Nonlinear mixed effect models for predicting relationships between total height and diameter of oriental beech trees in Kestel, Turkey

 

Modelos no lineales de efectos mixtos para predecir relaciones entre altura total y diámetro de árboles de haya oriental en Kestel, Turquía

 

Ilker Ercanli

 

Faculty of Forestry, Çankiri Karatekin University. 18200-Çankiri, Turkey. E-mail: ilkerercanli@karatekin.edu.tr, Phone: +90376 133301, Fax: +903762136983.

 

Received: February 20, 2015.
Accepted: April 24, 2015.

 

ABSTRACT

Statistical nonlinear mixed effect models were used to predict relationships between the total height and diameter at breast height of individual trees in Oriental beech (Fagus orientalis Lipsky) stands in Kestel, Bursa, Northwestern Turkey. 124 sample plots were selected to represent various stand conditions such as site quality, age, and stand density. Nine generalized nonlinear height–diameter models were fitted and evaluated based on Akaike's information criterion, Schwarz's Bayesian Information Criterion (BIC), Root Mean Square Error (RMSE), Absolute Bias and Adjusted Coefficient of Determination (R²_adj). The nonlinear Schnute's model was selected as the best predictive model. The height–diameter model based on the nonlinear mixed effect modeling approach accounted for 90.6 % of the total variance in height–diameter relationships and root mean square error (RMSE) values of 1.48 m. Various sampling scenarios that differed in sampling design and size of the selected sub-sample trees from the validation data set revealed that four randomly selected sub- sample trees in a given plot produced the best predictive results (43.3 % reduction of the sum of square errors, 98.4 % reduction of absolute bias, and 36.9 % reduction of the RMSE) in relation to the fixed effect predictions.

Keywords: Individual tree height–diameter, modeling, random parameter, calibration.

 

RESUMEN

Modelos estadísticos no lineales de efectos mixtos se utilizaron para predecir las relaciones entre la altura total y el diámetro a la altura del pecho (DAP) en rodales de árboles de haya oriental (Fagus orientalis Lipsky) en Kestel, Bursa, al noroeste de Turquía. Un total de 124 parcelas de muestreo se seleccionaron para representar la calidad de sitio, edad y densidad de rodal. Nueve modelos no lineales generalizados de altura-diámetro se ajustaron y evaluaron con base en el criterio de información de Akaike, el criterio de información bayesiana de Schwarz, la raíz del cuadrado medio del error (RMSE por sus siglas en inglés), el sesgo absoluto y el coeficiente de determinación ajustado (R²_adj). El modelo no lineal de Schnute se seleccionó como el mejor modelo predictivo. El modelo de altura-diámetro basado en el enfoque del modelo no lineal de efectos mixtos representó 90.6 % de la varianza total en las relaciones de altura-diámetro y los valores de RMSE de 1.48 m. Varios escenarios que difieren en el diseño de muestreo y el tamaño de los árboles submuestra, seleccionados del conjunto de datos de validación, revelaron que cuatro árboles submuestra seleccionados al azar produjeron los mejores resultados predictivos (reducción de 43.3 % de la suma de errores cuadrados, 98.4 % del sesgo absoluto y 36.9 % de la RMSE) en relación con las predicciones de los efectos fijos.

Palabras clave: altura-diámetro del árbol, modelo, parámetros aleatorios, calibración.

 

INTRODUCTION

Individual tree height and diameter at breast height (DBH) are important forest inventory measurements used for total and merchantable volume estimations, growth and yield modeling, and site index predictions (Calama & Montero, 2004; Soares & Tomé, 2002). In a forest inventory, tree heights can be frequently measured in a subset of trees in sample plots; however, DBH is recorded for all sampled trees. This is because measuring DBH is simpler, more accurate, and cheaper than tree height measurements (Laar & Akça, 1997). Statistical equations modeling the relationships between height and diameter are used to estimate the tree heights from DBH measurements in a forest inventory (Martin & Flewelling, 1998; Huang, Titus, & Wiens, 1992). The accurate height–diameter models are considered as effective forest inventory tools for the prediction of height using DBH as predictor variable (Nanos, Calama, Montero, & Gil, 2004).

