The aim of this paper is twofold: to introduce the mathematics of stochastic differential equations (SDEs) for forest dynamics modeling and to describe how such a model can be applied to aid our understanding of tree height distribution corresponding to a given diameter using the large dataset provided by the Lithuanian National Forest Inventory (LNFI). Tree height-diameter dynamics was examined with Ornstein-Uhlenbeck family mixed effects SDEs. Dynamics of a tree height, volume and their coefficients of variation, quantile regression curves of the tree height, and height-diameter ratio were demonstrated using newly developed tree height distributions for a given diameter. The parameters were estimated by considering a discrete sample of the diameter and height and by using an approximated maximum likelihood procedure. All models were evaluated using a validation dataset. The dataset provided by the LNFI (2006–2010) of Scots pine trees is used in this study to estimate parameters and validate our modeling technique. The verification indicated that the newly developed models are able to accurately capture the behavior of tree height distribution corresponding to a given diameter. All of the results were implemented in a MAPLE symbolic algebra system.

Abstract:The evolution of the 4-variate probability distribution of the diameter at the breast height, total height, crown base height, and crown width against the age in a forest stand is of great interest to forest management and the evaluation of forest resources. This paper focuses on the Vasicek type 4-variate fixed effect stochastic differential equation (SDE) to quantify the dynamic of tree size components distribution against the age. The new derived 4-variate probability density function and its marginal univariate, bivariate, trivariate, and conditional univariate distributions are applied for the modeling of stand attributes such as the mean diameter, height, crown base height, crown width, volume, and slenderness. All parameters were estimated by the maximum likelihood procedure using a dataset of 1630 Scots pine trees (12 stands). The results were validated using a dataset of 699 Scots pine trees (five stands). A newly developed 4-variate simultaneous system of SDEs incorporated covariance structure driving changes in tree size components and improved predictions in one tree size component given the other tree size components in the system.

This study focuses on the stochastic differential calculus of Itô, as an effective tool for the analysis of noise in forest growth and yield modeling. Idea of modeling state (tree size) variable in terms of univariate stochastic differential equation is exposed to a multivariate stochastic differential equation. The new developed multivariate probability density function and its marginal univariate, bivariate and trivariate distributions, and conditional univariate, bivariate and trivariate probability density functions can be applied for the modeling of tree size variables and various stand attributes such as the mean diameter, height, crown base height, crown width, volume, basal area, slenderness ratio, increments, and much more. This study introduces generalized multivariate interaction information measures based on the differential entropy to capture multivariate dependencies between state variables. The present study experimentally confirms the effectiveness of using multivariate interaction information measures to reconstruct multivariate relationships of state variables using measurements obtained from a real-world data set.

Research Highlights: Today’s approaches to modeling of forest stands are in most cases based on that the regression models and they are constructed as static sub-models describing individual stands variables. The disadvantages of this method; it is laborious because too many different equations need to be assessed and empirical choices of candidate equations make the results subjective; it does not relate to the stand variables dynamics against the age dimension (time); and does not consider the underlying covariance structure driving changes in the stand variables. In this study, the dynamical model defined by a fixed-and mixed effect parameters trivariate stochastic differential equation (SDE) is introduced and described how such a model can be used to model quadratic mean diameter, mean height, number of trees per hectare, self-thinning line, stand basal area, stand volume per hectare and much more. Background and Objectives: New developed marginal and conditional trivariate probability density functions, combining information generated from an age-dependent variance-covariance matrix of quadratic mean diameter, mean height and number of trees per hectare, improve stand growth prediction, and forecast (in forecast the future is completely unavailable and must only be estimated from historical patterns) accuracies. Materials and Methods: Fixed-and mixed effect parameters SDE models were harmonized to predict and forecast the dynamics of quadratic mean diameter, mean height, number of trees per hectare, basal area, stand volume per hectare, and their current and mean increments. The results and experience from applying the SDE concepts and techniques in an extensive whole stand growth and yield analysis are described using a Scots pine (Pinus sylvestris L.) experimental dataset in Lithuania. Results: The mixed effects scenario SDE model showed high accuracy, the percentage root mean square error values for quadratic mean diameter, mean height, number of trees per hectare, stand basal area and stand volume per hectare predictions (forecasts) were 3.37% (10.44%), 1.82% (2.07%), 1.76% (2.93%), 6.65% (10.41%) and 6.50% (8.93%), respectively. In the same way, the quadratic mean diameter, mean height, number of trees per hectare, stand basal area and stand volume per hectare prediction (forecast) relationships had high values of the coefficient of determination, R2, 0.998 (0.987), 0.997 (0.992), 0.997 (0.988), 0.968 (0.984) and 0.966 (0.980), respectively. Conclusions: The approach presented in this paper can be used for developing a new generation stand growth and yield models.

