A hyperbolastic type-I diffusion process: Parameter estimation bymeans of the firefly algorithm

TL;DR

The firefly metaheuristic optimization algorithm is applied after bounding the parametric space by a stagewise procedure to solve the problem of maximum likelihood estimation for the parameters of the stochastic diffusion process.

Abstract

A stochastic diffusion process, whose mean function is a hyperbolastic curve of type I, is presented. Themain characteristics of the process are studied and the problem of maximum likelihood estimation forthe parameters of the process is considered. To this end, the firefly metaheuristic optimization algo-rithm is applied after bounding the parametric space by a stagewise procedure. Some examples basedon simulated sample paths and real data illustrate this development.

Figures (10)

• Figure 1: Some simulated sample paths of the H1-type diffusion process.
• Figure 2: First simulation example: evolution (from light gray to black) of 80 generations of 40 fireflies for $\eta=0.5$, $\lambda=0.8$, $\mu=0.8$ and $\sigma=0.015$ (left to right, top to bottom). Original values in red.
• Figure 3: First simulation example: evolution of fireflies over the last 60 generations for every parameter (in the same order as in the previous figure). In green, evolution of best firefly; in red, evolution of worst one.
• Figure 4: First simulation example: for each pair of parameters, bidimensional path projection representing the evolution of the last firefly (the best) over successive generations (from light to dark red point).
• Figure 5: First simulation example: value of objective function at the best firefly. Comparison between observed and fitted mean (red).