Stochastic Mixed-Effects Parameters Bertalanffy Process, with Applications to Tree Growth Modeling

A stochastic modeling approach based on the Bertalanffy law gained interest due to its ability to produce more accurate results than the deterministic approaches. Additionally, the stochastic differential equation (SDE) method provides more sophisticated mathematical analysis tools compared to regression approaches. We examine tree crown width dynamic with the Bertalanffy type SDE and mixed-effects parameters. In this study, we demonstrate how this simple model can be used to calculate predictions of  crown width. This model allows us to estimate the parameters by considering discrete sampling of the diameter at breast height and crown width and by using maximum likelihood procedure. In this paper, we propose a parameter estimation method and computational guidelines. Performance statistics for the crown width equation include statistical indexes, Shapiro-Wilk test, normal probability plot and analysis of residuals. We use data provided by the Lithuanian National Forest Inventory from Scots pine trees to illustrate issues our modeling technique.