A new paradigm in modelling the evolution of a stand via the distribution of tree sizes

Our study focusses on investigating a modern modelling paradigm, a bivariate stochastic process, that allows us to link individual tree variables with growth and yield stand attributes. In this paper, our aim is to introduce the mathematics of mixed effect parameters in a bivariate stochastic differential equation and to describe how such a model can be used to aid our understanding of the bivariate height and diameter distribution in a stand using a large dataset provided by the Lithuanian National Forest Inventory (LNFI). We examine tree height and diameter evolution with a Vasicek-type bivariate stochastic differential equation and mixed effect parameters. It is focused on demonstrating how new developed bivariate conditional probability density functions allowed us to calculate the evolution, in the forward and backward directions, of the mean diameter, height, dominant height, assortments, stem volume of a stand and uncertainties in these attributes for a given stand age. We estimate the parameters by considering discrete samples of the diameter and height at a given age and by using an approximated maximum likelihood procedure. The model performance criteria for the height and diameter growth models include statistical indexes and an analysis of residuals.