Automatic fuzzy clustering for probability density functions using the genetic algorithm

Based on the genetic algorithm, this study develops the new fuzzy clustering method for probability density functions (pdfs) with the important improvements. First, the L1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^1$$\end{document}-distance is proposed as the measure to evaluate the level similar of the pdfs. It is surveyed to find the bounds and established the methods to compute. Second, we propose the new objective for the genetic algorithm. This objective measures both the similarity of elements in each group and the quality of clustering. Finally, the operators such as crossover, mutation, and selection of the traditional genetic algorithm are improved. Combining these improvements, we have an efficient cluster analysis algorithm for pdfs. In this algorithm, the proper number of groups, the specific pdfs in each cluster, and the fuzzy relationship between the pdf to the established clusters are determined at the same time. The convergence of the proposed algorithm is proved by theory and performed by the established MATLAB program. The experiments and applications show superiority of the developed algorithm in comparing to the existing algorithms. The proposed algorithm is also applied in recognizing images to certify the feasibility and applicability of the studied problem.