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Statistical models using stochastic differential equations (SDEs) to describe dynamically evolving natural systems are appearing in the scientific literature with some regularity in recent years. In this study, the SDE mixed-effects parameter models based on a Vasicek non-homogeneous diffusion process are formulated. The breast height diameter-dependent drift function additionally depends on deterministic function that describes the dynamic of certain exogenous stand variables (crown height, c h , crown width, c w , mean breast height diameter, d0, mean height, h0, age, A, soil fertility index, SFI, stocking level, S) versus breast height diameter. The mixed-effects parameters SDE models included a random parameter that affected the models asymptote. The parameter estimators are evaluated by maximum likelihood procedure. The objective of the research was to develop a generalized mixed-effects parameters SDE height–diameter models and to illustrate issues using dataset of Scots pine trees (Pinus sylvestris L.) in Lithuania with the breast height diameter outside the bark larger than 0 cm. The parameters of all used models were estimated using an estimation dataset and were evaluated using a validation dataset. The new developed height–diameter models are an improvement over exogenous stand variables, in that it can be calibrated to a new stand with observed height–diameter pairs, thus improving height prediction.

A stochastic modeling approach based on the Bertalanffy law gained interest due to its ability to produce more accurate results than the deterministic approaches. We examine tree crown width dynamic with the Bertalanffy type stochastic differential equation (SDE) and mixed-effects parameters. In this study, we demonstrate how this simple model can be used to calculate predictions of crown width. We propose a parameter estimation method and computational guidelines. The primary goal of the study was to estimate the parameters by considering discrete sampling of the diameter at breast height and crown width and by using maximum likelihood procedure. Performance statistics for the crown width equation include statistical indexes and analysis of residuals. We use data provided by the Lithuanian National Forest Inventory from Scots pine trees to illustrate issues of our modeling technique. Comparison of the predicted crown width values of mixed-effects parameters model with those obtained using fixedeffects parameters model demonstrates the predictive power of the stochastic differential equations model with mixed-effects parameters. All results were implemented in a symbolic algebra system MAPLE.

Our study focusses on investigating a modern modelling paradigm, a bivariate stochastic process, that allows us to link individual tree variables with growth and yield stand attributes. In this paper, our aim is to introduce the mathematics of mixed effect parameters in a bivariate stochastic differential equation and to describe how such a model can be used to aid our understanding of the bivariate height and diameter distribution in a stand using a large dataset provided by the Lithuanian National Forest Inventory (LNFI). We examine tree height and diameter evolution with a Vasicek-type bivariate stochastic differential equation and mixed effect parameters. It is focused on demonstrating how new developed bivariate conditional probability density functions allowed us to calculate the evolution, in the forward and backward directions, of the mean diameter, height, dominant height, assortments, stem volume of a stand and uncertainties in these attributes for a given stand age. We estimate the parameters by considering discrete samples of the diameter and height at a given age and by using an approximated maximum likelihood procedure. The model performance criteria for the height and diameter growth models include statistical indexes and an analysis of residuals.