A Linkage among Tree Diameter, Height, Crown Base Height, and Crown Width 4-Variate Distribution and Their Growth Models: A 4-Variate Diffusion Process Approach

Abstract

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.

Modeling Dynamics of Structural Components of Forest Stands Based on Trivariate Stochastic Differential Equation

“…. , M. Taking into account the transformation Y l (t) = e β i (t) ·X l i (t), i = 1, 2, 3 T [28], we can deduce that the conditional random vector X l (t) X l (t 0 ) = x 0 = X l i (t) X l i (t 0 ) = x i0 , i = 1, 2, 3 T has a 3-variate normal distribution N 3 µ l (t), Σ(t) with the mean vector µ l (t) defined by:…”

“…More recently, mixed effects univariate SDE models have provided the means to quantify and distinguish additional sources of variability in an observed dataset [26]. In addition to the inter-individual variability, multivariate SDE models also consider the covariance structure between size components [27][28][29].…”

Stochastic Models to Qualify Stem Tapers

“…This makes stochastic modeling a powerful tool in the hands of practitioners in fields for which population growth is a critical determinant of various outcomes. There is a general tendency in the population growth modeling literature to favor flexible techniques that represent features of multivariate data as well as possible [10,11]. Therefore, multivariate SDEs describing population growth models contain both the main effects and interaction effects involved in models using a variance-covariance matrix, improving the potential to interpret the data more informatively [12] and inferring the causality in a statistical sense as a type of dependence of the multivariate random variables [13].…”

Models for Tree Taper Form: The Gompertz and Vasicek Diffusion Processes Framework

“…Stochastic growth theories, for example, have been built around the Vasicek, Gompertz, Bertalanffy, and gamma SDEs in both univariate [14] and bivariate [15,16] forms. Multivariate diffusion processes allow us to further consider the underlying covariance structure driving the changes in the state variables; for example, a trivariate case [17], a quadrivariate Vasicek case [18], and a quadrivariate Bertalanffy case [19] have been investigated.The main goal of this paper is to develop a segmented mixed-effect parameters SDE stem taper model that uses one joining point to link two sections, where each section is modeled by a different type of mixed-effect parameters SDE, and to describe the maximum likelihood procedure for fixed-and mixed-effect parameter estimates. Another goal is to compare the developed stem taper models with well-known regression models for prediction of diameter at any specified height and volume.…”

“…Stochastic growth theories, for example, have been built around the Vasicek, Gompertz, Bertalanffy, and gamma SDEs in both univariate [14] and bivariate [15,16] forms. Multivariate diffusion processes allow us to further consider the underlying covariance structure driving the changes in the state variables; for example, a trivariate case [17], a quadrivariate Vasicek case [18], and a quadrivariate Bertalanffy case [19] have been investigated.…”