Construction of Reducible Stochastic Differential Equation Systems for Tree Height–Diameter Connections

Abstract

This study proposes a general bivariate stochastic differential equation model of population growth which includes random forces governing the dynamics of the bivariate distribution of size variables. The dynamics of the bivariate probability density function of the size variables in a population are described by the mixed-effect parameters Vasicek, Gompertz, Bertalanffy, and the gamma-type bivariate stochastic differential equations (SDEs). The newly derived bivariate probability density function and its marginal univariate, as well as the conditional univariate function, can be applied for the modeling of population attributes such as the mean value, quantiles, and much more. The models presented here are the basis for further developments toward the tree diameter–height and height–diameter relationships for general purpose in forest management. The present study experimentally confirms the effectiveness of using bivariate SDEs to reconstruct diameter–height and height–diameter relationships by using measurements obtained from mountain pine tree (Pinus mugo Turra) species dataset in Lithuania.

Analysis of Longitudinal Forest Data on Individual-Tree and Whole-Stand Attributes Using a Stochastic Differential Equation Model

“…Computed fixed-effect parameters estimates using the estimation dataset are given in Table 2. Estimates of standard errors were obtained as the inverse of the observed Fisher information matrix (see, for instance, [19]). Parameter estimates of all fitted models were significant (p < 0.05).…”

“…A stochastic differential equation model for height and diameter kinetics during a stand growth period was described in [19], regardless of the potentially available area dynamics. The potentially available area of an individual tree can be easily and reliably calculated using Voronoi diagrams [20].…”

A Multivariate Hybrid Stochastic Differential Equation Model for Whole-Stand Dynamics

“…Prior to the stand volume analysis, individual tree volume prediction was examined using a few general nonlinear models using power and q-exponential regression forms [37,38]. We incorporated the q-exponential function, which is defined as…”

“…Figure 7 shows the dynamics of the stand volume per hectare as a function of stand age using the mixed-effect scenario. Table 16 shows the predictive ability for both the newly-developed non-stationary and stationary mixed-effect scenario volume per hectare models defined by Equation (38). regression models to estimate tree volume from measurements of tree variables, such as the diameter at breast height and height, which are measured during inventory.…”

“…Figure 7 shows the dynamics of the stand volume per hectare as a function of stand age using the mixed-effect scenario. Table 16 shows the predictive ability for both the newly-developed nonstationary and stationary mixed-effect scenario volume per hectare models defined by Equation (38). Funding: This research received no external funding.…”