IntroductionTop
Researchers realize that volume, the primary variable that forest managers are interested in, is heavily dependent on both tree diameter and height. A traditional practice is to fit a marginal distribution to the diameter frequency data and then use an empirical height-diameter relationship to estimate the average height per diameter class and hence the volume (Clutter & Allison, 1974). Although this approach is satisfactory in many situations, it is often not appropriate because the approach tends to ignore the natural relationships between tree diameters and heights by treating them separately (Schreuder & Hafley, 1977; Tewari & von Gadow, 1999).
The traditional method does not quantify the distribution of heights for a given diameter and one approach for modelling the conditional height distribution for the different diameters is to use the height residuals (Gaffrey, 1996). However, it is very seldom that the height residuals are homoscedastic and normally distributed. In most forest stands the variance about the diameter-height regression is not homogeneous (Zucchini et al., 2001).
One of the most important elements of forest structure is the relationship between tree diameters and heights because information about size-class distributions of the trees within a forest stand is important for estimating product yields (Zucchini et al., 2001). The size-class distribution influences the growth potential and hence the current and future economic value of a forest stand (Knoebel & Burkhart, 1991). On the other hand, the social status of a tree, which reflects its further development in the aspects of growth and mortality, depends not only on its relative diameter, but also on its relative height in a stand (Siipilehto, 2000). Furthermore, knowledge of the height variation, both between and within diameter classes, improves the chances of successfully imitating different types of thinnings (Hafley & Buford, 1985).
For many years, the bivariate extension of the SB distribution, the SBB (Johnson, 1949a and 1949b), has been the most commonly bivariate distribution used for modeling bivariate tree diameter-height frequency data (e.g. Schreuder & Hafley, 1977; Hafley & Buford, 1985; Knoebel & Burkhart, 1991; Siipilehto, 1996; Tewari & von Gadow, 1997; Tewari & von Gadow, 1999; Zucchini et al., 2001; Li et al., 2002; Wang & Rennolls, 2007, Gorgoso-Varela et al., 2016; Ogana et al., 2018; Ogana, 2018a). The Johnson’s SBB is developed by applying a four-parameter logistic transformation to each of the component variables of a standard bivariate normal distribution (Johnson, 1949b; Wang, 2005; Gorgoso-Varela et al., 2016). The fitting of the joint distribution of diameters and heights is called bivariate distribution modeling (Ogana et al., 2018). Bivariate distribution modelling allows for the generation of bivariate frequencies of tree diameter and height (Tewari & von Gadow, 1997).
The parameters of the Johnson’s SBB distribution have been estimated with both the indirect and direct methods. The indirect method fits the bivariate Johnson’s SBB distribution by two-stage where the marginals are first fitted separately for the diameter and height; the estimates are then used to compute the parameter of association. Examples of the indirect method that have been reported in forestry literature include Conditional Maximum Likelihood (CML), moments, mode and Knoebel and Burkhart methods (Hafley & Buford, 1985; Tewari & von Gadow, 1997; Knoebel & Burkhart, 1991; Siipilehto, 1996; Tewari & von Gadow, 1999; Castedo-Dorado et al., 2001; Li et al., 2002; Ogana, 2018a). Of these indirect methods, CML and moments have been reported as the best methods (Ogana, 2018a). In the direct method, all the parameters in the bivariate distribution are estimated simultaneously (Wang, 2005; Mønness, 2015; Gorgoso-Varela et al., 2016; Ogana et al., 2018). However, these two methods (i.e., direct and indirect) of estimating the parameters of Johnson’s SBB distribution have never been compared using the same sample. The method used to calibrate the distribution matters in order to evaluate the accuracy of diameter and height predictions (Ogana, 2018a). Thus, the objective of the study was to compare the indirect fitting method for the Johnson’s SB based on two-stage method by the Conditional Maximum Likelihood and moments methods with the direct method where the parameters are estimated simultaneously.
Material and methodsTop
Data set
The data used for this study were obtained from three temperate species – Tasmanian blue gum (Eucalyptus globulus Labill) and two species of pine – Maritime pine (Pinus pinaster Ait) and Monterrey pine (Pinus radiata D. Don) in Spain. E. globulus stands occupy 320,774 ha in Galicia. The pure stands of P. pinaster cover 217,281 ha and 22,523 ha in the regions of Galicia and Asturias, respectively. The plantations of P. radiata occupy 96,177 ha and 25,385 ha in Galicia and Asturias, respectively (MMAMRM, 2011). A total of 308 permanent sample plots (PSPs) from E. globulus, 184 PSPs from P. pinaster stands and 96 PSPs from P. radiata stands were used for this study. The plot sizes ranged from 375 to 900 m2; to achieve a minimum of 30 trees per plot. Diameter at breast height (Dbh at 1.3 m above the ground) and total height were measured with calliper and hypsometer to a precision of the nearest 0.1 cm and 0.1 m, respectively. A total of 16382, 17845 and 12722 trees were measured from E. globulus, P. pinaster and P. radiata, respectively. The descriptive statistics of the inventory data are presented in Table 1.
