square Ratio

Combined regression

RSS 1.83884 1 1.83884 335.7xxx c 335.7*** Between slopes SSb 0.0'7080.01708 1 0.01700.017088 3.1 Between intercepts SSa 0.03887 1 0.03887 7.1*7.1* Residuals DSS 0.08764 16 0.005478 Total SSyc 1.98243 19

The asterisksasteriskB denote the significancesignifioance levellevel of the different variance ratios, determined by lookinglooking upup F values inin statisticalstatistical tablestables with 11 and 16 d.f. at thethe 5% (*),(*), 1% (**) and 0.1% (***)(***) probability levels.levels. In the above the regression is very highly significant (i.e.(i.e. not due to chance).ohanoe). The differenoedifference between slopes in the two separate regressions is not significant,signifioant, i.e.i.e. isis mostmost likelylikely due toto chanceohance samplingsampling effectseffects inin the two sets of data. On the other hand, the interceptsintercepts do differ significantly at the 5% level, suggesting that the regressions are inin factfact distinot although they have the samesame slope. Consequently, we may concludeoonclude forfor thethe exampleexample thatthat thethe data fromfrom thethe twotwo planta-plant.... tions cannotcannot bebe pooledpooled toto givegive aa combinedcombined regressionregression forfor predictionprediotion purposespurpo~es withoutwithout aa consequent lossloss ofof accuracy.acouraoy.

This oomparisoncomparison method canoan be extended toto any number of simple regressions byby noting that in the above calculations,caloulations, Whereverwherever separateseparate quantitiesquantities forfor thethe twotwo regressionsregressions areare added together,thentogether, then any number ofof quantitiesquantities maymay bebe added.added. Dawkins (see(see bibliography) presents a calculation 2E2~ ~forma for comparingcOmparing up toto 4 regressions. - 158-158

The use of conditional variables forfor comparisoncomparison ofof regressionregression lines.lines. ConditionalConditional variables are those which cancan only take values of zero or one, depending on whether aa parti­parti- oularcular observation is to enterenter intointo somesome partpert ofof aa regressionregression modelmodel oror not.not. They canoan be used for the comparisonoomparison of regressionregression lineslines andand forfor fittingfitting setssets ofof nestednested datadata (see(see sectionseotion 2.10).2.10).

Consider the case desoribeddescribed in thethe eXampleexample above,above, comparingoomparing volumevolume heightheight lineslines forfor two simple regressions fromfrom plantationsplantations II andand II.II. ForFbr eacheaoh plantation we havehave aa model:model:

log VV == a + b.log H

We can form a single model for both sets of data if we introduce an extra variablevariable ZZ which isis zerozero forfor data fromfrom forestforest I and one for data from forest II. This oombinedcombined model is: log V = a (1 + a .Z) + bb1(1(1 ++ bb2.Z)log.Z)log H a11 a20Z)2 1 2

For observations with Z = 00 (forest(forest I)I) thenthen thisthis amountsamoWltS to:to:

log VV= = a + blb loglog H al1 1

But whenwhen ZZ= = 11 (forest II) the coefficients are:

log VV= = (al(a ++ a2)a ) ++ (b1(b ++ b2)logb )log H 1 2 1 2 -(5) To fit equation (3) by line4;rlinear regressionregression thethe brackets are removedremoved giving:giving:

log V = + a1a2Z + b1 log H + b1b2Z.log H -(6) al which is equivalentequivalent toto thethe regressionregression equation:equation:

Y = c0+ c1X1 + c2X2 + c3X3

Using a multiple regression programmeprogramme toto fitfit thethe coefficients,coefficients, thethe valuesvalues obtainedobtained were: 1 e0 1.76640-1.76640 -12.312.3 1.00179 3.1 el 2.89657 25.2 02 c3 0.68884-0.68884 2.7-2.7

The t values for the ooefficientscoeffioients with 16 d.f. indioate a similar but not identical result to the covarianceoovarianoe analysis.analysis. The ol01 coefficient, corresponding to the term Z in equation (6), indicatesindioates a highly significant additional term forfor the interceptintercept for the second set of data. coefficient, corresponding to Z.log HI is also significant at second set of data. The 03 coeffioient, corresponding to Z.log H, is also significant at the 5% level, indicating that the slope of the two sets of data differs. In the covariancecovarianoe analysis it will be remembered, the slopesslopes did .!!.2!not differ significantlysignificantly although the inter-inter­ cepts did. This difference in the results of the two methods isis due to the differenoedifference in the statistical modelsmodels and,and. thethe hypotheses being tested.tested. The covariance analysis asks the question: "Do either or both of the regressions differ significantlysignifioantly fromfrom a pooledpooled regression?"regression'?" Cum,

- 159 -

whereas, thisthis techniqueteohnique asks:askB:

"Does the seoondsecond regression interoeptintercept (01) or slope (03)(c3) differ fromfrom that ofof thethe first regressionregression?' 1'1

It will be appreoiatedappreciated that the secondsecond typetype ofof testtest will bebe moremore sensitivesensitive toto differences than the first, but on the otherotherhand, hand, it is not so directlydirectly relevant if one wishes toto knowknow whether oneone cancan safelysafely poolpool regressions.

The coefficient values forfor thethe twotwo separateseparate lineslines cancan bebe workedworked outout fromfrom thethe aboveabove coefficients and will be found to be identical to the coefficients for the separate lines fittEdfitted in the oovariancecovariance analysis. They are:

Forest I:I:

Intercept Co0 -1.766401.76640 Slope c 2.89657 022 Forest II Intercept 0.76461-0.76461 00+01°0+c1 Slope c +o 2.20773 02032 3 In fact, the figures appear to differ in the third decimal place from those calculatedcalculated earlier, but this is an effect of roundingrounding error, as in the computer programme all calcula.­calcula- tions,tions9 including the log,log. transformations were carried out to 15 significant figures, whereaswhereas with the manual calculationcalculation usedused inin thethe covariancecovariance analysis,analysis, onlyonly 55 to 8 significant figures were used.used.

The use of conditional variables can becomebeoome quite oompleioomplex and is also quite flexible. For example, given any dependent variable Y and anyany predictorpredictor variable X with a conditional variable Z9Z, we can have a model with a commoncommon slopeslope andand onlyonly aa different interceptinteroept forfor the two sets of0 f data:

Y + c1Z + o X c0 2

With1'lith threethree setssets ofof data, twotwo conditionalconditional variablesvariables areare needed.needed. They are:are:

Z1 0 for data fromfrom setset 191, 11 for sets 2 and 3. Z1 0 " " 1 and 292, 1 forfor setset 3.3. Z2 " " "

And the model to testtest forfor different slopesslopes andand interceptsinteroepts is:is:

Y= co + ciZi + c2Z2+ 03X + c4Z1X +o5Z2X

FUrtherFurther informationinformation onon thethe waysways conditionalconditional variablesvariables maymay bebe usedused isis givengiven inin thethe next section. _160 __

2.9 Nest231112grassign_UsinEC9nditionälNested Regression Using Conditional Variables

Conditional variables whose value maynay be either zero or one have already beenbeen aalludedlluded to in the previous section. Apart from their use for comparing regressions they oancan bebe usedused to fit regression models toto nestednested data.data.

Nested data arises Whenwhen the measurements are grouped into sampling units or plotsplots where the within-plotwithinplot regressionregression maynay bebe differentdifferent from the between-plotbetweenplot regression.regressi on. The commonest situation in forest yield studies is with permanent sample plot measurementsmeasurements where oneone has data groupedgrouped byby plots.plots. A typical problem will illustrateillustrate thethe point:point:

Example Figure A.2.3 shows hypotheticalhypothetical height-ageheidhtage data from 5 permanent sample plotsplots each of which has been remeasured 3 or 4 times. The objective is to estimate thethe meanmean slopeslope ofof thethe height-ageheightage relation-relation­ ship, usingusing aa qiialiraticquadratic modelmodel ofof the form:

2 H = b + b A + bl H=b0O +b1A+1 b2A where b and b are assumed to bebe the saneRame forfor allall plotsplots butbut b0b b11 b22 O0 maynay differ (expressing(expressing aa sitesite effect).effect).

If a regression analysis isis mademade ofof thethe datadata inin figurefigure A.2.3A.2.3 usingusing equationequation (1)(1) asas the model and treating each observation as separate and independent, then the coefficient values obtainedobtained are:are:

b 7.807 b0O b 0.6939 b11 b 0.008652-0.008652 b22

, 2 . with a coefficientcoeffioient of determinationdetermina~on (R2(R ) ofof 0.53 and an F value of 4.94. The significance level of the regression isis 0.2%.0.2%.

It can be seen from figure A.2.3 that this regression,drawn as a broken line, obviously underestimates the mean slope of the plots.

To fit the nested model then 4 artificial variables are introducedintroduced Whichwhich can be called p 2 to P 5' These have thethe followingfollowing values:values: P2 P5.

