Monotonic variable consistency rough set approaches. (English) Zbl 1191.68673
Summary: We consider probabilistic rough set approaches based on different versions of the definition of rough approximation of a set. In these versions, consistency measures are used to control assignment of objects to lower and upper approximations. Inspired by some basic properties of rough sets, we find it reasonable to require from these measures several properties of monotonicity. We consider three types of monotonicity properties: monotonicity with respect to the set of attributes, monotonicity with respect to the set of objects, and monotonicity with respect to the dominance relation. We show that consistency measures used so far in the definition of rough approximation lack some of these monotonicity properties. This observation led us to propose new measures within two kinds of rough set approaches: Variable Consistency Indiscernibility-based Rough Set Approaches and Variable Consistency Dominance-based Rough Set Approaches (VC-DRSA). We investigate properties of these approaches and compare them to previously proposed Variable Precision Rough Set model, Rough Bayesian model, and previous versions of VC-DRSA.
MSC:
| 68T37 | Reasoning under uncertainty in the context of artificial intelligence |
| 68T30 | Knowledge representation |
Keywords:
rough sets; dominance-based rough set approach; monotonicity; Bayesian confirmation; rough membership; likelihood; accuracy of approximation; variable precision; variable consistencySoftware:
4eMka2References:
| [1] | Błaszczyński, J.; Greco, S.; Słowiński, R.; Szela¸g, M., On variable consistency dominance-based rough set approaches, (Greco, S.; Hata, Y.; Hirano, S.; Inuiguchi, M.; Miyamoto, S.; Nguyen, H. S.; Słowiński, R., Rough Sets and Current Trends in Computing. Rough Sets and Current Trends in Computing, LNAI, vol. 4259 (2006), Springer-Verlag), 191-202 · Zbl 1162.68520 |
| [2] | Błaszczyński, J.; Greco, S.; Słowiński, R.; Szela¸g, M., Monotonic variable consistency rough set approaches, (Yao, J.; Lingras, P.; Wu, W.; Szczuka, M.; Cercone, N. J.; Śle¸zak, D., Rough Sets and Knowledge Technology. Rough Sets and Knowledge Technology, LNAI, vol. 4481 (2007), Springer-Verlag), 126-133 |
| [3] | Brewka, G., Nonmonotonic Reasoning: Logical Foundations of Commonsense (1991), Cambridge University Press · Zbl 0723.68099 |
| [4] | Düntsch, I.; Gediga, G., Uncertainty measures of rough set prediction, Artificial Intelligence, 106, 1, 109-137 (1998) · Zbl 0909.68040 |
| [5] | B. Fitelson, Studies in Bayesian Confirmation Theory, Ph. D. Thesis, University of Wisconsin-Madison, 2001.; B. Fitelson, Studies in Bayesian Confirmation Theory, Ph. D. Thesis, University of Wisconsin-Madison, 2001. |
| [6] | Fitelson, B., Likelihoodism, Bayesianism, and relational confirmation, Synthese, 156, 473-489 (2007) · Zbl 1125.03302 |
| [7] | Ginsberg, M. L., Readings in Nonmonotonic Reasoning (1987), Morgan Kaufman: Morgan Kaufman Los Altos, CA |
| [8] | Greco, S.; Matarazzo, B.; Słowiński, R., The use of rough sets and fuzzy sets in MCDM, (Gal, T.; Hanne, T.; Stewart, T., Advances in Multiple Criteria Decision Making (1999), Kluwer Academic Publishers: Kluwer Academic Publishers Boston), 14.1-14.59, (Chapter 14) · Zbl 0948.90078 |
| [9] | Greco, S.; Matarazzo, B.; Słowiński, R., Rough sets theory for multicriteria decision analysis, European Journal of Operational Research, 129, 1, 1-47 (2001) · Zbl 1008.91016 |
| [10] | Greco, S.; Matarazzo, B.; Słowiński, R.; Stefanowski, J., Variable consistency model of dominance-based rough sets approach, (Ziarko, W.; Yao, Y., Rough Sets and Current Trends in Computing. Rough Sets and Current Trends in Computing, LNAI, vol. 2005 (2001), Springler-Verlag: Springler-Verlag Berlin), 170-181 · Zbl 1014.68544 |
| [11] | Greco, S.; Matarazzo, B.; Słowiński, R., Rough approximation by dominance relations, International Journal of Intelligent Systems, 17, 2, 153-171 (2002) · Zbl 0997.68135 |
