The Poisson regression is popularly used to model count data. However, real data often do not satisfy the assumption of equality of the mean and variance which is an important property of the Poisson distribution. The Poisson – Gamma (Negative binomial) distribution and the recent Conway-Maxwell-Poisson (COM-Poisson) distributions are some of the proposed models for over- and under-dispersion respectively. Nevertheless, the parameterization of the COM-Poisson distribution still remains a major challenge in practice as the location parameter of the original COM-Poisson distribution rarely represents the mean of the distribution. As a result, this paper proposes a new parameterization of the COM-Poisson distribution via the central location (mean) so that more easily-interpretable models and results can be obtained. The parameterization involves solving nonlinear equations which do not have analytical solutions. The nonlinear equations are solved using the efficient and fast derivative free spectral algorithm. Implementation of the parameterization in R (R Core Team, 2018) is used to present useful numerical results concerning the relationship between the mean of the COM-Poisson distribution and the location parameter in the original COM-Poisson parameterization. The proposed technique is further used to fit COM-Poisson probability models to real life datasets. It was found that obtaining estimates via this parameterization makes the estimation easier and faster compared to directly maximizing the likelihood function of the standard COM-Poisson distribution.
Obtaining parameter estimates for nonlinear regression model using gauss-newton and gradient-based methods present some complex analytical challenges. In this paper we investigated the effectiveness and simplicity of Particle Swarm Optimization (PSO) and Genetic Algorithm (GA) on five nonlinear regression models with varying level of complexities. We developed function in R-programming for each models and performed 30 independent runs for at least 100 iterations for both PSO and GA. We evaluated PSO and GA performance in view of computation time, residual error produced and compared our results with values published online. Based on the results obtained, PSO significantly outperform GA in view of computation time and quality of parameter estimates. Even so, GA required fewer iterations and produced fairly accurate results. Further investigation shows that PSO and GA are both competitive, effective, simple to implement, and can be considered reliable for obtaining the parameter estimates of different nonlinear regression tasks.
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