##
**MGRS: a multi-granulation rough set.**
*(English)*
Zbl 1185.68695

Summary: The original rough set model was developed by Pawlak, which is mainly concerned with the approximation of sets described by a single binary relation on the universe. In the view of granular computing, the classical rough set theory is established through a single granulation. This paper extends Pawlak’s rough set model to a multi-granulation rough set model (MGRS), where the set approximations are defined by using multi equivalence relations on the universe. A number of important properties of MGRS are obtained. It is shown that some of the properties of Pawlak’s rough set theory are special instances of those of MGRS.

Moreover, several important measures, such as accuracy measure \(\alpha \), quality of approximation \(\gamma \) and precision of approximation \(\pi \), are presented, which are re-interpreted in terms of a classic measure based on sets, the Marczewski-Steinhaus metric and the inclusion degree measure. A concept of approximation reduct is introduced to describe the smallest attribute subset that preserves the lower approximation and upper approximation of all decision classes in MGRS as well. Finally, we discuss how to extract decision rules using MGRS. Unlike the decision rules (“AND” rules) from Pawlak’s rough set model, the form of decision rules in MGRS is “OR”. Several pivotal algorithms are also designed, which are helpful for applying this theory to practical issues. The multi-granulation rough set model provides an effective approach for problem solving in the context of multi granulations.

Moreover, several important measures, such as accuracy measure \(\alpha \), quality of approximation \(\gamma \) and precision of approximation \(\pi \), are presented, which are re-interpreted in terms of a classic measure based on sets, the Marczewski-Steinhaus metric and the inclusion degree measure. A concept of approximation reduct is introduced to describe the smallest attribute subset that preserves the lower approximation and upper approximation of all decision classes in MGRS as well. Finally, we discuss how to extract decision rules using MGRS. Unlike the decision rules (“AND” rules) from Pawlak’s rough set model, the form of decision rules in MGRS is “OR”. Several pivotal algorithms are also designed, which are helpful for applying this theory to practical issues. The multi-granulation rough set model provides an effective approach for problem solving in the context of multi granulations.

### MSC:

68T30 | Knowledge representation |

68T37 | Reasoning under uncertainty in the context of artificial intelligence |

Full Text:
DOI

### References:

[1] | Dubois, D.; Prade, H., Rough fuzzy sets and fuzzy rough sets, International Journal of General Systems, 17, 191-209 (1990) · Zbl 0715.04006 |

[2] | Düntsch, I.; Gediga, G., Uncertainty measures of rough set prediction, Artificial Intelligence, 106, 109-137 (1998) · Zbl 0909.68040 |

[3] | Gediga, G.; Düntsch, I., Rough approximation quality revisited, Artificial Intelligence, 132, 219-234 (2001) · Zbl 0983.68194 |

[4] | Guan, J. W.; Bell, D. A., Rough computational methods for information systems, Artificial Intelligence, 105, 77-103 (1998) · Zbl 0909.68047 |

[5] | Jensen, R.; Shen, Q., Fuzzy-rough sets assisted attribute selection, IEEE Transactions on Fuzzy Systems, 15, 1, 73-89 (2007) |

[6] | Jeon, G.; Kim, D.; Jeong, J., Rough sets attributes reduction based expert system in interlaced video sequences, IEEE Transactions on Consumer Electronics, 52, 4, 1348-1355 (2006) |

[7] | Skowron, A.; Stepaniuk, J., Tolerance approximation spaces, Fundamenta Informaticae, 27, 2-3, 245-253 (1996) · Zbl 0868.68103 |

[8] | Kryszkiewicz, M., Rough set approach to incomplete information systems, Information Sciences, 112, 39-49 (1998) · Zbl 0951.68548 |

[9] | Kryszkiewicz, M., Rules in incomplete information systems, Information Sciences, 113, 271-292 (1999) · Zbl 0948.68214 |

[10] | Leung, Y.; Li, D. Y., Maximal consistent block technique for rule acquisition in incomplete information systems, Information Sciences, 153, 85-106 (2003) · Zbl 1069.68605 |

[11] | Liang, J. Y.; Dang, C. Y.; Chin, K. S.; Yam Richard, C. M., A new method for measuring uncertainty and fuzziness in rough set theory, International Journal of General Systems, 31, 4, 331-342 (2002) · Zbl 1010.94004 |

[12] | Liang, J. Y.; Li, D. Y., Uncertainty and Knowledge Acquisition in Information Systems (2005), Science Press: Science Press Beijing, China |

[13] | Liang, J. Y.; Qian, Y. H., Axiomatic approach of knowledge granulation in information systems, Lecture Notes in Artificial Intelligence, 4304, 1074-1078 (2006) |

[14] | Ma, J. M.; Zhang, W. X.; Leung, Y.; Song, X. X., Granular computing and dual Galois connection, Information Sciences, 177, 5365-5377 (2007) · Zbl 1126.68043 |

[15] | Mi, J. S.; Wu, W. Z.; Zhang, W. X., Approaches to knowledge reductions based on variable precision rough sets model, Information Sciences, 159, 255-272 (2004) · Zbl 1076.68089 |

