Parameter estimation and model selection for stochastic differential equations for biological growth

TL;DR

This paper develops methods for estimating parameters and selecting models among stochastic differential equations for biological growth, using maximum likelihood and information criteria, applicable even with sparse or incomplete data.

Contribution

It introduces a novel approach combining MLE, AIC, and EM algorithms for parameter estimation and model selection in stochastic growth models with various data sparsity levels.

Findings

The methodology accurately estimates parameters from simulated data.

AIC effectively identifies the best model among candidates.

The approach works with highly sparse and incomplete datasets.

Abstract

In this paper, we consider stochastic versions of three classical growth models given by ordinary differential equations (ODEs). Indeed we use stochastic versions of Von Bertalanffy, Gompertz, and Logistic differential equations as models. We assume that each stochastic differential equation (SDE) has some crucial parameters in the drift to be estimated and we use the Maximum Likelihood Estimator (MLE) to estimate them. For estimating the diffusion parameter, we use the MLE for two cases and the quadratic variation of the data for one of the SDEs. We apply the Akaike information criterion (AIC) to choose the best model for the simulated data. We consider that the AIC is a function of the drift parameter. We present a simulation study to validate our selection method. The proposed methodology could be applied to datasets with continuous and discrete observations, but also with highly sparse data. Indeed, we can use this method even in the extreme case where we have observed only one point for each path, under the condition that we observed a sufficient number of trajectories. For the last two cases, the data can be viewed as incomplete observations of a model with a tractable likelihood function; then, we propose a version of the Expectation Maximization (EM) algorithm to estimate these parameters. This type of datasets typically appears in fishery, for instance.