An interval set model for learning rules from incomplete information table.

*(English)*Zbl 1242.68235Summary: A novel interval set approach is proposed in this paper to induce classification rules from incomplete information table, in which an interval-set-based model to represent the uncertain concepts is presented. The extensions of the concepts in incomplete information table are represented by interval sets, which regulate the upper and lower bounds of the uncertain concepts. Interval set operations are discussed, and the connectives of concepts are represented by the operations on interval sets. Certain inclusion, possible inclusion, and weak inclusion relations between interval sets are presented, which are introduced to induce strong rules and weak rules from incomplete information table. The related properties of the inclusion relations are proved. It is concluded that the strong rules are always true whatever the missing values may be, while the weak rules may be true when missing values are replaced by some certain known values. Moreover, a confidence function is defined to evaluate the weak rule. The proposed approach presents a new view on rule induction from incomplete data based on interval set.

##### MSC:

68T05 | Learning and adaptive systems in artificial intelligence |

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

##### Software:

C4.5
PDF
BibTeX
XML
Cite

\textit{H. Li} et al., Int. J. Approx. Reasoning 53, No. 1, 24--37 (2012; Zbl 1242.68235)

Full Text:
DOI

**OpenURL**

##### References:

[1] | Bazan, J.G.; Latkowski, R.; Szczuka, M.S., Missing template decomposition method and its implementation in rough set exploration system, (), 254-263 · Zbl 1162.68684 |

[2] | Bustince, H., Indicator of inclusion grade for intervalvalued fuzzy sets application to approximate reasoning based on interval-valued fuzzy sets, International journal of approximate reasoning, 23, 137-209, (2000) · Zbl 1046.68646 |

[3] | Cendrowska, J., PRISM: an algorithm for inducing modular rules, International journal of man – machine studies, 27, 4, 349-370, (1987) · Zbl 0638.68110 |

[4] | Clark, P.; Niblett, T., The CN2 induction algorithm, Machine learning, 3, 4, 261-283, (1989) |

[5] | Cornelis, C.; Deschrijver, G.; Kerre, E.E., Implication in intuitionistic fuzzy and interval-valued fuzzy set theory: construction, classification, application, International journal of approximate reasoning, 35, 1, 55-95, (2004) · Zbl 1075.68089 |

[6] | Cui, W.; Blockley, D.I., Interval probability theory for evidential support, International journal of intelligent systems, 5, 2, 183-192, (1990) · Zbl 0706.68092 |

[7] | Denoeux, T.; Younes, Z.; Abdallah, F., Representing uncertainty on set-valued variables using belief functions, Artificial intelligence, 174, 7-8, 479-499, (2010) · Zbl 1209.68542 |

[8] | D. Dubois, H. Prade, Interval-valued fuzzy sets, possibility theory and imprecise probability, in: Proceedings of the 4th International Conference of the European Society for Fuzzy Logic and Technology, 2005, pp. 314-319. |

[9] | Gediga, G.; Düntsch, I., Maximum consistency of incomplete data via non-invasive imputation, Artificial intelligence review, 19, 4, 93-107, (2003) |

[10] | Ghahramani, Z.; Jordan, M.I., Supervised learning from incomplete data via an EM approach, (), 120-127 |

[11] | S. Greco, B. Matarazzo, R. Slowinski, Handling missing values in rough set analysis of multi-attribute and multi-criteria decision problems, in: N. Zhong, A. Skowron, S. Ohsuga, (Eds.), Proceedings of the 7th International Workshop on Rough Sets, Fuzzy Sets, Data Mining, and Granular-Soft Computing, LNAI 1711, Springer, Berlin, 1999, pp. 146-157. · Zbl 1037.91510 |

[12] | Grzymala-Busse, J.W., On the unknown attribute values in learning from examples, (), 368-377 |

[13] | Grzymala-Busse, J.W.; Hu, M., A comparison of several approches to missing attribute values in data mining, (), 378-385 · Zbl 1014.68558 |

[14] | Grzymala-Busse, J.W.; Grzymala-Busse, W.J., An experimental comparison of three rough set approaches to missing attribute values, (), 31-50 · Zbl 1058.68659 |

[15] | Grzymala-Busse, J.W., Data with missing attribute values: generalization of indiscernibility relation and rule induction, (), 78-95 · Zbl 1104.68759 |

[16] | Hall, J.W.; Blockley, D.I.; Davis, J.P., Uncertain inference using interval probability theory, International journal of approximate reasoning, 19, 3, 247-264, (1998) · Zbl 0944.68174 |

