Bivariate Distribution Modeling of Tree Diameters and Heights: Dependency Modeling Using Copulas
Models of diameter and height distributions play an important role in forest mensuration and inventory, and many models and fitting methods are available for fitting these distributions separately. Very few models and methods appropriate for fitting the joint distribution of diameters and heights are available. The S BB and distributions obtained using Plackett's method are the only appropriate models published in the forestry literature to date. A bivariate joint distribution can be represented in terms of its underlying marginals and a copula function that models the dependency structure. A copula is a function that joins or “couples” a multivariate distribution function to its one-dimensional marginal distributions. The copula approach provides a general way of constructing bivariate distributions, including previously used bivariate models. The basics of the copula representation, the most useful copula functions, and maximum likelihood estimation (MLE) of the copula function are introduced. The emphasis of this article is on the choice of copula function(s) appropriate for modeling the joint distribution of tree diameters and heights. Using an empirical dataset of 102 Chinese fir (Cunninghamia lanceolata Lamb.) plantation plots, we fit each of the copula functions that we considered to each sample-plot joint distribution of tree diameters and heights. A two-stage MLE approach with a nonparametric (empirical) approach to the marginal distributions is adopted. In this case study the normal copula proves to be the best for modeling the joint distribution (dependence) of tree diameters and heights.