Table 1. Summary statistics of the tree variables.
The Johnson’s Univariate (SB) and Bivariate Distribution (SBB)
The univariate Johnson’s SB distribution (Johnson, 1949a) is expressed as:
Where: ξ < x < ξ + λ, -∞ < ξ < + ∞, -∞ < y < +∞, λ > t0, and δ > 0.
ξ shape parameters (asymmetry and kurtosis parameters, respectively). The Johnson’s SBB (Johnson, 1949b) is the extension of the univariate Johnson’s SB distribution; given by:
Where:
Where: ρ is the correlation coefficient between Zd and Zh, the subscripts d and h in equations represent diameter and height, respectively.
Fitting Methods
Indirect method: this method fits the bivariate Johnson’s SBB distribution by a two-stage approach where the marginals are first fitted separately for the diameter and height. The estimates from the first stage are then used to compute the correlation parameter (as shown in Eq 3). Generally, fitting Johnson’s SBB distribution requires the location (ξ) and scale (λ) to be predetermined (related to minimum and range of the tree variable) while δ and γ can be analytically deduced through different approaches (Schreuder & Hafley, 1977). The method of CML and moments were the indirect methods considered in this study as recommended by Ogana (2018a).
Conditional Maximum Likelihood (CML): The values of the parameters were obtained with these expressions:
Where: xi (i = 1, 2, …, n) = tree diameters and heights.
The location (ξ) and scale (λ) parameters were set to equal minimum diameter times 0.75 (0.75*Dmin) and maximum diameter, respectively for the marginal distribution of diameter. These parameters were set at 1.3 and maximum height for the marginal height distribution. The factor 0.75 was adopted because Gorgoso et al. (2012) used 0.75 Dmin for this parameter ξ to achieved good result compared to other constraints in Pinus pinaster and Pinus radiata. The same species from the same region were used in the study at hand. The CML has been applied to fit the Johnson’s SBB by different authors, including Knoebel & Burkhart (1991); Siipilehto (1996); Tewari & von Gadow (1997); Tewari & von Gadow (1999); Castedo-Dorado et al. (2001); Ogana (2018a).
Method of Moments: this method is based on the functional relationship between the first and second moments of diameter and height and the parameters of the SBB distribution. Ogana (2018a) recently used this method to fit the SBB distribution. It is given by:
Where: X = mean diameters and heights of the plot, σx = plot diameter and height standard deviations and Sd(x) = modified standard deviation. The same procedure with CML was used for λ and ξ.
Direct method: in this method, all parameters in the bivariate Johnson’s SBB distribution were estimated simultaneously using maximum likelihood estimation. However, the location (ξ) and scale (λ) parameters were predetermined with the same procedure as the indirect method. This method was applied by Wang (2005), Wang et al. (2008) and Gorgoso-Varela et al. (2016). These authors used the normal copula for this distribution. The joint density of the SBB function derived from the normal copula is given by:
and the likelihood function is expressed as:
where f(d) and g(h) are marginal Johnson SB densities for diameter (d) and height (h), respectively defined in equation 1; θ parameters to the estimated and N is the number of observations.
The indirect method (CML and moments) estimations were obtained using SAS/STATTM software (SAS Institute 2003). The computation of the direct method was carried out using the ‘optim’ (optimal) function in R (R Core Team, 2017). The Nelder-Mead simplex algorithm (Nelder & Mead, 1965) was used for the ‘optim’.
Model Evaluation
The suitability of the direct and indirect methods for fitting SBB distribution was evaluated by tree height and volume predictions. Prediction of expected tree height given diameter and the estimation of stand volume are the main applications of bivariate distribution. Individual tree heights were predicted with the SBB median regression expressed as:
are the estimated parameters from the direct and indirect methods. h = total tree height (m) and d = tree diameter at 1.30 m (cm). The ϕ parameter determines the shape of the regression curve while ρ influences the slope. The relationship can only be linear if ρδd = δh and ργd = γh.
