New Insights into Tree Height Distribution Based on Mixed Effects Univariate Diffusion Processes

Abstract

The aim of this paper is twofold: to introduce the mathematics of stochastic differential equations (SDEs) for forest dynamics modeling and to describe how such a model can be applied to aid our understanding of tree height distribution corresponding to a given diameter using the large dataset provided by the Lithuanian National Forest Inventory (LNFI). Tree height-diameter dynamics was examined with Ornstein-Uhlenbeck family mixed effects SDEs. Dynamics of a tree height, volume and their coefficients of variation, quantile regression curves of the tree height, and height-diameter ratio were demonstrated using newly developed tree height distributions for a given diameter. The parameters were estimated by considering a discrete sample of the diameter and height and by using an approximated maximum likelihood procedure. All models were evaluated using a validation dataset. The dataset provided by the LNFI (2006–2010) of Scots pine trees is used in this study to estimate parameters and validate our modeling technique. The verification indicated that the newly developed models are able to accurately capture the behavior of tree height distribution corresponding to a given diameter. All of the results were implemented in a MAPLE symbolic algebra system.

Understanding the Evolution of Tree Size Diversity within the Multivariate Nonsymmetrical Diffusion Process and Information Measures

“…In the previous study [17], calibration relies on the mean trend Equation 9to predict the random effects in relation to fixed effects parametersθ m estimated by approximated maximum likelihood procedure (see Equations (41) and (42)). Both alternative techniques deal adequately with random effects calibration, whose are essential for analyzing large observed datasets.…”

“…In order to overcome these disadvantages, the multivariate stochastic differential equations have recently gained a lot of attention. Stochastic differential equations are often used in the modeling of population dynamic [10][11][12], tumor growth [13], chemical reaction networks [14], environmental pollution [15,16], forest growth and yield [17]. The deterministic differential equation carries its solution, which is completely determined in the value sense by knowledge of boundary and initial conditions.…”

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

“…Diameter and height dynamics via an average stand age as a bivariate stochastic process were modeled using fixed effects reducible SDEs [25]. 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].…”

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

“…The Gompertz, Verhulst, Bertalanffy, and Maltussian type SDEs were used to model the total tree height over age [10] and diameter [11,12], to model the stem taper [13], and to model tree crown width over diameter [14]. They found that the SDE models provided much more accurate predictions of individual tree height and crown width compared to nonlinear regression models.…”