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Fuzzy rough approximations of process data. (English) Zbl 1191.68694
Summary: This paper concerns the variable precision fuzzy rough set (VPFRS) model with asymmetric bounds. The discussion of the presented approach is preceded by a comparison of the original crisp rough set paradigm to the variable precision crisp rough set model. As a new aspect, a unified form of expressing the lower and upper crisp approximations is considered. It can be applied to defining new fuzzy rough set models. Crucial notions of the VPFRS model are redefined and explained. A new way of determining the upper variable precision fuzzy rough approximation is proposed. The VPFRS model is used for describing and analyzing the control actions which are accomplished by a human operator, who controls a complex dynamic system. The decision model is expressed by means of a decision table with fuzzy attributes. Decision tables are generated by the fuzzification of crisp data based on a set of fuzzy linguistic values of the attributes. A T-similarity relation is chosen for comparing elements of the universe. In an illustrative example, the task of stabilization of the aircraft’s bank angle during a turn maneuver is analyzed.

68T37Reasoning under uncertainty
93C42Fuzzy control systems
Full Text: DOI
[1] Bandler, W.; Kohout, L.: Fuzzy power sets and fuzzy implication operators, Fuzzy sets and systems 4, 13-30 (1980) · Zbl 0433.03013 · doi:10.1016/0165-0114(80)90060-3
[2] Bodjanova, S.: Approximation of fuzzy concepts in decision making, Fuzzy sets and systems 85, 23-29 (1997) · Zbl 0907.90003 · doi:10.1016/0165-0114(95)00404-1
[3] Burillo, P.; Frago, N.; Fuentes, R.: Inclusion grade and fuzzy implication operators, Fuzzy sets and systems 114, 417-429 (2000) · Zbl 0962.03050 · doi:10.1016/S0165-0114(98)00128-6
[4] Chen, S. -M.; Yeh, M. -S.; Hsiao, P. -Y.: A comparison of similarity measures of fuzzy values, Fuzzy sets and systems 72, 79-89 (1995)
[5] Cornelis, C.; Van Der Donck, C.; Kerre, E.: Sinha -- dougherty approach to the fuzzification of set inclusion revisited, Fuzzy sets and systems 134, 283-295 (2003) · Zbl 1013.03063 · doi:10.1016/S0165-0114(02)00225-7
[6] Cross, V. V.; Sudkamp, T. A.: Similarity and compatibility in fuzzy set theory, (2002) · Zbl 0992.03066
[7] D. Dubois, H. Prade, Putting rough sets and fuzzy sets together, in: Słowiński [29], pp. 203 -- 232.
[8] Salido, J. M. Fernández; Murakami, S.: Rough set analysis of a general type of fuzzy data using transitive aggregations of fuzzy similarity relations, Fuzzy sets and systems 139, 635-660 (2003) · Zbl 1047.68139 · doi:10.1016/S0165-0114(03)00124-6
[9] Greco, S.; Matarazzo, B.; Słowiński, R.: The use of rough sets and fuzzy sets in MCDM, Advances in multiple criteria decision making, 14.1-14.59 (1999) · Zbl 0948.90078
[10] Greco, S.; Matarazzo, B.; Słowiński, R.: Rough set processing of vague information using fuzzy similarity relations, Finite versus infinite --- contributions to an eternal dilemma, 149-173 (2000)
[11] Greco, S.; Matarazzo, B.; Słowiński, R.; Stefanowski, J.: Variable consistency model of dominance-based rough set approach, Lecture notes in artificial intelligence 2005, 170-181 (2001) · Zbl 1014.68544 · http://link.springer.de/link/service/series/0558/bibs/2005/20050170.htm
[12] S. Greco, M. Inuiguchi, R. Słowiński, Rough sets and gradual decision rules, in: Wang et al. [31], pp. 156 -- 164. · Zbl 1026.68627 · http://link.springer.de/link/service/series/0558/bibs/2639/26390156.htm
[13] S. Greco, B. Matarazzo, R. Słowiński, Rough membership and bayesian confirmation measures for parametrized rough sets, in: Śle¸zak et al. [28], pp. 314 -- 324.
[14] M. Inuiguchi, Generalizations of rough sets: From crisp to fuzzy cases, in: Tsumoto et al. [30], pp. 26 -- 37. · Zbl 1103.03048
[15] Katzberg, J. D.; Ziarko, W.: Variable precision extension of rough sets, Fundamenta informaticae 27, 155-168 (1996) · Zbl 0858.90084
[16] T.Y. Lin, Topological and fuzzy rough sets, in: Słowiński [29], pp. 287 -- 304.
[17] T.Y. Lin, Coping with Imprecision Information --- Fuzzy Logic, Downsizing Expo, Santa Clara Convention Center, 1993.
[18] Mieszkowicz-Rolka, A.; Rolka, L.: Variable precision rough sets in analysis of inconsistent decision tables, Advances in soft computing, 304-309 (2003) · Zbl 1033.68657
[19] Mieszkowicz-Rolka, A.; Rolka, L.: Fuzziness in information systems, Electronic notes in theoretical computer science 82 (2003) · Zbl 1270.68318
[20] Mieszkowicz-Rolka, A.; Rolka, L.: Variable precision fuzzy rough sets, Lecture notes in computer science (Journal subline) 3100, 144-160 (2004) · Zbl 1104.68767 · doi:10.1007/b98175
[21] A. Mrózek, Rough sets in computer implementation of rule-based control of industrial processes, in: Słowiński [29], pp. 19 -- 31.
[22] Pawlak, Z.: Rough sets: theoretical aspects of reasoning about data, (1991) · Zbl 0758.68054
[23] Pawlak, Z.: AI and intelligent industrial applications: the rough set perspective, Cybernetics and systems: an international journal 31, 227-252 (2000) · Zbl 1061.68558 · doi:10.1080/019697200124801
[24] Peters, J. F.; Skowron, A.; Suraj, Z.: An application of rough sets methods in control design, Fundamenta informaticae 43, 269-290 (2000) · Zbl 0971.93052
[25] L. Polkowski, Toward rough set foundations. Mereological approach, in: Tsumoto et al. [30], pp. 8 -- 25. · Zbl 1103.03049
[26] Radzikowska, A. M.; Kerre, E. E.: A comparative study of fuzzy rough sets, Fuzzy sets and systems 126, 137-155 (2002) · Zbl 1004.03043 · doi:10.1016/S0165-0114(01)00032-X
[27] D. Śle¸zak, W. Ziarko, Variable precision bayesian rough set model, in: Wang et al. [31], pp. 312 -- 315. · Zbl 1026.68655
[28] , Lecture notes in artificial intelligence 3641 (2005)
[29] , Intelligent decision support: handbook of applications and advances of the rough sets theory (1992) · Zbl 0820.68001
[30] , Lecture notes in artificial intelligence 3066 (2004)
[31] , Lecture notes in computer science 2639 (2003)
[32] W.N. Liu, J. Yao, Y. Yao, Rough approximations under level fuzzy sets, in: Tsumoto et al. [30], pp. 78 -- 83. · Zbl 1103.68858
[33] Ziarko, W.: Variable precision rough sets model, Journal of computer and system sciences 46, No. 1, 39-59 (1993) · Zbl 0764.68162 · doi:10.1016/0022-0000(93)90048-2
[34] W. Ziarko, Probabilistic rough sets, in: Śle¸zak et al. [28], pp. 283 -- 293.