[16] | Pawlak, Z., Rough Sets: Theoretical Aspects of Reasoning about Data, System Theory, Knowledge Engineering and Problem Solving, vol. 9 (1991), Kluwer: Kluwer Dordrecht · Zbl 0758.68054 |

[17] | Pawlak, Z.; Skowron, A., Rudiments of rough sets, Information Sciences, 177, 3-27 (2007) · Zbl 1142.68549 |

[18] | Qian, Y. H.; Liang, J. Y., Rough set method based on multi-granulations, Proceedings of 5th IEEE Conference on Cognitive Informatics, 1, 297-304 (2006) |

[20] | Qian, Y. H.; Liang, J. Y.; Li, D. Y.; Zhang, H. Y.; Dang, C. Y., Measures for evaluating the decision performance of a decision table in rough set theory, Information Sciences, 178, 181-202 (2008) · Zbl 1128.68102 |

[21] | Qian, Y. H.; Liang, J. Y.; Dang, C. Y., Converse approximation and rule extraction from decision tables in rough set theory, Computers and Mathematics with Applications, 55, 1754-1765 (2008) · Zbl 1147.68736 |

[22] | Qian, Y. H.; Liang, J. Y.; Dang, C. Y., Knowledge structure, knowledge granulation and knowledge distance in a knowledge base, International Journal of Approximate Reasoning, 50, 1, 174-188 (2009) · Zbl 1191.68660 |

[23] | Qian, Y. H.; Dang, C. Y.; Liang, J. Y.; Zhang, H. Y.; Ma, J. M., On the evaluation of the decision performance of an incomplete decision table, Data& Knowledge Engineering, 65, 3, 373-400 (2008) |

[24] | Qian, Y. H.; Dang, C. Y.; Liang, J. Y., Consistency measure, inclusion degree and fuzzy measure in decision tables, Fuzzy Sets and Systems, 159, 2353-2377 (2008) · Zbl 1187.68614 |

[25] | Rasiowa, H., Mechanical proof systems for logic of reaching consensus by groups of intelligent agents, International Journal of Approximate Reasoning, 5, 4, 415-432 (1991) · Zbl 0738.68073 |

[26] | Rasiowa, H.; Marek, V., Mechanical proof systems for logic II, consensus programs and their processing, Journal of Intelligent Information Systems, 2, 2, 149-164 (1992) |

[27] | Rauszer, C., Distributive knowledge representation systems, Foundations of Computing and Decision Sciences, 18, 307-332 (1993) |

[28] | Rauszer, C., Communication systems in distributed information systems, Proceedings of the Intelligent Information Systems, 15-29 (1993) |

[29] | Rauszer, C., Approximation methods for knowledge representation systems, Lecture Notes in Computer Science, 689, 326-337 (1993) |

[30] | Rauszer, C., Rough logic for multiagent systems, Lecture Notes in Artificial Intelligence, 808, 161-181 (1994) |

[31] | Rebolledo, M., Rough intervals-enhancing intervals for qualitative modeling of technical systems, Artificial Intelligence, 170, 667-685 (2006) |

[32] | Shen, Q.; Chouchoulas, A., A rough-fuzzy approach for generating classification rules, Pattern Recognition, 35, 2425-2438 (2002) · Zbl 1006.68902 |

[33] | Ślezak, D.; Ziarko, W., The investigation of the Bayesian rough set model, International Journal of Approximate Reasoning, 40, 81-91 (2005) · Zbl 1099.68089 |

[34] | Wu, W. Z.; Zhang, W. X., Constructive and axiomatic approaches of fuzzy approximation operators, Information Sciences, 159, 233-254 (2004) · Zbl 1071.68095 |

[35] | Wu, W. Z.; Mi, J. S.; Zhang, W. X., Generalized fuzzy rough sets, Information Sciences, 152, 263-282 (2003) · Zbl 1019.03037 |

[36] | Xu, Z. B.; Liang, J. Y.; Dang, C. Y.; Chin, K. S., Inclusion degree: a perspective on measures for rough set data analysis, Information Sciences, 141, 227-236 (2002) · Zbl 1008.68134 |

[38] | Yao, Y. Y., Information granulation and rough set approximation, International Journal of Intelligent Systems, 16, 87-104 (2001) · Zbl 0969.68079 |

[40] | Zadeh, L. A., Toward a theory of fuzzy information granulation and its centrality in human reasoning and fuzzy logic, Fuzzy Sets and Systems, 90, 111-127 (1997) · Zbl 0988.03040 |

[41] | Zeng, A.; Pan, D.; Zheng, Q. L.; Peng, H., Knowledge acquisition based on rough set theory and principal component analysis, IEEE Intelligent Systems, 78-85 (2006) |

[42] | Zhang, W. X.; Mi, J. S.; Wu, W. Z., Approaches to knowledge reductions in inconsistent systems, International Journal of Intelligent Systems, 21, 9, 989-1000 (2003) · Zbl 1069.68606 |

[43] | Ziarko, W., Variable precision rough sets model, Journal of Computer System Science, 46, 1, 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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.