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

Martynas Narmontas,Petras Rupšys,Edmundas Petrauskas

doi:10.3390/math8081363

Martynas Narmontas, Petras Rupšys + Show 1 more

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https://doi.org/10.3390/math8081363

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Journal: Mathematics	Publication Date: Aug 14, 2020
Citations: 7	License type: CC BY 4.0

Affiliation: Vytautas Magnus University

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.

Highlights

The structural class hierarchy in a forest stand has interested researchers for more than a century.This study shows that the framework of the stochastic differential equations (SDEs) can be generalized to incorporate symmetric or non-symmetric size component distributions to explain empirical datasets in a theoretically consistent way.Tree height (h) and tree diameter at breast height (d) are traditionally used variables for tree volume calculation, aboveground carbon-stock estimation, and modeling of tree growth and yield
Tree height and diameter are traditionally formalized by using their regression relationship [1], the artificial neural network (ANN) [2,3], or stochastic differential equations [4,5]
This study evaluated the use of SDEs to model the tree diameter and height at any given age in mountain pine tree (Pinus mugo Turra) species of Lithuania

Summary

Introduction

Tree height (h) and tree diameter at breast height (d) (in the sequel—diameter) are traditionally used variables for tree volume calculation, aboveground carbon-stock estimation, and modeling of tree growth and yield. Tree height and diameter are traditionally formalized by using their regression relationship [1], the artificial neural network (ANN) [2,3], or stochastic differential equations [4,5]. Height–diameter regression equations have been developed for the local (stand) level and the generalized (ecoregional) level, by introducing additional stand variables, as well as random parameters [6,7]. The bias of tree-height predictions may affect the accuracy of the estimation of the stand volume and aboveground carbon stocks. To achieve reliable predictions of tree height, European forest statisticians have applied different techniques, such as generalized height–diameter models that include additional stand

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