Robust Regression Analysis of Ochroma Pyramidale (Balsa-Tree)

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Article Improving the Modeling of the Height–Diameter Relationship of Tree Species with High Growth Variability: Robust Regression Analysis of Ochroma pyramidale (Balsa-Tree)

Jorge Danilo Zea-Camaño 1,*, José R. Soto 2, Julio Eduardo Arce 1, Allan Libanio Pelissari 1 , Alexandre Behling 1, Gabriel Agostini Orso 1, Marcelino Santiago Guachambala 3 and Rozane de Loyola Eisfeld 1

1 Department of Forest Sciences, Federal University of Parana, Campus III, 3400 Lothário Meissner Av., Curitiba 80000-000, Paraná State, Brazil; [email protected] (J.E.A.); [email protected] (A.L.P.); [email protected] (A.B.); [email protected] (G.A.O.); [email protected] (R.d.L.E.) 2 School of Natural Resources and the Environmental, The University of Arizona, 1064 E. Lowell St., Tucson, AZ 85721, USA; [email protected] 3 PLANTABAL S.A.—3ACOREMATERIALS, Oﬁc. 102 Mz. CC3 SL. 120, Complejo Industrial Quevedo, 1 Km 4 2 vía a Valencia, Quevedo 120501, Ecuador; [email protected] * Correspondence: [email protected]; Tel.: +55-83-99926-8487

Received: 23 January 2020; Accepted: 9 March 2020; Published: 12 March 2020

Abstract: Ochroma pyramidale (Cav. ex. Lam.) Urb. (balsa-tree) is a commercially important tree species that ranges from Mexico to northern Brazil. Due to its low weight and mechanical endurance, the wood is particularly well-suited for wind turbine blades, sporting equipment, boats and aircrafts; as such, it is in high market demand and plays an important role in many regional economies. This tree species is also well-known to exhibit a high degree of variation in growth. Researchers interested in modeling the height–diameter relationship typically resort to using ordinary least squares (OLS) to fit linear models; however, this method is known to suffer from sensitivity to outliers. Given the latter, the application of these models may yield potentially biased tree height estimates. The use of robust regression with iteratively reweighted least squares (IRLS) has been proposed as an alternative to mitigate the influence of outliers. This study aims to improve the modeling of height–diameter relationships of tree species with high growth variation, by using robust regressions with IRLS for data-sets stratified by site-index and age-classes. We implement a split sample approach to assess the model performance using data from Ecuador’s continuous forest inventory (n = 32,279 trees). A sensitivity analysis of six outlier scenarios is also conducted using a subsample of the former (n = 26). Our results indicate that IRLS regression methods can give unbiased height predictions. At face value, the sensitivity analysis indicates that OLS performs better in terms of standard error of estimate. However, we found that OLS suffers from skewed residual distributions (i.e., unreliable estimations); conversely, IRLS seems to be less affected by this source of bias and the fitted parameters indicate lower standard errors. Overall, we recommend using robust regression methods with IRLS to produce consistent height predictions for O. pyramidale and other tree species showing high growth variation.

Keywords: statistical regression; robust methods; M-estimators; site-index curves; balsa-wood

Forests 2020, 11, 313; doi:10.3390/f11030313 www.mdpi.com/journal/forests Forests 2020, 11, 313 2 of 20

1. Introduction Ochroma pyramidale (Cav. ex. Lam.) Urb. (balsa-tree) is a large Malvaceae pioneer tree species native from tropical forests. This species can reach 30–40 m in height, 60–120 cm in diameter at breast height (DBH), and crowns up to 40 m at the age of 15 years. O. pyramidale is popularly known as balsa-tree and balsa-wood, with natural distribution throughout America, from southern Mexico to Bolivia, northern Brazil and the Antilles [1,2]. 3 Due to its low weight (the density ranges between 0.06 and 0.38 g cm− ) and mechanical resistance, the wood of O. pyramidale is commonly used in wind turbine propellers, sports equipment, ships, and aircrafts [3]. Balsa-wood has been in high demand on the international market, mainly Europe, China, and United States. Therefore, the species is planted in commercial reforestation programs and in mixed plantations to restorage degraded areas [2,4,5]. This species is characterized by high growth rates and is managed in a clear-cutting and replanting system with harvesting of three- to five-year-old stands [5,6]. However, the fast growth contributes to considerable tree diameter and height variability in even-aged stands. The latter is largely due to high competition for water, light and nutrient resources among neighboring trees of similar age [7]. Other important factors such as genetic characteristics, edaphoclimatic conditions, as well as forest management practices may contribute to growth variability; these also have been reported for Eucalyptus spp. (eucalypts) stands, another species with high growth rates [8]. Regarding the management of O. pyramidale stands a lack of practical information exists about the behavior of height–diameter relationships on different site-index classes. The stand dynamics and individual tree characteristics are tightly linked to site-index conditions (local yield-capacity) [9]. O. pyramidale stands management needs reliable information to assess its development [5,6]. The height–diameter relationship involves two very important variables in forest allometrics and modeling. This relationship is not only used to characterize the vertical structure but is also fundamental for many growth and production models [10]. Due to the costs of measuring all tree heights, it is common to use statistical models to describe the height–diameter relationship [6,11,12], which may be performed by site-index and/or age classes to improve predictions. Ordinary least squares (OLS) is the most widely used estimation method for linear statistical models. However, this method is sensitive to outliers, in which a single atypical value may influence model fits and predictions [13]. In this context and considering that O. pyramidale data may present high variability, it is necessary to consider other regression methods that minimize the influence of outliers. As concerns OLS regression methods, Montgomery et al. [14] emphasize the importance of the distribution of values in relation to the independent variable. Namely, all values in a data-set are given equal weight, however, each value can affect the relationship between the dependent and the independent variables. That is, the slope of an OLS regression can be disproportionally influenced by extreme values of independent variables which are far from the point cloud (i.e., outliers). Given the above, our study is motivated by the need for improved tree height predictions for O. pyramidale given the high probability of outlier occurrence. The use of robust regressions is widely known to mitigate the influence of outlier observations [15]. The approach was first developed by Huber [16] and improved by Hampel [17] Rousseeuw [18], Yohai [19]; Yohai and Zamar [20], Maronna and Yohai [21], Aelst et al. [22] among others. Over more than half a century of development, several robust estimators have been proposed: maximum likelihood estimator (M), maximum likelihood estimator modified (MM), generalized M-estimators (GM), least median of squares (LMS), least trimmed squares (LTS), scale estimator (S), generalized S-estimates (GS), and robust and efficient weighted least square estimator (REWLSE), each with their particular characteristics [13,15,23]. Here we focus on M-estimators which are characterized by estimation procedures that use likelihood functions to minimize error [16,24]. The fit of the models by robust regression is performed by iteratively reweighted least squares (IRLS), in which weights are applied to the observations through minimization functions, aiming Forests 2020, 11, 313 3 of 20 to reduce the influence of outliers, whose residuals do not meet the traditional linear regression assumptions, especially normality [25,26]. Our study uses a large Ecuadorian data-set of O. pyramidale stands (containing outliers; n = 32,279 trees), in order to compare the performance of OLS and IRLS—when modeling height–diameter relationships. We assess model fitness and validate results using a split-sampling strategy. The overall aim is to derive best-modeling-practices to inform researchers and practitioners. The specific objectives of our study are to: (i) model height–diameter relationship of Ochroma pyramidale stands from Ecuador by site-index and age classes; (ii) compare the model performance of the regression methods using OLS and IRLS with robust M-estimators, by (iii) employing large data-sets for model fitting and validation (Approach 1) and a smaller data-set representing six outlier scenarios (Approach 2), and (iv) examine the result consistency of performed comparisons by the adjusted determination coefficient, the standard error of estimate (SEE%), and graphical residual analysis. We test the hypothesis that robust regressions provide greater accuracy of height–diameter models than OLS by site-index and age classes in O. pyramidale stands.

