Similarity, inclusion and entropy measures between type-2 fuzzy sets based on the Sugeno integral

  • Authors:

  • Chao-Ming Hwang;Miin-Shen Yang;Wen-Liang Hung;E. Stanley Lee

  • Affiliations:

  • Department of Applied Mathematics, Chinese Culture University Yangminshan, Taipei, Taiwan;Department of Applied Mathematics, Chung Yung Christian University, Chung-Li 32023, Taiwan;Graduate Institute of Computer Science, National Hsinchu University of Education, Hsin-Chu, Taiwan;Department of Industrial and Manufacturing Systems Engineering, Kansas State University, KS 66506, USA

  • Venue:
  • Mathematical and Computer Modelling: An International Journal
  • Year:
  • 2011

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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.