A Bertalanffy-Richards growth model perturbed by a time-dependent pattern, statistical analysis and applications

TL;DR

This paper introduces a modified Richards growth model with a time-dependent perturbation, analyzes its features, compares it to the original, and applies stochastic processes and optimization algorithms for parameter estimation, including a real oil production case study.

Contribution

It presents a novel modification of the Richards growth model with a switching time, along with stochastic process representations and advanced estimation techniques.

Findings

The modified growth model captures time-dependent perturbations effectively.

Maximum likelihood estimation via meta-heuristics is validated through simulations.

Application to French oil production demonstrates practical relevance.

Abstract

We analyze a modification of the Richards growth model by introducing a time-dependent perturbation in the growth rate. This modification becomes effective at a special switching time, which represents the first-crossing-time of the Richards growth curve through a given constant boundary. The relevant features of the modified growth model are studied and compared with those of the original one. A sensitivity analysis on the switching time is also performed. Then, we define two different stochastic processes, i.e. a non-homogeneous linear birth-death process and a lognormal diffusion process, such that their means identify to the growth curve under investigation. For the diffusion process, we address the problem of parameters estimation through the maximum likelihood method. The estimates are obtained via meta-heuristic algorithms (namely, Simulated Annealing and Ant Lion Optimizer). A simulation study to validate the estimation procedure is also presented, together with a real application to oil production in France. Special attention is devoted to the approximation of switching time density, viewed as the first-passage-time density for the lognormal process.