On Equivalence of Fuzzy Bipolar Similarity Relations and Correlation Functions

Ildar Batyrshin

Fellow of IFSA, Mexican Society for Artificial Intelligence and Russian Association for Fuzzy Systems and Soft Computing
Professor, The Center for Computing Research (CIC), National Polytechnic Institute (IPN), Mexico


Correlation and association coefficients play an important role in statistics and in knowledge extraction from data. The definition of the correlation coefficient for different generalizations of fuzzy sets is also an important topic for fuzzy set theory. Recently, we introduced the concept of correlation function (association measure) as a function defined on an involutive set and satisfying some simple properties. It was shown that correlation functions could be constructed using fuzzy relations called dissimilarity and similarity functions or, for brevity, (dis)similarity functions. This approach gives a possibility to represent most of the known correlation and association coefficients introduced in statistics during more than one hundred years using suitable (dis)similarity functions or distances. These results pave the way for constructing correlation coefficients for different generalizations of fuzzy sets if one can introduce an involutive complement of such sets and suitable (dis)similarity functions. In a recent paper, we showed the equivalence of correlation functions and fuzzy bipolar similarity relations. In this talk, we exploit this property for constructing fuzzy bipolar (dis)similarity relations from fuzzy relations satisfying more simple properties. It gives new methods for creating correlation coefficients from (dis)similarity functions. We will discuss the methods and examples of construction of correlation coefficients from fuzzy similarity relations for different data types.


Ildar Batyrshin graduated from the Moscow Physical-Technical Institute. He received Ph.D. from the Moscow Power Engineering Institute and Dr. Sc. from the Higher Attestation Committee of the Russia. He occupied professor and research positions in Applied Mathematics and Computer Science at the National Research Technological University, Kazan, Russia, the Institute of Problems of Informatics of Academy of Sciences of the Republic of Tatarstan, Russia, and Mexican Petroleum Institute (IMP). Since 2014 he is the Titular Professor C of the Center for Computing Research (CIC) of National Polytechnic Institute (IPN), Mexico. He is the Vice-President of the Mexican Society for Artificial Intelligence (SMIA), Past-President of the Russian Association for Fuzzy Systems and Soft Computing (RAFSSoftCom), a member of the NAFIPS Board of Directors, a member of BoG of IEEE SMCS, and Senior Member of IEEE. His awards: Fellow of IFSA, SMIA and RAFSSoftCom, Level 3 (highest) Researcher of the National System of Researchers of Mexico; 1st Prize for the Best Research of IMP for the Development of Expert System in Diagnostics of Water Production (SMART-Agua) in 2007; Honorary Professor of Obuda University, Budapest, Hungary; Honorary Researcher of the Republic of Tatarstan, Russia; State Research Fellowship of the Presidium of Russian Academy of Sciences for Distinguished Researchers (1997-2003). He is an associate editor of several scientific journals. He served as a Co-Chair of Program or Organizing Committees of more than 10 International Conferences on Soft Computing, Artificial Intelligence, Computational Intelligence, and Data Mining. He is a co-author and co-editor of more than 20 books and special volumes of journals.