A Mixed Effects Height-Diameter Model with the Vasicek Type Stochastic Differential Equation

Abstract

Our aim is twofold: to introduce the mathematics of stochastic differential equations and to explain how such model can be used to aid our understanding of height-diameter process of an individual tree. In this study, we examine tree height dynamics with the Vasicek type stochastic differential equation and mixed-effects parameters. We demonstrate how this simple model can be used to calculate stochastic predictions of tree heights. We estimate the parameters by considering discrete sampling of the diameter and height and by using maximum likelihood procedure. Performance statistics for the height-diameter equation include statistical indexes, Shapiro-Wilk test, normal probability plot and analysis of residuals. We use the data provided by the National Forest Inventory (2006-2010) of Scots pine trees to validate our modeling technique. Results indicate that our model is able to capture the behavior of tree height quite accurately. All results were implemented in a symbolic algebra system MAPLE