In this work, we employ stochastic differential equations (SDEs) to model tree stem taper. SDE stem taper models have some theoretical advantages over the commonly employed regression-based stem taper modeling techniques, as SDE models have both simple analytic forms and a high level of accuracy. We perform fixed- and mixed-effect parameters estimation for the stem taper models by developing an approximated maximum likelihood procedure and using a data set of longitudinal measurements from 319 mountain pine trees. The symmetric Vasicek- and asymmetric Gompertz-type diffusion processes used adequately describe stem taper evolution. The proposed SDE stem taper models are compared to four regression stem taper equations and four volume equations. Overall, the best goodness-of-fit statistics are produced by the mixed-effect parameters SDEs stem taper models. All results are obtained in the Maple computer algebra system.

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.

The growth and yield modeling of a forest stand has progressed rapidly, starting from the generalized nonlinear regression models of uneven/even-aged stands, and continuing to stochastic differential equation (SDE) models. We focus on the adaptation of the SDEs for the modeling of forest stand dynamics, and relate the tree and stand size variables to the age dimension (time). Two different types of diffusion processes are incorporated into a hybrid model in which the shortcomings of each variable types can be overcome to some extent. This paper presents the hybrid multivariate SDE regarding stand basal area and volume models in a forest stand. We estimate the fixed- and mixed-effect parameters for the multivariate hybrid stochastic differential equation using a maximum likelihood procedure. The results are illustrated using a dataset of measurements from Mountain pine tree (Pinus mugo Turra).

This study examines the performance of 11 tree taper models to predict the diameter of bark at any given height and the total stem volume of eight dominant tree species in the boreal forests of Lithuania. Here, we develop eight new models using stochastic differential equations (SDEs). The symmetrical Vasicek model and asymmetrical Gompertz model are used to describe tree taper evolution, as well as geometric-type diffusion processes. These models are compared with those traditionally used for four tree taper models by using performance statistics and residual analysis. The observed dataset consists of longitudinal measurements of 3703 trees, representing the eight dominant tree species in Lithuania (pine, spruce, oak, ash, birch, black alder, white alder, and aspen). Overall, the best goodness of fit statistics of diameter predictions produced the SDE taper models. All results have been implemented in the Maple computer algebra system using the "Statistics" and "VectorCalculus" packages.

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