Evaluation of Different Modeling Approaches to Estimating Total Bole Volume for Pinus occidentalis, Swartz in Different Ecological Zones

1. Introduction

Accurate measurements of tree dimensions like diameter, height, stem form, and volume are not just academic exercises, but crucial for estimating forest attributes. This research provides practical tools like individual-tree and species-specific volume equations, which are necessary to predict inventory levels and ensure long-term wood yields. Volume equations use DBH and height to calculate stem volume, providing reliable estimates of tree volumes by accounting for size and shape variations. These practical tools are essential in forestry practices, empowering forest managers and timber industry professionals with accurate and reliable data.

Stem volume models estimate timber volume, biomass, and carbon sequestration potential for individual trees and larger areas. They are useful in forest management planning and growth simulations, driven by the need for accurate estimation of tree bole volume. Because the same regression may not be equally suitable for predicting total volume estimates in different ecological conditions [1], specific details and techniques for developing statistical models for stem volume estimation may vary.

The history of excurrent tree bole volume estimation dates back several decades. Various statistical techniques can be used to develop stem volume models, including linear regression, nonlinear regression, mixed-effects procedures, and machine learning algorithms. The choice depends on the dataset’s underlying assumptions and characteristics [2]. They are developed in a two-step process: model fitting and validation. The model is fitted to the dataset using a statistical technique and then validated using an independent dataset. Evaluation uses measures such as mean squared error, root mean squared error, bias, and coefficient of determination.

The most common statistical procedure for developing volume tables is ordinary least squares (OLS) regression analyses, which relate bole volume to explanatory variables such as diameter-at-breast-height (DBH), height (H), and sometimes stem form. When dealing with biological data, the constant variance assumption is often violated whenever a direct measure of stem content is used as the dependent variable in a regression equation [3]. If the assumption of constant variance is unmet, the equation must be weighted by a factor proportional to the standard deviation of the dependent variable.

Weighted least squares can be used when the ordinary least squares assumption of constant variance in the errors is violated It will produce a new regression model which results in the dependent variable having constant variance [3]. In the weighted linear regression model, each observation is assigned a weight $W_i$. The weighted sum of squared residuals, $Q= {\sum }_{i=1}^n{W}_i . {\left({\widehat{\epsilon }}_i\right)}^2$ is minimized.

Assumptions related to linear and nonlinear statistical models involve linearity, independence, homoscedasticity, normality, and zero mean of residuals. Additionally, nonlinear models must have a correct expectation function [4]. These assumptions guide the development and application of these equations in forestry practices, and their violation may lead to biased predictions and incorrect inferences.

The scarcity of data on volume models for tropical and sub-tropical tree species negatively impacts the accuracy of estimating tree volume for both research and operational purposes [5]. Even when developed, stem volume models should be periodically updated to account for changes in forest structure, climate conditions, or management practices to ensure accurate predictions and relevant decision support. P. occidentalis is the main timber species in the DR, growing on approximately 302,500 hectares and comprising approximately 95% of all timber harvested [6]. Despite its economic importance, estimating inventory levels and accounting for harvested volume is difficult because no standardized system is used for volume appraisal and inventory purposes.

The Schumacher and Hall [7] equation (SH) has been widely used in forestry to estimate tree bole volumes. Other researchers have shown that this equation and the Combined Variable equation developed by Bennett et al. [8] provide accurate results for estimating tree volumes, good performance in graphical analysis, and reliable estimates without bias [9].

The objectives of this study were: (1) to fit two commonly used volume equations, the combined variable (CV) and the SH equations with nine different modeling alternatives (MV), to estimate total bole volume content of P. occidentalis trees growing in three ecological zones (EZ) within La Sierra, DR; (2) to evaluate goodness-of-fit statistics of these nine MVs in each EZ; (3) to conduct and indicator variable analysis to determine if separate equations were needed for each EZ; and (4); based on ranking of performance in terms of accuracy and precision, recommend the best MV alternative for estimating total bole volume in individual P. occidentalis trees.

2. Materials and Methods

The study area is located in the north-central part of Cordillera Central, DR, covering approximately 1,800 km² (Figure 1). Even-aged natural stands of P. occidentalis are located within three ecological zones (EZs) according to the Holdridge [10] classification: Subtropical Dry Forest (DEZ), Subtropical Humid Forest (IEZ), and Subtropical Very Humid Forest (HEZ). Average elevations above sea level are 500, 650, and 800 m, respectively [11]. The climate varies depending on the altitude and precipitation. The average annual temperature is between 12ºC and 24ºC. These forests usually develop in shallow, carbonate, lateritic, low-producing soils with rugged topography.