2. Materials and Methods We structure the analysis in two distinct approaches: Approach 1 implements a split-sample strategy to first (split one; n = 24,850) assess the modeling fitness of OLS and IRLS (from hereon, all IRLS estimations are assumed to use M-estimators), and then validate these results using a separate sampling split (n = 7429); Approach 2 uses a sub-sample strategy (one plot; n = 26) to conduct a sensitivity analysis of six artificially constructed outlier scenarios by systematically varying the arrangement or positioning of the outlier in order to assess the effects of these arrangements in both OLS and IRLS methods. Note that in Approach 1, we stratify the data-set by age and site-index:

(i) Stratiﬁcation into age classes. Given the aggregate summary statistics from Table1 (i.e., ages varied from 1.1 to 4.8 years), we determined that strata should not include ranges greater than one year. Therefore, four age classes were determined. (ii) Stratiﬁcation into site-index classes. Similarly, using aggregate results from Table1 (i.e., dominant heights [hdom] in the plots varied from 8.6 to 36.9 m), we decided on three site-index strata groupings.

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Table 1.TableDescriptive 1. Descriptive statisticsstatistics of height–diameter of height–diameter variables in variables the fit and invalidation the ﬁt data-sets and validation of data-sets of OchromaOchroma pyramidale pyramidale(balsa-tree) (balsa-tree) stands stands in Ecuador. in Ecuador. Fit Data-Set Validation Data-Set Classes Fit Data-Set Validation Data-Set Classes n Min. Avg. Max. cv% n Min. Avg. Max. cv% n Min.DBH Avg. Max. cv% n Min. Avg. Max. cv% Age I 1 2 y. 3514 3.8 17.4 29.8 DBH 21.3 975 6.2 18.0 27.7 18.4 AgeAge I II 21 ˧ 23 y.y. 26633514 9.8 3.8 22.2 17.4 39.1 19.3 29.8 716 21.3 9.8 975 21.7 6.236.0 19.1 18.0 27.7 18.4 AgeAge IIIII 32 ˧ 34 y.y. 21532663 11.8 9.8 25.6 22.2 43.2 19.7 39.1 487 19.3 10.9 716 24.0 9.843.3 20.0 21.7 36.0 19.1 AgeAge III IV 43 ˧ 45 y.y. 10182153 14.7 11.8 27.1 25.6 43.2 19.7 43.2 312 19.7 13.5 487 25.0 10.9 40.1 20.4 24.0 43.3 20.0 Age IV 4 5 y. 1018 14.7 27.1 43.2 19.7 312 13.5 25.0 40.1 20.4 ˧ AllAll data-setdata-set 1 5 y.y. 93489348 3.8 3.8 21.7 21.7 43.2 26.7 43.2 2.490 26.7 6.2 2.490 21.1 6.243.3 23.4 21.1 43.3 23.4 Site-indexSite-index III III 20 mm 2997 2997 10.0 10.0 23.0 23.0 43.2 24.4 43.2 24.4 Site-indexSite-index II II 25 mm 4508 4508 7.1 7.1 22.1 22.1 43.2 24.6 43.2 24.6 Site-indexSite-index I I 30 mm 1843 1843 3.8 3.8 18.6 18.6 37.4 30.0 37.4 30.0 Height (h)Height (h) AgeAge I I 1 ˧ 2 y.y. 35143514 3.0 3.0 15.3 15.3 24.0 19.9 24.0 975 19.9 6.8 975 16.3 6.824.5 16.4 16.3 24.5 16.4 AgeAge II II 2 ˧ 3 y.y. 26632663 7.8 7.8 21.0 21.0 31.2 16.3 31.2 716 16.3 9.3 716 21.2 9.331.0 16.5 21.2 31.0 16.5 AgeAge III III 3 ˧ 4 y.y. 21532153 14.0 14.0 24.5 24.5 37.0 14.7 37.0 487 14.7 14.3 487 23.6 14.3 33.3 14.2 23.6 33.3 14.2 Age IV 4 5 y. 1018 12.5 26.3 37.5 15.6 312 11.8 24.6 35.7 17.0 Age IV 4 ˧ 5 y. 1018 12.5 26.3 37.5 15.6 312 11.8 24.6 35.7 17.0 All data-set 1 5 y. 9348 3.0 20.2 37.5 26.7 2.490 6.8 20.2 35.7 23.1 Site-indexAll data-set III 1 20˧ 5 m y. 9348 2997 3.0 9.5 20.2 22.7 37.5 26.7 37.5 2.490 23.6 6.8 20.2 35.7 23.1 Site-index II III 2025 mm 2997 4508 9.5 7.2 22.7 20.3 37.5 23.6 34.0 22.3 Site-index I II 2530 mm 4508 1843 7.2 3.0 20.3 15.9 34.0 22.3 31.7 29.2 DBH = diameterSite-index at I breast 30 m height 1843 (cm), 3.0 h = 15.9total 31.7 height 29.2 (m), Min. = minimum value, Max. = maximum value, Avg. = average,DBH = diameter s = standard at breast height deviation (cm), (h =), total cv% he=ightcoe (m),ﬃcient Min. = of minimum variation value, in percentage,Max. = maximum y. = years. value, Avg. = average, s = standard deviation± (±), cv% = coefficient of variation in percentage, y. = years.

2.1.2. Site-Index The effects of site-index on height–diameter relationship were only assessed for the fit data-set; namely, site-index was only used to assess model fit and not validation. Measurements of dominant height—hdom (m) of the 100 largest trees per hectare were done following Assmann [28]. See Table 2 for descriptive statistics.

Table 2. Descriptive statistics of variables used in modelling of hdom for Ochroma pyramidale (balsa-tree) stands in Ecuador.

Variable Minimum Average Maximum s cv% hdom 8.6 21.2 36.9 5.2 24.3 DBH 3.8 25.7 43.2 5.8 26.5

DBH = diameter at breast height (cm), hdom = dominant height (m), s = standard deviation (±), cv% = coefficient of variation in percentage.

Table 3 details the two statistical models used to predict hdom as a function of age; these were used to determine site-index and the subsequent classification of plots. Model (1) uses linear regression for fit, whereas Model (2) uses non-linear.

Table 3. Statistical models used to predict dominant height.

Author Model 1 Schumacher (1939) [29] ln(h ) = β +β + ε (1) t Chapman and Richards (1959) [30] h = β (1 − e ) + ε (2)

hdom: dominant height (m), t: age (years), and β0, β1 and β2: regression parameters, ε: random error.

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2.1. Approach 1

2.1.1. Study Area Characterization and Data Collection Data collection was performed in O. pyramidale stands located in five provinces of Ecuador: Los Ríos, Cotopaxi, Guayas, Manabí, and Santo Domingo de Tsachilas, owned by PLANTABAL S.A.—3ACOREMATERIALS®. The altitude ranges from 75–500 m a.s.l. and, according to the Holdridge classification, the region is situated in Tropical Dry Forest and Tropical Moist Forest, with 1 800–3000 mm year− precipitation, average annual temperature around 25 ◦C and average annual air humidity of about 85%, with slightly uneven topography and soils composed mainly of silt and clay and pH around 6.5–7.0 [5,27]. As previously indicated, Approach 1 splits the full sample of 1414 permanent plots and 32,279 trees into two data-sets: the first data-set denoted as fit data-set was used to fit the equations described below (call this fit data-set) and determine which of these is the most representative model by site-index and age classes (i.e., model with the closest fit); the second validation data-set sample aimed at validating the performance of the most representative model (called validation data-set). On all plots the DBH was measured for all trees and the height of a sub-sample of trees. The DBH was measured with a diameter tape at 1.3 m from the ground and recorded in cm, and the height was determined with a Haglöf clinometers and recorded in m. The fit data-set contains information collected on 1112 permanent plots, with 24,850 trees Forests(1.1–4.8 2020 years, 11, x old),FOR PEER of which REVIEW 9348 have paired measurements of DBH and h. The validation data-set4 of 21 contains information on 7429 trees (1.1 to 4.7 years; 302 permanent plots), of which 2429 trees have dispersionpaired measurements for both data-sets. of DBH and The h. dotted Figure 1line features corresponds a scatter to plot the which average illustrates trend theof the data fit dispersion data-set, whereasfor both data-sets.the solid relates The dotted to the linevalidation corresponds data-set. to the average trend of the fit data-set, whereas the solid relates to the validation data-set.

Figure 1.1. DispersionDispersion of of paired paired diameter diameter at breastat breast height height (DBH) (DBH) and h and data h in data ﬁt and in validationfit and validation data-sets data-setsused to represent used to represent the height–diameter the height–diameter relationship relationship (Approach (Approach 1). 1).

