IntroductionTop

Stand volume, one of the most important variables in forest management, is usually estimated based on sampled tree diameters and heights (Wang & Rennolls, 2007). The common practice is to obtain the height data from a subsample of trees for which diameters are available, and to fit an empirical height-diameter relationship to estimate the average height per diameter class. Tree volume is then estimated using an individual-tree volume equation. Although this approach may appear satisfactory, it is often not appropriate because one tends to ignore the fact that height may vary considerably for a given diameter due to genetic, environmental or silvicultural factors. Therefore, the height residuals are seldom homoscedastic and normally distributed and in many forests the variance about the diameter-height regression is heterogeneous (Zucchini et al., 2001).

An alternative approach for improving stand volume estimation, which takes into account those variations, involves the use of a bivariate distribution (Zucchini et al., 2001; Wang et al., 2008; Mønness, 2015). The joint bivariate distribution of tree diameter and height provides a detailed impression of the relationship between the two variables, which is not given by the two marginal distributions (Rupsys & Petrauskas, 2010). Moreover, bivariate distributions of diameter and height are also useful for assessing timber value based on price sizes (Schreuder & Hafley, 1977) and stand structural diversity (Staudhammer & LeMay, 2001).

Hence there has been considerable interest in identifying suitable bivariate distributions to describe diameter-height frequency data. For many years, the bivariate extension of the S_B distribution, the S_BB (Johnson, 1949), has been the only bivariate distribution used for modeling bivariate tree diameter-height frequency data (e.g. Hafley & Schreuder, 1977; Knoebel & Burkhart, 1991; Tewari & Gadow, 1999; Castedo Dorado et al., 2001; Zucchini et al., 2001). Johnson’s S_BB is developed by applying a four-parameter logistic transformation to each of the component variables of a standard bivariate normal distribution (Johnson, 1949; Rennolls & Wang, 2005). The construction of any other analytic bivariate distribution without resorting to a transformation of a bivariate normal distribution is complicated (Wang & Rennolls, 2007). However, the use of a copula function has provided a general way of constructing multivariate distributions. During recent years, several authors made use of the approach described by Sklar (1973) joining a multivariate distribution based on their one dimensional marginal distributions (e.g. Li et al., 2002; Wang & Rennolls, 2007; Wang et al., 2008).

The objective of the present study is to fit and compare the accuracy of three bivariate distributions: Johnson’s S_BB, Weibull and Logit-Logistic fitted to diameter-height data from pure and even-aged stands of Eucalyptus globulus in Northwestern Spain. The Weibull and the Logit-Logistic (LL) bivariate distributions, denoted as Weibull-2^P and LL-2^P, were obtained from Weibull and LL marginal distributions by using the Plackett copula whereas the S_BB bivariate distribution was obtained from S_B marginal distributions using the Normal copula, i.e., a four-parameter logistic transformation to each of the component variables.

Material and methodsTop

Data

All tree diameters and heights were measured in 128 field plots in Tasmanian blue gum(Eucalyptus globulus Labill.) stands in Galicia. The plots had been re-measured 1, 2 or 3 times resulting in a total of 308 inventories. The plots were established in pure and even-aged stands covering a wide variety of combinations of age, number of trees per hectare, site quality and method of regeneration. The sample plot size ranged from 375 to 900 m², depending on stand density. The objective was to assess a minimum of 30 trees per plot.

All trees in each plot were numbered; diameters at breast height were measured with a caliper, to the nearest 0.1 cm, and heights were measured with hypsometer to the nearest 0.1 m. The stand variables calculated in each inventory included the quadratic mean diameter, the number of trees per hectare, dominant height, basal area and mean height. A total of 17,588 trees were measured. The summary statistics of the main stand variables are presented in Table 1.

Table 1. Summary of the main descriptive statistics of stand variables.

Copula functions

Wang et al. (2008) presented an exhaustive review of different one-parameter copulas which are useful for modeling bivariate tree diameter and height distributions. A copula is a function that joins a multivariate distribution function based on its one-dimensional marginal distributions. Suppose X and Y are two continuous random variables and F(x) = Pr(X ≤ x) and G(y) = Pr(Y ≤ y) are their marginal cumulative distribution functions, respectively. The copula function C combines these two marginal to give the joint distribution function H(x, y) as H(x, y) = C(F(x), G(y)) . If both marginal distribution functions and the copula are differentiable, the joint density function can be expressed as:

where f(x) and g(y) are the marginal density functions, and is the used copula density.

Frequently used copulas are the Normal (Mardia, 1970) and the Plackett copula (Plackett, 1965). Their densities are given by (Wang et al., 2008):

Plackett copula

where z_x and z_y are specific transformations of x and y, respectively and ω, defined as the cross-product ratio or odds-ratio, is a positive constant for all (x,y) for which neither F nor G assumes the value 0 or 1; ρ is a measure of the degree of association.

