On Similarity, Inclusion Measure and Entropy between Type-2 Fuzzy Sets

In this paper, new similarity, inclusion measure and entropy between type-2 fuzzy sets corresponding to grades of memberships are proposed. We also create the relationships among these measures between type-2 fuzzy sets. Several examples are used to present the calculation of these similarity, inclusion measure and entropy between type-2 fuzzy sets. The comparison results show that the proposed similarity measure presents better than those of Hung and Yang (2004) and Yang and Lin (2009). Moreover, measuring the similarity between type-2 fuzzy sets is important in clustering. We also use the proposed similarity measure as a clustering method for type-2 fuzzy sets.

[1]  Virginia R. Young,et al.  Fuzzy subsethood , 1996, Fuzzy Sets Syst..

[2]  Miin-Shen Yang,et al.  A Similarity Measure between Type-2 Fuzzy Sets with Its Application to Clustering , 2007, Fourth International Conference on Fuzzy Systems and Knowledge Discovery (FSKD 2007).

[3]  Asok K. Nanda,et al.  Some results on generalized residual entropy , 2006, Inf. Sci..

[4]  Miin-Shen Yang,et al.  Similarity Measures Between Type-2 Fuzzy Sets , 2004, Int. J. Uncertain. Fuzziness Knowl. Based Syst..

[5]  Liu Xuecheng,et al.  Entropy, distance measure and similarity measure of fuzzy sets and their relations , 1992 .

[6]  Leonid Kitainik,et al.  Fuzzy Decision Procedures with Binary Relations , 1993, Theory and Decision Library.

[7]  Miin-Shen Yang,et al.  On similarity and inclusion measures between type-2 fuzzy sets with an application to clustering , 2009, Comput. Math. Appl..

[8]  H. B. Mitchell.  Pattern recognition using type-II fuzzy sets , 2005, Inf. Sci..

[9]  Wenyi Zeng,et al.  Inclusion measures, similarity measures, and the fuzziness of fuzzy sets and their relations , 2006, Int. J. Intell. Syst..

[10]  B. Kosko.  Fuzziness vs. probability , 1990 .

[11]  Chris Cornelis,et al.  Sinha-Dougherty approach to the fuzzification of set inclusion revisited , 2003, Fuzzy Sets Syst..

[12]  L. A. ZADEH,et al.  The concept of a linguistic variable and its application to approximate reasoning - I , 1975, Inf. Sci..

[13]  Humberto Bustince,et al.  Weak Fuzzy S-Subsethood Measures: Overlap Index , 2006, Int. J. Uncertain. Fuzziness Knowl. Based Syst..

[14]  Settimo Termini,et al.  A Definition of a Nonprobabilistic Entropy in the Setting of Fuzzy Sets Theory , 1972, Inf. Control..

[15]  M. Mizumoto,et al.  Fuzzy sets and type 2 under algebraic product and algebraic sum , 1981 .

[16]  Jerry M. Mendel,et al.  Type-2 fuzzy sets made simple , 2002, IEEE Trans. Fuzzy Syst..

[17]  John T. Rickard,et al.  Fuzzy Subsethood for Fuzzy Sets of Type-2 and Generalized Type- ${n}$ , 2009, IEEE Transactions on Fuzzy Systems.

[18]  E. Dougherty,et al.  Fuzzification of set inclusion: theory and applications , 1993 .

[19]  E. Walker,et al.  Sets With Type-2 Operations , 2006, NAFIPS 2006.

[20]  I. Burhan Türksen,et al.  An approximate analogical reasoning approach based on similarity measures , 1988, IEEE Trans. Syst. Man Cybern..

[21]  Miin-Shen Yang,et al.  Cluster analysis based on fuzzy relations , 2001, Fuzzy Sets Syst..

[22]  Lotfi A. Zadeh,et al.  Fuzzy Sets , 1996, Inf. Control..

[23]  K. Selçuk Candan,et al.  Similarity-based ranking and query processing in multimedia databases , 2000, Data Knowl. Eng..

[24]  C. Pappis,et al.  A comparative assessment of measures of similarity of fuzzy values , 1993 .

[25]  Masaharu Mizumoto,et al.  Some Properties of Fuzzy Sets of Type 2 , 1976, Inf. Control..

[26]  L. Kitainik,et al.  Fuzzy Inclusions and Fuzzy Dichotomous Decision Procedures , 1987 .

[27]  Humberto Bustince,et al.  Definition and construction of fuzzy DI-subsethood measures , 2006, Inf. Sci..

[28]  Humberto Bustince,et al.  Construction of fuzzy indices from fuzzy DI-subsethood measures: Application to the global comparison of images , 2007, Inf. Sci..

[29]  Athanasios Kehagias,et al.  L-fuzzy valued inclusion measure, L-fuzzy similarity and L-fuzzy distance , 2001, Fuzzy Sets Syst..

[30]  Bart Kosko,et al.  Fuzzy entropy and conditioning , 1986, Inf. Sci..

[31]  Lotfi A. Zadeh,et al.  The Concepts of a Linguistic Variable and its Application to Approximate Reasoning , 1975 .

[32]  RONALD R. YAGER,et al.  Fuzzy Subsets of Type Ii in Decisions , 1980, Cybern. Syst..

[33]  Rami Zwick,et al.  Measures of similarity among fuzzy concepts: A comparative analysis , 1987, Int. J. Approx. Reason..

[34]  Humberto Bustince,et al.  Indicator of inclusion grade for interval-valued fuzzy sets. Application to approximate reasoning based on interval-valued fuzzy sets , 2000, Int. J. Approx. Reason..