The Tale of Tail Dependence: Modeling height–diameter relationships with elliptical copulas

Modeling height–diameter (H–D) relationships is fundamental to forest management. Copula could be used to simulate H–D distributions that mirror the underlying dependence structures of an observed relationship. Tail dependence measures the co-movements of extreme events. It is an intrinsic property of a copula, but it has rarely been examined in forestry studies that applied a copula. The overall goal of this paper was to introduce tail dependence in H–D modeling using a small dataset of two species groups and the elliptical copulas (Normal and t copula). The H–D observations were first transformed into pseudo-observations by ranking, which were used to compute tail dependence. Elliptical copulas were fitted and used to simulate distributions of pseudo-observations. The simulated pseudo-observations were transformed into simulated H–D distributions by the fitted marginals. Results showed that the empirical tail dependence of the two species groups was asymmetrical, more variable in the lower tail, but more similar in the upper tail. The elliptical copulas failed to capture the asymmetric empirical tail dependence because of the symmetry imposed by the copulas. This led to inadequate representation of the simulated H–D distributions to the observed ones by producing extreme outliers that were not biologically meaningful. For example, the Normal copula simulated H–D pairs with D ≤ 2 cm but H up to 18 m for one species group. Our assessments of the lack of fit of the elliptical copulas are made possible by studying the distributions of pseudo-observations and computing tail dependence. They unravel dependence structures that are not immediately apparent in the corresponding H–D relationships. Lastly, we recommend future studies to consider reporting pseudo-observations and tail dependence and to explore alternative copulas that accommodate asymmetric tail dependence.