2.1. Tree Data Sets

Three data sources were used for fitting total bole volume outside bark ($V_{SOB}$) models, one for each ecological zone (EZ). Trees from either dominant, intermediate, or overtopped classes in each zone were selected for destructive sampling, given that their phytosanitary status would allow it. After recording DBH outside bark at 1.30 m from the ground and measured with a diameter tape (0.1 cm precision), trees were felled as close to the ground as possible and H was measured with open-reel tape (0.1 cm precision). Besides DBH, other diameters were recorded at every meter along the bole from the base (10 cm above ground) up to the apex. Stem analysis data for model fitting purposes included 37 trees from the DEZ, 48 from the IEZ, and 72 from the HEZ. In addition, an independent data set collected from each zone was available to validate the resulting best total $V_{SOB}$ model. These data included 85, 90, and 75 independently measured P. occidentalis trees for the DEZ, IEZ, and HEZ. Table 1 showcases descriptive statistics for both the fitting and validation data set.

2.1. Data Exploration

To visualize the form of the relation between de dependent variable $V_{SOB}$ and the chosen explanatory variables DBH and total tree height (H), we proceeded to merge corresponding data from all three EZs and plot the $V_{SOB}$ against DBH first and later against D²H.

3. Results

3.1. Data Exploration

On average, sampled trees were smaller in terms of DBH, H, and $V_{SOB}$ in the DEZ, and largest in the HEZ. DBH ranged from 11.50 to 42.00 cm in the DEZ, from 16.50 to 53.50 cm in the IEZ, and from 21.50 to 46.50 cm in the HEZ. Following the same zone order, H from 10.30 to 24.00 m, 14.00 to 25.10 m, and 14.50 to 35.00 m. Volume outside bark ranged from 0.06 to 1.10 m³, 0.32 to 1.30 m³, and 0.20 to 2.73 m³, respectively (Table 1).

The cluster of points plotted to visualize the form of the relation between $V_{SOB}$ and DBH is depicted in the left side panel of Figure 2. It shows that the relation is not linear, and the volume variance increases with DBH. The relationship between $V_{SOB}$ against the combined effect variable D²H shown on the right side of Figure 2 shows that the relationship between these two variables is linear, but as before, the volume variance increases with D²H.

and DBH (left panel) and D²H. On the left, the relationship is nonlinear and presents heteroscedasticity. On the right, the relationship is linear, but heteroscedasticity persists.

3.1. Indicator Variables Analysis (IVA)

The results from IVA show that the HEZ and DEZ have different intercepts and the same slope. HEZ and IEZ have the same intercept and slope, and DEZ and IEZ zones have the same intercept but different slopes (Table 2). Therefore, the HEZ and IEZ can have the same equations to estimate $V_{SOB}$ of individual P. occidentalis trees, while the DEZ requires its own equations. Tree stem form has been observed to change in different zones with variation attributable to environmental factors [1]. Based on these IVA results, we combined the observed data from the HEZ and IEZ and fitted the tested models to the combined data.

Merged fitting data from the HEZ and IEZ totaled 121 observations. DBH ranged from 16.50 to 53.50 cm; H ranged from 14.00 to 35.00 m; and ${Vol}_{Sob}$ ranged from 0.20 to 2.76 m³. Respective averages and sample standard deviations (within parenthesis) were 31.77 $\left(\pm 7.05\right)$ cm, 22.67 $\left(\pm 4.73\right)$m and 0.84 $\left(\pm 0.48\right)$m³. After merging data from both zones, the validation data set had 165 observations with DBH ranging from 10.60 to 54.20 cm; H ranging from 9.40 to 27.80 m; and ${Vol}_{Sob}$ from 0.11 to 2.22 m³. Respective averages and sample standard deviations were 28.53 $\left(\pm 8.15\right)$ cm, 20.27 $\left(\pm 3.78\right)$ m, and 0.59 $\left(\pm 0.36\right)$ m³.

3.1. Total Bole Volume Model Fitting in the DEZ and the Combined IEZ and HEZ Ecological Zone (CIHEZ)

The SH02 variant fitted by non-linear least square (NLS) procedures, was ranked number one for estimating $V_{SOB}$ P. occidentalis in the DEZ; while in the CIHEZ, the best variant was SH03, which was fitted by weighted non-linear least squares (WNLS) procedures. Based on the ranks of goodness-of-fit statistics criteria applied to the data fitting and validation phases, these results indicate the superiority of the SH model variants in this study. The fitting and validation statistics values, along with rank sums and overall ranks for each MV in each EZ, are presented in Table 3 and Table 4.