Individual tree volume equation developed by García-Villabrille (2015) for E. globulus and Diéguez-Aranda et al. (2009) for P. pinaster and P. radiata were used to obtain observed and predicted tree volume. The procedure of Li et al. (2002) and Ogana (2018a) was used in this study. Observed tree volume was computed by substituting the observed diameter and height into the individual tree volume equations. The predicted individual tree volumes were obtained from the observed diameter and predicted tree height by SBB height-diameter median regression using direct and indirect methods:
Where: v = total volume of the stem with bark (m3); d = tree diameter (cm) and h = tree height (m). The root mean square error reported by the authors for equation 18, 19 and 20 were 0.0116, 0.0156 and 0.0131, respectively.
Different fit indices including coefficient of determination (R2), root mean square error (RMSE), model efficiency (MEF) (Mayer & Butler, 1993), Bayesian Information Criterion (BIC) and Hannan-Quinn Criterion (HQC) were used to assess the adequacy of the methods for height and volume predictions. In addition, significant difference between observed and predicted tree heights and volumes were tested using paired sample t-test at 5% level for each plot by species. Plot was rejected if significant difference occurs between observed height/volume and predicted height/volume for that plot at p<0.05. Pairwise comparison was done with t-test because outlier was not detected in the data.
Where: RSS = residual sum of square, n = sample size, p = number of parameters; Yi = average tree height or volume; Yi is the observed value and Yi is the theoretical value predicted by the model.
ResultsTop
The descriptive statistics i.e., the mean, maximum, minimum and standard deviation of the estimated parameters of the bivariate Johnson’s SBB distribution are presented in Table 2. The location parameter of the marginal height distribution was set to 1.3 for both direct and indirect methods across the three species. The shape, scale and rho (ρ) parameters of the indirect methods (CML and moments) show little variations. In fact, the standard deviations of the parameters’ values were the same at least up to 1 decimal place. This indicates that both CML and moments are closely related. However, estimates from the direct method were different. The standard deviations of the estimated parameters are smaller than that of CML and moments of the indirect method.
Table 2. Descriptive statistic of the Johnson’s SBB parameters for the three species.
The assessment of the suitability of indirect (CML and moments) and direct methods for tree height prediction showed that the indirect method, especially the CML method, had the highest R2 and the lowest RMSE, MEF, BIC and HQC in E. globulus, P. pinaster and P. radiata (shown in Table 3). The R2, RMSE, MEF, BIC and HQC of CML ranged from 0.994 – 0.820, 1.454 – 1.676, 0.055 – 0.179, 4559 – 13400 and 4492 – 13335, respectively. However, the direct method had relatively lower R2 and larger RMSE, MEF, BIC and HQC in E. globulus, P. pinaster and P. radiata relative to the indirect methods. The number of plot in which the observed and predicted values were significant at 5% level (t-test) was 1 out of 308 plots (equivalent to 0.32%) in E. globulus for CML. Eight (2.59%) and 14 (4.45%) plots were significant for moments and the direct methods, respectively in E. globulus. There was no significant difference between the observed and predicted tree height by CML and moments across the 184 and 96 plots in P. pinaster and P. radiata, respectively. But 4 and 5 plots were significant with the direct method in P. pinaster and P. radiata, respectively at the specified level.
Table 3. Fit indices and number of plot rejection by t-test at α = 0.05 for height prediction.
In the case of tree volume prediction, indirect (CML and moments) and direct methods performed relatively the same (Table 4). The methods had the same R2, RMSE and MEF values up to 2 decimal places. The direct method had lowest BIC and HQC in E. globulus and P. radiata. The number of plot in which the observed and predicted values was significant at 5% level for CML was like the result in tree height prediction (i.e., 1, 0, 0). However, 6, 6, and 3 plots were significant in E. globulus, P. pinaster and P. radiata, respectively for the direct method.
Table 4. Fit indices and number of plot rejection by t-test at α = 0.05 for volume prediction.
The mean of the residuals from the height-diameter median regression was computed across DBH classes and plotted accordingly for the three species. The data were group into DBH classes of 5 cm interval and the mean residual prediction for each class by species was assessed. The graphs showed that CML and moments over- and under-estimated the tree height in the larger diameter classes (> 42.5 cm) in E. globulus and P. pinaster (Fig. 1a to c). Nevertheless, their prediction errors lie within the range of – 0.5 to 0.5, except in the larger DBH classes. Conversely, the direct method did not only over- and under-estimate tree height in the larger diameter classes but also over-estimate tree heights in the lower diameter classes in E. globulus. The mean residual height prediction for both direct and indirect method were relatively poor in P. radiata.
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Furthermore, the graph of residual against predicted tree volume showed that CML and moments and the direct methods occupied the same horizontal band (Fig. 2). Their mean residuals plot lied within the range of -0.6 to 0.6, -0.6 to 0.6 and -0.4 to 0.2 in E. globulus, P. pinaster and P. randiata, respectively. Such plot is used to assess whether the residual is homoscedastic (i.e. constant variance) or heteroscedastic. The residual of the methods assessed were relatively constant.
