×

zbMATH — the first resource for mathematics

Quantitative rough sets based on subsethood measures. (English) Zbl 1339.68249
Summary: Subsethood measures, also known as set-inclusion measures, inclusion degrees, rough inclusions, and rough-inclusion functions, are generalizations of the set-inclusion relation for representing graded inclusion. This paper proposes a framework of quantitative rough sets based on subsethood measures. A specific quantitative rough set model is defined by a particular class of subsethood measures satisfying a set of axioms. Consequently, the framework enables us to classify and unify existing generalized rough set models (e.g., decision-theoretic rough sets, probabilistic rough sets, and variable precision rough sets), to investigate limitations of existing models, and to develop new models. Various models of quantitative rough sets are constructed from different classes of subsethood measures. Since subsethood measures play a fundamental role in the proposed framework, we review existing methods and introduce new methods for constructing and interpreting subsethood measures.

MSC:
68T37 Reasoning under uncertainty in the context of artificial intelligence
03E72 Theory of fuzzy sets, etc.
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Azam, N.; Yao, J. T., Analyzing uncertainties of probabilistic rough set regions with game-theoretic rough sets, Int. J. Approx. Reason., 55, 142-155, (2014) · Zbl 1316.68168
[2] W. Bandler, L. Kohout, Fuzzy Relational Products and Fuzzy Implication Operators, Report No. FRP-1, Department of Mathematics, University of Essex, Colchester, 1978. · Zbl 0435.68042
[3] Bandler, W.; Kohout, L., Fuzzy power sets and fuzzy implication operators, Fuzzy Sets Syst., 4, 13-30, (1980) · Zbl 0433.03013
[4] Burillo, P.; Frago, N.; Fuentes, R., Inclusion grade and fuzzy implication operators, Fuzzy Sets Syst., 114, 417-429, (2000) · Zbl 0962.03050
[5] Bustince, H.; Mohedano, V.; Barrenechea, E.; Pagola, M., Definition and construction of fuzzy DI-subsethood measures, Inform. Sci., 176, 3190-3231, (2006) · Zbl 1104.03052
[6] Bustince, H.; Barrenechea, E.; Pagola, M., A method for constructing V. young’s fuzzy subsethood measures and fuzzy entropies, (Chountas, P.; Petrounias, I.; Kacprzyk, J., Intelligent Techniques and Tools for Novel System Architectures, Studies in Computational Intelligence (SCI) 109, (2008), Springer Berlin), 123-138
[7] De Baets, B.; De Meyer, H.; Naessens, H., On rational cardinality-based inclusion measures, Fuzzy Sets Syst., 128, 169-183, (2002) · Zbl 1001.03048
[8] X.F. Deng, Y.Y. Yao, An information-theoretic interpretation of thresholds in probabilistic rough sets, in: Proceedings of the 7th International Conference on Rough Sets and Knowledge Technology, RSKT 2012, LNCS (LNAI), vol. 7414, 2012, pp. 369-378.
[9] Dubois, D.; Prade, H., Fuzzy sets and systems: theory and applications, (1980), Academic press New York · Zbl 0444.94049
[10] Duda, R. O.; Hart, P. E., Pattern classification and scene analysis, (1973), Wiley New York · Zbl 0277.68056
[11] Fan, J.; Xie, W.; Pei, J., Subsethood measure: new definitions, Fuzzy Sets Syst., 106, 201-209, (1999) · Zbl 0968.94026
[12] Fodor, J.; Yager, R., Fuzzy sets theoretic operators, (Dubois, D.; Prade, H., Foundations of Fuzzy Sets, (2000), Kluwer Boston, MA), 125-193 · Zbl 0973.03072
[13] Goguen, J. A., The logic of inexact concepts, Systhese, 19, 325-373, (1969) · Zbl 0184.00903
[14] A. Gomolińska, On certain rough inclusion functions, in: LNCS Transactions on Rough Sets IX, LNCS, vol. 5390, 2008, pp. 35-55.
[15] A. Gomolińska, Rough approximations based on weak q-RIFs, in: LNCS Transactions on Rough Sets X, LNCS, vol. 5656, 2009, pp. 117-135.
[16] Gomolińska, A., A logic-algebraic approach to graded inclusion, Fundam. Inform., 109, 265-279, (2011) · Zbl 1243.68278
[17] A. Gomolińska, M. Wolski, Rough inclusion functions and similarity indices, in: Proceedings of the 22nd International Workshop on Concurrency, Specification and Programming, CS&P 2013, CEUR Workshop Proceedings (CEUR-WS), vol. 1032, 2013, pp. 145-156.
[18] S. Greco, B. Matarazzo, R. Słowiński, Rough membership and Bayesian confirmation measures for parameterized rough sets, in: Proceedings of the 10th International Conference on Rough Sets, Fuzzy Sets, Data Mining, and Granular Computing, RSFDGrC 2005, LNCS (LNAI), vo. 3641, 2005, pp. 314-324. · Zbl 1134.68531
[19] S. Greco, R. Słowiński, I. Szcze¸ch, Assessing the quality of rules with a new monotonic interestingness measure Z, in: Proceedings of the 9th International Conference on Artificial Intelligence and Soft Computing, ICAISC 2008, LNCS (LNAI), vol. 5097, 2008, pp. 556-565.