Figure2 2 features features the the stratification stratification of theof the fit data-set,fit data-set, by age by classes.age classes. Visual Visual inspection inspection of this of figure this figurehighlights highlights noticeable noticeable dispersions dispersions between between DBH and DB h,H along and withh, along some with possible some outliers; possible it outliers; is important it is importantto note that to such note dispersions that such dispersions are visually are noticeable visually in noticeable these short in one-year these short periods, one-year with periods, amplitude with in amplitudeheight greater in height than 20 greater m in allthan cases. 20 m in all cases. Table1 features descriptive statistics of DBH and h variables for the fit data-set and the validation data-set. The coefficient of variation (cv%) indicates the variability which appears heights in the aggregate rows (labeled ‘all data-set’). Note: some of the rows of validation data-set were intentionally

Figure 2. Dispersion of paired DBH and h data by age classes used to fit models to represent the height–diameter relationship (Approach 1).

Table 1 features descriptive statistics of DBH and h variables for the fit data-set and the validation data-set. The coefficient of variation (cv%) indicates the variability which appears heights in the aggregate rows (labeled ‘all data-set’). Note: some of the rows of validation data-set were intentionally excluded (i.e., stratification by site-index class) given that these were deemed unnecessary as the validation was only performed for the aggregate data (i.e., all data-sets).

Forests 2020, 11, x FOR PEER REVIEW 4 of 21 dispersion for both data-sets. The dotted line corresponds to the average trend of the fit data-set, whereas the solid relates to the validation data-set.

Figure 1. Dispersion of paired diameter at breast height (DBH) and h data in fit and validation data-sets used to represent the height–diameter relationship (Approach 1).

Forests 2020, 11, 313 5 of 20 Figure 2 features the stratification of the fit data-set, by age classes. Visual inspection of this figure highlights noticeable dispersions between DBH and h, along with some possible outliers; it is importantexcluded (i.e.,to note stratiﬁcation that such dispersions by site-index are class) visually given noticeable that these in these were short deemed one-year unnecessary periods, as with the amplitudevalidation wasin height only performedgreater than for 20 the m in aggregate all cases. data (i.e., all data-sets).

Figure 2. DispersionDispersion of of paired paired DBH DBH and and h h data data by by age age classes classes used used to to fit fit models to represent the height–diameter relationship (Approach 1). 2.1.2. Site-Index Table 1 features descriptive statistics of DBH and h variables for the fit data-set and the validationThe e ffdata-set.ects of site-index The coefficient on height–diameter of variation (cv%) relationship indicates werethe variability only assessed which for appears the fit data-set;heights innamely, the aggregate site-index rows was (labeled only used ‘all to data-set’). assess model Note fit: andsome not of validation. the rows of Measurements validation data-set of dominant were intentionallyheight—hdom excluded(m) of the 100(i.e., largest stratification trees per by hectare site-index were doneclass) following given that Assmann these [28were]. See deemed Table2 unnecessaryfor descriptive as statistics.the validation was only performed for the aggregate data (i.e., all data-sets).

Table 2. Descriptive statistics of variables used in modelling of hdom for Ochroma pyramidale (balsa-tree) stands in Ecuador.

Variable Minimum Average Maximum s cv%

hdom 8.6 21.2 36.9 5.2 24.3 DBH 3.8 25.7 43.2 5.8 26.5

DBH = diameter at breast height (cm), hdom = dominant height (m), s = standard deviation ( ), cv% = coeﬃcient of variation in percentage. ±

Table3 details the two statistical models used to predict h dom as a function of age; these were used to determine site-index and the subsequent classiﬁcation of plots. Model (1) uses linear regression for ﬁt, whereas Model (2) uses non-linear.

Table 3. Statistical models used to predict dominant height.

Author Model 1 Schumacher (1939) [29] ln(hdom) = β0 + β1 t + ε (1) β β2 Chapman and Richards (1959) [30] h = β (1 e 1 t) + ε (2) dom 0 − − hdom: dominant height (m), t: age (years), and β0, β1 and β2: regression parameters, ε: random error.

The selection criteria used to determine the most representative hdom model is based on: the 2 adjusted determination coeﬃcient (R adj.); the standard error of estimate in percent (SEE%); along with a graphical analysis of standardized residuals. These measures were also used to determine the most appropriate height–diameter model (more on this below). Forests 2020, 11, 313 6 of 20

For Model (1), we use a logarithmic transformation (Meyer correction factor [MCF]) [31] to correct for the log back-transformation regarding the hdom predicted values (see Equation (1)); more speciﬁcally, the following was used to recalculate the SEE%:

MCF = e0.5.MSE (1) where MSE is mean square error. The corresponding values of the guide curve come from the dominant heights of the 1112 permanent plots (i.e., aggregate ﬁt data-set). Once the most representative model was selected, the guide curve method was used to predict the site-index curves. Three site-index classes were considered based on the amplitude of the hdom data at the reference age (aRef). The aRef was set at 3.5 years since it was close to the O. pyramidale age of clear-cutting and replanting system with harvesting (5.0 years old stand).

2.1.3. Height–Diameter Relationship Table4 features the four classic models used to describe the height–diameter relationship. These models were ﬁtted in the following manner in three ways: (i) using the aggregate set of observations (i.e., ‘all data-set model’); (ii) stratifying by age; and (iii) stratifying by site-index. These various model speciﬁcations were used to determine the most appropriate method to predict the heights.

Table 4.ForestsStatistical 2020, 11 models, x FOR for PEER estimating REVIEW total height. 5 of 21

Author (Year) Statistical Model Table 1. Descriptive statistics of height–diameter variables in the fit and validation data-sets of Henriksen (1950) [32]Ochroma h =pyramidaleβ0 + β1 (balsa-tree)ln(DBH) + standsε in Ecuador. (3) Curtis (1967) [33] ln h = β0 + β11/(DBH) + ε (4) Stoﬀels and Soest (1953) [34] ln h = β0 + β1ln (DBH) + ε Fit Data-Set (5) Validation Data-Set Classes2 2 Näslund (1936) [35] h = (DBH /(β0 + β1 DBHn ) Min.) + 1.3 Avg.+ ε Max.(6) cv% n Min. Avg. Max. cv%

h: total height (m), DBH: diameter at breast height (cm), β0, β1 and β2: regression parameters, DBHε: statistical error. Age I 1 2 y. 3514 3.8 17.4 29.8 21.3 975 6.2 18.0 27.7 18.4 The data were stratified into four ageAge classes II (I: 12 ˧ 32 y. years, 2663 II: 2 9.83 years, 22.2 III: 39.1 3 4 19.3 years 716 and 9.8 21.7 36.0 19.1 IV: 4 5 years) and three site-index classesAge (I, IIIII e III). 3 In˧ 4 eachy. 2153 class, 11.8 two modeling 25.6 43.2 methods 19.7 were 487 10.9 24.0 43.3 20.0 tested: traditional linear regression withAge ordinary IV least 4 ˧ 5 squares y. 1018 ( OLS 14.7) and 27.1 robust 43.2 regression 19.7 312 by 13.5 25.0 40.1 20.4 iteratively reweighted least squares (IRLS),All in data-set which three 1 ˧maximum 5 y. 9348 likelihood 3.8 21.7 estimators 43.2 were26.7 tested 2.490 6.2 21.1 43.3 23.4 (M-estimators): Huber, Hampel, and BiweigthSite-index (Tukey), III 20 whose m 2997 objective 10.0 and 23.0 weight 43.2 functions 24.4 are described in Table5. Site-index II 25 m 4508 7.1 22.1 43.2 24.6 According to the condition of each estimator,Site-index the I value 30 m of breakdown 1843 3.8 point 18.6 (k) 37.4 in the 30.0 case of the Huber and Biweight estimators, and the values of a, b, and c of the Hampel estimator,Height (h) correspond to the restrictions that determine which weightAge function I is 1applied ˧ 2 y. to 3514 the residuals. 3.0 15.3 In the 24.0 first 19.9 case, small 975 6.8 16.3 24.5 16.4 values of k penalize more outliers, but reduceAge accuracy II 2 when˧ 3 y. the 2663 residuals 7.8 are 21.0 normally 31.2 distributed. 16.3 716 9.3 21.2 31.0 16.5 Because of this, it is suggested to apply k = 1.345Age III which guarantees3 ˧ 4 y. 2153 a loss 14.0 of effi 24.5ciency 37.0 of up 14.7 to 5% 487 over 14.3 23.6 33.3 14.2 OLS [13,18,20]. Age IV 4 ˧ 5 y. 1018 12.5 26.3 37.5 15.6 312 11.8 24.6 35.7 17.0 The ranking of the models fitted to representAll data-set height–diameter 1 ˧ 5 y. 9348 relationship 3.0 20.2 by site-index 37.5 26.7 and 2.490 age 6.8 20.2 35.7 23.1 classes was based on their accuracy, for whichSite-index the relative III 20 change m 2997 of SEE% 9.5 was 22.7 quantified 37.5 by 23.6 applying both estimation methods (IRLS and OLS).Site-index II 25 m 4508 7.2 20.3 34.0 22.3 To verify the linear regression assumptions,Site-index Lilliefors’s I 30 testm (D) 1843 with 3.095% probability 15.9 31.7 level 29.2 was used to assess normality, as well as the graphicDBH analysis = diameter of theat breast residuals height using (cm), theh = normaltotal height quantile-quantile (m), Min. = minimum value, Max. = maximum plot (QQ-plot). To evaluate the homoscedasticity,value, Avg. = Breusch-Pagan’saverage, s = standard (BP) deviation test [36] with(±), cv% 95% = probabilitycoefficient of variation in percentage, y. = level was applied and the standardizedyears. residuals graphs were analyzed. The tests were used to examine for violations: given that no violations could be identified, we proceeded to evaluate the significance of the corresponding2.1.2. estimated Site-Index parameters. The effects of site-index on height–diameter relationship were only assessed for the fit data-set; namely, site-index was only used to assess model fit and not validation. Measurements of dominant height—hdom (m) of the 100 largest trees per hectare were done following Assmann [28]. See Table 2 for descriptive statistics.