P 1 for data fromfrom plotplot 2,2, zero for all other data P22

11 il II 30 11 11 11 P 1 " " " " 3, " " " " " P33 II It P - 1 " " " " 4, " " " " " P414

11 P - 1 " " " " 5, " " " " " 5 - 161161 -

Figure A.2.3 Hypothetical height-age datadata toto exemplifyexemplify effect of fitting nested regressionregression

Stand Height (m.)(m.)

25 11

20

5 15

10

11111--40• • ObservationsObservations onon 1 PSP Ungrouped datadata regressionregression 5 Nested regression forfor thethe mean sitesite

Age oO 5 1010 15 20 25

The model fittedfitted is:is: 2 H=a +aP +aP+aP +aP +aP +bA+bA+bA+bA2 -(2) 11 22 3333 44 5555 11 2 wheretheaiand b .are the regression coefficients. This is the quadratic model as in where the a~ and bi are the regression coefficients. This is the quadratiC model as in equation (1(1), ), but withwith a differentdifferent interoeptintercept for every plot. The intercept valuesvalues equivalent toto bb0 in equation (1)(1) are: O E!2.!Plot InterceptIntercept

11 a a11 2 a1al + aa2-2 3 a1al + a3

4 ala1 + a4 5 ala1 + a5 - 162 -

When fitted toto thethe data fromfrom figurefigure A.2.3A.2.3 thethe coefficientcoefficient valuesvalues obtainedobtained were:were:

a 7.432 al1 a2 -70:11339'2-0.391515 a3 -1.755 a4 -3.427 a5 -6.833 1:11b1 0.9192 b2b2 -0.01157

with a coefficient of determination of 0.98 and an F valuevalue ofof 70.88 with 6 and 9 degrees of freedom. This is significant at lessless thanthan 0.0001%.0.0001%. The slope of thisthis line,line, given byby 11b1 andand b2'b2, is drawn on figure A.2.3 for the mean intercept valuesvalues ofof the plateplots asas aa heavyheavy solidsolid line. It can be seen that this gives a much better repre6entationrepresentation ofof thethe meanmean slopeslope ofof thethe plots than the regression fittedfitted toto ungroupedungrouped data.data.

This technique can be extended to give different intercepts and differentdifferent slopes for each plot or a common interceptintercept and differentdifferent slopes.slopes. The latterlatter casecase isis mostmost likelylikely toto bebe useful with models that result inin asymptoticasymptotic graphsgraphs suchsuch asas thethe SchumacherSchumacher equationequation (see(see main text).text). For a single X and Y variable andand withwith 33 plots,plots, forfor example,example, wewe have:have:

Common slope,slope. different intercepts:intercepts:

Y = al + a2P2 + a3P3+biX

Different slopeslope andand intercepts:intercepts: Y =a +aP +aP +bX+bPX+bPX 1 22 33 1 22 33 Different sloslope, e common interceintercept: t:

Y =a1 +bX+b2PX+b3P3X1 2 In the above the ai and bi are intercept and slope coefficients to be fifivted,ted by regression; the X and Y are normal variablesvariables andand thethe PiP j areare variablesvariables Whichwhich areare 11 for plot jj and zerozero forfor otherother plots.plots.

These ideas can be extended to multivariate models, although the number of coeffi­coeffi- cients becomes fairlyfairly large.large. Nested regressions ofof thisthis typetype cancan bebe calculatedcaloulated byby hand,hand, as although thethe number of variables maymay bebe large,large, thethe 0-1 naturenature of mostmost of the variables means that many short-cuts are possible in the calculations. If this is done,done, then the Jordan elimination techniauetechnique forfor invertinginverting thethe correctedcorrected cross-productcross-product matrix, given inin seotionsection 2.5, is not the fastest or easiest method; Seber (1977(1977 - see bibliography) givesgives details of more efficienteffioient methods forfor thosethose acquaintedacquainted withwith matrixmatrix algebra. However, forfor computer programmes the relative inefficiencyineffioienoy of conventionaloonventional caloulation techniques is unimportant since the processing times involvedinvolved are in any case very short. - 163 -

3. SOLUTION OFOF EQUATIONSEQUATIONS

3.1 Solution of thethe QuadraticQuadratic Equation

The quadratioquadratic equation is widely used as a regression modelmodel forfor datadata exhibitingexhibiting aa slight curvature inin thethe X-YX-Y trend.trend. Having fitted an equation to predict Y,Y, it maymay some­some- times be necessary toto solvesolve forfor XX givengiven aa knownknown valuevalue forfor Y.Y. To do this,onethispone uses the standard formulaformula forfor thethe rootsroots ofof aa quadraticquadratic equation.equation. If the regression modelmodel is:is:

Y = b0 + blX + b2X2

then rewrite itit as:as: 2 aX2aX + bXbX + cc = 0

where aa= = b2,b , bb= = bb and c = b0-Y,bO-Y ' andand obtainobtain XX from:from: 2 1

2 XX= = -b-b ± Vb2V b - 4ac

There are normally 2 solutions, dependindependingg on \metherwhether a ++ or a -- sign is takentaken beforebefore the square rootroot sign.sign. It will usually be obvious that only oneone ofof thethe twotwo solutionssolutions calcu­calcu, lated in a particular casecase isis possible.possible. For example, one might derive diameters ofof+20 +20 and -15 as solutionssolutions toto aa problem;problem; only the positivepositive diameterdiameter hashas anyany meaning.meaning. 2 When b2b is less than 4ac,4ac, thethe quantityquantity under the square rootroot signsign isis negativenegative andand there is no real solutionsolution to the equation.equation. For most foreetryforestry applicationsapplications this is likely to imply an error inin the coefficientcoefficient values oror aa Y value that isis too large or too small. Suppose, for example, thatthat one has a height-age functionfunction that reaches a maximum at 42 m height. If one tries to solve it to see at what age the. stand will reach 50 m, then this condition will arise; there isie no solution.

3.2 Graphical Solution ofof EquationsEquations

There are quite aa number ofof complexcomplex modelsmodels thatthat cannotcannot readilyreadily bebe manipulatedmanipulated byby algebra and which mustmust bebe solvedsolved eithereither byby aa graphicalgraphioal oror aa numericalnumerical method.method. The graphical method of solution isis self-evidentself-evident andand cancan bestbest bebe illustratedillustrated byby anan example.example. It consists simply of plotting thethe valuevalue ofof YY forfor anan equationequation ofof thethe type:type:

Y = f(X)f(X)

for selected values ofof X;X; J01n1ngjoining the points by a smooth ourvecurve and then estimating the value of XX forfor thethe desireddesired YY atat whichwhich aa solutionsolution isis required.req~red.

Suppose,Suppes e, forfor example,exampla, thatthat oneone hashas fittedfitted thethe equation:equation:

log MAlMAI == -0.8892-0.8892 - 0.03055 A + 2.0972.097 loglog A relating mean annual volumevolume incrementincrement toto ageage forfor aa plantationplantation andand itit isis necessarynecessary toto solvesolve it to determine the age whenwhen MAlMAI firstfirst reachesreaches 1515 mmYha/yr.3/ha/yr. For a graphical solution use - 164 - the equationequation toto calcula-tecaloulate MAlMAI forfor aa seriesseries ofof agesages whichwhich itit isis hopedhoped willwill bracketbracket thethe desired answer.answer. Below are tabulated a seriesseries ofof MA'sMAls calculatedcalculated forfor agesages 1010 toto 30:

Lge~ ~MAI 10 7.99 15 13.15 20 16.91 25 18.99 30 19.58

These are then plotted, as shown by the +s in the graph below, andand connectedconnected byby aa smooth curve.curve. It is then possible to read off the x valusvalue correspondingoorresponding toto thethe knownknown yy valuevalus which in this case is, for an MAIMAl of 25 m3/ha/yr,m3/ha/yr, an age of 17.2 years.

MAlMI (m3/ha/yr)(m3/ha/yr) 20

15

10

5 Age (yrs)(yrs) 10 15 20 25 30

The main disadvantages ofof graphicalgraphical solutionsolution methodsmetl-.ods are:are:

(i) The acouraoyaccuracy is restricted to perhapsperhaps:!; ± t% of the range of thethe x scale.scale.

(ii) The method is slow and essentially manual. It cannot be used within a oomputercomputer programme or be programmed into a calculator.oaloulator. 165

3.3 Numerical Solution of EquationsEquations

There are a number of numerical solution methods for equations, butbut oneone ofof thethe simplest andand most reliable isis thethe so-calledsocalled 'Bisection'Biseotion Method'.Method'. This proceeds through the following stages:stages:

(i) Define the functionfunotion to be solved as:

Yy ~= f(x)f(x)

where y isis a known value, but x is the unknowntmknown to be determined by the method. There isis a range of values within which x must be known to lie, from Xaxa to xb'xb, suohsuch that:

(1) f(xa)f(xa) is1s lessless thanthan f(xb);f(~);

(2) f(xa)f(Xa) -y Y has thethe oppositeopposite sisign gn to f(xb)f(Xb) - y.

