Automatic Classification Method Based on a Fuzzy Similarity Relation
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AUTOMATIC CLASSIFICATION METHOD BASED ON A FUZZY SIMILARITY RELATION L. F. Hulianytskyi1† and I. I. Riasna1‡ Abstract. To solve problems of automatic classification, the IFC fuzzy clusterization method is proposed that uses new fuzzy logical operators, namely, threshold triangular norms and conorms. This method differs from clusterization methods based on a fuzzy equivalence relation in that it allows one to develop faster algorithms for constructing clusters. In this case, data on relationships between elements of the set being investigated are not distorted, which provides the transparency of interpretation of the results of investigations. Examples of application of the method to some well-known problems are given.
Keywords: fuzzy cluster, classification, cluster analysis.
INTRODUCTION After the publication of the fundamental works of L. Zadeh [1] and S. Tamura et al. [2], fuzzy clusterization methods are frequently used in solving problems of automatic classification whose component part is the operation of transitive closure of the similarity matrix for a set of objects. The result of this operation is a fuzzy equivalence relation. An advantage of this approach is that any set of this relation specified by a threshold is a collection of nonintersecting subsets or equivalence classes. An ordered set of thresholds allows one to determine the influence the threshold level on the number and structure of equivalence classes being formed. This approach or its modifications are applied both in using immediate expert judgment on similarity when a dual dissimilarity relation is not a metric (for example, in the problem of recognition of portraits [2, 3]) and also in the cases when the initial data are represented in the form of a matrix of distances between objects [4]. However, a complement of a fuzzy equivalence relation is an ultrametric whose properties cannot be interpreted in some applied problems [5]. Another problem arising in fuzzy cluster analysis in using the operation of transitive closure of the similarity matrix for given N objects requires large computational expenditures (of the order of O ( N 4 )) for computing a fuzzy equivalence relation. In the present article, the fuzzy cluster analysis method, which is called Interpretable Fuzzy Clusterization (IFC) and in which the operation of transitive closure is not applied, is used to solve automatic classification problems. In this case, the results of analysis have an obvious interpretation, and the algorithm of analysis uses a considerably smaller number of operations. In Sec 1, the proposed IFC method is substantiated. In Sec. 2, based on data from [2, 4], the results of practical application of the IFC method are presented. For fuzzy sets, the symbols U and I are used for the fuzzy operations of union (max) and intersection (min) of these sets. The symbol + denotes the end of the proof of a lemma or a theorem. 1
V. M. Glushkov Institute of Cybernetics, National Academy of Sciences of Ukraine, Kyiv, Ukraine, [email protected]; ‡[email protected]. Tr
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