Stochastic differential equation approach of height-diameter equations of individual trees

Abstract

In this paper we use a stochastic differential equation to describe the dynamic evolution of the height of an individual tree. The first model is defined by Gompertz shape stochastic differential equation. The second model is defined by Gompertz stochastic differential equation with a threshold parameter. This model can be considered as an extension of the three parameter stochastic Gompertz model with the addition of a fourth parameter. The parameters are estimated by considering discrete sampling of the diameter and height and by using maximum likelihood procedure. Two developed models were employed to compare predicted values with observed values of a height. Performance statistics for developed height-diameter equations included statistical indexes, Shapiro-Wilk test and normal probability plot. We used the data of tropical Atlantic moist forest trees in southeastern Brazil (Scaranello et al., 2012) to validate our modelling technique. Results indicated that our model is able to capture the behaviour of tree height quite accurately. All results were implemented in a symbolic algebra system MAPLE.