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DiscussionTop
The direct and indirect methods of fitting Johnson’s SBB have been evaluated. The methods follow similar trend across E. globulus, P. pinaster, and P. radiata with CML being the best for tree height and volume predictions. The number of plots rejected by t-test increased from CML to moments and to the direct method. The percentage of rejections in E. globulus was higher than the other species because some plots of coppice stands were considered, in which the minimum diameter considered was less than 5 cm (value of the minimum diameter considered for Pinus pinaster and Pinus radiata stands). There were also biases in the larger diameter classes (> 42.5 cm) for all methods. Kalbi et al. (2017) attributed this bias to the small number of trees in the larger diameter class. Few trees were found in the diameter classes > 42 cm across the three species. The Johnson’s height-diameter function often perform well if parameter ϕ i.e., the determinant of the regression curve is greater than one (Tewari & von Gadow, 1999). In this study, ϕ had values greater than one in some of the sample plots for CML, moments and direct methods. The model performed relatively well in these plots.
Furthermore, the values of the BIC and HQC difference show that the method of CML fits the diameter-height data better than direct methods for height prediction in E. globulus, P. pinaster and P. radiata. As a rule of thumb, a minimum ∆AIC/∆BIC/∆HQC of ≤ 2 is required for two models to be indistinguishable (Ogana, 2018b; Tewari & Singh, 2018). Though, the measures of the accuracy of the estimates (RMSE) and the relative measures of the model performance (MEF) look quite similar for all methods, the CML seems to be slightly better. Parallel observation was reported in Ogana (2018a), who evaluated the performance of CML, moments, Knoebel and Burkhart (KB) and mode methods for height and volume predictions. The author found CML and moment to be the best methods for predicting tree height and volume in E. camaldulensis Dehn.
It is obvious from this study that when a complex fitting approach does not outperform another simpler one it is convenient to take the latter. Estimating the parameters of Johnson’s SBB by CML or moments is much easier and pose little computational difficulty. The CML and moments estimation procedure can even be carried out on Microsoft excel platform. However, the direct method involves complicated algorithm using maximum likelihood technique. This often requires writing the log-likelihood function and the specification of initial values for the parameters which may not achieve convergence (Wang, 2005). Given the complexity of the likelihood function, its gradient is not tractable and then numerical optimization methods have to be used (i.e. through the ‘optim’ function), which may be less efficient than those based on derivatives. The computation procedure gets more complex as the number of parameters in the distribution increases. Bivariate Johnson’s SBB has nine parameters which makes fitting somewhat complicated. Often constraints are imposed on the two location and scale parameters of the bivariate distribution to make the estimates more plausible (Wang & Rennolls, 2007). Mønness (2015) used maximum likelihood estimation to fit the Johnson’s SBB distribution and compared the result with power-normal and hyperbolic height prediction model. The prediction from power-normal and hyperbolic models were better than bivariate Johnson SBB fitted with maximum likelihood. Other studies that have used maximum likelihood technique to fit the bivariate Johnson’s SBB distributions include Wang et al. (2008), Gorgoso-Varela et al. (2016) and Ogana et al. (2018). Satisfactory results were obtained by the authors for the Johnson’s SBB relative to other bivariate functions evaluated in their study.
The bivariate Johnson’s SBB height-diameter median regression remains the most applied function amidst several bivariate distributions in forestry literature. This is because of its flexibility and, more importantly, because its implied relationship between height and diameter is biologically reasonable (Burkhart & Tome, 2012). The closed-form median regression of the Johnson’s SBB height-diameter model has also enhanced its continuous application to forestry. This median regression is frequently used for computing percentile lines wherein bounds on height are set. These percentile lines can be used to show how the variation in height decreases with increasing tree size for a specified diameter (Tewari & von Gadow, 1999). The scatter plots of the Johnson’s SBB median regression using indirect method fitted by CML from some sample plots in E. globulus, P. pinaster and P. radiata are presented in Figures S1 to S3 [suppl.].
ConclusionsTop
Modelling the joint distribution of diameter and height by Johnson SBB remains an important tool for assessing the variation of tree height for a given diameter, detailed stand structure and volume estimation. Its accuracy is affected by the method of estimating the parameters of the distribution. In this study, we found the indirect (two-stage) method, especially by conditional maximum likelihood, to be the most suitable method for the bivariate Johnson’s SBB distribution. This method predicted tree height and volume in E. globulus, P. pinaster and P. radiata better than the direct method.
ReferencesTop
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