Table 2. Descriptive statistics of variables used in modelling of hdom for Ochroma pyramidale (balsa-tree) stands in Ecuador.

Variable Minimum Average Maximum s cv% hdom 8.6 21.2 36.9 5.2 24.3 DBH 3.8 25.7 43.2 5.8 26.5

DBH = diameter at breast height (cm), hdom = dominant height (m), s = standard deviation (±), cv% = coefficient of variation in percentage.

Table 3. Statistical models used to predict dominant height.

Author Model 1 Schumacher (1939) [29] ln(h ) = β +β + ε (1) t Chapman and Richards (1959) [30] h = β (1 − e ) + ε (2)

hdom: dominant height (m), t: age (years), and β0, β1 and β2: regression parameters, ε: random error.

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Complementing the analysis, the coeﬃcient of variation (cv%) was calculated and the signiﬁcance of regression parameters was evaluated by Student’s t-test with 95% probability level. Data processing were performed using R® software, version 3.6.1. [37]—the MASS package for R [38] (with the rlm function) was used to perform robust regression.

Table 5. Functions and condition of maximum likelihood estimators (M-estimators).

Estimator Objective Function ρ(e) Weight Function w(e) Condition OLS e2 1 e2 1 For |e| k (7) 2 ≤ Huber 2 k e k k/ e For |e| > k (8) | | − 2 | | ( 3) 2 2 22 k 1 1 e 1 e For |e| k (9) Biweight (Tukey) 6 − − k − k ≤ 2 k 0 For |e| > k (10) 6 e2 1 For |e| a (11) 2 ≤ a2 a For a e b (12) a e 2 e < | | Hampel | | − | | ≤ 2 a c e e a(c e ) 2 −| | For b < |e| c (13) | |− 7 a2 e (c b) c b 6 | | − ≤ − − a(b + c a) 0 For |e| > c (14) − k: breakdown point constant, a = 1.7, b = 3.4, c = 8.5, e: error of residuals. Source: Montgomery et al. [14].

In summary, all the above model specifications and tests for assumptions were used to select the most representative model for the aggregate fit data-set (i.e., all data-set) and by site-index and age classes; this fit model for all data was subsequently validated using the validation data-set. Our validation strategy follows [26,39]; as such, a data-set with 2500 observations (~20%) was randomly sampled (from the full data-set), and the adherence between the observed and predicted data was determined by using the Chi-square test (χ2) with 95% probability level; thus, we test the hypothesis that predicted estimates by OLS or IRLS are not different than the observed values. This analysis was complemented with the density histogram performed using the geom_density function of the ggplot2 package [40] of R software [37]. Additionally, the standardized residuals plot was evaluated, and the SEE% was calculated. It is important to note that all equations featured by this study (i.e., assessed for performance) were validated using data from the same sampling regions of Ecuador. Thus, data for validation were not independent. However, the goal of independent data validation to assess the generalizability of performance of these equations, in other region (or stands), was not the focus of this study. Our contribution, as initially noted, was to demonstrate empirically an improvement to the standard research approach (using OLS with species with high variation). Lastly, we further note that our sampling region contains roughly 80%–90% of all O. pyramidale stands worldwide, therefore, our results carry representative strength in terms of this particular species.

2.2. Approach 2 Approach 2 uses a smaller data-set of O. pyramidale (1 plot test case; n = 26 trees), to conduct a sensitivity analysis which compares the Henriksen’s model (fitted by OLS; henceforth OLS) with the Huber estimator of IRLS (henceforth IRLS). The raw data from this approach come from 1 permanent plot that was selected from the aggregate fit data-set (1112 plots). This plot includes 26 trees of the same age (4.7 years). We note that the number of trees in this particular plot is slightly higher than the average number of trees-per-plot from the entire sample used in Approach 1 (~22 trees-per-plot). The following six sensitivity analysis scenarios were artificially constructed: 1 scenario with no outliers (S1); and 5 other scenarios containing two different outliers (S2, S3, S4, S5, and S6; see Figure3). Forests 2020, 11, x FOR PEER REVIEW 8 of 21

The raw data from this approach come from 1 permanent plot that was selected from the aggregate fit data-set (1112 plots). This plot includes 26 trees of the same age (4.7 years). We note that the number of trees in this particular plot is slightly higher than the average number of trees-per-plot from the entire sample used in Approach 1 (~22 trees-per-plot). The following six sensitivity analysis scenarios were artificially constructed: 1 scenario with no outliers (S1); and 5 other scenarios containing two different outliers (S2, S3, S4, S5, and S6; see Figure 3). The artificially created outliers included in scenarios 2–6 were added visually. This visual strategy was guided by two main considerations: (1) the definition of an outlier provided by Montgomery et al. [14]; and (2) the working experience of field teams re-measuring the sample plots. Montgomery et al. [14] describe similar scenarios; the authors also note that such situations (similar to Figure 3) occur fairly often in practice, adding that, in general, researchers should be awareForests 2020 that, 11 in, 313 some data-sets, one point (or a small cluster of points) may influence key modeling8 of 20 properties. These scenarios were artificially developed in order to examine model performance using typicalThe artificially outlier observations created outliers that included could be in found scenarios in fo 2–6rest were allometric added databases; visually. This the visualcapacity strategy to control was forguided the placement by two main of considerations:these also simplified (1) the the definition analysis of which an outlier enables provided the understanding by Montgomery the et specific al. [14]; objectivesand (2) the of working this research experience (i.e., difference of field teams between re-measuring (OLS and the IRLS). sample plots.

Figure 3. SensitivitySensitivity analysis analysis of of 6 6 scenarios scenarios (S1 (S1 to to S6) S6) fr fromom a subsample with 26 trees of the same (4.7 years), 55 scenariosscenarios containing containing artiﬁcially artificially created create outliersd outliers (Approach (Approach 2) for modeling2) for modeling of height–diameter of height– diameterrelationship relationship of O. pyramidale of O. pyramidalein Ecuador. in Ecuador.