[20] S.M. Gu, W.Z. Wu, H.T. Chen, A classification approach of granules based on variable precision rough sets, in: Proceedings of the 6th IEEE International Conference on Cognitive Informatics, ICCI 2007, 2007, pp. 163-168.
[21] Herbert, J. P.; Yao, J. T., Game-theoretic rough sets, Fundam. Inform., 108, 267-286, (2011) · Zbl 1243.91016
[22] G. Huang, Y.S. Liu, New subsethood measures and similarity measures of fuzzy sets, in: Proceedings of 2005 International Conference on Communications, Circuits and Systems, 2005, pp. 999-1002.
[23] Kehagias, A.; Konstantinidou, M., L-fuzzy valued inclusion measure, L-fuzzy similarity and L-fuzzy distance, Fuzzy Sets Syst., 136, 313-332, (2003) · Zbl 1020.03055
[24] Kitainik, L., Fuzzy inclusions and fuzzy dichotomous decision procedures, (Kacprzyk, J.; Orlovski, S., Optimization Models Using Fuzzy Sets and Possibility Theory, (1987), Reidel Dordrecht), 154-170 · Zbl 0638.90004
[25] Kosko, B., Fuzzy entropy and conditioning, Inform. Sci., 40, 165-174, (1986) · Zbl 0623.94005
[26] Kosko, B., Fuzziness vs. probability, Int. J. Gen. Syst., 17, 211-240, (1990) · Zbl 0706.62010
[27] Kuncheva, L. I., Fuzzy rough sets: application to feature selection, Fuzzy Sets Syst., 51, 147-153, (1992)
[28] X.Y. Jia, W.W. Li, L. Shang, J.J. Chen, An optimization viewpoint of decision-theoretic rough set model, in: Proceedings of the 6th International Conference on Rough Sets and Knowledge Technology, RSKT 2011, LNCS (LNAI), vol. 6954, 2011, pp. 457-465.
[29] Li, H. X.; Zhou, X. Z., Risk decision making based on decision-theoretic rough set: a three-way view decision model, Int. J. Comput. Intell. Syst., 4, 1-11, (2011)
[30] Liu, B., Uncertainty theory, (2007), Springer Berlin
[31] Liu, D.; Li, T. R.; Li, H. X., A multiple-category classification approach with decision-theoretic rough sets, Fundam. Inform., 115, 173-188, (2012) · Zbl 1248.68492
[32] Liu, D.; Li, T. R.; Ruan, D., Probabilistic model criteria with decision-theoretic rough sets, Inform. Sci., 181, 3709-3722, (2011)
[33] Marczewski, E.; Steinhaus, H., On a certain distance of sets and the corresponding distance of functions, Colloq. Math., 6, 319-327, (1958) · Zbl 0089.03502
[34] Pawlak, Z., Rough sets, Int. J. Comput. Inform. Sci., 11, 341-356, (1982) · Zbl 0501.68053
[35] Pawlak, Z., Rough sets: theoretical aspects of resoning about data, (1991), Kluwer Academic Boston
[36] Pawlak, Z.; Wong, S. K.M.; Ziarko, W., Rough sets: probabilistic versus deterministic approach, Int. J. Man-Mach. Stud., 29, 81-95, (1988) · Zbl 0663.68094
[37] Polkowski, L.; Skowron, A., Rough mereology: a new paradigm for approximate reasoning, Int. J. Approx. Reason., 15, 333-365, (1996) · Zbl 0938.68860
[38] Sanchez, E., Inverses of fuzzy relations: application to possibility distributions and medical diagnosis, Fuzzy Sets Syst., 2, 75-86, (1979) · Zbl 0399.03040
[39] Sinha, D.; Dougherty, E. R., Fuzzification of set inclusion: theory and applications, Fuzzy Sets Syst., 55, 15-42, (1993) · Zbl 0788.04007
[40] Skowron, A.; Polkowski, L., Rough mereological foundations for design, analysis, synthesis, and control in distributed systems, Inform. Sci., 104, 129-156, (1998) · Zbl 0947.68133
[41] Skowron, A.; Stepaniuk, J., Tolerance approximation spaces, Fundam. Inform., 27, 245-253, (1996) · Zbl 0868.68103
[42] Tian, D.; Wang, L.; Wu, J.; Hu, M. H., Rough set model based on uncertain measure, J. Uncertain Syst., 3, 252-256, (2009)
[43] Tversky, A., Features of similarity, Psychol. Rev., 84, 327-352, (1977)
[44] Wang, C. C.; Don, H. S., A modified measure for fuzzy subsethood, Inform. Sci., 79, 223-232, (1994) · Zbl 0799.94016
[45] S.K.M. Wong, W. Ziarko, A Probabilistic Model of Approximate Classification and Decision Rules with Uncertainty in Inductive Learning, Technical Report CS-85-23, Department of Computer Science, University of Regina, 1985.