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

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

[19] | Laurikkala, J.; Kentala, E.; Juhola, M.; Pyykkö, I.; Lammi, S., Usefulness of imputation for the analysis of incomplete otoneurologic data, International journal of medical informatics, 58-59, 235-242, (2000) |

[20] | Leung, Y.; Fischer, M.M.; Wu, W.Z.; Mi, J.S., A rough set approach for the discovery of classification rules in interval-valued information systems, International journal of approximate reasoning, 47, 2, 233-246, (2008) · Zbl 1184.68409 |

[21] | Li, H.X.; Yao, Y.Y.; Zhou, X.Z.; Huang, B., A two-phase model for learning rules from incomplete data, Fundamenta informaticae, 94, 2, 219-232, (2009) · Zbl 1192.68527 |

[22] | Li, H.X.; Yao, Y.Y.; Zhou, X.Z.; Huang, B., Two-phase rule induction from incomplete data, (), 47-54 |

[23] | Li, H.X.; Zhou, X.Z.; Yao, Y.Y., Missing values imputation hypothesis: an experimental evaluation, (), 275-280 |

[24] | Mitchell, T.M., Generalization as search, Artificial intelligence, 18, 203-226, (1982) |

[25] | Mitchell, T.M., Machine learning, (1997), McGraw-Hill New York · Zbl 0913.68167 |

[26] | Moore, R.E., Interval analysis, (1966), Prentice-Hall Englewood Cliffs, NJ · Zbl 0176.13301 |

[27] | Pawlak, Z., Rough sets: theoretical aspects of reasoning about data, (1991), Kluwer Academic Publishers Dordrecht, The Netherlands · Zbl 0758.68054 |

[28] | Polkowski, L.; Artiemjew, P., Granular classifiers and missing values, (), 186-194 |

[29] | 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 and knowledge engineering, 65, 3, 373-400, (2008) |

[30] | Qian, Y.H.; Liang, J.Y., Combination entropy and combination granulation in incomplete information systems, Lecture notes in artificial intelligence, 4062, 184-190, (2006) · Zbl 1196.68269 |

[31] | Quinlan, J.R., C4.5: programs for machine learning, (1993), Morgan Kaufmann San Mateo, CA |

[32] | Stefanowski, J.; Tsoukiàs, A., On the extension of rough sets under incomplete information, International journal of intelligent system, 16, 29-38, (1999) |

[33] | Wang, G.Y., Extension of rough set under incomplete information systems, Journal of computer research and development, 39, 10, 1238-1243, (2002), (in Chinese) |

[34] | Wong, S.K.M.; Wang, L.S.; Yao, Y.Y., On modeling uncertainty with interval structures, Computational intelligence, 11, 2, 406-426, (1995) |

[35] | Yager, R.R.; Kreinovich, V., Decision making under interval probabilities, International journal of approximate reasoning, 22, 3, 195-215, (1999) · Zbl 1041.91500 |

[36] | Yao, J.T.; Yao, Y.Y., Induction of classification rules by granular computing, (), 331-338 · Zbl 1013.68514 |

[37] | Yao, Y.Y., Concept formation and learning: a cognitive informatics perspective, (), 42-51 |

[38] | Y.Y. Yao, Interval-set algebra for qualitative knowledge representation, in: Proceedings of the 5th International Conference on Computing and Information, 1993, pp. 370-374. |

[39] | Y.Y. Yao, Interval sets and interval-set algebras, in: Proceedings of the 8th IEEE International Conference on Cognitive Informatics, 2009, pp. 307-314. |

[40] | Yao, Y.Y.; Zhong, N., An analysis of quantitative measures associated with rules, (), 479-488 |

[41] | Zhang, H.Y.; Zhang, W.X.; Wu, W.Z., On characterization of generalized interval-valued fuzzy rough sets on two universes of discourse, International journal of approximate reasoning, 51, 1, 56-70, (2009) · Zbl 1209.68552 |

[42] | Zhang, S.C.; Qin, Z.X.; Ling, C.X.; Sheng, S.L., Missing is useful: missing values in cost-sensitive decision trees, IEEE transactions on knowledge and data engineering, 17, 1689-1693, (2005) |

[43] | Zadeh, L.A., The concept of a linguistic variable and its application to approximate reasoning (part I), Information science, 8, 3, 199-249, (1975) · Zbl 0397.68071 |

[44] | Zadeh, L.A., The concept of a linguistic variable and its application to approximate reasoning (part II), Information science, 8, 4, 301-357, (1975) · Zbl 0404.68074 |

[45] | Zeng, W.Y.; Li, H.X., Relationship between similarity measure and entropy of interval valued fuzzy sets, Fuzzy sets and systems, 157, 11, 1477-1484, (2006) · Zbl 1093.94038 |

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.