(ii) Calculate aa trialtrial valuexlvalue Xi- from:

x.=¡--(xX. = t(x + xx ) ) 1a. a b

(iii) Calculate the value yiYi forfor xi:Xi:

y. = f(x.)

(ir)(iv) If: Yiy. is greater than y, letlet xb~ =~ x.;xi; Yiyi is less than y, letlet xa = xi;xi; Yiyi is exactly equal toto y,y, thenthen thethe solutionsolution isi6 xi;XiI this case is unlikely to occuroccur inin practice.practice.

(v) Go back and repeat steps (ii) to (iv) n times, where n is determined by the accuracy to which one wishes to approximate the correctoorrect solutsolution..ion. Given initial values of xaXa and xb then the accuracyaocuracy ofof the solutionsolution will be:be:

accuracy = 2n(xax,b)

After thethe n iterationsiterations have been completed,oompleted, thenthen xiXi isis thethe desired solution to the accuracyaocuraoy stipulated above. In practice, since the iterations are carried out at high speed by computer or programmable calculator,caloulator, itit isis usuallyususlly sufficientliufficient toto useuse a fixed value of n, say 15 or 20 for a range of different problems.problems. We have forfor example:example:

No. of Iterations Accuracy as % of Range

5 3% 8 0.3% 10 0.1% 15 0.003% 20 0.0001% 30 10-10-7%7% - 166-166 -

Fbr most forestryforestry problems 15 iterations are sufficientsuffioient to give aa realisticrealistic degreedegree of aocuracy.accuracy.

As an exampleexample of thethe bisectionbisection method,method, supposesuppose thatthat wewe have,have, asas inin thethe previousprevious section, anan equationequation relatingrelating ageage (A)(A) andand meanmean annualannual incrementinorement (MAI)(MAl) ofof thethe form:form:

log MAIMAl == -0.8892 - 0.030550.03055 AA ++ 2.0972.097 log A which we wish to solve to determine the age at whichwhidh MA!MAI reaches 15 m3/ha/yr.m3/ha/yr. Taking xa as 1 year, xb as 30 years and y as 15 years, then applying stages (ii)(ii) to (iv)(iv) enumeratedenumerated above,above, the solution goesgoes as follows:follows:

i x x.X. Yi xaa ~ 1 Yi

1 1.000 30.000 15.500 13.595 2 15.500 30.000 22.750 18.254 3 15.500 22.750 19.125 16.370 4 15.500 19.125 17.312 15.091 5 15.500 17.312 16.406 14.370 6 16.406 17.312 16.859 14.737 7 16.859 1717.312 .312 17.085 14.916 8 17.085 17.312 17.199 15.004 9 17.08517 .085 17.199 1717.142 .142 14.960 10 1717.142 .142 17.199 1717.170 .170 14.982 11 17.170 1717.199 .199 17.185 14.993 12 1717.185 .185 17.199 17.192 14.999 13 17.192 17.199 17.195 15.001 14 17.192 1717.195 .195 1717.193 .193 15.000 15 1717.192 .192 1717.193 .193 17.193 14.999

The main disadvantages of the bisectionbiseotion method is that compe.redcompared to some numerioalnumerical solution methods (notably Newton's method), the oonvergenoeconvergence withwith each iteration is quitequite slow; the method also fails when there are two (or(or any other even number) solutions withinwithin the initial range given, andand ofof course,course, oneone .tAst bebe ableable toto stipulatestipulate aa rangerange for the solu­solu tion. ForFbr most forestryforestry workwork thesethese disadvantagesdisadvantages seldomseldom becomebecome important.important. If they do appear to be critical,oritioal, then the worker should refer to some textbook on numerical methods, sunhsuch as, for example, Stark (1970) for more detailed guidance.guidanoe.

4. FITTINGFI'l'l'ING NONLINEAR MODELSMODElS

A nonlinear model is any model whichwhioh cannot be transformed into a form whoseWhose parameters can be estimated direotlydirectly by linear leastleast squares analysis, as desoribeddescribed in sectionseotion 2.

FOrFbr example, thethe equation:equation: b Yy ~= a.X can be transformed by taking logarithms of both sides to give the linear model:

log YY= c loglog A +4. b log X - 167 -

so that loglog A andand bb cancan bebe directlydirectly estimatedestinRted byby linearlinear regression.regression. On the otherother hand:hand: c Y=Y = aa ++ b.Xcb.X

where a, bband and c are coefficients to be determined,determined, car~otcannot be directlydirectly transformedtransformed intointo such a linear form and hence itit isis saidsaid toto bebe aa nonlinearnonlinear modelmodel oror equation.equation.

In this manual it is only possible to dealdeal withwith fitting methodsmethods forfor aa limitedlimited rangerange of nonlinear functions in common use in growth and yield studies characterised by thethe following features:features:

There is onlonlyy one predictor (X)(X) and one dependent (Y)(Y) variable;

(ii) There are three parametersparameters toto bebe estimatedestimated (which(which wewe willwill callcall a,a, bb andand k)k) by the fittingfitting process;process;

(iii) Provided that parameter k is assumed to be a known value,value, aa andand bb cancan bebe estirnated,usingestimatedlusing appropriate transformations,bytransformetions,by simplesimple linearlinear regression.regression.

The types of equations which fall within these restrictions include thethe following widely usedused forms:forms: k Y=Y = aa + bb.Xk.X

k Y=Y = a.a.exp(b.Xk) exp(b.X )

Y = a(1a( 1 _ exp(-k.X))exp(k.X))b b

There are a number of other models besides these thattr~t also can be fitted, but these are less commonly usedused inin forestryforestry workwork andand hencehence areare notnot listedlisted here.here.

The main problem isis thethe estimationestimation ofof thethe parameterparameter k.k. Fbr a knownknown value ofof kk wewe can derive linearlinear formsforma forfor thethe threethree equationsequations asas follows:follows:

YY= = aa + b.(Xk)b.(J2<) where Xk is the predictor variable.variable. k log YY= = logelog a + b.(Xk)b.(X ) logee e where again the predictor variable isis XkXk andand thethe dependent variable isis log Y. logeY.e

log Y = loglog a + b.loge(1b.10g (1 - exp(-k.X))exp(k.X)) logee e e where the predictor variable isis loge(1log (1 - exp(k.X))exp(-k.X)) andand thethe dependentdependent e variable is logeY.log Y. e These three linearizedlinearized formsforms cancan allall bebe fittedfitted usingusing simplesimple linearlinear regressionregression exceptexcept for the k parameter. The value of k which gives the best fitfit cancan be estimatedestimated graphically.

The essential principal of thethe graphicalgraphical approachapproach isis toto calculatecalculate thethe residualresidual sum of sqvAressquares for the linearlinear forms,forms, usingusing thethe conventionalconventional methodsmethods describeddescribed in section 2.1,2.1, for a series of trial values of k.k. These residual sums of squares are plotted graphically - 16168- 8 -

against the trial valuesvalues ofof kk andand aa smoothsmooth curveourve isis drawndrawn throughthrough thethe points.points. The value of k at whichwhioh a minimumminimtun sumstun ofof squaressquares isis observedobserved toto occurooour providesprovides thethe bestbest estimateestimate ofof k;k; the regression is recalculatedreca10ulated using this value of k to give the oorrespondingcorresponding bestbest estimates of the a and b parameters.parameters.

FbrFor example9example, suppose we have the data relating plot ageage and heightheight shownshown onon figurefigure A.4.1 toto whichwhioh wewe wishwish toto fitfit thethe modelmodel (3):(3):

b Y = a(1a(l - exp(-k.X»exp(-k.X))b where plotplot heightheight isis YY andand ageage isis X.X. We use the linearlinear form:form:

log Y = loglog a + b.log(1b.10g(1 exp(-k.X))exp(-k. X» and estimate log a as the interceptinteroept and b as the slope in a simple linear regressionregression ofof log Y on log(1-exp(-k.X)).10g(1-exp(-k.X» . In this regression we are mainly conoernedconcerned at this stagestage withwith the residuals or deviations, sumsum ofof squares,squares, symboli3edsymboli3ed asas DSS.DSS. This canoan be obtained as shown on FormFbrm A.2 inin sectionseotion 2.1.2.1. With thethe data from figure A.4.1 and using the trial values of kk shownshown belowbelow wewe obtainobtain thethe deviationdeviation sumsBumS ofof squares:squares:

.!s Deviation Sums of Squares 00.05. 05 0.2802 00.15. 15 0.03020 . 0302 0.25 0.0732 0.35 0.2657 0.45 0.4885

The best fitfit (minimum(minimtun DSS)DSS) appearsappears toto bebe nearnear aa k k valuevalue ofof 0.15.0.15. Taking some addi­addi- tional trial values we have:

1s ~DSS 0.11 00.0916. 0916 0.13 0.0543 0.17 0.0187 0.19 0.0187 0.21 0.0287 0.23 0.0474

Plotting these values graphically (figure(figure A.4.2) we can see that the minimumminimtun appears to be oloseclose toto kk= = 0.18.0.18. When thethe coefficient valuesvalues are calculatedoa10ulated forfor thisthis value of k,k, we find:find: logea - 3.290 ( .6.'.. aa= = 26.83)26.83) b = 5.199 and henoehence our resultantresultant equationequation is:is: 1 H = 26.83(1_exp(-O.18.A»5.26.83(1-exp(-0.18.A))5°19999 whiohwhich is plotted on figure A.4.1 as a broken line.line. - 169-169

.EllreFigure A04.1A.4.1 Hypothetical height-ageheightage datadata forfor aa plot,plot, together with a modelmodel (broken(broken line)line) fittedfitted by the nonlinear regressionregression methodmethod desoribeddescribed inin thethe teattext

Height (me)(m.)

