Models for Tree Taper Form: The Gompertz and Vasicek Diffusion Processes Framework

Abstract

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.

Stochastic Models to Qualify Stem Tapers

“…Nowadays, the available computational power has substantially increased, so it becomes feasible to use more advanced SDE models and to introduce some kind of noise in the stem taper form. In this study, the developed models are nondeterministic models that consider both the symmetric and asymmetric diffusions of stem diameter at a particular height [21]. Multiple regression models that describe stem tapers have been proposed, traditionally incorporating the diameter at breast height and tree height as the independent variables [14][15][16][17][18][19] or using additional independent variables such as the density [18], crown ratio [33], and many more [34,35].…”

“…However, the most published models are deterministic regression models. It is advisable to divide taper equations into five categories, namely, (1) segmented polynomial taper equations [14], (2) variable exponential equations [15][16][17], (3) artificial neural network (ANN) equations [18], (4) fundamental mechanisms of stem profile formation equations [19,20], and (5) diffusion process equations [21]. The first four categories include deterministic taper models that describe the variation of the stem form by means of a single equation, where the diameter at any specified stem height is the dependent variable, while the diameter at breast height and overall height are the independent variables.…”

“…The shape of a tree stem is related to the growth environment of a tree, where different types of irregularities, such as wounds, coarse bark areas, and branches vary among trees of the same species [21]. A goodness of fit of a model is related with the number of incorporated independent variables, such as the planting density, fertilization, thinning, and crown height.…”

“…Diffusion processes based on stem taper models are not affected by the mathematical form, because we start from the evolution of the probability density function of the relative diameter [21]. Another characteristic of newly developed stem taper models is that the derived normal and log-normal probability density functions of the solutions of the SDEs enable us to define the evolution of any central moment (mean, variance, and so on), quartile, p-quantile (0 < p < 1), and coefficient of variation of the diameter at any specified height when both stem diameter and height are given.…”

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

“…One advantage of the stationary conditional distribution is that the conditional quantile functions, and different measures of central tendency and variability can be obtained, providing a more comprehensive analysis of the relationship between diameter and height. Until now, the quantile regression has been widely conducted in tree-variable modeling [10,16,37]. Table 10.…”