3. ResultsMontgomery et al. [14] describe similar scenarios; the authors also note that such situations (similar to Figure3) occur fairly often in practice, adding that, in general, researchers should be aware 3.1.that Approach in some data-sets,1 one point (or a small cluster of points) may influence key modeling properties. These scenarios were artificially developed in order to examine model performance using typical 3.1.1.outlier Site-Index observations that could be found in forest allometric databases; the capacity to control for the placement of these also simplified the analysis which enables the understanding the specific objectives According to the fit statistics evaluated, the Champan–Richards’ model (2) in Table 3 was of this research (i.e., difference between (OLS and IRLS). selected to describe the behavior of hdom, which presented R2adj. = 0.709 and SEE% = 12.71%. The regression3. Results parameters for Equation (2) were significant (p < 0.01). h = 29.85162 (1 − e. ). (2) 3.1. Approach 1

3.1.1. Site-Index According to the ﬁt statistics evaluated, the Champan–Richards’ model (2) in Table3 was selected 2 to describe the behavior of hdom, which presented R adj. = 0.709 and SEE% = 12.71%. The regression parameters for Equation (2) were signiﬁcant (p < 0.01).

0.67151 age 1.51484 h = 29.85162 (1 e− ) (2) dom − Graphical analysis of the residuals (Figure4a) shows that most of them are distributed between 2 − to 2 standard deviations along the predicted line. The site-index curves from Equation (2) (Figure4b) allowed to classify all plots in diﬀerent classes, resulting in 33.7% in site-index class I, 48.1% in class II and 18.2% in class III. Forests 2020, 11, x FOR PEER REVIEW 9 of 21

Graphical analysis of the residuals (Figure 4a) shows that most of them are distributed between −2 to 2 standard deviations along the predicted line. The site-index curves from Equation

Forests(2) (Figure2020, 11, 4b) 313 allowed to classify all plots in different classes, resulting in 33.7% in site-index 9class of 20 I, 48.1% in class II and 18.2% in class III.

(a) (b)

FigureFigure 4. Standardized4. Standardized residuals residuals for predicted for predicted hdom ( ah)dom and ( site-indexa) and site-index curves (b )curves for Ochroma (b) for pyramidale Ochroma (balsa-tree)pyramidale stands (balsa-tree) in Ecuador. stands in Ecuador.

3.1.2.3.1.2. Height–Diameter Height–Diameter Relationship Relationship TheThe most most representative representative height–diameter height–diameter models mode fittedls byfitted site-index by site-index and age classesand age are presentedclasses are inpresented Table6. Näslund’s in Table model6. Näslund’s presented model better presented fit in site-index better fit classes in site-index and all data-set classes throughand all robustdata-set regressionthrough robust using Biweightregression estimator. using Biweight For age estimator. classes, Curtis’s For age model classes, appears Curtis’s to omodelffer the appears closest to fit offer (as comparedthe closest to fit the (as observed compared data), to thisthe observed is followed data), by Henriksen; this is followed however, by Henriksen; its fits were however, better through its fits ForestsOLSwere 2020 method, , 11better, x FOR PEER exceptthrough REVIEW for OLS class method, III, where except IRLS for was class more II statisticallyI, where IRLS accurate was5 of 21more with Hampel’sstatistically estimator. accurate with Hampel’s estimator. TableTable 1. Descriptive 6. Statistics statistics of theof height–diameter most representative variables in fit the models fit and forvalidation estimating data-sets the of total height of Ochroma Ochroma pyramidale (balsa-tree) stands in Ecuador. pyramidale Table 6. (balsa-tree)Statistics of treesthe most by site-index representative and age fit classes models in Ecuador.for estimating the total height of Ochroma Fit Data-Set Validation Data-Set Classes Pyramidale (balsa-tree) trees by site-indexTraditional and age classes in Ecuador. n Min. Avg. Max. cv% n Min. Avg. Max. cv%Robust Regression (IRLS) Classes Selected Model DBHRegression (OLS) 2 Traditional 2 Age I 1 2 y. 3514 3.8 17.4 29.8R adj. 21.3 975SEE% 6.2 Best 18.0 Method 27.7 18.4 R adj. SEE% Change in SEE% AgeAge I II 21 ˧ 23 y.y. 2663 Curtis 9.8 22.2 39.1 0.556 19.3Regression 71613.272 9.8 21.7 Hampel 36.0Robust 19.1 0.556 Regression 13.273 (IRLS) 0.0130% Selected AgeAge II IIIClasses 32 ˧ 34 y.y. 2153 Curtis 11.8 25.6 43.2 0.434 19.7(OLS) 48712.247 10.9 24.0 Hampel 43.3 20.0 0.434 12.247 0.0038% Age IV 4 ˧ 5 y. 1018 14.7Model 27.1 43.2 19.7 312 13.5 25.0 40.1 20.4 Age III 3 4 y. Henriksen 0.353 11.852 HampelBest 0.354 11.852Change 0.0003% in All data-set 1 ˧ 5 y. 9348 3.8 21.7 43.2 26.72adj. 2.490 6.2 21.1 43.3 23.4 2adj. Age IV 4 5 y. Curtis 0.330R 12.748SEE% HampelR 0.330 SEE% 12.747 0.0018% Site-index III 20 m 2997 10.0 23.0 43.2 24.4 Method −SEE% All data-set Näslund 0.659 15.580 Biweight 0.662 15.516 0.4101% Site-index II 25 m1 ˧4508 2 7.1 22.1 43.2 24.6 − Site-index I 30 m Näslund 0.646 14.047 Biweight 0.648 14.003 0.3084% Site-indexAge I I 30 m 1843 3.8Curtis 18.6 37.4 0.556 30.0 13.272 Hampel 0.556 13.273 0.0130%− Site-index II 25 my. Näslund 0.624 13.673 Biweight 0.626 13.629 0.3219% Height (h) − Site-index III 20 m Näslund 0.701 15.975 Biweight 0.704 15.903 0.4481% Age I 1 ˧ 2 y.2 ˧ 3514 3 3.0 15.3 24.0 19.9 975 6.8 16.3 24.5 16.4 − 2 Age II Curtis 0.434 12.247 Hampel 0.434 12.247 0.0038% R ajd.Age: adjusted II 2 determination˧ 3 y. y. 2663 7.8 coe ffi 21.0cient, 31.2SEE%: 16.3 standard 716 error 9.3 of estimate 21.2 31.0 in percent, 16.5 cv%: coefficient of variation in percent,Age III y.: years.3 ˧ 4 y. It 2153 corresponds 14.0 24.5 to an 37.0 age close 14.7 to 487 the moment 14.3 23.6 when 33.3 the clear-cutting 14.2 and replanting system with harvesting is performed.3 ˧ 4 This age is used as a reference to estimate the site-index curves. AgeAge IV III 4 ˧ 5 y. 1018 12.5Henriksen 26.3 37.5 0.353 15.6 31211.852 11.8 24.6Hampel 35.7 17.00.354 11.852 0.0003% All data-set 1 ˧ 5 y. y. 9348 3.0 20.2 37.5 26.7 2.490 6.8 20.2 35.7 23.1 Site-index III 20 m4 ˧2997 5 9.5 22.7 37.5 23.6 TheAge stratified IV model resultsCurtis for site-index0.330 12.748 classes Hampel indicate that0.330 SEE% 12.747 reduces −0.0018% in smaller Site-index II 25 m y. 4508 7.2 20.3 34.0 22.3 classificationsSite-index I (goes 30 m from 1843 0.3084% 3.0 15.9 to 31.7 0.4481%)—this 29.2 applies to models fitted by IRLS with the All data-set Näslund 0.659 15.580 Biweight 0.662 15.516 −0.4101% BiweightDBH = diameter estimator at breast in height comparison (cm), h = total with height the (m), OLS Min. estimator= minimum value, (Table Max.6). = maximum In the fits by age classes there was bettervalue,Site-index accuracy Avg. = average, byI thes 30= standard OLSm method, deviation Näslund (±), except cv% 0.646= coefficient for class 14.047 of IV, variation where Biweight in percentage, the reduction 0.648 y. = in 14.003 SEE% of 0.0018%−0.3084% was observedyears.Site-index by applying II 25 robustm Näslund regression through 0.624 Hampel’s 13.673 Biweight estimator. 0.626 13.629 −0.3219% 2.1.2. Site-IndexSite-indexThe regression III 20 parameters m Näslund and their 0.7 respective01 15.975 standardBiweight errors 0.704 of the 15.903 most representative−0.4481% R2ajd.: adjusted determination coefficient, SEE%: standard error of estimate in percent, cv%: coefficient height–diameterThe effects of site-index models on height–diameter fitted by site-index relationship and were age only classes assessed are for shown the fit data-set; in Table 7, in which they were namely,significant site-indexof variation (p < was0.05). onlyin usedpercent, to assess y.: modelyears. fit andIt notcorresponds validation. Measurementsto an age close of dominant to the moment when the height—hdom (m) of the 100 largest trees per hectare were done following Assmann [28]. See Table 2 for descriptive statistics.