30

25

; ; 20 , / " I / / I

15 I I I I I I I o 10 I. ••----.. Original datadata Fitted model model

5

o 5 10 15 20 25 Age (years) --170-170-

Figure A.4.2 Graphical determination of kk parameterparameter value which minimizes residual sums ofof sauaressquares in a nonlinear regression

Residual sum of sauaressquares 0.10

0.08

0.06

InterpolatedInterpolated kk valuevalue at thethe minim=minimum residualresidual sum of,oftsquares squares 0.04

0.02

o.oo+------r------~------~~--~--~------,0.00 0.11 0.13 0.15 0.17 0.19 0.21 k parameter value --171171 -

This approachapproaoh cancan be easilyeasily extendederlended in two ways, although a detailed disoussiondiscussion lies outside the scopescope ofof thisthis manual.manual.

(1) Any model inin which only £!!!!one of the parameters to be fitted is nonlinearnonlinear can be fittedfitted inin anan analogousanalog

Y. b0+b1X1k +bX+bXkk 2 2 3 1 2

and fit the bi using multiplemultiple linearlinear regressionregression withwith aa seriesseries ofof trialtrial values of k;k; plot the resultant residual sums of squaressquares forfor eacheach trial;trial; estimate the resultant kk atat thethe minimum;minimum; andand thenthen fitfit thethe bibi usingusing thatthat value of k.k.

(ii) The search for the value of k whiohwhich minimizes the residual sum ofof squaressquares can be automated so that all the parameters includinginclu1ing the nonlinear para.­para- meter can be calculatedcalculated byby aa computeroomputer programme.programme. This is quite easy to do with a single nonlinear parameter with the resultant programmes beingbeing compact enoughenough eveneven forfor somesome programmableprogrammable calculators.calculators. Suitable minimiza.­minimiza- tion methods are suggested in Sadler (1975).

The above methods are designed forfor thethe worker who does not have access to a large or medium sized computer. Given good computingoomputing facilities,facilities, the best and most flexible approach to nonlinear curve fittingfitting isis byby one of the modifications of the so-called Gauss-Newton procedure, as disousseddiscussed in such standard textbookBtextbooks as Draper & Smith (1966). These methods are quite difficult to understand, although the ultimate result, namely a set of fitted coefficients, may be easyeasy enoughenough toto use.use. Consequsntly,Consequently, mostmost workersworkers willwill needneed accessaccess toto a computer programme that can bebe adapted to their machine.machine. There are a number of such programmes available, includinginclu1ing for example, SNIFTASNIFl'A (Small(Small Nonlinear Fitting Alg

Biometrics Section Unit of Tropical SilvidultureSilvioulture Commonwealth FbrestryForestry Institute South ParksParkB Road, Oxford OX10X1 3RB United Kingdom.Kingdom.

," - 173-173 -

43.12penclixBAppendix B

TABLESTAllLES OFOF COMMON COMMON TRANSFORMATIONSTRANSFORJ.IATIONS USEDUSED FOR

Iill:lIilllSIONREGRESSION MODRTSMODElS DESCRIBED DESCRIBED n,IN THETHE: TEXTTEXT --175-175 -

Table 11 Natural logarithms (base(base e)e) forfor valuesvalues 1 to 100100

X Inln X X Inln X X Inln X ------1 00.00000. 00000 36 3.58351 71 4.26267 2 0.69314 37 3.61091 72 4.27666 3 11.09861. 09861 38 3.63758 73 4.29045 4 11.38629. 38629 39 3.66356 74 4.30406

5 1.60943 40 3.68887 75 4.31748 6 1.79175 41 3.71357 76 4.34.330733073 7 1.945911.94591 42 33.73766.73766 77 4.34380 8 22.07944. 07944 43 3.76120 78 44.35670. 35670 9 2.19722 44 3.7843.7841818 79 4.369444.369 44

10 2.30258 45 3.3.8066680666 80 4.4.3820238 202 11 2.397892.39789 46 33.82864.82864 81 4.39444 12 22.48490. 48490 47 3.3.8501485014 82 4.40671 13 2.56494 48 3.87120 83 4.41884 14 2.63905 49 3.89182 84 4.43081

15 2.70805 50 3.91202 85 4.442654.44265 16 2.77258 51 3.93182 86 4.45434 17 2.83321 52 3.95124 87 4.46590 18 2.89037 53 3.97029 88 44.47733. 47733 19 2.94443 54 33.98898. 98898 8989 4.488634. 48 863

20 2.99573 55 4.007334. 00733 90 4.49980 21 3.04452 56 44.02535. 02535 91 4.510854. 51085 22 3.09104 57 4.04305 92 4.521784.52178 23 3.13549 58 4.06044 93 4.532594.53259 24 3.17805 59 4.07753 94 4.54329

25 3.21887 60 44.09434.09434 95 4.55387 26 3.258093. 25809 61 4.110874. 11087 96 4.56434 27 3.29583 62 4.12713 97 4.57471 28 33.33220. 33220 63 4.14313 98 4.58496 29 3.367293. 36729 64 4.15888 99 4.595114.59511

30 3.40119 65 4.17438 100 4.605174 . 60517 31 3.43398 66 4.18965 101 4.61512 32 3.46573 67 4.20469 102 4.62497 33 3.49650 68 44.21950. 21950 103 4.634724. 63472 34 3.52636 69 4.23410 104 4.644394. 64439

35 3.55534 70 4.24849 105 4.653964. 65396 - 176176 _- Table 22 Exponential functionfunction exp(X)exp(X) for for valuesvalues 00 to 55

X .00.00 .01.01 .02 .03 .04.04 .05 .06 .07 .08 .09 0.0 1.001.00 1.01 1.02 1.031.03 1.04 1.05 1.06 1.07 1.08 1.09 00.1. 1 1.1.10 10 1.1.11 11 1.1.12 12 1.13 1.1.15 15 1.161. 16 1.1.17 17 1.1.18 18 1.19 1.20 0.2 1.221.22 1.231.23 1.24 1.251.25 1.27 1.28 1.29 1.1.30 30 1.32 1.33 0.3 1.1.34 34 1.1.36 36 1.1.37 37 1.391.39 1.1.40 40 1'.411.41 1.1.43 43 1.1.44 44 1.46 1.47 0.4 1.491.49 1.50 1.52 1.531.53 1.55 1.56 1.58 1.59 1.1.61 61 1.63 0.5 1.641.64 1.66 1.1.68 68 1.691.69 1.71 1.73 1.75 1.76 1.78 1.80 0.6 1.821.82 1.841.84 1.85 1.87 1.89 1.1.91 91 1.93 1.95 1.97 1.99 0.7 2.01 2.03 2.05 22.07. 07 2.09 2.112. 11 2.13 2.15 2.18 2.20