Since height–diameter relationships vary from one stand to another with various stand structures, a simple statistical model cannot be used to estimate tree heights in all stand situations within a forest (Castedo, Diéguez-Aranda, Barrio, Sánchez, & von Gadow, 2006). Generalized height–diameter models have been recognized as an alternative approach to estimate height–diameter relationships in different stands within a forest (Adame, del Río & Cañellas, 2008; Crecente-Campo, Tomé, Soares, & Diéguez- Aranda, 2010; Paulo, Tomé, & Tomé, 2011). Therefore, a generalized height–diameter model has provided a solution to predict region-wide height–diameter relationships. However, height–diameter data are commonly obtained from trees in stands with different growing conditions (Schmidt, Kiviste, & von Gadow, 2010). Such a clustered and hierarchical data structure results in highly correlated data, also called an auto-correlation problem (Gregoire, 1987; West, Ratkowsky, & Davis, 1984). When the height-diameter models are fitted using these clustered data, the use of ordinary least squares (OLS) for linear models or nonlinear least squares (NLS) technique for nonlinear models leads to biased estimates of the confidence interval for model parameters (Searle, Casella, & McCulloch, 1992).

The linear or nonlinear mixed effect modeling approach has been increasingly used as an alternative statistical method to deal with the auto-correlation problems caused by a hierarchical data structure (Gregoire, 1987; Lappi, 1997; Calama & Montero, 2004). In linear or nonlinear mixed effect modeling, the fixed parameter for population-specific and random parameter for sampling unit-specific effect are simultaneously estimated by defining a covariance matrix in the same model structure (Calama & Montero, 2004). The inclusion of random parameters into the model structure enables the estimation of the residual variance of height–diameter relationships among clustered or nested sample units located in different stands (Calama & Montero, 2004). Furthermore, the mixed effect height–diameter models can be calibrated for locations that have not been sampled previously and further applied without calibration in which the height prediction models were the simple fixed effect models. These useful characteristics in the mixed effect modeling technique provide more efficient and more accurate height predictions for nested and clustered sample units in different forest stands than the OLS or NLS techniques.

The objectives of the present study were 1) to use statistical nonlinear mixed effect models to predict the relationships between total height and diameter of Oriental beech (F. orientalis Lipsky) trees located in Kestel-Bursa forests in Northwestern Turkey and 2) to evaluate calibration strategies based on different sampling scenarios, including the selection of a different number of sub-sample trees for predicting random parameters in describing unit-specific effect of sample units.

 

MATERIALS AND METHODS

Data

The data used in this study were obtained from even- aged and pure Oriental beech (Fagus orientalis Lipsky) stands located in the Kestel forests in northwestern Turkey (40º 00' 00" - 40º 12' 10" N, 29º 13' 00" - 29º 21' 54" E). In the even-aged Oriental beech stands, 124 sample plots were selected in the summer of 2005 to represent various stand conditions such as site quality, age and stand density.

The size of circular sample plots ranged from 400 to 800 m² to include a minimum of 30 to 35 trees in sample plots; the number of trees was dependent on stand crown closure. In each sample plot, DBH was measured to 0.1 cm precision using calipers for every living tree with a DBH > 8 cm. Total tree height (h) was measured on a subset of trees created by selecting two trees for each 4-cm diameter class using a Blume-Leiss Altimeter (0.1 m precision, Trigonometric Measurement Model originated from Germany).

In addition, the height and diameter measurements were obtained from dominant and co-dominant trees that were selected based on the 100 dominant and co- dominant highest trees per unit area level (e. g. four highest trees in a 0.04-ha plot). In addition to the tree level measurements, the number of stems per hectare (N·ha^-1), stand basal area (m²·ha^-1), dominant diameter (cm) and dominant height (m) were calculated as part of the plot level information for each sample plot. Dominant height and diameter were calculated by averaging the height and diameter of the dominant and co-dominant trees.

