Abstract
In this work, we make a review of the concepts of overlap and grouping functions, mainly from a theoretical point of view. In particular, we summarize some of the most relevant works that have been published in recent years about this topic.
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- 1.
In this paper, an increasing (decreasing) function does not need to be strictly increasing (decreasing).
- 2.
This kind of negation is called in the Literature strict.
References
Alsina, C., Frank, M.J., Schweizer, B.: Associative Functions. Triangular Norms and Copulas. World Scientific, Hackensack (2006)
Amo, A., Montero, J., Molina, E.: Representation of consistent recursive rules. Eur. J. Oper. Res. 130, 29–53 (2001)
Amo, A., Montero, J., Biging, G., Cutello, V.: Fuzzy classification systems. Eur. J. Oper. Res. 156, 459–507 (2004)
Bedregal, B., Dimuro, G.P., Bustince, H., Barrenechea, E.: New results on overlap and grouping functions. Inf. Sci. 249, 148–170 (2013)
Bustince, H., Barrenechea, E., Pagola, M., et al.: Weak fuzzy S-subsethood measures. Overlap index. Int. J. Uncertainty Fuzziness Knowl.-Based Syst. 14(5), 537–560 (2006)
Bustince, H., Mohedano, V., Barrenechea, E., et al.: Definition and construction of fuzzy DI-subsethood measures. Inf. Sci. 176(21), 3190–3231 (2006)
Bustince, H., Pagola, M., Barrenechea, E.: Construction of fuzzy indices from fuzzy DI-subsethood measures: application to the global comparison of images. Inf. Sci. 177(3), 906–929 (2007)
Bustince, H., Montero, J., Barrenechea, E., Pagola, M.: Semiautoduality in a restricted family of aggregation operators. Fuzzy Sets Syst. 158(12), 1360–1377 (2007)
Bustince, H., Fernandez, J., Mesiar, R., Montero, J., Orduna, R.: Overlap functions. Nonlin. Anal.-Theory Meth. Appl. 72, 1488–1499 (2010)
Bustince, H., Pagola, M., Mesiar, R., Hullermeier, E., Herrera, F., Montero, J., Orduna, R.: Grouping, overlap, and generalized bientropic functions for fuzzy modeling of pairwise comparisons. IEEE Trans. Fuzzy Syst. 20, 405–415 (2012)
Calvo, T., Kolesárová, A., Komorníkova, M., Mesiar, R.: Aggregation operators: properties, classes and construction methods. In: Aggregation Operators New Trends and Applications. Physica-Verlag, Heidelberg (2002)
Cutello, V., Montero, J.: Recursive connective rules. Int. J. Intell. Syst. 14, 3–20 (1999)
Dimuro, G.P., Bedregal, B., Bustince, H., Asiaín, M.J., Mesiar, R.: On additive generators of overlap functions. Fuzzy Sets Syst. (in press) doi:10.1016/j.fss.2015.02.008
Dubois, D., Koning, J.L.: Social choice axioms for fuzzy set aggregation. Fuzzy Sets Syst. 58, 339–342 (1991)
Dubois, D., Ostasiewicz, W., Prade, H.: Fuzzy Sets: History and Basic Notions. In: Fundamentals of Fuzzy Sets. Kluwer, Boston (2000)
Fodor J., Roubens M., Fuzzy preference modelling and multicriteria decision support. In: Theory and Decision Library, Kluwer Academic Publishers (1994)
Gómez, D., Montero, J.: A discussion on aggregation operators. Kybernetika 40, 107–120 (2004)
Gómez, D., Rodríguez, J.T., Montero, J., Bustince, H., Barrenechea, E.: n-dimensional overlap functions. Fuzzy Sets Syst. (in press). doi:10.1016/j.fss.2014.11.023
Hosseini, M.S., Eftekhari-Moghadam, A.M.: Fuzzy rule-based reasoning approach for event detection and annotation of broadcast soccer video. Appl. Soft Comput. 13(2), 846–866 (2013)
Ishibuchi, H., Nakashima, T., Nii, M.: Classification and Modeling with Linguistic Information Granules: Advanced Approaches to Linguistic Data Mining. Springer, Berlin (2004)
Ishibuchi, H., Yamamoto, T.: Rule weight specification in fuzzy rule-based classification systems. IEEE Trans. Fuzzy Syst. 13, 428–435 (2005)
Jurio, A., Bustince, H., Pagola, M., Pradera, A., Yager, R.: Some properties of overlap and grouping functions and their application to image thresholding. Fuzzy Sets Syst. 229, 69–90 (2013)
Klement, E.P., Mesiar, R., Pap, E.: Triangular Norms, Trends in Logic, Studia Logica Library, vol. 8. Kluwer Academic Publishers, Dordrecht (2000)
Klir, G.J., Folger, T.A.: Fuzzy Sets. Uncertainty and Information, Prentice Hall, Englewood Cliffs (1988)
Madrid, N., Burusco, A., Bustince, H., Fernandez, J., Perfilieva, I.: Upper bounding overlaps by groupings. Fuzzy Sets Syst. 264, 76–99 (2015)
Montero, J., Gomez, D., Bustince, H.: On the relevance of some families of fuzzy sets. Fuzzy Sets Syst. 158, 2429–2442 (2007)
Nelsen, R.B.: An introduction to Copulas. Lecture Notes in Statistics, vol. 139. Springer, New York (1999)
Sanz, J., Galar, M., Jurio, A., Brugos, A., Pagola, M., Bustince, H.: Medical diagnosis of cardiovascular diseases using an interval-valued fuzzy rule-based classification system. Appl. Soft Comput. J. 20, 103–111 (2014)
Schweizer, B., Sklar, A.: Probabilistic Metric Spaces. North-Holland, Amsterdam (1983)
Viceník, P.: Additive generators of non-continuous triangular norms, Topological an Algebraic Structures in Fuzzy Sets. Kluwer (2003)
Viceník, P.: Additive generators of associative functions. Fuzzy Sets Syst. 153, 137–160 (2005)
Zadeh, L.A.: Fuzzy sets as a basis for a theory of possibility. Fuzzy Sets Syst. 1, 3–28 (1978)
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The authors have been supported by project TIN2013-40765-P.
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Bustince, H., Barrenechea, E., Pagola, M., Fernandez, J. (2016). The Notions of Overlap and Grouping Functions. In: Saminger-Platz, S., Mesiar, R. (eds) On Logical, Algebraic, and Probabilistic Aspects of Fuzzy Set Theory. Studies in Fuzziness and Soft Computing, vol 336. Springer, Cham. https://doi.org/10.1007/978-3-319-28808-6_8
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