00.8.8 2.22 2.24 2.27 2.29 2.31 22.33. 33 2.36 2.38 2.41 2.432.14 0.9 2.45 2.48 2.50 2.53 2.552.55 2.58 2.61 2.63 2.66 2.69 1.0 2.71 . 2.74 2.77 2.80 2.82 2.85 2.88 2.91 2.94 2.97 1.11.1 3.00 3.033.03 3.06 3.09 3.123.12 3.15 3.18 3.22 3.25 3.28 1.2 3.32 3.35 3.38 3.42 3.45 3.49 3.52 3.56 3.59 3.63 1.3 3.66 3.70 3.74 33.78. 78 3.81 3.85 3.89 3.93 3.97 4.01 1.4 4.05 4.09 4.13 4.17 44.22.22 4.26 4.304 .30 4.34 4.39 4.43 1.5 4.48 4.52 4.57 4.61 4.66 4.71 4.75 4.80 4.85 4.90 1.6 4.95 5.00 5.05 5.10 5.15 5.20 55.25.25 5.31 5.36 5.41 1.7 5.47 5.52 5.58 5.64 5.69 5.75 5.81 55.87.87 5.92 55.98.98 1.81.8 6.04 6.6.11 11 6.17 6.23 6.29 6.35 6.42 6.48 6.55 6.61 1.91.9 6.68 6.75 6.826.82 6.88 6.95 7.02 7.09 7.17 7.24 7.31 2.0 7.38 7.46 7.53 7.61 77.69.69 7.76 7.84 7.92 8.00 8.08 2.1 8.16 8.24 8.33 8.41 8.49 8.58 8.67 8.75 8.84 8.93 2.2 9.02 9.11 9.20 9.29 9.39 9.48 9.58 9.679 . 67 9.77 9.87 2.3 9.97 10.07 10.17 10.27 10.38 10.48 10.59 10.69 10.80 10.91 2.4 1111.02 .02 11.1311 . 13 11.2411 .24 11.35 11.47 1111.58 .58 11.7011 .70 11.8211 .82 11.11.94 94 12.06 22.5. 5 12.18 12.30 12.42 12.55 12.67 12.80 12.93 13.06 13.19 13.32 2.6 13.46 13.59 13.73 13.87 14.01 14.15 14.29 14.43 14.58 14.7314 .73 2.7 14.87 15.02 15.18 15.33 15.48 15.64 15.79 15.95 16.1116. 1 1 16.28 2.8 16.44 16.60 16.77 1618.94.94 1717.11 .11 17.28 17.46 17.63 1717.81 .81 17.9917 .99 2.9 18.17 18.35 18.54 18.72 18.91 19.19.10 10 19.29 19.49 19.68 19.88 3.0 20.08 20.28 20.49 20.69 20.90 21.1121 • 11 21.32 21.54 21.75 21.97 3.3.1 1 22.19 22.42 22.64 22.87 23.10 23.33 23.57 23.8023.80 24.04 24.28 3.2 24.5324 . 53 24.77 25.02 25.27 25.53 25.79 26.04 26.3126 . 31 26.57 26.84 3.3 27.27.11 11 2727.38. 38 27.66 27.93 28.21 28.50 28.78 29.07 29.37 29.66 3.4 29.96 30.26 30.56 30.87 3131.18 . 18 31.50 31.8131 .81 32.13 32.45 32.78 3.5 33.33.11 11 33.44 33.78 34.12 34.46 34.81 35.16 35.51 35.87 36.23 3.6 36.59 36.96 37.33 37.71 38.09 38.47 38.86 39.25 39.64 40.04 3.7 4040.44. 44 40.85 41.26 41.67 42.09 42.52. 42.94 43.38 43.81 44.25 3.8 44.70 45.15 45.60 46.06 46.52 46.99 47.46in .46 47.94 48.42 48.91 3.9 49.40 49.89 50.40 50.90 51.51.41 4.1 51.93 52.45 52.98 53.51 54.05

4.0 54.59 55.14 55.70 56.26 56.82 .57.3957.39 57.97 58.55 59.14 59-73 4.1 60.3460 •.3 4 60.94 61.55 62.17 62.80 63.4383.43 64.07 64.71 65.36 66.02 4.2 66.68 67.35 68.03 68.71 69.40 70.10 70.80 71.5271.5? 72.24 72.96 4.3 73.69 74.44 75.18 75.94 76.70 77.4777 .47 78.25 79.04 79.83 80.64 4.4 81.45 82.26 83.09 83.93 84.77 85.62 86.48 87.35 88.23 89.12 4.5 90.01 90.92 91.83 92.75 93.69 94.63 95.58 96.54 97.51 98.49 4.6 99.48 100.48100.48 101.49101 .49 102.51102.51 103.54103.54 104.58104 . 58 105.63105.63 106.69106.69 107.77107.77 108.85108.85 4.7 109.94 111.0511 1. 05 112.16112.16 113.29113.29 114.43114.43 115.58115 .58 116.74116.74 117.91117.91 119.10119.10 120.30120.30 4.8 121.51 122.73 123.96123.96 125.21125.21 126.46126.46 127.741?7.74 129.02129.02 130.32130.32 131.63131.63 132.95132.95 4.9 134.28 135.63 137.00137 .00 138.37 139.77139.77 141.17141.17 142.59142.59 144.02144.02 145.47145.47 146.93146.93 - 177177 -- Table 33 Powers of of reciprocalsreciprocals forfor vaìuevalues 2 tOto 5050

Value ofof PowerPower X 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 ------2 .87055 .75785 .65975 .57434 .50000 .43527 .37892 .32987.32987 .28717 .25000 3 .80274 .64439.64439 .51728 .41524 .33333.33333 .26758 .21479 .17242 .13841• 13841 .11111 4 .75785 .57434 .43527 .32987 .25000 .18946 .14358 ..10881 10881 .08246 .06250 5 .72477 .52530 .38073 .27594.27594 .20000 .14495 .10506 .07614 .05518 .04000 6 ..6988269882 .48835 .34127 .23849 ..1666616666 .11647 .08139 .05687 .03974 .02777 7 .67761 .45915 .31112 .21082 •.14285 14285 .09680 .06559 .04444 .03011 .02040 8 .65975 .43527 .28717 .18946 .12500 .08246 .05440 .03589 .02368 .01562 9 .64439 .41524 .26758 .17242 .11111.11111 .07159 .04613 .02973 .01915 .01234

10 .63095 .398.39810-10 .25118 ..15848 15848 .10000 .06309 .03981 .02511 .01584 .01000 11 .61904 .38321 .23722 .14685 .09090 .05627 .03483 .02156 .01335 .00826 12 .60836 .37010 .22516 .13697 .08333 .05069.0506<> .03084 .01876 .01141 .00694 13 .59870 .35844 .21460. 21460 .12848 .07692 .04605 .02757 .01650.01650 .00988. 00988 .00591.005Q1 14 .58989.58989 ..3479734797 .20526 .12108.12108 .07142 .04213 .02485 .01466 .00864 .00510 15 .58181 .33850 .196911.19694 .11458. 11458 .06666.06666 .03878 .02256. 02256 .01312 .00763 .00444 16 .57434 .32987 .18946· 18946 .10881.10881 .06250 .03589 ..0206102061 .01184 .00680 .00390 17 .56742 .32197 .18269 .10366 .05882 .03337 .01893 .01074 .00609 .00346 18 .56097 .31469 .17653 .09903 .05555.05555 .03116 .01748 .00980 .00550 .00308 19 .55494 .30796 .17090 .09484 .05263 .02920 .01620 .00899 .00499 .00277 20 .54928 .30170 .16572 .09102 .05000 .02746 .01508 .00828 .00455.00455 .00250 21 .54394 .29587 ·.16094 16094 .08754 .04761 .02590 .01408.01408 .00766 .00416 .00226 22 ..5389053890 .29042 ·.15651 15651 .08434 .04545.04545 .02449 .01320 .00711 .00383 .00206 23 .53413 .28530 ·.15239 15239 .08139 .04347.04347 '-:0232202322 .01240 .00662 .00353 .00189 24 .52961 .28048 .14855 .07867 .04166 .02206 .01168 .00618 .00327 .00173.00173 25 .52530 .27594 ·.14495 14495 .07614 .04000 .02101 .01103 .00579 .00304 .00160 26 .52120 .27165. 27165 .14158.14158 .07379.07379 .03846 .02004 .01044 .00544 .00283 .00147 27 .51728 .26758 .13841 ..0715907159 .03703 .01915 .00991 .00512 .00265 .00137 28 .51353 .26371 .13542.13542 .06954 .03571 .01834 .00941 .00483 .00248 ..0012700127 29 .50994 .26004 .13260.13260 .06762 .03448 .01758 .00896 .00457 .00233 .00118. 00118

30 .50649 .25653 ·.12993 12993 .06581 .03333 .01688 .00855. 00855 .00433 .00219 .00111.00111 31 .50318 .25319 .12740 .06410. 06410 .03225 .01623 .00816 .00410. 00410 .00206. 00206 .00104.00104 32 .50000 .25000 .12500.12500 .06250 .03125 .01562 .00781. 00781 .00390 .00195 .00097.00097 33 .49693 .24694 .12271 .06098 .03030 .01509.01505 .00748 .00371 .00184 .00091 34 .49397 .24401 .12053 .05954 .02941 .01452 .00717.00717 .00354 ..0017500175 .00086 35 .4911149111 .24119 ·.11845 11845 .05817 .02857 .01401.01403 .00689 .00338 .00166 .00081 36 .48835 .23849 .11647 .05687 .02777 .01356 .00552.00662 .00323 .00157 .00077 37 .48569 .23589 ·.11457 11457 .05554.05564 .02702 .01312.0131~ .00637.00537 .00309 .00150 .00073 38 .48310 .23339 .11275.11275 .05447 .02531.02631 .01271 - .00514.00614 .00295.00296 ..0014300143 ..0006900059 39 .48050.48060 .23097 ·.11100 11100 .05335 .02564 .01212.0123~ .00592 .00284 .00136 .00065.00055

