On the Construction of Growth Models via Symmetric Copulas and Stochastic Differential Equations

Abstract

By nature, growth regulatory networks in biology are dynamic and stochastic, and feedback regulates their growth function at different ages. In this study, we carried out a stochastic modeling of growth networks and demonstrated this method using three mixed effect four-parameter Gompertz-type diffusion processes and a combination thereof using the conditional normal copula function. Using the conditional normal copula, newly derived univariate distributions can be combined into trivariate and bivariate distributions, and their corresponding conditional bivariate and univariate distributions. The link between the predictor variable and the remaining one or two explanatory variables can be formalized using copula-type densities and a numerical integration procedure. In this study, for parameter estimation, we used a semiparametric maximum pseudo-likelihood estimator procedure, which was characterized by a two-step technique, namely, separately estimating the parameters of the marginal distributions and the parameters of the copula. The results were illustrated using two observed longitudinal datasets, the first of which included the age, diameter, and potentially available area of 39,437 trees (48 stands), while the second included the age, diameter, potentially available area, and height of 8604 trees (47 stands) covering uneven mixed-species (pine, spruce, and birch) stands. All results were implemented using the MAPLE symbolic algebra system.