In total, 1,057 pairs of height–diameter measurements in 124 sample plots were used to analyze the relationships among tree heights, diameter and stand attributes, and to develop tree height prediction models. The sample plots were randomly split into two data sets, the model fitting and the validation data sets, using the random number function RANUNI implemented in the SAS statistical package (Statistical Analysis System [SAS Institute], 2009). Of those, about 85 % (907 trees in 104 sample plots) were used to fit model parameters, and the remaining 150 trees in 20 sample plots were reserved for the evaluation of the best predictive model and also used to determine the calibration response for the nonlinear mixed effect models. The summary statistics, such as the mean, standard deviations, minimum and maximum for tree and stand variables used for model fitting and validation data set, are given in Table 1.

Selection of the best statistical regression models

In this study, nine nonlinear functions were used to develop generalized height–diameter models proposed in recent studies (Table 2). These statistical models were chosen because they display better fitting results for predicting the relationships between height and DBH than other models.

First, the nonlinear functions were fitted using the model fitting data (907 trees in 104 sample plots). The estimation of the parameters of these functions was done with the NLIN procedure available in SAS/STAT® 9 software (SAS Institute, 2004). These functions were compared based on evaluations of the magnitudes and distributions of models' residual and five goodness- of-fit statistics: Akaike's information criterion (AIC), Schwarz's Bayesian Information Criterion (BIC), Root Mean Square Error (RMSE), Absolute Bias and Adjusted Coefficient of Determination (R²_adj). The expressions for these statistical criteria were defined as follows:

Adjusted coefficient of determination (R²_adj)=

where:

L = Maximum value of the log likelihood function

q = Number of parameters in the model

p = Number of coefficients in the model

N = Number of sample plots

N_i = Number of trees in the ith plot

ĥ_i = Mean height in the ith sample plot

h_ij and ĥ_ij = jth observed and estimated h in ith sample plot, respectively.

Smaller values of AIC, BIC, RMSE and the absolute Bias indicate better model fit results. Higher values of R²_adj (the adjusted coefficient of determination) give the predictable proportion of the variance of the dependent variable, height, from the independent variables.

The nonlinear mixed effect modeling approach was then used to estimate simultaneously fixed- and random-effects parameters of the height–diameter model that was selected as the best predictive model based on these statistical criteria.

Statistical nonlinear mixed effect modeling approach

The nonlinear height–diameter functions were fitted using dataset with multiple measurements taken in sample plots from different forest stands. In such nested data, structure measurements are not independent; the data is highly correlated, which will consequently result in unexplained variation of height–diameter relationships among clustered or nested sample units. To deal with this auto-correlation problem, a nonlinear mixed effect modeling procedure was applied to the best predictive height–diameter model by simultaneously fitting both fixed and random parameters in its model structure.

The nonlinear mixed effect modeling approach has some basic assumptions including the multivariate normal distribution of the residual terms and random- effects parameter (Calama & Montero, 2004).

u_i and v ∼ N(0, D_i) ε_ij ∼ N(0, R_i)

where:

D = Positive-definite variance-covariance matrix q x q for the random-effects representing the among- plot variability

R_i = Intra plot variance-covariance matrix.

The variances-covariance structures were defined by D_i and R_i matrixes to model random variability existing within and among plot levels (Calama & Montero, 2004). The D matrix is common to all plots and typically assumed to be an unstructured covariance matrix and identical for all plots, which describes variability for among-plot level (Huang, Meng, & Yang, 2009). In this study, the D matrix with two random parameters, u_j and v_j, was considered to model the variability among sampling plots, and the variance-covariance matrix structure is defined as below;

where:

σ_u = Variance for the random effect u

σ_v = Variance for the random effect v

σ_uv = Covariance among random effects (Castedo et al., 2006).

The other variance-covariance component, R_i matrix, is important to account for the variability observed for within-sample plots. Castedo et al. (2006) and Paulo et al. (2011) used the simplified structures for the within- sample plots variance-covariance R_i matrix, which assumed that the variance in within-sample plots is homogenous and residuals are uncorrelated;

R_i = σ² G_i

where:

σ² = Value for scaling error variance of the model

G_i = n x n diagonal matrix describing the non-constant variances of errors by predicting model (Castedo et al., 2006).

The variance components and fixed parameters of the best predictive height–diameter nonlinear model were estimated with PROC NLMIXED procedures of the SAS/ ETS 9 package (SAS Institute, 2004). The maximum- likelihood (ML) method was used to fit the nonlinear mixed effect regression. The adaptive Gaussian quadrature was used in the computation of the integral over the random effects as described by Pinheiro and Bates (2000). Furthermore, the PROC NLMIXED procedure was performed assuming the homogenous within-tree variance and uncorrelated residuals.