40 .47817 .22865 .10933.10933 .05228 .025Qll.02500 _ .01199.0119~ .00571 .00273 .00130 .00062.00052 41 .47582 ·.22640 .10772.10772 .05125 .02439 .01160.01150 .00552 .00252.00262 .00125 .00059 42 .47353 .22423 ·.10618 10618 .05028.05028 .02380 .01121 .00533 .00252.00252 .00119 .00056 43 .47130 .22213 .10459.10469 .04934 .02325 .01096 .00516 .00243 .00114 .00054 44 .46914 .22009 .10325 .04844 .02272 .01066.01056 .00500 .00234.00234 .00110 .000~1.000q1 45 .46704 .21812 .10187.10187 .04758 .02222 .01037 .00484. 00484 .00226.00226 .00105 .00049.0004Q 46 .46499 .21622 .10054.10054 .04575.04675 .02173 .01010 .00470 .00218 .00101 .00047 47 .46299 .21436 .09925.09925 .04595 .02127 .00935 .00456.00455 .00211 .00097 .00045 48 .46105 .21257 .09800.09800 .04518 .02083 ..0096000960 .00442 .00204 .00094 .00043 49 .45915 .21082 .09680.09680 .04444 ;02040.02040 .00937 .00430 .00197 .00090 .00041

50 .45730 .20912 .09563.09563 .04373 .02000 .00914 .00418 .00191 .00087.00087 .00040 - 178-178 --

Table 44 SpacintSpacing andand relativerelative spacingssacint % % from from height het andand stockingstocking

Spacing 2.02.0 2.5 3.0 3.5 4.0 4.5 5.0 6.0 7.0 B8.0. O m. N/ha.N/ha . 2500 1600 1111 816B16 625 494 400 278 204 256

Height ------Relative spacing %%------

5 40 50 60 70 80 89 100 119 140 160 6 33 41 50 58 66 747~ 83 99 116 133 7 28 35 42 50 57 64 71 85 100 114 8 25 31 37 43 50 56 62 74 87 100 9 22 27 33 38 44 49 55 66 77 88

10 20 25 30 35 40 44 50 59 70 80 11 18 22 27 31 36 40 45 54 63 72 12 16 20 25 29 33 37 41 49 58 66 13 15 19 23 26 30 34 38 46 53 61 14 14 17 21 25 28 32 35 42 50 57

15 13 16 20 23 26 29 33 39 46 53 16 12 15 18 21 25 28 31 37 43 50 17 11 14 17 20 23 26 29 35 41 47 18 11 13 16 19 22 24 27 33 38 44 19 10 13 15 18 21 23 26 31 36 42

20 10 12 15 17 20 22 25 29 35 40 21 9 11 14 16 19 21 23 28 33 38 22 9 11 13 15 18 20 22 27 31 36 23 8 10 13 15 17 19 21 26 30 34 24 8 10 12 14 16 1813 20 24 29 33

25 8 10 12 14 16 17 20 23 28 32 26 7 9 11 13 15 17 19 23 26 30 27 7 9 11 12 14 16 18 22 25 29 28 7 8 10 12 14 16 17 21 25 28 29 6 8 10 12 13 15 17 20 24 27

30 6 8 10 11 13 14 16 19 23 26 31 6 8 9 11 12 14 16 19 22 25 32 6 7 9 10 12 14 15 18 21 25 33 6 7 9 10 12 13 15 18 21 24 34 5 7 8 10 11 13 14 17 20 23

35 5 7 8 10 11 12 14 17 20 22 36 5 6 8 9 11 121 13 16 19 22 37 5 6 8 9 10 12 13 16 18 21 38 5 6 7 9 10 11 13 15 18 21 39 5 6 7 8 10 111 1 12 15 17 20

40 5 6 7 8 10 11 12 14 17 20 41 4 6 7 8 9 10 12 14 17 19 42 4 5 7 8 9 10 11 14 16 19 43 4 5 6 8 9 10 11 13 16 18 44 4 5 6 7 9 1C10 11 13 15 18

45245 4 5 6 7 8 9 11 13 15 17 - 17179 9 --

LmallE_2APJ)!lndix C

BLANK COPIESCOP:rn:; OFOF CALODIATIONCALCUlATION ANDAND nATADATA RECORDINGREX:ORDING FORMSFORME DESCRI1!E:D DESCRIBED IN IN PI-JETHE MAINMAIN TEXT - 181-

Form 3.1 Sample Plot Assessment Form - Plantations Sheet ...of...•••ot ••• Forest District Compartment Plot number Species Office LI I I I CD use Plot Slope Assessment area I CDR. date OJ IT]II month year no. of cards Tree no. Diameter bhob Height ------Codes --

Notes ......

• • • • • • • • • 10 ••• •• 0 ••••••••••••••••••••••••••••••••••••••••••••••••• ...... Assessed by •••••••••••••••••••••••••• Date •••• / •.•• / •••• USE BACK OF FCIDI FOR HEIGHT CALCULATIONS - 1~-182' -

Form 4.1 COmmonmm n slopeone andand common common intercept interagression regression models Part 1 Plot data summarization. Use ' as many part 1 sheets as necessary for aallll plots.plots. Data transformations used:used: xX = yY = ------fPiotlot fray;Raw data Transformed data X yY X2 yyXY

TotalsTotals n n =IC= Plot Raw datadata Transformed datudata

-4- xq. Yy X2 yy]

Totals n Totals n _

(PlotPlot Raw datadata /rranEransformedsformed data X Yy X2 XYXY

Totals n - 18118} -

FormForui 44.1.1 Common slopeslone and commoncomm on intercelIxeLreaL.312n_m2d.L1Lintercept regression models Part 22 Totals betwbetweeneen plots and coefficientcoefficient calculationcalculation

EXEXYrXrXY . (ELY'(rx) 2 rx2_(rx)22 n EY Plot rXYIXY-TY-gg x -() rx2EX2 2 rY n n rxEX2

I

;

,

Totalsrtals W W l2.I ~ ~ ~ _

Common slopeslope b = (1)/(2)(1)/(2) = = ------Common intercept a = ((6)-(3))/((5)-(4))«6)-(3))/«5)-(4)) =- ______

= ------Form 4.24.2 Transformations to age and height data from PSF'sPSP's toto fit k parameter in SchumacherSchuma cher equationequation by common slope estimatorestimator

1 Ii [ Jj A · a b C Xx.. H··H d e f 9g h y. IJ IJ IJJ IJJ ~~\ .~,\ ~ o~ Age Sum Diffarff 2/2 In c Height Sum Diff e% dA Vg In hh ~ Va ~ ~ ~ . ,{., ...... -., ~ ., . . - . :0 •• ......

-",. "~ ~' ~ .,.~.'. ~ ..00 I

~-At: a::l~" ':-,;~ - ... erf.... ::.:.;:;;,r: c' - • •••"""'-., ....' · "'0 . :"":,:... : .. -. ... ~ ...... ~ ..... 'l~ ~"_ V ~ ~' ,,~: I' I I I r :!-r '. 'J e'- · l ·· · ·- · ~ t -.- r' '-:r '-'~ f

1....toemarasovmaceasw=-rrensaler-mdeman. IJIIII Transfer to form 4.14. 1 - 185 -

Form A.1A.l Coefficients for an approximating quadratic or cubic curve

Objective To calculatecalculr.te the coefficients bbi in eithere·i ther of thethe equations:

Y =bo +.X + b2.X2 + b3.X3 (cubic curve) or Y = b+ b .Y + b .X-2 (quadratic(qua dra.tic curve)curve) o 1 - 2 Items marked <><> are not requiredrequired forfor thethe quadraticqU8.dratic curvecurve calculations.

(1) TabulTabulatea te 33 or 4 datadata points forfor thethe functionfunction andand computecompute squares andand cubes.cubes.

I 1 2 3 4 <> Y;Yi ...<> XXiI X? <> 3 <> <> <> <> XI

(2) FromFroe tablet a ble (1)(1) calculatecalcula te differencesdifferen ces asa s shown:sh own :

.... 3 I X2 _X2 X _X3 \+1- 1 1 'li+1\II+1 -'li Xi 'i+1i-41 i i +1 i <><> 1 <> 2 <> ...<> <>,. .<> 3

From tabletable (2)(2) constructconstruct thethe termsterms shownshown inin thethe tabletable below,below, accordingaccording toto the followingfollowing defini- defini­ v. wi tions: I u·I v·I w·I <> u = (Yi+l-Yi)/(Xi+l-Xi ) 1 i (Yd.4-1Yi)/(xi+1xi) <> x 2. +1 VVi= . = (X2 y- 2 1_ (X~+l-X~)1+1 i"/ (Xi+l--1+1 X`11i) <> <> <> W . = 3 Wi= (x3( X1+1~+l-X~)/(Xi+l-Xi)x3)/(xi 1+1x.)i - 186 -

Form A.1A.l (continued....)(continued ••. • )

(4) From tabletable (3) calculate the terms inin thethe followingfollowing table,table, from these definitions:definitions:

I1 Vi+1-Vj P. q. pi= (ui+i-ui)/(vi+i-vi) P i v1.+1-vi 1L I <>.0. 1 ql.. = (VI. l-W. )/(v. l-v.) qi= 1.+ 1. 1.+ 1. 2 <> <>.c. .0.<>

(5) CCalculatealculate the coefficients according toto thethe followingfollowing formulae:

Coefficient Cubic eqn.eqn. QuadraticQuadratic eqn.egn.

