Fuzzy Classifier Design
Fuzzy sets were first proposed by Lotfi Zadeh in his seminal paper [366] in 1965, and ever since have been a center of many discussions, fervently admired and condemned. Both proponents and opponents consider the argu ments pointless because none of them
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1.1 What are fuzzy classifiers? Fuzzy pattern recognition is sometimes identified with fuzzy clustering or with fuzzy if-then systems used as classifiers. In this book we adopt a broader view: fuzzy pattern recognition is about any pattern classification paradigm that involves fuzzy sets. To a certain extent fuzzy pattern recognition is dual to classical pattern recognition, as delineated in the early seventies by Duda and Hart [87], Fukunaga [100], Tou and Gonzalez [324], and thereby consists of three basic components: clustering, classifier design and feature selection [39]. Fuzzy clustering has been the most successful offspring offuzzy pattern recognition so far. The fuzzy c-means algorithm devised by Bezdek [34] has admirable popularity in agreat number of fields, both engineering and non-engineering. Fuzzy feature selection is virtually absent, or disguised as something else. This book is about the third component fuzzy classifier design.
The diversity of applications in the studies retrieved upon the keyword "fuzzy classifier" is amazing. Remote sensing; environmental studies; geoscience; satellite and medical image analysis; speech, signature and face recognit ion are few examples of highly active areas. Even more curious are the concrete applications such as grading fish products and student writing samples; analysis of seasonal variat ion of cloud parameters; speeding up fractal image compression; development of metric-based software; classification of odours, road accidents, military targets and milling tool ware; estimat ing a crowding level in a scene; tactile sensing; glaucoma monitoring; and even quality evaluation of biscuits during baking. It seems that applications of fuzzy pattern recognition are far ahead of the theory on the matter. This book aims at systematizing and hopefully a better understanding of the theoretical side of fuzzy classifiers. 1.1.1 Three "fuzzy" definitions of a fuzzy classifier What are fuzzy classifiers? It is difficult to propose a clear-cut definition. Let x be a vector in an n-dimensional real space ~n (the feature space),
L. I. Kuncheva, Fuzzy Classifier Design © Springer-Verlag Berlin Heidelberg 2000
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1. Introduction
and let n = {Wl' .. . ,wc } be a set of class labels 1 . A (crisp) classifier is any mapping D:!R n -t n (1.1) In a broad sense, we can define a fuzzy classifier as follows Definition 1.1.1. A fuzzy classifier is any classifier which uses fuzzy sets either during its training or during its operation Bezdek et al. [38] define a possibilistic classifier as the mapping (1.2) i.e., instead of assigning a class labeI from n, D p assigns to x E lRn a soft class labeI with degrees of membership in each class (by convention, the zero vector is excluded from the set of possible soft labels). We can think of the components of the output vector as degrees of support for the hypothesis that x belongs to the respective class. Denote by J..L(x) = [JLl (x), ... ,JLc(x)]T the classifier output calculated via (1.2). Then, according to [38], Definition 1.1.2. A fuzz
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