Calibration response

A vector of random parameters was predicted as calibration response for height–diameter models and the prior measurements of sub-sample trees were used to obtain the predictions for specific stands. To calibrate the nonlinear mixed height–diameter models for specific sample plots, random parameters, u_i and v_i for a given plot, were predicted using the best linear unbiased predictors, BLUPs (Lappi, 1991; Mehtätalo, 2004);

where:

= Estimated random parameters for specific plot

= q x q variance-covariance matrix for the among- plot variability (common for all plots)

= estimated k x k variance-covariance matrix for within-plot variability

= k x q matrix of partial derivatives of the nonlinear function with respect to random parameters

= Residual value defined as the difference between observed and predicted heights by the model including only fixed parameters (Crecente-Campo et al., 2010).

The comprehensive formula and explanations for components of BLUPs equation are specified by Calama and Montero (2004) and Castedo et al. (2006).

Evaluation of the calibration response for nonlinear mixed effect height–diameter models was done by comparing the sampling scenarios with different height sampling design and sizes from the validation data set with 150 trees in 20 sample plots. These scenarios were based on the selection of prior trees according to stand size categories, e.g. the largest, the smallest, and the medium-size trees per plot in a given number of trees (Crecente-Campo et al., 2010). The sub- sampling scenarios with more than five trees, e.g. nine to 10 trees, were excluded from the evaluation because they involve more laborious and time-consuming tree measurements than other sampling tree techniques. The sub-sampling procedures, including the selection of three to five trees, were preferred in other studies (Adame et al., 2008; Paulo et al., 2011). Thus, the sampling scenarios used here are as follows: 1) Total height of three to five randomly selected trees per plot, 2) Total height of three to five highest trees per plot, 3) Total height of three to five medium-size trees, considered as closest to the quadratic mean diameter at breast height per plot and 4) Total height of three to five lowest trees in a plot.

Random parameters were calculated using BLUP estimators on a subsample of trees selected by the sampling scenarios. They were integrated into the height–diameter model structure by adding them to the fixed effect parameters. Such calibrated height– diameter models were used to predict height values of trees in 20 sample plots. The calibration responses in sampling scenarios were compared using statistics criteria such as Sum of Squared Error (SSE), Absolute Bias and RMSE, where the observed heights were the heights obtained from sample plots and the predicted heights were obtained from the calibrated height– diameter model for the validation data set.

 

RESULTS AND DISCUSSION

Height–diameter model selection

The parameter estimates with probability levels and the goodness-of-fit statistics, including AIC, BIC, RMSE, Absolute Bias and R², for the studied nonlinear height– diameter models are given in Table 3. All parameters of estimates for these nonlinear models were found to be significant (P = 0.05). RMSE was between 1.79 m and 2.04 m, absolute bias between 3.23 m and 4.18 m, and R² between 0.86 m and 0.82 m in all tested models. Fitting results in the studied nonlinear models accounted for more than 82.3 % of total variance in height–diameter relationships, whereas RMSE was less than 2.04 m and Absolute bias less than 4.18 m.

On the basis of the goodness-of-fit statistics, the Schnute (1981) model (Eq. 9, Table 3), showed better fitting ability with AIC (1,068.11), BIC (1,077.73), R2 (0.86), RMSE (1.79), and Absolute bias (3.23) than the other studied functions. Therefore, the Schnute (1981) model was selected as the best height–diameter model for use in further analyses. This model was used to predict tree height based on dominant height and dominant diameter with tree diameter as the independent variable.

Stand dominant height and diameter are important stand quality attributes that affect the relationship between the growth in height and diameter, which in turn can be further modeled using Eichorn's rule (Eichorn, 1902). The use of site quality indices, e. g. stand dominant height and diameter, in modeling the relationships between height and diameter have been previously proposed (Adame et al., 2008; Crecente- Campo et al., 2010; Paulo et al., 2011; Sánchez-González, Cañellas, & Montero, 2007).

