Applying particle swarm optimization to determine the bandwidth parameter in probability density estimation

The determination of bandwidth parameter is a critical factor for the performance of probability density estimation method. The advanced parameter selection methods, such as the bootstrap method, the least-squares cross-validation (LSCV) method and the biased cross-validation (BCV) method, always need the help of the brute-force search or exhaustive search to find the optimal bandwidth parameters. In this paper, we apply five particle swarm optimization (PSO) algorithms-standard PSO (SPSO), PSO with a constriction factor (PSOCF), Gaussian PSO (GPSO), Gaussian PSO with Gaussian jump (GPSOGJ) and Gaussian PSO with Cauchy jump (GPSOCJ)-to determine the optimal bandwidths. In order to experimentally validate the feasibility and effectiveness of selecting the optimal parameters by using PSO algorithms, we carry out some numerical simulations on four univariate artificial datasets: Uniform dataset, Normal dataset, Exponential dataset and Rayleigh dataset. The finally comparative results show that our strategies are well-performed and Gaussian PSO with jump methods can obtain the better estimations than other PSO algorithms.