Mixed effect parameters 4-variate Gompertz type diffusion process: Computational aspects and information measures

Abstract

A 4-variate diffusion process is used to parameterize tree size variables data in forest stands. A diffusion process is defined by a mixed effectparameters 4-variate stochastic differential equation (SDE). We propose a system of the 4-variate mixed effect parameters Gompertz type SDE toquantify the dynamics via stand age of the probability density function of the tree size variables with a sigmoid form trend for the mean values.The SDE model allows us a better understanding of biological processes driving the dynamics of natural phenomena. The new derived 4-variateprobability density function and its marginal univariate, bivariate and trivariate, and conditional univariate, bivariate and trivariate distributionscan be applied for the modeling of stand attributes such as the mean diameter, height, stem and stand volumes. All parameters are estimated bythe maximum likelihood procedure using real-life dataset. We have introduced a general mutual information to capture multivariate interactionsbetween tree size variables. We have experimentally confirmed that 4-variate mutual information quantities are appropriate measures to reconstructmultivariate interactions in tree size variables. The results are implemented in the symbolic algebra system MAPLE