Considering both the fitting and validation phases, SH02 in the DEZ was ranked No. 1 in terms of RMSE (0.0146) and RVE (0.0002) and No. 2 in AIC criteria (-199.86) in the fitting phase (Table 3). Likewise, it was ranked No. 1 in terms of RMSE (0.0702), Bias (-0.0099), and RVE (0.0030) in the validation phase. Percent bias calculated as $\left[V_{SOB}- {\widehat{V}}_{SOB}/V_{SOB}\right]*100$in the fitting phase indicated that SH02 tends to overestimate slightly $V_{SOB}$ by approximately -0.059 %. Percent RMSE was calculated at 4.77% for SH02, suggesting that predictions deviate by about 4.77% from the mean of the actual values. We consider these values to be adequate for estimating $V_{SOB}$for individual trees of P. occidentalis in this EZ.

SH02 had an overall sum rank of 31, being the lowest ranked of all volume variants tested in the DEZ. It was followed by SH04, the fourth variant of model (3) with a sum rank of 33 and fitted by WNLS procedure, where exponent “c” in the variance model ($Var\left(\epsilon \right)={\left(k{DBH}^c\right)}^2$), was a parameter that needed to be estimated.

SH03 in the CIHEZ was ranked No. 1, achieving the lowest sum in the ranking (23) among the nine variants tested in this combined EZ (Table 4). For weights, the conditional standard deviation of $V_{SOB}$ derived from DBH is proportional to DBH⁴. In the fitting phase, SH03 was first regarding AIC criteria (-311.66). Similarly, it was ranked No. 1 in terms of RMSE (0.0943), Bias (-0.0614), and RVE (0.0059) in the validation phase. Percent bias calculated as $\left[V_{SOB}- {\widehat{V}}_{SOB}/V_{SOB}\right]*100$in the fitting phase indicated that SH03 tends to underestimate on average $V_{SOB}$ by approximately 0.025%. Percentage RMSE was calculated at 8.96%, suggesting predictions that deviate by about 8.96% from the mean of the actual values for estimating $V_{SOB}$for individual trees of P. occidentalis in the CIHEZ. Regarding the sum rank, SH03 was followed by MV SH01, the unweighted nonlinear version of the SHMV, with a sum rank of 35.

The parameterization of SH02 and SH03 are, respectively:

$$V_{SOB-DEZ}=0.00005815{*D}^{1.7802}{H}^{1.0787}$$

$$V_{SOB-CIHEZ}=0.00005669{*D}^{1.7019}{H}^{1.0786}$$

Although different fitting procedures estimated them, the intercept and coefficient corresponding to the DEZ and CIHEZ tree height are very similar. The intercepts differed by 2.54%, and the tree height (H) coefficients differed by 0.009%. The coefficients for D differed by 4.49%.

In checking regression assumptions, our plots show no curvature, although there is one point beyond the cutoff limit of two standard deviations in the top left panel corresponding to the DEZ. We assumed that MVs SH02 and SH03 do not seriously contradict the constant variance assumption. Even though the quantile–quantile plots of the residuals appear to have a slight structure in both zones, most of the points are aligned along a straight line and remain relatively consistent across all levels of the predictor variables, indicating that the constant variance assumption is sustained.

In Figure 3, scatter plots show randomly clustered values around the 1:1 line for outer bark volumes in the DEZ (top left) and the CIHEZ (bottom left). The points around the line remain relatively constant across all levels of the predictor variables, confirming the constant variance assumption and closely following the observed values, suggesting that these models capture the variability in the data well.

In Figure 4, the top right panel shows predictions made by MV SH02 against predictors DBH and H in the DEZ. Likewise, the bottom right panel shows the predictions made by MV SH03 against the same predictors in the CIHEZ. These plots to the right support observations in the left panel graphs by showing the agreement between predictions and measured variables. SH02 and SH03 efficiently estimated stem volume outside bark for P. occidentalis in both EZs and effectively captured the variability in the data.

4. Discussion

Alvarado-Segura et al. [17] used a nonlinear version of the SH model to estimate the stem volume of Pinus patula Schiede ex Schlechtendal et Chamisso var. Patula, in Hidalgo, Mexico and reported an RMSE of 0.0914. Our RMSE statistic value is much better for our SH02 MV (0.0146), while the corresponding value for SH03 (0.0943) is close to that reported by these authors. The SH model has been found to perform well in predicting stem volume for many species and genera in diverse environments [18]. Castillo-López et al. [19] and Valerio-Hernández et al. [19] used this model in studying volume equations for pine species in Mexico and Nicaragua. Based on the evaluation criteria, this study’s nonlinear version (SH02) and the nonlinear weighted variant (SH03) of the SH model performed best.

All nine variants tested from models (2) and (3) in both EZs resulted in parameter estimates that are statistically significant and logically consistent for outside bark volume estimation (Table 5 and Table 6). The estimates for intercept, interpreted as the average value of total stem volume accumulated by a tree until it reaches breast height (when DBH is zero and total height is 1.3 m) were positive and significantly different from zero. All other parameters were consistent in terms of sign and magnitude.

