Abstract
The evolution of the 4-variate probability distribution of the diameter at the breast height, total height, crown base height, and crown width against the age in a forest stand is of great interest to forest management and the evaluation of forest resources. This paper focuses on the Vasicek type 4-variate fixed effect stochastic differential equation (SDE) to quantify the dynamic of tree size components distribution against the age. The new derived 4-variate probability density function and its marginal univariate, bivariate, trivariate, and conditional univariate distributions are applied for the modeling of stand attributes such as the mean diameter, height, crown base height, crown width, volume, and slenderness. All parameters were estimated by the maximum likelihood procedure using a dataset of 1630 Scots pine trees (12 stands). The results were validated using a dataset of 699 Scots pine trees (five stands). A newly developed 4-variate simultaneous system of SDEs incorporated covariance structure driving changes in tree size components and improved predictions in one tree size component given the other tree size components in the system.
Highlights
In forestry literature, different mechanisms have been proposed to describe relationships between components of a tree size in a stand including the diameter at breast height, total height, crown base height, crown width, and age
In the modeling of the tree diameter, the largest contribution arises from the tree height and from the tree height and crown width
In the modeling of the tree height, the largest contribution arises from the tree diameter and from the tree diameter and crown base height
Summary
In forestry literature, different mechanisms have been proposed to describe relationships between components of a tree size in a stand including the diameter at breast height (in the sequel-diameter), total height (in the sequel-height), crown base height, crown width, and age. Note that the linear or nonlinear regression relationships presented in previous studies provide information about the stem and crown dimensions that can be predicted at a given set of predictor variables; they do not relate stem and crown dynamics to the age dimension, and do not consider the underlying covariance structure driving changes in the tree diameter, height, crown base height, and crown width. The problem of constructing age-diameter regression relationships defies a precise solution as the rate of the stem and crown size components is highly variable. There is another basic issue to keep in mind, in that the growth
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