[46] S.K.M. Wong, W. Ziarko, INFER: an adaptative decision support system based on the probabilistic approximate classification, in: Proceedings of the 6th International Workshop on Expert Systems & Their Applications, 1987, pp. 713-726.
[47] Wong, S. K.M.; Ziarko, W., Comparison of the probabilistic approximate classification and the fuzzy set model, Fuzzy Sets Syst., 21, 357-362, (1987) · Zbl 0618.60002
[48] Willmott, R., Two fuzzier implication operators in the theory of fuzzy power sets, Fuzzy Sets Syst., 4, 31-36, (1980) · Zbl 0433.03014
[49] Willmott, R., On the transitivity of containment and equivalence in fuzzy power set theory, J. Math. Anal. Appl., 120, 384-396, (1986) · Zbl 0609.03018
[50] Xu, Z. B.; Liang, J. Y.; Dang, C. Y.; Chin, K. S., Inclusion degree: a perspective on measures for rough set data analysis, Inform. Sci., 141, 227-236, (2002) · Zbl 1008.68134
[51] Yao, Y. Y., Two views of the theory of rough sets in finite universes, Int. J. Approx. Reason., 15, 291-317, (1996) · Zbl 0935.03063
[52] Yao, Y. Y., Constructive and algebraic methods of the theory of rough sets, Inform. Sci., 109, 21-47, (1998) · Zbl 0934.03071
[53] Yao, Y. Y., Probabilistic approaches to rough sets, Expert Syst., 20, 287-297, (2003)
[54] Yao, Y. Y., Probabilistic rough set approximations, Int. J. Approx. Reason., 49, 255-271, (2008) · Zbl 1191.68702
[55] Yao, Y. Y., Three-way decisions with probabilistic rough sets, Inform. Sci., 180, 341-353, (2010)
[56] Yao, Y. Y., The superiority of three-way decisions in probabilistic rough set models, Inform. Sci., 181, 1080-1096, (2011) · Zbl 1211.68442
[57] Y.Y. Yao, An outline of a theory of three-way decisions, in: Proceedings of the 7th International Conference on Rough Sets and Knowledge Technology, RSCTC 2012, LNCS (LNAI), vol. 7413, 2012, pp. 1-17. · Zbl 1404.68177
[58] Y.Y. Yao, R. Fu, Partitions, coverings, reducts and rule learning in rough set theory, in: Proceedings of the 6th International Conference on Rough Sets and Knowledge Technology, RSKT 2011, LNCS (LNAI), vol. 6954, 2011, pp. 101-109.
[59] Yao, Y. Y.; Lin, T. Y., Generalization of rough sets using modal logic, Intell. Automat. Soft Comput., Int. J., 2, 103-120, (1996)
[60] Yao, Y. Y.; Wong, S. K.M., A decision theoretic framework for approximating concepts, Int. J. Man-Mach. Stud., 37, 793-809, (1992)
[61] Yao, Y. Y.; Wong, S. K.M.; Lingras, P. J., A decision-theoretic rough set model, (Ras, Z. W.; Zemankova, M.; Emrich, M. L., Methodologies for Intelligent Systems, vol. 5, (1990), North-Holland New York), 17-24
[62] Y.Y. Yao, N. Zhong, An analysis of quantitative measures associated with rules, in: Proceedings of the Third Pacific-Asia Conference on Knowledge Discovery and Data Mining, PAKDD 1999, LNCS (LNAI), vol. 1574, 1999, pp. 479-488.
[63] Young, V. R., Fuzzy subsethood, Fuzzy Sets Syst., 77, 371-384, (1996) · Zbl 0872.94062
[64] Zhang, H. Y.; Leung, Y.; Zhou, L., Variable-precision-dominance-based rough set approach to interval-valued information systems, Inform. Sci., 244, 75-91, (2013) · Zbl 1355.68268
[65] Zhang, H. Y.; Zhang, W. X., Hybrid monotonic inclusion measure and its use in measuring similarity and distance between fuzzy sets, Fuzzy Sets Syst., 160, 107-118, (2009) · Zbl 1183.03059
[66] Zhang, M.; Xu, L. D.; Zhang, W. X.; Li, H. Z., A rough set approach to knowledge reduction based on inclusion degree and evidence reasoning theory, Expert Syst., 20, 298-304, (2003)
[67] Zhang, W. X.; Leung, Y., The uncertainty reasoning principles (in Chinese), (1996), Xi’an Jiaotong University Press Xi’an, China
[68] W.X. Zhang, Y. Leung, Theory of including degrees and its applications to uncertainty inferences, in: Proceedings of the 1996 Asian Fuzzy Systems Symposium, Soft Computing in Intelligent Systems and Information Processing, 1996, pp. 496-501.
[69] Ziarko, W., Variable precision rough set model, J. Comput. Syst. Sci., 46, 39-59, (1993) · Zbl 0764.68162
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.