b3 (p2-P1)/(c12-q1) None =

b2 p1-b3. q1 pl

= =

ul -b2.vi b1 u1-b2ev1 -b3owl

= =

3 Y -b .X -b.X2 bo Y1 -b1 .X1 -b2* 1X2-b3-1 1112 1

= =

" " " " " " " " --187-187 -

Form A.3 Coefficients and statisticsstatistics of a regression with _T2_redjc-ples(Etarttwo predictor variables (part 1) 1)

Descri~tionDescription ofof problem:problem: ______------

(1) Regression data ((4) 4) Corrected rroductsproducts v fy7-:-EY72;51-/nrye. = I.Ye._( H) e. /n = Y A Z = ryqfExy = .EXY-XX.EY/nrXY-I.X.I.Y/n = rzyEzy = EZY-EZ.ZY/nI.ZY-rZo rY/n == 2 rx2Xx.2 = rxEX2-(Y,X)2/n_(LX)2/n =- zxzrx~ == EXZ-YX.EZ/n I.Xz-rx.rz/n == 2 2 rz2 =Ez2-(IZrz •. pz)2Ln 9n = I = = - ((5) 5) Determinant &8: GaussGauss multipliersmulti-_iers D == Yx-.Ez--(Ixz)2xx22..rz"-(rxz)" = rz2/DEz2/D = gl=°.0"1 = = g2=o9.0-= -1.xz.jD-yxz/D- = 2 g3= rxEx2/D/D =

((6) 6) Regression coefficientscoefficients bi= bl = g10Exy-t-g9.1.zygl .ryqf+g2· rzy =- b2= .-Zy) = b2= g2g2·rxy+g3orzy .Exydrbcx3- = b0=bO= Y-b1°7-b20ZY-bl oX-b2oZ = ______

1--(77)(7) Regression sumsum ofof squaressquares rye.-2 = = b1..ncy+b20Izyblo ryqf+ b2.lZY = (8)(8 ) Residual sumsum ofof squaresSf!.uares 1.d"=Ed-= EylY"-ry~ -Ly777- = I La....Residual( 9) Residual degreesdegrees of freedom ( 2) Totals and products idf=df= n-3 =- 1.y2 1.Y rXY zZYr/,Y

n rx 1.ZZZ 1.X 2 EXZrxz (3) Means y = I.Y/n = 1.<12 B A,. = i.:X/n = " = rZ/n = - 188188 -

Form A.3 ((...Fart ••• Pe.rt 2)2)

(10(10) ) CorrelationsCorr el a tions betweenbetween variables r 2 r yx = Exy//(Ex2.zy2)r. xy/{( r.x • ry2) = 2 ryzyz = r.yz/Eyz//(Ez2.Ey2){( r.z .r.y2) = f 2 22, =zxz//kEx .zz ) r xz = r. xz/{(r.x . r.z ) =

(11) MultipleMultiple correlationcorrelation coefficientcoefficient RR andand R-R ~ r?R2 = r.Y/zyr.Y/ry == ,RR =/R2= { R2 =

(12 ) AnalysisAnal ysis ofof variancevariance sum ofof mean variance so.upressQua.res d.f. square rratioatio

Regression Ty-2r.y-2 2 7 ResidualResidua.l r.dId« df =7.= s"S- TotTotalal Zy2r.y" n-1n-l

(13) Residual standards tandard deviation

(14) Standard error andand t-valuet-value ofof coefficientscoefficients

CCoefficientoefficient b bb1l b22

Standard error sbsb sq-gi=s·{gl= s·{g3= ~t = b/sbb/s b

(15) Confidence intervalsintervals forfor predictionspredictions For a series of m estimatesestimates ofof Y atat aa givengiven levellevel ofof XX andand Z,Z, the standard error ofof thethe mean predictionprediction isis givengiven by:by:

, = s.\r(l/m+1/n + g1.x2 +2.g20x.z +g3.z2)

where x = (X(X - X)7) and z = (Z -7).Z). FerFor a large sample,sample, l/mlim will be zero; for a singlesingle point itit willwill bebe unity. - 189 -

Appendix D

ANNOTATEDANNOTATTID BIBLIOGRAPHYBIBLIOGRAPHY

This bibliograPhybibliography is !!2!not a comprehensive bibliography of forest mensuration litera.­litera- ture. It simply suggestssuggests a setset of referencereferenoe books to acoompany this manual, whichwhioh may be used to research or extend the teohniques suggested.

Technical articles referred to in the text are given as footnotes andand areare not represented here. - 191191 -

Assmann, E.,E •• TheThe PrinciplesPrinoiples ofof ForestForest Yield Study.Stuiy. Pergamon Press. pp 506506 1970 .!A very comprehensive treaty onon the theory of forest growth, involving discussiondiscussion of numerousnumerous examples.examples. Most of of the ooneeptsoonoepts are generalized fromfrom NorthNorth European,EUropean even--ag

Avery, T.E., Forest Measurements.Measurements. NoGra.MoGraw-Hill...... Hl .ll. pp 290. 1967 Emphasises practicalpractical aspectsaspeo·te ofof fieldfield workoork andand primaryJlZ'1ma.ry allalysisanalysis ofof data,data, asas opposedopposed to·to model construotion,oonstruotion. Oriented to North American practices and terminology, soso not entirely suited for the tropics,tropios. but stillstill aa usefuluaeful textbook.textbook.

Burley J.J. andand WoodWood, P.J. P.J. (Editors),(DUtors), A Manual onon SpeoiesSpecies andand ProvenenceProvenanoe ResearchReee ....oh with 1976 PartioularParticular Referenoe toto thethe Tropics.Tropics. TropicalTropioal Forestry PaperPa.per 10,10, Commonwealth Forestry Institute.Institute. Oxford.

A uasfuluseful companionoompanion volume volume ·to tothe the present present work, work, "overing covering practica.l practical a.apeotsanpeots ofof research requirements forfor the exploratoryexpJ.oratory phasesphases of the introductionintrod uctioll of eYoticexotic plantation species.species. PraotioalPrao~ioal in orientation andand eimplysimply written.written. Separate special appendixappendix (by(by J.F.J.F. HughesHughes enriand R.A.R.A0 Plumtre)Plumtre) ooverscovers woodwood qualityquality stndies.studies.

Cab, Computerized Methods inin Forest Planning andend Forecasting.Foreoasting. Annotated Bibliography F14.F140 1977 Commonwealth Agrioultural Agricultural Bureau,Bureau, MC. UK.

Covers published world literatureliterature 1973-1976.1973-1976. Gives a full abetl'a.otabstract on each reference. Most entries are directly-re/eventdireotly relevell"t to the field of yield predictionprediotion and control.

Carron, L.T.,L.T., JUIAn Outline of ForestFo,.... st Mensuration. Australian National University Press.Press. 1968 pp 224.

Better oriented toto yieldyield andand volumevolume table construotionconstruction in widely spaced plantations than AmericanAmerioan oror European texts.texts. Brief ande.;od clear summary of methods, but does not discussniscuss problemsproblems of model fitting or construotion.construction.

Daniel, C. C. and and Wood. Wood, F.S.,F0S02 Fitting Equations toto Data.Data. loIileyo-Interso;.enoe.Wileyeintersoienoe. pp 342. 1971 Covers practicalpraotical use of multiple regression for model fitting, fitting, inoledingiltoluding residualresidual analysis, nested regression models, stepwise regression and non-linaarnon-linear fitting.fitting. Includes examplesexamples andand documentationdooumentation forfor aa computeroomputer package,package, butbut no sourcesource listing.listing.

Day, A.C.,A.G., FUR'fRANFORTRAN Techniques. Cambridge University Press. pp 96.96. 1972 A good supplement to any introduotoryintroductory FORTRANFURTRAN course.oourse. Covers methods of drawing graphs, sorting, pointers,pointers, stacks, openopen subroutines, recursion and other teohniques.techniques. - 192 -

Draper, N. and Smith, HogH., Applied Regression Analysis. Wiley-Interscienoe.Wile;?,-Intersoienoe. pp 407.407. 1966 Now almost a classicalolassical textbook on multiplemultiple and non-linear regression.regression. Covers theory of simple regression, matrix algebra forfor multiple regression, residualresidual analysis.analysis, stecwisestepwise regression,regression. use of regression to analyse controlledcontrolled experi-experi­ ments,ments. and introductoryintroductory non-linearnon-linear regression.regression. Elementary matrixmatrix algebraalgebra isis required.required, together with differential calculus.calculus.

FAO.FAO, Manual of Forest Inventory with Special Reference to Mixed Tropical Fbrests,Forests. Food 1973 and Agriculture Organization,Organization. Rome.Rome. pp 200.200.

An eesentialessential reference forfor yieldyield studystudy work.work. Much discussiondisoussion of design of forestforest sampling and appropriate formulae.fonnulae. Practical guideguidelines lines for executing a forestfor"st inventory and data recording.reoording.

