On some new relations between copulas and fuzzy implication functions

Copulas are a special kind of aggregation functions that have been deeply investigated because of their applications in many fields, specially in Statistics and Economy. An important research topic from the theoretical point of view is the study of new construction methods of copulas. In this line, this paper presents two construction methods based on probabilistic implications and survival implications. From these construction methods, the axiomatic characterization of these families of fuzzy implication functions, which are in fact the same, is presented.

[1]  Joan Torrens,et al.  On two construction methods of copulas from fuzzy implication functions , 2015, Progress in Artificial Intelligence.

[2]  S. Gottwald.  A Treatise on Many-Valued Logics , 2001 .

[3]  R. Mesiar,et al.  Aggregation Functions (Encyclopedia of Mathematics and its Applications) , 2009 .

[4]  Bernard De Baets,et al.  Residual operators of uninorms , 1999, Soft Comput..

[5]  Haris N. Koutsopoulos,et al.  Models for route choice behavior in the presence of information using concepts from fuzzy set theory and approximate reasoning , 1993 .

[6]  Michal Baczynski,et al.  Laws of Contraposition and Law of Importation for Probabilistic Implications and Probabilistic S-implications , 2014, IPMU.

[7]  Joan Torrens,et al.  A Survey on Fuzzy Implication Functions , 2007, IEEE Transactions on Fuzzy Systems.

[8]  Przemyslaw Grzegorzewski,et al.  Probabilistic implications , 2013, Fuzzy Sets Syst..

[9]  Balasubramaniam Jayaram,et al.  Rule reduction for efficient inferencing in similarity based reasoning , 2008, Int. J. Approx. Reason..

[10]  Joan Torrens,et al.  A characterization of residual implications derived from left-continuous uninorms , 2010, Inf. Sci..

[11]  Humberto Bustince Sola,et al.  Advances in Fuzzy Implication Functions , 2013 .

[12]  Marc Roubens,et al.  Fuzzy Preference Modelling and Multicriteria Decision Support , 1994, Theory and Decision Library.

[13]  Przemyslaw Grzegorzewski,et al.  On the Properties of Probabilistic Implications , 2011, Eurofuse.

[14]  Manuel González Hidalgo,et al.  On the Choice of the Pair Conjunction–Implication Into the Fuzzy Morphological Edge Detector , 2015, IEEE Transactions on Fuzzy Systems.

[15]  Radko Mesiar,et al.  Invariant copulas , 2002, Kybernetika.

[16]  Etienne E. Kerre,et al.  Fuzzy If-Then Rules in Computational Intelligence , 2000 .

[17]  Michal Baczynski,et al.  Properties of the probabilistic implications and S-implications , 2016, Inf. Sci..

[18]  R. Mesiar,et al.  Conjunctors and their Residual Implicators: Characterizations and Construction Methods , 2007 .

[19]  Joan Torrens,et al.  An Overview of Construction Methods of Fuzzy Implications , 2013 .

[20]  R. Mesiar,et al.  Aggregation operators: new trends and applications , 2002 .

[21]  Ronald R. Yager,et al.  Modeling holistic fuzzy implication using co-copulas , 2006, Fuzzy Optim. Decis. Mak..

[22]  R. Nelsen.  An Introduction to Copulas , 1998 .

[23]  Mariano Eriz.  Aggregation Functions: A Guide for Practitioners , 2010 .

[24]  Radko Mesiar,et al.  Triangular Norms , 2000, Trends in Logic.

[25]  Przemyslaw Grzegorzewski,et al.  Survival Implications , 2012, IPMU.

[26]  Etienne Kerre,et al.  Fuzzy techniques in image processing , 2000 .

[27]  Michal Baczynski,et al.  Fuzzy Implications , 2008, Studies in Fuzziness and Soft Computing.

[28]  Michal Baczynski,et al.  Properties of the survival implications and S-implications , 2015, IFSA-EUSFLAT.

[29]  Joan Torrens,et al.  From three to one: Equivalence and characterization of material implications derived from co-copulas, probabilistic S-implications and survival S-implications , 2017, Fuzzy Sets Syst..

[30]  Humberto Bustince,et al.  A review of the relationships between implication, negation and aggregation functions from the point of view of material implication , 2016, Inf. Sci..

[31]  Etienne E. Kerre,et al.  Fuzzy If-Then Rules in Computational Intelligence: Theory and Applications , 2012 .

[32]  Michal Baczynski,et al.  Fuzzy Implications: Past, Present, and Future , 2015, Handbook of Computational Intelligence.