A Framework for Analyzing Individual-Tree and Whole-Stand Growth by Fusing Multilevel Data: Stochastic Differential Equation and Copula Network

Abstract

In forestry, growth functions form the basis of research and are widely used for the mathematical modeling of stand variables, e.g., tree or stand basal area, stand height, stand volume, site index, and many more. In this study, to estimate five-dimensional dependencies between tree diameter at breast height, potentially available area, height, crown area and crown base height, we used a normal copula approach whereby the growths of individual variables are described using a stochastic differential equation with mixed-effect parameters. The normal copula combines the marginal distributions of tree diameter at breast height, potentially available area, height, crown area, and crown base height into a joint multivariate probability distribution. Copula models have the advantage of being able to use collected longitudinal, multivariate, and discrete data for which the number of measurements of individual variables does not match. This study introduced a normalized multivariate interaction information measure based on differential entropy to assess the causality between tree size variables. In order to accurately and quantitatively assess the stochastic processes of the tree size variables’ growth and to provide a scientific basis for the formalization of models, an analysis method of the synergetic theory of information entropy has been proposed. Theoretical findings are illustrated using an uneven-aged, mixed-species empirical dataset of permanent experimental plots in Lithuania.