A min-max approach to fuzzy clustering, estimation, and identification

Abstract

This study, for any unknown physical process y=f(x <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> ,...,x <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sub> ), is concerned with the: 1) fuzzy partition of n-dimensional input space X=X <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> timesmiddotmiddotmiddottimesX <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sub> into K different clusters, 2) estimating the process behavior ycirc=f(xcirc) for a given input xcirc=(xcirc <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> ,middotmiddotmiddot,xcirc <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n </sub> )isinX, and 3) fuzzy approximation of the process, with uncertain input-output identification data {(x(k)plusmndeltax <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k </sub> ),(y(k)plusmnv <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</sub> )} <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k=1,...</sub> , using a Sugeno type fuzzy inference system. A unified min-max approach (that attempts to minimize the worst-case effect of data uncertainties and modeling errors on estimation performance), is suggested to provide robustness against data uncertainties and modeling errors. The proposed method of min-max fuzzy parameters estimation does not make any assumption and does not require a priori knowledge of upper bounds, statistics, and distribution of data uncertainties and modeling errors. To show the feasibility of the approach, simulation studies and a real-world application of physical fitness classification based on the fuzzy interpretation of physiological parameters, have been provided