Table 2. Descriptive statistics of variables used in modelling of hdom for Ochroma pyramidale (balsa-tree) stands in Ecuador.

Variable Minimum Average Maximum s cv% hdom 8.6 21.2 36.9 5.2 24.3 DBH 3.8 25.7 43.2 5.8 26.5

DBH = diameter at breast height (cm), hdom = dominant height (m), s = standard deviation (±), cv% = coefficient of variation in percentage.

Table 3. Statistical models used to predict dominant height.

Author Model 1 Schumacher (1939) [29] ln(h ) = β +β + ε (1) t Chapman and Richards (1959) [30] h = β (1 − e ) + ε (2)

hdom: dominant height (m), t: age (years), and β0, β1 and β2: regression parameters, ε: random error.

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Table 7. Parameters standard errors of the most representative ﬁt models for estimating the total height of Ochroma pyramidale (balsa-tree) trees by site-index and age classes in Ecuador.

Class Selected Model Estimator Parameters S.E. dif. S.E. OLS 3.3512 * 0.0090260055 β0 IRLS (Hampel) 3.3538 * 0.0089116163 Age I Curtis 1.27% OLS 10.6435 * 0.1431225471 − β1 − IRLS (Hampel) 10.6439 * 0.1413087132 − OLS 3.5606 * 0.0114515061 β0 IRLS (Hampel) 3.5557 * 0.0112676466 Age II Curtis 1.61% OLS 11.3617 * 0.2385131519 − β1 − IRLS (Hampel) 11.215 * 0.2346837088 − OLS 9.6536 * 0.9962025945 β0 − IRLS (Hampel) 9.781 * 1.0009075994 Age III Henriksen − 0.47% OLS 10.5866 * 0.3086080742 β1 IRLS (Hampel) 10.6257 * 0.3100656116 OLS 3.7058 * 0.0200450964 β0 IRLS (Hampel) 3.7021 * 0.0199973835 Age IV Curtis 0.24% OLS 11.7049 * 0.5095871877 − β1 − IRLS (Hampel) 11.5539 * 0.5083742299 − OLS 2.2102 * 0.0170358260 β0 IRLS (Biweight) 2.2078 * 0.0170576048 All data-set Näslund 0.13% OLS 0.1282 * 0.0007591147 β1 IRLS (Biweight) 0.1275 * 0.0007600852 OLS 2.1079 * 0.0288437402 β0 IRLS (Biweight) 2.1066 * 0.0294409095 Site-index I Näslund 2.07% OLS 0.1246 * 0.0012161606 β1 IRLS (Biweight) 0.1241 * 0.0012413395 OLS 1.9520 * 0.0232523838 β0 IRLS (Biweight) 1.9676 * 0.0235008560 Site-index II Näslund 1.07% OLS 0.1404 * 0.0010234409 β1 IRLS (Biweight) 0.139 * 0.0010343773 OLS 2.1596 * 0.0346975551 β0 IRLS (Biweight) 2.1696 * 0.0341338255 Site-index III Näslund 1.62% OLS 0.1456 * 0.0017865903 − β1 IRLS (Biweight) 0.1441 * 0.0017575637

βˆ i: regression parameters, * signiﬁcant with 95% probability level, S.E.: standard error, dif. S.E.: Relative diﬀerence between the standard errors of regression parameters.

Approach 1 presented relative difference between standard errors by OLS and IRLS less than 3% (Table7). The negative percentage values in dif. S.E. indicate that the regression parameters standard error by IRLS was higher than OLS. The quality of fit and consequently the predictions by classes of site-index and age were graphically verified (Figure5). Data stratification resulted in homogeneous distribution of observations, where the best site-index (I) and the highest age (IV) presented the highest total tree heights. When evaluating the assumptions of the regression analysis, it was observed that, in most cases, the normality of the residuals was rejected by the Lilliefors test (p < 0.05). However, the normal QQ-plot showed that the residuals follow the normal distribution trend, with some outliers on two-tailed (Figure6). Forests 2020, 11, x FOR PEER REVIEW 11 of 21

Approach 1 presented relative difference between standard errors by OLS and IRLS less than 3% (Table 7). The negative percentage values in dif. S.E. indicate that the regression parameters standard error by IRLS was higher than OLS. The quality of fit and consequently the predictions by classes of site-index and age were graphically verified (Figure 5). Data stratification resulted in homogeneous distribution of observations, where the best site-index (I) and the highest age (IV) presented the highest total tree Forestsheights.2020, 11, 313 11 of 20

FigureFigure 5. Predicted 5. Predicted curves curves for for the the most most representativerepresentative model model by by classes classes of site-index of site-index and andage for age for OchromaOchroma pyramidale pyramidale(balsa-tree) (balsa-tree) stands stands in in Ecuador. Ecuador. Forests 2020, 11, x FOR PEER REVIEW 12 of 21 When evaluating the assumptions of the regression analysis, it was observed that, in most cases, the normality of the residuals was rejected by the Lilliefors test (p < 0.05). However, the normal QQ-plot showed that the residuals follow the normal distribution trend, with some outliers on two-tailed (Figure 6).

Figure 6. Graphical analysis of normality for the most representative model fitted by site-index and age Figure 6. Graphical analysis of normality for the most representative model fitted by site-index and classes for Ochroma pyramidale (balsa-tree) stands in Ecuador. D: Lilliefors’ test value. * is statistically age classes for Ochroma pyramidale (balsa-tree) stands in Ecuador. D: Lilliefors’ test value. * is significant at p < 0.05, ns is not statistically significant at p < 0.05. statistically significant at p < 0.05, ns is not statistically significant at p < 0.05.

The graphicalThe graphical analysis analysis of of residuals residuals of of modelsmodels adjusted adjusted by by site-index site-index and and age ageclasses classes (Figure (Figure 7), 7), doesdoes not seemnot seem to indicateto indicate the the presence presence of of aa greatergreater bias along along the the midline. midline. This This is in is spite in spite of the of the Breusch–Pagan test rejecting the homogeneity hypothesis in almost all adjustments (p < 0.05), except for age classes III and IV.

Forests 2020, 11, 313 12 of 20

Breusch–Pagan test rejecting the homogeneity hypothesis in almost all adjustments (p < 0.05), except for age classesForests 2020 III, and 11, x FOR IV. PEER REVIEW 13 of 21

Forests 2020, 11, x FOR PEER REVIEW 13 of 21

Figure 7. Standardized residual plot for the most representative models fitted by site-index and age Figure 7. Standardized residual plot for the most representative models fitted by site-index and age classes for Ochroma pyramidale (balsa-tree) stands in Ecuador. BP: Breusch–Pagan’s test value. * is classes forFigurestatisticallyOchroma 7. Standardized significant pyramidale atresidual p (balsa-tree)< 0.05, plot ns is for not the standsstatistically most repres in significant Ecuador.entative atmodels BP:p < 0.05. Breusch–Pagan’sfitted by site-index and test age value. * is statisticallyclasses significant for Ochroma at p pyramidale< 0.05, ns (balsa-tree)is not statistically stands in Ecuador. significant BP: Breusch–Pagan’s at p < 0.05. test value. * is statisticallyThe results significant from the at p validation< 0.05, ns is not data-set statistically confirm significant the atNäslund p < 0.05. model adjusted by IRLS The results(Biweight) from as the the most validation representative data-set fitted confirm model for the all data-sets Näslund (i.e., model no stratification). adjusted byThisIRLS result(Biweight) The results from the validation data-set2 confirm the Näslund model adjusted by IRLS as the mostcan representative be seen in Figure fitted 8a; namely, model the for χ all test data-sets did not reveal (i.e., nosufficient stratification). evidence to This reject result the null can be seen (Biweight)hypothesis as(at the the most 95% representativelevel); furthermore, fitted modelthe SEE% for forall data-setsthis model (i.e., was no similar stratification). to that obtained This result in χ2 2 in Figure8canthea; namely,fitbe data-setseen in the Figure(15.516%). 8a;test namely,In didsummary, not the revealχ this test appears did su ffinot cientto reveal show evidence sufficientthat the equation toevidence reject obtained to the reject null fromthe hypothesis null the (at the 95% level);hypothesisNäslund furthermore, model (at the adjusted 95% level); the by SEE% IRLSfurthermore, (Biweight) for this the model canSEE% be forused was this to similar model appropriately was to thatsimilar represent obtained to that obtainedthe in height– the in fit data-set the fit data-set (15.516%). In summary, this appears to show that the equation obtained from the (15.516%).diameter In summary, relationship this in appearsO. pyramidale to using show the that non-stratified the equation data-set. obtained from the Näslund model Näslund model adjusted by IRLS (Biweight) can be used to appropriately represent the height– adjusted bydiameterIRLS (Biweight)relationship in can O. pyramidale be used using to appropriately the non-stratified represent data-set. the height–diameter relationship in O. pyramidale using the non-stratified data-set.