Fitting the SBB, Weibull-2P and Logit-logistic (LL-2^P) bivariate distributions

The S_BB distribution was obtained from S_B marginal distributions using the normal copula. In this case, the variables x and y were defined as: x = (D — ε₁)/λ₁ and y = (H — ε₂)/λ₂ where ε₁ and ε₂ are the location parameters and λ₁ and λ₂ are the observed ranges of diameter (D) and height (H), respectively. The values of z_x and z_y were obtained from a four-parameter logistic transformation of x and y z_x = γ₁ + δ₁ log(x/1 — x) and z_y = γ₂ + δ₂ log(y/1 — y). These variables have a joint normal bivariate distribution with correlation ρ:

The parameters ε were predetermined as d_min-0.5 and h_min-0.5 for diameter and height, respectively, whereas the parameter λ was set equal to the range of diameters and the range of heights plus one, for the two marginal distributions, respectively.

The Weibull-2^P and the LL-2^P bivariate distributionswere obtained using the Plackett copula and the marginal Weibull and Logit-Logistic density functions:

Logit-logistic density function

where x is the diameter (D), y is the height (H), ε is the location parameter, b and c are the scale and shape parameters of the Weibull distribution, with b, c > 0; λ is the scale parameter and μ and σ are the shape parameters of the Logit-logistic distribution, with ε < x < ε + λ; - ∞ < ε < ∞; - ∞ < μ < ∞; λ > 0; σ > 0.

The parameters were estimated by minimizing the negative log-likelihood function of equations (1) for Weibull-2^P and LL-2^P and (4) for S_BB using the R function optim (R Core Team, 2014). Assuming that the sample observations are independent with identical distributions, the negative log-likelihood function is the sum of single-tree terms (Wang & Rennolls, 2007). Both univariate distributions considered in this study to develop bivariate distributions using the Plackett copula (Weibull-2^P and LL-2^P) have a closed form of their cumulative distributions. If this is not the case, numerical methods should be used for evaluating the cumulative distribution in the model-fitting process.

Results and discussionTop

The means, maxima, minima and standard deviations of the estimated parameters for the three bivariate distributions (bivariate Johnson’s S_BB, Weibull-2^P and LL-2^P) are presented in table 2. The maximum likelihood estimation converged for all plots and for all three bivariate distributions. In a study in Chinese fir plantations (Cunninghamia lanceolata Lamb.) a number of sample plots did not converge for the LL-2^P bivariate and bivariate beta distributions, probably due to these plots having J-shaped marginal distributions (Wang & Rennolls, 2007). Our good results could be due to the fact that all sample plots were installed in even-aged forests.

Table 2. Mean values, maximum, minimum and standard deviation of the parameters for the three bivariate distributions compared.

Table 3 presents the between-model comparative performance of the three bivariate distributions in terms of their goodness-of-fit statistics and the Wilcoxon rank test. The best results were obtained with the Logit-logistic marginal distribution combined with the Plackett copula. Wang et al. (2008), in a study for Chinese fir plantations, comparing five different copulas including the Normal and the Plackett, found that the normal copula showed the best results. In this study, we cannot compare directly the copulas because we are using different marginal distributions with each copula. Moreover, as the authors pointed out, the age range of the Chinese fir plantations used in their study was very limited and older stands had different structures influencing the observed outcomes.

Table 3. Between-model comparative performance of these three models in terms of their goodness-of-fit statistics and the Wilcoxon test. Ratio is the proportion of cases in which the row distribution model had a lower value of the negative log-likelihood function than the column distribution.

The S_BB distribution showed better results in terms of goodness-of-fit statistics than the Weibull distribution (Table 3), although the differences were not significant. The better performance of LL-2^P over S_BB and Weibull was expected since the logit-logistic univariate distribution is more flexible than the other two, covering a wide range of skewness-kurtosis combinations. However, the good results of the Weibull distribution were unexpected, because the Weibull univariate turned out to be the least flexible of the three univariate distributions used. The reason for this may be the very regular shape of the marginal diameter and height distributions of our even-aged stands. Moreover, it also should be taken into account that the locations (ε) and the ranges (λ) of diameters (D) and heights (H) were fixed, affecting especially the performance of LL-2^P and S_BB bivariate distributions.

Both the normal and the Plackett copulas have shown a good performance for modeling the joint distribution of tree diameters and heights. They could be easily extended for modelling multivariate distributions involving other tree variables. However, it should be noted that the normal copula, in general, does not have a closed form for its joint density, except for the Normal or Johnson’s marginal distributions. Another point to consider is the fact that the Plackett copula requires that the marginal has a closed form for its cumulative distribution (F (x) and G( y) in equation (3)), to avoid numerical methods for evaluating the cumulative distribution in the model-fitting process.