Comparing Optimization Methods, in Continuous Space, for Modelling with a Diffusion Process

Abstract

Many probabilistic models are frequently used for natural growth-patterns modelling and their forecasting such as the diffusion processes. The maximum likelihood estimation of the parameters of a diffusion process requires a system of equations that, for some cases, has no explicit solution to be solved. Facing that situation, we can approximate the solution using an optimization method. In this paper we compare five optimization methods: an Iterative Method, an algorithm based on Newton-Raphson solver, a Variable Neighbourhood Search method, a Simulated Annealing algorithm and an Evolutionary Algorithm. We generate four data sets following a Gompertz-lognormal diffusion process using different noise level. The methods are applied with these data sets for estimating the parameters which are present into the diffusion process. Results show that bio-inspired methods gain suitable solutions for the problem every time, even when the noise level increase. On the other hand, some analytical methods as Newton-Raphson or the Iterative Method do not always solve the problem whether their scores depend on the starting point for initial solution or the noise level hinders the resolution of the problem. In these cases, the bio-inspired algorithms remain as a suitable and reliable approach.