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Some Statistical Aspects of Spatial Distribution Models for Plants and Trees

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Some Statistical Aspects of Spatial Distribution Models for Plants and Trees STUDIA FORESTALIA SUECICA Some Statistical Aspects of Spatial Distribution Models for Plants and Trees PETER DIGGLE Department of Operational Efficiency Swedish University of Agricultural Sciences S-770 73 Garpenberg. Sweden SWEDISH UNIVERSITY OF AGRICULTURAL SCIENCES COLLEGE OF FORESTRY UPPSALA SWEDEN Abstract ODC 523.31:562.6/46 The paper is an account of some stcctistical ur1alysrs carried out in coniur~tionwith a .silvicultural research project in the department of Oi_leruriotzal Efficiency. College of Forestry, Gcwpenberg. Sweden (Erikssorl, L & Eriksson. 0 in prepuratiotz 1981). 1~ Section 2, u statbtic due ro Morurz (19501 is irsed to detect spafiul intercccriorl amongst coutlts of the ~u~rnbersof yo14119 trees in stimple~~lo~slaid out in a systelnntic grid arrangement. Section 3 discusses the corlstruction of hivuriate disrrtbutions for the numbers of~:ild and plunted trees in a sample plot. Sectiorc 4 comiders the relutionship between the distributions of the number oj trees in a plot arld their rota1 busal area. Secrion 5 is a review of statistical nlerhodc for xte in cot~tlectiorz w~thpreliminai~ surveys of large areas of forest. In purticular, Secrion 5 dzscusses tests or spurrnl randotnness for a sillgle species, tests of independence betwren two species, and estimation of the number of trees per unit urea. Manuscr~ptrecelved 1981-10-15 LFIALLF 203 82 003 ISBN 9 1-38-07077-4 ISSN 0039-3150 Berllngs, Arlov 1982 Preface The origin of this report dates back to 1978. At this time Dr Peter Diggle was employed as a guest-researcher at the College. The aim of Dr Diggle's work at the College was to introduce advanced mathematical and statistical models in forestry research. Dr Diggle was specially employed to develop some biological models for a planning system for silvicultural treatments and for various cutting sys- tems. To tackle this problem a research-team has been working to develop an advanced system for planning the above mentioned activities. Since Dr Diggle left Sweden as a guest-researcher he has been working as a consul- tant to the project. And we see in this report the results of his effort in this field. College of Forestry Garpenberg, March 1981 Sten Nilsson Contents 1 Introduction ............................ 1.1 General introduction ................. 1.2 Sources of data ...................... 1.3 Plan of report ....................... 2 Spatial interaction ....................... 2.1 Introduction ........................ 2.2 A test for spatial interaction .......... 2.3 Application to PTAX data ............ 2.4 Results ............................. 3 Quadrat count distributions ............... 3.1 Introduction ....................... 3.2 Univariate quadrat count distributions . 3.2.1 The Poisson distribution ............ 3.2.2 The negative binomial distribution ... 3.2.3 The binomial distribution ........... 3.2.4 The beta-binomial distribution ....... 3.3 Bivariate distributions ................ 3.4 Likelihood analysis .................. 3.4.1 Poisson ........................... 3.4.2 Negative binomial ................. 3.4.3 Binomial ......................... 3.4.4 Beta-binomial ..................... 3.5 Application to PTAX data ........... 3.6 Results ............................. 4 A relationship between two measures of variation in yield within a stand 4.1 Introduction ............................................... 4.2 Relationships for plots of a single size ........................ 4.3 Relationships for plots of differtent sizes ...................... 4.4 Application to typestand data ................................ 5 Statistical methods for sparsely sampled point patterns .................. 5.1 Introduction ................................................... 5.2 The Poisson process ............................................ 5.3 The quadrat sampling method ............................ .. .... 5.4 Distance sampling methods ...................................... 5.4.1 Further results for the Poisson process .......................... 5.4.2 Tests based on univariate sampling procedures ................... 5.4.3 Tests based on bivariate sampling procedures .................... 5.4.4 Tests for heterogeneity ..................................... ... 5.5 Robust intensity estimation ...................................... 5.6 Spatial association .............................................. 5.7 Sampling in the field ............................................ 6 Concludingremarks ............................................... Acknowledgement ................................................ 