Similarity,inclusion and entropy measures between type-2 fuzzy sets based on the Sugeno integral |
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| Authors: | Chao-Ming Hwang Miin-Shen Yang Wen-Liang Hung E. Stanley Lee |
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| Affiliation: | 1. Department of Applied Mathematics, Chinese Culture University Yangminshan, Taipei, Taiwan;2. Department of Applied Mathematics, Chung Yung Christian University, Chung-Li 32023, Taiwan;3. Graduate Institute of Computer Science, National Hsinchu University of Education, Hsin-Chu, Taiwan;4. Department of Industrial and Manufacturing Systems Engineering, Kansas State University, KS 66506, USA |
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| Abstract: | Similarity measures of type-2 fuzzy sets are used to indicate the similarity degree between type-2 fuzzy sets. Inclusion measures for type-2 fuzzy sets are the degrees to which a type-2 fuzzy set is a subset of another type-2 fuzzy set. The entropy of type-2 fuzzy sets is the measure of fuzziness between type-2 fuzzy sets. Although several similarity, inclusion and entropy measures for type-2 fuzzy sets have been proposed in the literatures, no one has considered the use of the Sugeno integral to define those for type-2 fuzzy sets. In this paper, new similarity, inclusion and entropy measure formulas between type-2 fuzzy sets based on the Sugeno integral are proposed. Several examples are used to present the calculation and to compare these proposed measures with several existing methods for type-2 fuzzy sets. Numerical results show that the proposed measures are more reasonable than existing measures. On the other hand, measuring the similarity between type-2 fuzzy sets is important in clustering for type-2 fuzzy data. We finally use the proposed similarity measure with a robust clustering method for clustering the patterns of type-2 fuzzy sets. |
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