Nonlinear mixed effect modeling

In a nonlinear mixed effect modeling, the parameters of the model must be set to random or fixed effect parameters in a given model structure. Table 4 shows the AIC and BIC statistics that were calculated using none (all parameters are fixed-effect parameters), one (b₀ or b₁ is a random parameter), and two (both b₀ and b₁ are random parameters) random-effects parameters of the Schnute (1981) (Eq. 9, Table 2). For random or fixed parameters, the lowest values of AIC (730.48) and BIC (737.05), and the highest value for R² (0.90) were obtained in a nonlinear mixed effect modeling assuming two random parameters, b₀ and b₁ together, in a model structure. It was concluded that the best predictive height–diameter model could be obtained by the nonlinear mixed-effect model in Eq. 9 (Table 2) with two random parameters. The general expression of the nonlinear mixed effect in Eq. 9 (Table 2), including b₀ and b₁ as random parameters, is as follows:

where:

b₀, b₁ = Fixed parameters

u₀, v₁ = Random parameters

h_ij = Height of the jth tree in the ith plot

d_ij = DBH of the jth tree in the ith plot

H₀ = Dominant height in the ith plot

D₀ = Dominant diameter in the ith plot

The estimates of fixed parameters and variance components with standard errors and probability levels for the nonlinear mixed effect in the Schnute equation, including b0 and b1 as random parameters, are given in Table 5.

The height–diameter model based on the nonlinear mixed effect modeling approach accounted for about 90 % of the total variance in height–diameter relationships and the RMSE value of 1.48 m. Various other fit statistics for modeling the height–diameter relationships by using nonlinear mixed effect modeling were found in previous studies (R² = 0.94 and RMSE = 1.50 m in Castedo et al., 2006; R² = 0.88 and RMSE = 2.39 m in Crecente-Campo et al., 2010; R² = 0.82 and RMSE = 1.37 m in Adame et al., 2008; R² = 0.82 and RMSE = 1.15 m in Sánchez-González et al., 2007). The satisfactory predictive results from these studies and the present study suggest that modeling of the height–diameter relationships from hierarchical and clustered sample units located in different stands can be acquired using the nonlinear mixed effect modeling procedure.

The standardized residuals against predicted heights for the fixed effect nonlinear Schnute (1981) model are shown in Figure 1. In addition, the standardized residuals against predicted heights (a) and predicted heights against observed heights (b) for a nonlinear mixed effect Schnute (1981) model are shown in Figure 2. Figure 2a showed no heteroscedastic pattern in residuals. Correspondingly, White's test (White, 1980) that used x² table as critical values was performed to evaluate heteroscedasticity problem and this test revealed that there is no heteroscedasticity problem in residuals (test value = 39.01, P > 0.05). Therefore, a weighting factor with alternative transformations is not necessary to remove probable heteroscedastic variance error. In addition, the results of the simultaneous F-test (Figure 2b) indicate that the null-hypothesis of intercept = 0 and slope = 1 was not rejected, meaning that there were no systematic over-and underestimates in the height–diameter model. Eventually, the nonlinear mixed effect model based on the Schnute (1981) model with two random parameters provided the homoscedastic prediction variance (Figure 2).

Calibration responses in new observations

The nonlinear mixed effect height–diameter model based on Schnute (1981) (Eq. 9, Table 2) with two random parameters was validated with the validation dataset, which included 150 trees in 20 sample plots, representing about 15 % of the total data. The sum of squares of the errors (SSE), absolute bias, and RMSE values for different sub-sample alternatives, and the percentage of reduction in these values were compared to the predictions of the height–diameter model based on fixed effect parameters (Table 6). The nonlinear height– diameter mixed effect model with merely fixed effect model without any random parameters produced higher values for SSE (637.99), absolute bias (0.86) and RMSE (2.06) than the models involving random parameters.

The calibration protocol based on four randomly selected sub-sample trees in a sample plot gave the best predictive results, with 43.34 % reduction of SSE, 98.37 % reduction of absolute bias, and 36.85 % reduction of RMSE as compared to the fixed effect predictions. The subsequent better predictive performance for calibration was obtained using sampling scenarios based on selection of three to five medium size sub-sample trees in sample plots. The calibration procedure, including five medium- size trees in a sample plot, achieved 39.1 %, 95.1 %, and 21.9 % reduction in values for SSE, absolute bias, and RMSE, respectively. However, the sub-sampling alternatives that selected the smallest and the largest trees gave the poorest predictive results and higher values of SSE, absolute bias, and RMSE (Table 6).