The best-ranked Combined Variable model variant was CV02 which achieved an overall ranking of 3 in the DEZ and 4 in the CIHEZ. Values for intercept coefficients of variants in model (2) resemble in a certain way the shape and form of a geometric solid, with higher numbers representing better and more cylindrical form and thus greater volume [12]. On average, model (2) variants intercepts are lower (105.77%) in the DEZ than their counterparts in the CIHEZ, indicating that stem volume accumulated by individual P. occidentalis trees below breast height is lower in the former zone. A t-test assuming unequal variances indicates that intercepts from the two zones are statistically and significantly different (P=0.0026). Slope coefficients for D²H in the same model (2) variants are also statistically and significantly different (P=0.000126).

In mathematics, the volume of a circular base solid with base diameter D and height H is expressed as: $V=\beta D^{2}H$ [21]. If the diameter is measured in cm, $\beta = \raisebox{1ex}{$\pi $}\!\left/ \!\raisebox{-1ex}{$40000$}\right.$ for a cylinder, $\raisebox{1ex}{$\pi $}\!\left/ \!\raisebox{-1ex}{$80000$}\right.$ for a paraboloid, $\raisebox{1ex}{$\pi $}\!\left/ \!\raisebox{-1ex}{$120000$}\right.$ for a cone, and $\raisebox{1ex}{$\pi $}\!\left/ \!\raisebox{-1ex}{$160000$}\right.$ for a neiloid. This $\beta$ is equivalent to ${\beta }_1$, the coefficient in the second term of the CV. The averages of the five ${\beta }_1$ values from our results are equivalent to 0.0000347 in the DEZ, and 0.0000321 in the CIHEZ, differing by about 12.4 % and 20.47% off the perfect paraboloid, respectively. Similar values for ${\beta }_1$ using the CV equation in fitting data from 150 P. patula trees were found by Alvarado-Segura et al. [17] in Mexico. Therefore, tree shapes in our study approximate a paraboloid solid, which is described mathematically by the expression π/80000 = 0.0000393 (diameters are in cm units). This contrasts with the results of Sharma [21] who found that the shape of the trees is not a solid described by a cylinder, paraboloid, cone, or neiloid while assessing tree volume of several species in natural stands of Canada and northeastern United States.

Differences in the parameters of the CV models between EZs may result from differences in the height-dbh relationship, tree form, tree taper, or a combination of these factors [1], although all were logically consistent. These results agree with the findings reported by Sharma [22], who employed the CV equation to compare goodness-of-fit statistics, logical consistency, and predictive accuracy of several models in Pinus resinosa Sol. ex Aiton. Exploratory analyses of variance not reported in the study showed that the form and quotient coefficients of P. occidentalis trees in these zones were statistically significant. That, combined with higher values for the ${\beta }_0$ coefficient in the HEZ Zone may indicate that the form may be better there.

5. Conclusions

Nine model variants (MVs), five from the CV and four from the SH, were evaluated in their capacity to estimate stem volume outside bark (SVOB) of individual P. occidentalis trees growing in natural stands within three EZs in La Sierra, D.R. All variants produced results consistent with those of other pine species. All MVs gave good results on the fitting dataset, but the performance was poorer when used in the validation dataset. Ercanli et al. [23] and Sahin [24] encounter the same situation while modeling tree volume of conifer species in Turkey.

An indicator variable analysis indicated that only one equation was required to estimate SVOB in the IEZ and HEZ. Therefore, observations from these two EZs were combined to evaluate the different MVs further.

The CV model variants were logically consistent in parameter estimation. They complied with regression assumptions but were inferior to the chosen SHVE model variants regarding a ranking based on goodness-of-fit statistics and predictive ability. MVs SH02 and SH03, strategies for modeling $V_{SOB}$ individual P. occidentalis trees were efficient and selected as the volume equations for the DEZ and CIHEZ, respectively.

The maximum bias among all MVs in the DEZ was 0.20% and 5.54% in the fitting and validation phases, respectively. Likewise, in the CIHEZ, bias was 0.007% and 8.84%, respectively. According to Sharma [21], volume equations resulting in a bias larger than 10% should not be recommended for forest management decision-making. The minimum observed variability explained by all variants in both zones was 94% for outside bark volume estimates.

Developing forest management strategies requires accurate tree volume estimates, employing tools such as tree volume models. These volume models are simpler to use if the entire stem volume is of interest [25]. As the assumptions underlying the regression methods employed in the study for constructing these nine volume model variants were validated, and the results showed that these MVs performed well in both the fitting and validation phases, we propose their use as useful tools for predicting total stem volume outside bark in these EZs within La Sierra, D.R. The equations presented here provide a more scientifically accurate and consistent predictions of individual P. occidentalis trees. They are a valuable tool for supporting present and future management decisions.