(a) (b)

Figure 8. Density histogram(a) (a) and standardized residuals (b) in the (validation.b) X2: chi-square test; ns: not significant with 95% probability level; SEE%: standard error of estimate in percent. 2 Figure 8. DensityFigure 8. Density histogram histogram (a) and (a) standardizedand standardized residuals residuals ( (bb)) in in the the validation. validation. X : chi-square X2: chi-square test; test; ns: ns: not significant with 95% probability level; SEE%: standard error of estimate in percent. not signiﬁcant with 95% probability level; SEE%: standard error of estimate in percent.

Additionally, the density histogram shows that the distribution of observed and predicted data is very similar, with greater symmetry in the predicted values (Figure8a). For standardized residuals, they did not show apparent bias (Figure8b). Forests 2020, 11, 313 13 of 20

3.2. Approach 2 Statistics for the Henriksen model ﬁtted by OLS and IRLS (Hampel) in the six scenarios evaluated 2 in this approach are presented in Table8. In all cases, there were higher values of R adj. and lower 2 SEE% values using the OLS method. The negative value of R adj. in scenario 3 is due to the high value of the sum of squares of the residual due to the outlier.

Table 8. Statistics for Henriksen’s model ﬁtted to predict total height in Ochroma pyramidale trees of age 4.7 in Ecuador (Approach 2).

OLS IRLS (Hampel) n Scenarios cv% 2 2 R adj. SEE% R adj. SEE% Change in SEE% 1 8.26 0.3693 6.5604 0.3693 6.5604 0.0000% 2 22.98 0.4337 17.2913 0.2763 19.5469 13.0448% 3 21.40 0.0040 21.3597 0.0674 22.1113 3.5188% 26 − 4 21.40 0.5819 13.8394 0.4175 16.3342 18.0272% 5 24.48 0.3665 19.4828 0.2327 21.4422 10.0574% Forests 2020, 116, x FOR PEER REVIEW 21.86 0.0383 21.4387 0.0365 21.4585 0.0926% 15 of 21 2 n: number of trees, cv%: coefficient of variation in percent, R adj.: adjusted coefficient of determination, SEE%: standard error of estimateIRLS in (Hampel) percent. 7.6725 * 1.7832962517 OLS 0.5067 17.1923557670 β0 IRLS (Hampel) 3.2965 6.0638269675 The parameters6 of the fitted equations from the Henriksen model for the six scenarios−64.73% of Approach 2, OLS 7.4412 5.2690135673 β1 as well as their standardIRLS errors (Hampel) and the difference 6.5448 between * them, 1.8584065497 are shown in Table9. Except for scenario 1 (withoutβi: regression outlier), parameters, in all * significant cases the with IRLS 95% (Hampel) probability, methodS.E.: standard resulted error, dif. in theS.E.: lowestRelative parameters standard error,difference whose between difference the standard with errors OLS of ranged regression from parameters. 38.11%–82.81%. In Figure9, the OLS fit curves are compared with the IRLS fit using the three M-estimators. It is In Figure 9, the OLS fit curves are compared with the IRLS fit using the three M-estimators. It evident in S2 to S6 that the outliers influenced the results, distancing the OLS fitted curves from those is evident in S2 to S6 that the outliers influenced the results, distancing the OLS fitted curves from fitted bythoseIRLS fitted. by IRLS.

Figure 9.FigurePredicted 9. Predicted curves curves for thefor the Henriksen’s Henriksen’s model model ﬁtted fitted forfor Ochroma pyramidale pyramidale (balsa-tree)(balsa-tree) aged aged 4.7 years in4.7 Ecuador years in forEcuador six di forﬀerent six different scenarios scenarios (S1 to(S1 S6). to S6).

In the evaluation of absolute residuals for Approach 2, it was observed in Figure 10 that in scenarios with outliers, there were biases in S2, S3, S4 and S5 using OLS, something not observed in the residuals generated by IRLS (Hampel) method.

Forests 2020, 11, 313 14 of 20

In the evaluation of absolute residuals for Approach 2, it was observed in Figure 10 that in scenariosForests 2020, with11, x FOR outliers, PEER REVIEW there were biases in S2, S3, S4 and S5 using OLS, something not observed16 of in21 the residuals generated by IRLS (Hampel) method.

Figure 10.10. Residuals of Henriksen’s model ﬁttedfitted for Ochroma pyramidale (balsa-tree) agedaged 4.74.7 yearsyears inin Ecuador forfor sixsix didifferentﬀerent scenariosscenarios (S1(S1 toto S6).S6).

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Table9. Parameters standard errors in the Henriksen’s model ﬁtted to predict total height of O. pyramidale from a subsample of data with 4.7-year-old trees in Ecuador (Approach 2).

Scenario Estimator Parameters S.E. dif. S.E. (%) OLS 2.7299 5.5221824914 β0 IRLS (Hampel) 2.7299 5.5221824914 1 0.00% OLS 6.7208 * 1.6996089315 β1 IRLS (Hampel) 6.7208 * 1.6996089315 OLS 30.4084 * 11.9541359308 β0 − IRLS (Hampel) 2.6277 5.8315357825 2 − 51.22% OLS 16.7109 * 3.7234077967 − β1 IRLS (Hampel) 8.3517 * 1.8163743432 OLS 5.6502 19.4107232002 β0 IRLS (Hampel) 2.2176 6.3255220839 3 67.41% OLS 6.2855 5.9939661492 − β1 IRLS (Hampel) 6.9074 * 1.9532999804 OLS 32.3614 * 9.7764176808 β0 − IRLS (Hampel) 5.9246 5.5350291450 4 − 43.38% OLS 17.7930 * 2.9742162885 − β1 IRLS (Hampel) 9.4590 * 1.6838861000 OLS 33.3318 * 14.7482194296 β0 − IRLS (Hampel) 0.3436 5.7722317805 5 − 60.86% OLS 17.9185 * 4.5563735879 − β1 IRLS (Hampel) 7.6725 * 1.7832962517 OLS 0.5067 17.1923557670 β0 IRLS (Hampel) 3.2965 6.0638269675 6 64.73% OLS 7.4412 5.2690135673 − β1 IRLS (Hampel) 6.5448 * 1.8584065497

βˆ i: regression parameters, * signiﬁcant with 95% probability, S.E.: standard error, dif. S.E.: Relative diﬀerence between the standard errors of regression parameters.

