Univariate and bivariate diffusion models: computational aspects and applications to forestry

Abstract

This tutorial is concerned with modeling principles and methods inspired by the properties of diffusion processes. The goal of this chapter is to introduce a methodology for examining continuous-time Ornstein-Uhlenbech family processes defined by stochastic differential equations (SDEs). A practical viewpoint is adopted to establish a proper foundation for growth modeling using SDEs. This study introduces the mathematics of mixed effect parameters in univariate and bivariate SDEs and describes how such a model can be used to aid our understanding of growth processes using real world datasets. The Vasicek, Gompertz, von Bertalanffy and gamma type diffusion processes are examined. We introduce a detailed discussion of parameter estimation by the approximated maximum likelihood procedure for the non-random parameters and the random effects. Results and experience from applying the concepts and techniques in an extensive individual tree and stand growth modeling program in Lithuania are described as examples