| [12] | Greco, S.; Pawlak, Z.; Słowiński, R., Can Bayesian confirmation measures be useful for rough set decision rules?, Engineering Applications of Artificial Intelligence, 17, 345-361 (2004) |
| [13] | Greco, S.; Słowiński, R.; Yao, Y. Y., Bayesian Decision theory for dominance-based rough set approach, (Yao, J.; Lingras, P.; Wu, W.; Szczuka, M.; Cercone, N. J.; Śle¸zak, D., Rough Sets and Knowledge Technology. Rough Sets and Knowledge Technology, LNAI, vol. 4481 (2007), Springer-Verlag), 134-141 |
| [14] | Greco, S.; Matarazzo, B.; Słowiński, R., Rough membership and Bayesian confirmation measures for parameterized rough sets, International Journal of Approximate Reasoning, 49, 2, 285-300 (2008) · Zbl 1191.68678 |
| [15] | Hempel, C. G., Studies in the logic of confirmation (I), Mind, 54, 1-26 (1945) · Zbl 0060.01910 |
| [16] | Hu, Q.; Daren, Y.; Zongxia, X.; Liu, J., Fuzzy probabilistic approximation spaces and their information measures, IEEE Transactions on Fuzzy Systems, 14, 2, 191-201 (2006) |
| [17] | M. Inuiguchi, Structure-based approaches to attribute reduction in variable precision rough set models, in: Granular Computing, 2005 IEEE International Conference, vol. 1, pp. 34-39.; M. Inuiguchi, Structure-based approaches to attribute reduction in variable precision rough set models, in: Granular Computing, 2005 IEEE International Conference, vol. 1, pp. 34-39. |
| [18] | Makinson, D., Bridges from Classical to Nonmonotonic Logic (2005), College Publications · Zbl 1084.03001 |
| [19] | Pawlak, Z., Rough sets, International Journal of Information and Computer Sciences, 11, 341-356 (1982) · Zbl 0501.68053 |
| [20] | Pawlak, Z., Rough Sets: Theoretical Aspects of Reasoning about Data (1991), Kluwer Academic Publishers: Kluwer Academic Publishers Norwell, MA, USA · Zbl 0758.68054 |
| [21] | Pawlak, Z.; Skowron, A., Rough membership functions, (Yager, R. R.; Fedrizzi, M.; Kacprzyk, J., Advances in the Dempster-Shafer Theory of Evidence (1994), Wiley: Wiley New York), 251-271 · Zbl 0794.03045 |
| [22] | Qian, Y.; Liang, J.; Dang, Ch.; Zhang, H.; Jianmin, M., On the evaluation of the decision performance of an incomplete decision table, Data and Knowledge Engineering, 65, 3, 373-400 (2008) |
| [23] | Qian, Y.; Liang, J.; Dang, Ch., Consistency measure, inclusion degree and fuzzy measure in decision tables, Fuzzy Sets and Systems, 159, 2353-2377 (2008) · Zbl 1187.68614 |
| [24] | Słowiński, R.; Greco, S.; Matarazzo, B., Rough set based decision support, (Burke, E. K.; Kendall, G., Search Methodologies: Introductory Tutorials in Optimization and Decision Support Techniques (2005), Springer-Verlag: Springer-Verlag New York), 475-527, (Chapter 16) |
| [25] | Śle¸zak, D., Rough Sets and Bayes Factor, Transactions on Rough Sets III. Transactions on Rough Sets III, LNCS, vol. 3400 (2005), Springer-Verlag, pp. 202-229 · Zbl 1117.68072 |
| [26] | Śle¸zak, D.; Ziarko, W., The investigation of the Bayesian rough set model, International Journal of Approximate Reasoning, 40, 81-91 (2005) · Zbl 1099.68089 |
| [27] | Wong, S. K.M.; Ziarko, W., Comparison of the probabilistic approximate classification and the fuzzy set model, Fuzzy Sets and Systems, 21, 357-362 (1987) · Zbl 0618.60002 |
| [28] | Ziarko, W., Variable precision rough set model, Journal of Computer and System Sciences, 46, 1, 39-59 (1993) · Zbl 0764.68162 |
| [29] | Ziarko, W., Stochastic approach to rough set theory, (Greco, S.; Hata, Y.; Hirano, S.; Inuiguchi, M.; Miyamoto, S.; Nguyen, H. S.; Słowiński, R., Rough Sets and Current Trends in Computing. Rough Sets and Current Trends in Computing, LNAI, vol. 4259 (2006), Springer-Verlag), 38-48 · Zbl 1162.68705 |
| [30] | Yao, Y. Y., Decision-theoretic rough set models, (Yao, J.; Lingras, P.; Wu, W.; Szczuka, M.; Cercone, N. J.; Śle¸zak, D., Rough Sets and Knowledge Technology. Rough Sets and Knowledge Technology, LNAI, vol. 4481 (2007), Springer-Verlag), 11-12 |
| [31] | Yao, Y. Y., Probabilistic rough set approximations, International Journal of Approximate Reasoning, 49, 2, 255-271 (2008) · Zbl 1191.68702 |
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