Trivariate Distribution Modeling of Tree Diameter, Height, and Volume

Abstract

Trivariate distribution models were constructed for tree diameter (D), height (H), and volume (V) using a normal copula (a mathematical function that joins or links marginal distribution functions to form multivariate distribution functions). The model parameters were estimated by a two-stage semiparametric method as follows: univariate distribution models were fitted separately for each tree variable, and then the copula dependency (correlation) parameters were independently estimated using empirical distribution functions to approximate the marginal models. Our results indicated that, for the available trivariate data in this study, the log-logistic (a limiting form of the logit-logistic distribution) best described both D and V distributions, whereas the logit-logistic, Johnson's SB, and Weibull marginal models provided roughly the same fitting for the H distribution. The trivariate distribution models produced satisfactory estimates of trivariate frequencies for the three variables, as well as the volume predictions from the resulting median regression models, which were comparable with commonly used volume regression models. The advantage of the normal copula method is that one can use any marginal models suitable for modeling tree diameter, height, and volume distributions separately, rather than being restricted to the SB marginal model in Johnson's SBBB system.