4. Discussion According to the site-index results for O. pyramidale evaluated in Approach 1, the Chapman–Richards model was the most representative for hdom predicted, although it showed a slight tendency to underestimate at the lowest heights and overestimate at the highest (Figure4a). Because it is a non-linear biological model with three parameters, it is very flexible and can properly represent the growth of an allometric variable like hdom over time [5,39,41]. 2 Based on the result of R adj. (0.709) of the Chapman–Richards fitted model, it is possible to state that, in O. pyramidale stands, the independent variable (age) explained the behavior of the dependent variable (hdom) very well. On the other hand, regarding the SEE% (12.71%), although it can be considered high (>10%) [42]), the result was satisfactory for the estimation of hdom in the context of the variability presented by the data (overall cv% 24.3%, Table2). It was observed that hdom amplitude over age was greater than 10 m (Figure4b), which proves to be a feature of the development of O. pyramidale, especially when seminally propagated, in which there is greater genetic variability [43]. Our study does not control for this type of variability in growth; we note that this is a limitation in our study which could influence the way we classified plots in terms of site-index. Forests 2020, 11, 313 16 of 20

Since the site-index curves from Figure4b do not include all observations, our site-index classification, pertaining to intervals upper-class I and lower-class III, may have been ‘kept open’—as described by Cañadas-López et al. [5]. Through site-index classification, it was possible to group the plots of the study into three productivity classes, with approximately half of them allocated to the intermediate class and the others to the lower and upper classes. This allowed for reducing the variability of observations by class and improving height–diameter predictions. The dominant height of O. pyramidale is closely related to the site-index, as with other species [10,44]. Future work is suggested related to the effects of site-quality, which ideally ought to also account for the effects of climate, soil and forest management. Results from Approach 1 suggest that DBH growth and h of O. pyramidale is related to site-index and age classes (Figure5). Our findings also suggest that models that do not stratify are out-performed by models that stratified by age—these results were consistent in both OLS and IRLS. The above provides evidence of a relationship between data variability and fit performance; namely, models that do not stratify and models that only stratify by site-index appear to display a higher percentages of variation (i.e., cv%; see Table1). Our results (Table6) also show better performance in models fitted by IRLS (e.g., Näslund; using Biweight)—this result appears consistent in models that do not stratify and those that only stratify by site-index. These results using the Näslund model are consistent with other empirical studies e.g., [45], which suggests a certain flexibility of this model to describe the height–diameter relationships; namely, the mathematical design of this model appears to allow for better residual distribution—when evaluated directly from a linear model fit that does not isolate for the influence of the variable h (Figure7). The higher accuracy of IRLS predictions of data with greater variability is due to the application of lower weights to outliers, minimizing error and improving the quality of fit [24,26,46]. This result appears to be supported by models that stratify for site-index classes and those that do not stratify; namely, these two model specifications appear to exhibit greater variability for h and DBH (cv% >20%, Table1)—this suggests a relative gain in accuracy (SEE%) when compared to models that stratify by site and age classes (Table6). In contrast, the gain in accuracy of OLS models that stratify by age classes appears to be partially explained by the better compliance of regression assumptions and lower variability for h and DBH (cv% <20% in most cases, Table1). In fact, when the linear regression assumptions are fulfilled, the OLS method enables more accurate predictions than robust estimators [15,18,20,22]. A desirable feature when fitting linear models is the smallest parameter standard error. Table7, shows for Approach 1 that, in this regard, both fitting methods presented less than 3% difference when analyzing the standard errors in the different classes of site-index and age. This suggests, a priori, that the performance of the four models adjusted by OLS and IRLS is very similar and this reflects the 2 results in SEE% and R adj. that had little difference. Regarding the apparent violation, in some cases, of the normality assumption, it was demonstrated for Approach 1 that the residuals of the fitted models follow the typical behavior of a normal distribution (Figure6) and this proves that OLS performs best in most age classes. On the other hand, robust regression has the characteristic that it can be used even when there is violation of regression assumptions [18,20,22,25,26]. The distribution of standardized residuals (Figure7) appears to be related to data variability (this result was found in both in DBH and h). This variability seems to account for a portion of bias in the predictions of the height–diameter relationship; this was identified in particular in models that models stratified by age. We note that these models included several observations that were considered outliers (i.e., values with magnitudes greater than 2 standard deviations). 2 In Approach 2, there was also a similar result of SEE% and R adj. for the Henriksen model adjusted by both methods in the six scenarios evaluated; however, the IRLS (Hampel estimator) technique Forests 2020, 11, 313 17 of 20 allowed a decrease in the parameters standard error from 43.38%–67.41% (Table9) compared to OLS in extreme scenarios (S2–S6). 2 Previous results suggest that, although important, the R adj. and SEE% statistics are not appropriate to compare OLS and IRLS methods with robust M-estimators. The parameter standard error is appropriate to compare the methods since it is directly related to the confidence interval of the predictions, and consequently to the efficiency of the model, which is something that actually differs in the fitting of the two methods. M-estimators are resistant to outliers in the dependent variable but are sensitive to the high number of leverage points in the independent variable [13]. In this sense and based on the results obtained for Approach 1 by the three M-estimators tested, it was found that the data-set used in modeling the height–diameter relationship by site-index and age classes presents some variation with some possible outliers in both dependent variable h and independent variable DBH (Table1). This characteristic of the data-set conditions the efficiency of the M-estimators. Outliers that do not relate to data collection errors should not be removed from the analysis [47]. When outliers are present, robust regression is indicated due to its similar precision with the least squares method; however, the robust regression can decrease the parameter standard error, as demonstrated in the present study and which is directly related to the confidence interval of the predictions. However, in the absence of outliers the loss of efficiency of robust estimators compared to the OLS can be a maximum of 5% [13,18,20].

5. Conclusions Approach 1 of this study demonstrates the improvements in the modeling of height–diameter relationships, by site-index and age-classes, using a large sample of forest inventory plots in O. pyramidale stands in five Ecuadorian provinces (n = 9348); our findings suggest that non-stratified models are out-performed by models stratified by age-classes. The results suggest consistent performance improvements when using robust regressions with iteratively reweighted least squares—as compared to OLS. Notwithstanding this, we note that there is no single model that represents all the different site-index and age classes, since development in diameter at breast height and total height of this species features high variability over time. In Approach 2, we use a sub-sample of sample tree data (n = 26) to engage in a sensitivity analysis which indicates that IRLS decreases the standard error of regression parameters and improves the confidence of the predicted values. Robust regression with the M-estimators for models fitted by iteratively weighted least-squares can improve more accurate predictions of the total height of O. pyramidale relative to the ordinary least squares method; these were particularly noted when comparing models which failed past tests of classical linear assumptions, as well as those with a cv% that is greater than 20%. Furthermore, the performance comparison of the fitted models by both methods, based on the 2 adjusted determination coefficient (R adj.) and standard error of estimate in percent (SEE%), is not consistent since the results with outliers do not show a clear trend. Thus, the efficiency of these methods can be better analyzed based on the standard errors of regression parameters and the graphic distribution of the residuals. In summary, the results of this study are timely, given that many practitioners and researchers in the private sector typically use OLS to analyze a relatively small forest operation (i.e., relatively small number of plots). Given the results of this study, we advise researchers who are interested in analyzing data with high variability (particularly those using a relatively small sample size) to consider the benefits of improving estimations using robust methods; namely, our results using large data-sets (i.e., Approach 1) appear to indicate performance parity of robust methods to OLS, whereas Approach 2 shows evidence of performance improvements in the presence of outliers and in smaller data-sets.

Author Contributions: Conceptualization, J.D.Z.-C., J.E.A., A.L.P., A.B. and J.R.S.; methodology, J.D.Z.-C., A.L.P., A.B. and J.R.S.; software, J.D.Z.-C., A.L.P. and G.A.O.; validation, J.D.Z.-C., A.L.P., A.B. and G.A.O.; formal analysis, J.E.A., J.R.S., M.S.G. and R.d.L.E.; investigation, J.D.Z.-C., J.E.A., A.L.P. and A.B.; resources, J.R.S. and M.S.G.; data Forests 2020, 11, 313 18 of 20 curation, J.D.Z.-C. and R.d.L.E.; writing—original draft preparation, J.D.Z.-C. and A.L.P.; writing—review and editing, J.E.A., A.B., G.A.O., J.R.S., M.S.G. and R.d.L.E.; visualization, J.D.Z.-C. and G.A.O.; supervision, J.E.A., A.L.P. and A.B.; project administration, J.E.A. and M.S.G.; funding acquisition, J.E.A., M.S.G. and J.R.S. All authors have read and agreed to the published version of the manuscript. Please turn to the CRediT taxonomy for the term explanation. Authorship must be limited to those who have contributed substantially to the work reported. Funding: This research was funded by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior—Brasil (CAPES)—Finance Code 001. Acknowledgments: The authors would like to extend their sincere appreciation to PLANTABAL S.A.—3ACOREMATERIALS® for their support in developing this research. Conﬂicts of Interest: The authors declare no conﬂict of interest.

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