Symmetric and Asymmetric Diffusions through Age-Varying Mixed-Species Stand Parameters

Petras Rupšys,Edmundas Petrauskas

doi:10.3390/sym13081457

Abstract

(1) Background: This paper deals with unevenly aged, whole-stand models from mixed-effect parameters diffusion processes and Voronoi diagram points of view and concentrates on the mixed-species stands in Lithuania. We focus on the Voronoi diagram of potentially available areas to tree positions as the measure of the competition effect of individual trees and the tree diameter at breast height to relate their evolution through time. (2) Methods: We consider a bivariate hybrid mixed-effect parameters stochastic differential equation for the parameterization of the diameter and available polygon area at age to ensure a proper description of the link between them during the age (time) span of a forest stand. In this study, the Voronoi diagram was used as a mathematical tool for the quantitative characterization of inter-tree competition. (3) Results: The newly derived model considers bivariate correlated observations, tree diameter, and polygon area arising from a particular stand and enables defining equations for calculating diameter, polygon-area, and stand-density predictions and forecasts. (4) Conclusions: From a statistical point of view, the newly developed models produced acceptable statistical measures of predictions and forecasts. All the results were implemented in the Maple computer algebra system.

Highlights

Forest statisticians often need to address complex issues in important whole-stand, unevenly aged growth models with high levels of uncertainty that affect individual trees.Providing scientific evidence for effective decision processes in forest areas is a key issue, and stochastic calculus is an essential tool [1].In the past few decades, mathematical models based on diffusion processes were fruitfully applied to describe stochasticity phenomena belonging to even extremely different disciplinary fields that range in scale from human population in biology [2] to cryptocurrency in finance [3] and from infectious disease in medicine [4] to networks in neuroscience [5]
This study considers a stochastic approach for the parameterization of diameter D and polygon area P at age to ensure the proper descriptions of their link during the age span of a forest stand
The likelihood function maximization technique with respect to parameter vector θ for a given set of discretely observed diameter and polygon area data is defined by a two-step procedure

Summary

Introduction

Providing scientific evidence for effective decision processes in forest areas is a key issue, and stochastic calculus is an essential tool [1]. Mixed-effect parameters diffusion processes provide a convenient tool to account for differences between several experiments or several subjects [6]. Results are large datasets of observations and statistical analysis acquired by using competition indices to both direct and indirect stand measurements [7]. To operate with these data, it is necessary to investigate information in a convenient form for the mathematical modeling of individual-based tree growth and yield. Of the type of an individual tree or whole-stand stochastic modelling

Methods

Results

Discussion

Conclusion