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Overlap functions. (English) Zbl 1182.26076

Summary: We address a key issue in scenario classification, where classifying concepts show a natural overlapping. In fact, overlapping needs to be evaluated whenever classes are not crisp, in order to be able to check if a certain classification structure fits reality and still can be useful for our declared decision making purposes. In this paper, we address an object recognition problem, where the best classification with respect to the background is the one with less overlapping between the class object and the class background. In particular, we present the basic properties that must be fulfilled by overlap functions, associated to the degree of overlapping between two classes. In order to define these overlap functions, we take as reference properties like migrativity, homogeneity of order 1 and homogeneity of order 2. Hence, we define overlap functions, proposing a construction method and analyzing the conditions ensuring that t-norms are overlap functions. In addition, we present a characterization of migrative and strict overlap functions by means of automorphisms.

MSC:

26E50 Fuzzy real analysis
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[1] Amo, A.; Montero, J.; Biging, G.; Cutello, V., Fuzzy classification systems, European Journal of Operational Research, 156, 459-507 (2004) · Zbl 1056.90077
[2] Bustince, H.; Pagola, M.; Barrenechea, E., Construction of fuzzy indices from fuzzy DI-subsethood measures: Application to the global comparison of images, Information Sciences, 177, 3, 906-929 (2007) · Zbl 1112.94033
[3] Bustince, H.; Barrenechea, E.; Pagola, M., Weak fuzzy S-subsethood measures. Overlap index, International Journal of Uncertainty Fuzziness and Knowledge-Based Systems, 14, 5, 537-560 (2006) · Zbl 1160.28311
[4] Bustince, H.; Mohedano, V.; Barrenechea, E., Definition and construction of fuzzy DI-subsethood measures, Information Sciences, 176, 21, 3190-3231 (2006) · Zbl 1104.03052
[5] Dubois, D.; Koning, J. L., Social choice axioms for fuzzy set aggregation, Fuzzy Sets and Systems, 58, 339-342 (1991)
[6] Dubois, D.; Ostasiewicz, W.; Prade, H., Fuzzy sets: History and basic notions, (Fundamentals of Fuzzy Sets (2000), Kluwer: Kluwer Boston, MA) · Zbl 0967.03047
[7] Zadeh, L. A., Fuzzy sets as a basis for a theory of possibility, Fuzzy Sets and Systems, 1, 3-28 (1978) · Zbl 0377.04002
[8] Bustince, H.; Montero, J.; Barrenechea, E.; Pagola, M., Semiautoduality in a restricted family of aggregation operators, Fuzzy Sets and Systems, 158, 12, 1360-1377 (2007) · Zbl 1123.68125
[9] Calvo, T.; Kolesárová, A.; Komorníkova, M.; Mesiar, R., Aggregation operators: Properties, classes and construction ethods, (Aggregation Operators New Trends and Applications (2002), Physica-Verlag: Physica-Verlag Heidelberg)
[10] Klir, G. J.; Folger, T. A., Fuzzy Sets, Uncertainty and Information (1988), Prentice Hall: Prentice Hall Englewood Cliffs, NJ · Zbl 0675.94025
[11] Gómez, D.; Montero, J., A discussion on aggregation operators, Kybernetika, 40, 107-120 (2004) · Zbl 1249.68229
[12] Amo, A.; Montero, J.; Molina, E., Representation of consistent recursive rules, European Journal of Operational Research, 130, 29-53 (2001) · Zbl 1137.03322
[13] Cutello, V.; Montero, J., Recursive connective rules, International Journal of Intelligent Systems, 14, 3-20 (1999) · Zbl 0955.68103
[14] Klement, E. P.; Mesiar, R.; Pap, E., (Triangular Norms. Triangular Norms, Trends in Logic, Studia Logica Library, vol. 8 (2000), Kluwer Academic Publishers: Kluwer Academic Publishers Dordrecht) · Zbl 0972.03002
[15] Schweizer, B.; Sklar, A., Probabilistic Metric Spaces (1983), North-Holland: North-Holland Amsterdam · Zbl 0546.60010
[16] Fodor, J.; Roubens, M., Fuzzy preference modelling and multicriteria decision support, (Theory and Decision Library (1994), Kluwer Academic Publishers) · Zbl 0827.90002
[17] Alsina, C.; Frank, M. J.; Schweizer, B., Associative Functions, Triangular Norms and Copulas (2006), World Scientific: World Scientific Hackensack · Zbl 1100.39023
[18] Durante, F.; Sarkoci, P., A note on the convex combinations of triangular norms, Fuzzy Sets and Systems, 159, 77-80 (2008) · Zbl 1173.03042
[19] Mesiar, R.; Novák, V., Open problems from the 2nd International conference of fuzzy sets theory and its applications, Fuzzy Sets and Systems, 81, 185-190 (1996) · Zbl 0877.04003
[20] Fodor, J.; Rudas, I. J., On continuous triangular norms that are migrative, Fuzzy Sets and Systems, 158, 1692-1697 (2007) · Zbl 1120.03035
[21] Bustince, H.; Montero, J.; Mesiar, R., Migrativity of aggregation operators, Fuzzy Sets and Systems, 160, 6, 766-777 (2009) · Zbl 1186.68459
[22] Baczynski, M.; Jayaram, B., \((S, N)\)- and \(R\)-implications: A state-of-the-art survey, Fuzzy Sets and Systems, 159, 14, 1836-1859 (2008) · Zbl 1175.03013
[23] Nelsen, R. B., (An introduction to Copulas. An introduction to Copulas, Lecture Notes in Statistics, vol. 139 (1999), Springer: Springer New York) · Zbl 0909.62052
[24] Mesiarová, A., A note on two open problems of Alsina, Frank and Schweizer, Aequationes Mathematicae, 72, 1-2, 41-46 (2006) · Zbl 1101.39011
[25] Mesiarová, A., \(k - l_p\)-Lipschitz \(t\)-norms, International Journal of Approximate Reasoning, 46, 596-604 (2007) · Zbl 1189.03058
[26] Bustince, H.; Barrenechea, E.; Pagola, M., Restricted equivalence functions, Fuzzy Sets and Systems, 157, 17, 2333-2346 (2006) · Zbl 1110.68158
[27] Bustince, H.; Barrenechea, E.; Pagola, M., Image thresholding using restricted equivalence functions and maximizing the measures of similarity, Fuzzy Sets and Systems, 158, 5, 496-516 (2007) · Zbl 1111.68139
[28] Montero, J.; Gomez, D.; Bustince, H., On the relevance of some families of fuzzy sets, Fuzzy Sets and Systems, 158, 2429-2442 (2007) · Zbl 1157.03319
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