Fries, J. (Editor),(Editor). Growth Models~Iodels forfor Tree and Stand Simulation,Simulation. ResearohResearch Note No. 30. 1974 Department ofof FbrestForest YieldYiDld Research,Researoh. RoyalRoyal CollegeCollege ofof Forestry,Forestry, Stockholm,Stookholm. Sweden,Sweden. pp 379.379. (Available from J. Fries.Fries, University of Agricultural Sciences,Scienoes. S-7505-750 0707 Uppsala,Uppsala. Sweden)

Collection of papers from IlIFROIUFRO. Working Party 54.01-.454.01-4 msetings.meetings. Many useful ideas.

Fries, J.,J •• Brukhart,lIrukhart. H.H. andand Max,Max, To,T •• (Editors),(Editors). Growth ModelaModels forfor LongLong TermTenn FbrecastingForeoasting ofof 1978 Timber Yields. -F\/S-1-78.-FWS-1-78, SohoolSchool of Forestry and WildlifeWildlife Resouroes.Resouroes, VirginiaVirginia PolyteohnicPolytechnic Institute and State University.University, Blacksburg.Blacksburg, VAVA 24061,24061. USA.reA. pp 249.249.

CoColleotion lleotion of papers from TUFRO,IUFRO. SubjeotSubject Group 54.01 meeting. Many useful idear.ideal!!.

Green,Green. P.E. and Carroll,Carroll. J.D.2J .D.. MathematicalMathematioal ToolsTools forfor Applied MultivariateMultivariate Analysis.Analysis. 1976 AcademicAoedemio Press.Press. pp 376.

A good self-teaching text forfor matrix algebra as it relates to multiple regressionregression,, principalprinoipal component analysis,analysis. factorfactor analysis and ordination analysis.

Husch.Husch, BB.,•• Miller.Miller, C.I.C.I. andand Beers.Beers, T.W T.W. •• FbrestForest Mensuration.Mensuration. 1972 A tclassicale'olassical' forest mensuration text,-("xt. emPhasisingemphasising measurementmeasurement ratherrather thanthan models,modela. and with onlyonly veryvery limitedlimited treatmenttreatment ofof groarth gronh prediction.prediotion. Oriented to North AmerioanAmerican conditions,oonditions. so some practicespractioes are inappropriate for tropicaltropioal use,use. whileWhile specifioally tropical problems are not considered.oonsidered. A useful textbook of basic mens.uration.mensuration.

Land.Land, A. and Powell,Powell. S.,S.. FORTRAN Codes forfor MathematicalMathematioal Programming.Programming. Wiley. PPpp 249. 1973 Gives completeoomplete programmesprogrammes andand fullfull dooumentationdooumentation forfor linear,linear. integerinteger and. and quadraticquadratio programming using FORTRAN.FORTRAN, with a brief explanation of underlying theory.theory. - 193 ·-'

Parde,Farde, J., Dendrometrie. 1961 A classioalclassical text of forest mensuration,mensura.tion, butbut withwith ratherrather moremore emphNlisemPhasis onon yieldyield table construction than Husoh et al.0.1. Very good on instrumenta/instrumental measurements.measurements, espeoiallyespecially Relaskop,Relsskop, and on the development of simplesimple modelsmodels forfor thethe inter-inter­ relationship of standstlillld variables. Probably the·the best currentourrent text book on forest mensuration. Not oriented to tropicaltropioal problems. In French,French.

PiePielou, l ou, E.C.,E.G., Mathema.tica.lMathematical Ecology. li'iley--Intersciencfl.Wiley-interscience, pp 385.385. 1977 An excellentexcellent referenoereference book for mathematicalmathematioal modelling of population dynamics,dyea.mios, population dispersion and association analysis. Of more interest to the eoologist/ecologist/ silvioulturistsilviculturist than to the forest manager. The association analysis section is important for studiesstuiies in tropicaltropioal rainforests.rainforents.

Poole, R.W., An IntroductionIntroduction toto QuantitativeQuantitative Edology.Ecology. McGraw-Hill,MoGra_Hill. pp 532.532. 1974197A Covers a wide range of teohniqussteohniquas inin eoologYeoology withwith an emphasisemphasis on model building.building. Less theoretioal,theoretical, and with more examplesera.mplos and more basiobasic andand introductoryintroductory informa­informa- tion than PielouPie lou (Bee(see above). Assumes a littleli'tUe caloulusoaloulus and matrizmatrix algebra.

Prod=,Prodan M.M., FOrestForest Biometrics. Biometrics. Pergamon Press. PPpp 447. 1968 Very usefulueeful treatmenttreatment ofof thethe statisticalstatistioal backgroundbaokground toto forestforest growthgrowth studies.studies. Requires moderatemoderate mathematicalmathematioal competence.oompetence. Good treatment of fitting non-linear geserwthgr'G'Wth funotions, functions, but but the the rook book isis notnot direotlydirectly orientedoriented to practical application. Mainly influencedinfluenoed by European forestry practioes.

Royce-Sadler, D.9D., Numerioal Methods forfor Non-linearNon-linear Regression.Regression. University of Queensland 1975 Press. pp 89.89.

A summarysUJll!llal'Y of thethe mainmain practicalpractioal methodsmethods forfor fittingfitting the coefficientscoeffioients of non-non­ linear models. Assumes somasome calculuscaloulUE andand matrixma.trix algebra.algebra. Very olear and briafbrief exposition ofof thethe essentialessential algorithns.algorithms.

Seber, G.A,F.,G.A.F. Linear Regression Analysis. Wiley-interscience.li'iley-Interscienoe. pp 465.465. 19771977 Morel~ore theoretical,theoretioa.l, moremore comprehensiveoomprehensive andand moremore upup toto date thanthan Drapernr-d.per & Smith. Good understandingunderlltanding of matrixmatrix algebra required,required, butbut aa varyvery usefuluseful referencerefe:renoe book forfor thethe statistician.statistician.

Shannon, R.E..R.E., Systems Simulation: The AxtArt and Science.Soienoe. Prentice4fall.Prentioe-Hall. pppp 387.387. 1975 A useful referencereferenoe book forfor workers interestedinterested inin thetho oonstructionoonstruotion ofof simelationsimulation models. CovtrsCovers random number generation,generation. model fitting,fitting, systems analysis,e.n.a.lysis. experiments onon modele,models, deoisione-makingdeoisio,.....eldng usingusing modelsmodels and gives several oasecase studies.stuiies. - 191119,4 -

Snedecor, 0.W.G.W. and Cochran, W.G.,N.G., StatistioalStatistical Methods.~p.thodB. Iowa State UP,UP. 1967 Exoa11entEkoallent referenoe book for statisticalRt~tistical methods. Does notnot requirerequire an e~cessivelyeTcessively highhigh level of mathematicalmathematiQa1 ability (calculus(oaloulus andand matrixm"trix algebraalgebra notnot needed).needed). Subjeots covered includeinolude linearlinear and non-linear regression, multiplemultiple regression,regression, e~p"rim9ntalexpArimental andend sampling'alysifJ. AllAI). teohniqvteohnives.ea areare wellwen illustratedillustrated with worked examples andand therethere areere numerousnumerous testtest questionsquestions onon eacheach seotion.seotion.

Stark, P.A.,P. A., Introduc-tionIntroduction to NumerioalNumerical Methods,Methods. Macmillan.~lacmills:l. ppFP 334. 1970 BasicBasio textbook on numerical methodsmethods forfor computers.oomputers. InoluiesIncludes solution of equations, simultaneous equations, numerioa1numerical integration and differentiation, andand interpola-interpol,.,.. tion with polynomials. Most methods are illustrated "lithwith short 10RTRANFORTRAN programmes. Also gives introduction to matrixmatrix algebra.algebra. ReviresRequirM mathematicalmathematioal abili*yability toto aboutabout university entrance standard.stlUldard. Explains matrix algebra, but assumes knowledge of calouloo.calculus.

Universal ancyclopaediaEncyolo]:aedia ofof Mathematics.Jf,athematics. PanPan Books.Books. pp 715.115.

An inexpensive (panerback)(Jlf\perbaok) and extremelyext"emely useful referenoereference book for formulae inin algebra, trigonometry,trlgonometry, geometry, calculus.calculus, etc.,etc., mathematioalmathematical tablestables andand terminology.termino logy.

Wagner, H.M., PrinoiplesPrinoiples of Management Science.Soienoe. Prentice-Ha.ll.Prentice-Hall. pp 612.612. 1975 An excellontexcellent textbook of quantitative decision-makingdecision-ma.king methods for managers, including linear andand mathematical programming of all types, inventory control, simulation and queueing models.models. Does not assumeassume muohmuch mathematioalmathematioal abilityability (no(no oalculuscalculus or matrix algebra).

Wright, H.H. (Editor), Planning, PerformanoePerformance and Evaluation of Growth and Yield St,ul.ies.Studies, 1980 Canmon"ealthCommonwealth Forestry Institnte,Institute, Oxford,Oxford, GreatGreat BritanBrit"" (in(in print).print).

Collection ofof paperspapers franfrom IUFRO,=Rot SubjectSubject GroupGroup 54.0154.01 meeting.meeting. Many useful ideas.idas.