Blyuss O. Entropy methods in fuzzy clustering problems

The objectives are: fuzzy clustering problems, including multi-criterion clustering, which also depends on the entropy function. The aims of this work are: formulation and study of new fuzzy clustering problems with particular account for the entropy function, which allows one to improve the quality of clustering; development and justification of algorithms for solving such problems. Methods used are: cluster analysis, theory of functional inequalities, penalty functions and multi-criterion optimization. The problems of fuzzy clustering with bounds on the entropy function and compactness of clusters are formulated, theorems about solvability of such problems are proven, and algorithms for their solution are designed. The method for choosing an exponential weight in the fuzzy c-means is developed. An algorithm of fuzzy clustering with an adaptive choice of the exponential weight is developed and numerically implemented. A new problem multi-criterion fuzzy clustering is formulated, existence of its solutions is proven, and an algorithm of its solution is developed. Applications include teaching, assessment of mining safety due to possible explosions, and analysis of crystallograms of biological liquids.