7 References 1 Introduction 1.1 General introduction Research workers in the department of Operational Efficiency have been working for several years to develop a planning system for silvicultural treatments and for various cutting options. The goal of this system is to provide a long-term silvicultural strategy for a forest company. Today in Sweden, the practical consequences of, for example. regenera- tion or pre-commercial thinning operations are evaluated by such simple criteria as number of living plants one year after planting, number of stems after pre-commercial thinning, etc. However, the first income from a stand comes some 50 to 80 years after regeneration. It is therefore necessary to analyse the effects of a silvicultural treatment on the complete life-cycle of a stand. No such complete description is currently available. To tackle this problem, a computerised planning system has been devel- oped. (Eriksson, L & Eriksson, O in preparation 1981.) The two principle components of this system are (i) a biological model (ii) an economic model. The biological model simulates the decelopment of a stand from regen- eration to final felling. Parameters in the model can be varied to accommo- date various policies for regeneration. pre-commercial cleaning. thinning, etc. Principally because of the long time-scale involved, no data are avail- able concerning the complete development of a stand. Several different kinds of basic biological research data have therefore been used in the model-building process. These data-sets relate to investigations carried out in different departments of the university, and each such investigation has been directed towards answering relatively specific questions concerned with particular aspects of stand development. The need to link these aspects together in the biological model has generated a number of loosely related statistical problems. This report presents a number of "sub-models" which are now used within the full biological simulation model. and it should therefore be read in conjunction with Eriksson, L & Eriksson, O (in preparation). However, we hope that some of the statistical problems discussed in the report will prove of wider interest in the general context of forestry research. 1.2 Sources of data We make extensive use of data collected in the PTAX investigations (HultCn & Jansson 1972). These data include information on the survival or death of young planted trees and on the incidence of naturally regenerated trees in circular plots of radius 2 m. laid out systematically in a number of stands distributed throughout Sweden. The number of plots per stand BERGSHULT DEGERFORS Figure I. Typical exarnple of sampling design for PTAX data varies considerably. but is typically of the order of 40 per stand, excluding plots which are unsuitable for planting. The spacing between adjacent plots also varies according to the overall area of the stand, but is typically of the order of 30 m. Figure 1 shows a typical example of the layout of sampling plots within a stand. We also use typestand data, in particular the typestands for cleaning described in Gustafsson (1974). Each typestand consists of a 25 m by 20 m rectangle within which the locations and species of each stem are recorded, together with additional information including the cleaning grade of each stem, from 1 to 7; where: grade 1 indicates trees to be removed to get a stand of 2800 stemsiha grade 1-2 indicates trees to be removed to get a stand of 2400 stemslha grade 1-3 indicates trees to be removed to get a stand of 2000 stemslha grade 14indicates trees to be removed to get a stand of 1600 stemslha grade 1-5 indicates trees to be removed to get a stand of 1200 stemslha grade 1-6 indicates trees to be removed to get a stand of SO0 stemslha grade 7 indicates all remaining trees 1.3 Plan of report In Section 2 we define the concept of spatial interaction for a set of observations recorded at fixed spatial locations, and show how a test for spatial interaction can be applied to the PTAX data in an exploratory way. to identify stands with interesting spatial structure. In Section 3 we describe a number of probability distributions which arise in the analysis of quadrat count data. We motivate the various distributions by considering both the mode of regeneration (natural or planted) and the possible existence of environmental heterogeneity. We then discuss an extension to the bivariate case, based on a class of bivarate distributions introduced by Plackett (1965). The PTAX data again provide an illustrative application, and we note the difficulties which arise in making a combined interpretation of the results from a large number of stands. Section 4 investigates a specific problem concerning variation in yield within a stand. In young stands. this is conventionally measured by the variance-to-mean ratio of the number of individual stems within a sample plot. In older stands, the accepted measure is the coefficient
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