The calibrated height–diameter models further improved the height predictions by applying the sub- sampling scenarios such as the selection of randomly chosen three to five sub-sample trees in the plots. Calama and Montero (2004) and Adame et al. (2008) proposed the use of randomly selected height measurements for calibration of height–diameter models. However, other studies (Castedo et al., 2006; Crecente-Campo et al., 2010; Paulo et al., 2011) found that the best predictive results for calibration response were obtained by selecting the smallest trees in sample plots. Thus, different sub-sampling scenarios may be successful in calibrating height–diameter models. Attributes, such as the largest, medium-size, or the smallest tree measurements, when randomly chosen among sample plots, may provide additional information and give a successful predictive performance (Castedo et al., 2006). In this regard, the calibration responses for a given mixed-effect model depend on the model structure and characteristics of a species growing in different regional and local forest conditions. Certain sampling scenarios with different sub-sample selection will provide better information for calibration response than other sampling alternatives at various forest sites and for various tree species.

The graphical representation of the height trends in the nonlinear least squared model and the calibrated nonlinear mixed effect height–diameter models based on four randomly selected sub-sample trees from validation data set are given in two plots, plot No. 1 and No. 5, (Figure 3). The height predictions of the nonlinear least squared model in these two sample plots were obtained using the parameters of nonlinear height–diameter model, Eq. 9, given in Table 3. However, the height predictions of the nonlinear mixed effect models were achieved using the calibrated model based on four randomly selected sub-sample trees in these plots. The nonlinear least squared height–diameter model presented larger residual variance with biased predictions, whereas the calibrated nonlinear mixed effect model produces height predictions that are consistent with observed values in these plots (Figure 3).

The Schnute (1981) function has been used successfully to model height–diameter relationships in other studies (Castedo et al., 2006; Huang et al., 1992; Lei & Parresol, 2001; Lei & Zhang, 2006; Zhang, 1997). Schnute initially developed his growth equation for fishery research (Lei & Zhang, 2006), and later it was shown that this function can produce the most flexible growth trends for describing height–diameter relationships with quickly fitting convergence (Bredenkampn & Gregoire, 1988; Lei & Parresol, 2001). This function includes two parameters that differentiate between curve shapes (S-shaped or concave-shaped curves), b0 and b1, and its upper asymptotic values represent the height growth potential.

As a desirable attribute of this function, Eq. 9, Schnute's (1981) height–diameter model predicts tree height to be 1.3 m, if diameter at breast equals zero (d^1.30 = 0). These attributes of height–diameter models were previously explained by Paulo et al. (2011) and Sánchez-González et al. (2007). The height–diameter prediction model based on Schnute (1981) can produce appropriate height predictions in the biological growth trend of Oriental beech trees in studied stands.

 

CONCLUSIONS

In this study, various nonlinear growth models were developed and evaluated to quantify the relationships between height and diameter of trees. Among the studied models, Schnute's growth model with dominant height and diameter variables as stand parameters provided the best predictive results. Additionally, the nonlinear mixed effect modeling procedure in the Schnute's model was used to simultaneously estimate fixed and random parameters in a model structure. The nonlinear mixed effect modeling procedure improved predictive results in Height-Dbh relationships as compared with fixed effect models. Schnute's height– diameter model that was developed in this study is suitable for forest inventories using height and diameter measurements from three to five randomly selected trees per plot. It produces unbiased predictions using the calibrated model in specific sample plots. This model will help forest planners and silviculturists to obtain tree height predictions in Oriental beech stands. In this study, we emphasized the ability of nonlinear mixed effect modeling for predicting the relationships between height and diameter in Oriental beech (Fagus orientalis Lipsky) stands. The height–diameter models may present an important tool in forest management planning and site quality evaluations of these studied stands located in Turkey.

 

ACKNOWLEDGMENTS

The authors would like to thank the Head of the Forest Management and Planning Department, General Directorate of Forestry, Republic of Turkey, for providing study data.

 

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