Advertisement

A Rough Set Approach to Incomplete Data

  • Jerzy W. Grzymala-Busse
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9436)

Abstract

This paper presents main directions of research on a rough set approach to incomplete data. First, three different types of lower and upper approximations, based on the characteristic relation, are defined. Then an idea of the probabilistic approximation, an extension of lower and upper approximations, is presented. Local probabilistic approximations are also discussed. Finally, some special topics such as consistency of incomplete data and a problem of increasing data set incompleteness to improve rule set quality, in terms of an error rate, are discussed.

Keywords

Incomplete data Characteristic relation Singleton concept and subset approximations Probabilistic approximations Local probabilistic approximations 

References

  1. 1.
    Breiman, L., Friedman, J.H., Olshen, R.A., Stone, C.J.: Classification and Regression Trees. Wadsworth & Brooks, Monterey (1984)zbMATHGoogle Scholar
  2. 2.
    Chan, C.C., Grzymala-Busse, J.W.: On the attribute redundancy and the learning programs ID3, PRISM, and LEM2. Technical report, Department of Computer Science, University of Kansas (1991)Google Scholar
  3. 3.
    Clark, P.G., Grzymala-Busse, J.W.: Experiments on probabilistic approximations. In: Proceedings of the 2011 IEEE International Conference on Granular Computing, pp. 144–149 (2011)Google Scholar
  4. 4.
    Clark, P.G., Grzymala-Busse, J.W.: Consistency of incomplete data. In: Proceedings of the Second International Conference on Data Technologies and Applications, pp. 80–87 (2013)Google Scholar
  5. 5.
    Clark, P.G., Grzymala-Busse, J.W.: A comparison of two versions of the MLEM2 rule induction algorithm extended to probabilistic approximations. In: Cornelis, C., Kryszkiewicz, M., Ślȩzak, D., Ruiz, E.M., Bello, R., Shang, L. (eds.) RSCTC 2014. LNCS, vol. 8536, pp. 109–119. Springer, Heidelberg (2014) Google Scholar
  6. 6.
    Clark, P.G., Grzymala-Busse, J.W., Hippe, Z.S.: An analysis of probabilistic approximations for rule induction from incomplete data sets. Fundam. Informaticae 55, 365–379 (2014)Google Scholar
  7. 7.
    Clark, P.G., Grzymala-Busse, J.W., Kuehnhausen, M.: Local probabilistic approximations for incomplete data. In: Chen, L., Felfernig, A., Liu, J., Raś, Z.W. (eds.) ISMIS 2012. LNCS, vol. 7661, pp. 93–98. Springer, Heidelberg (2012) CrossRefGoogle Scholar
  8. 8.
    Clark, P.G., Grzymala-Busse, J.W., Kuehnhausen, M.: Mining incomplete data with many missing attribute values. A comparison of probabilistic and rough set approaches. In: Proceedings of the Second International Conference on Intelligent Systems and Applications, pp. 12–17 (2013)Google Scholar
  9. 9.
    Clark, P.G., Grzymala-Busse, J.W., Rzasa, W.: Mining incomplete data with singleton, subset and concept approximations. Inf. Sci. 280, 368–384 (2014)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Cyran, K.A.: Modified indiscernibility relation in the theory of rough sets with real-valued attributes: application to recognition of fraunhofer diffraction patterns. Trans. Rough Sets 9, 14–34 (2008)Google Scholar
  11. 11.
    Dai, J.: Rough set approach to incomplete numerical data. Inf. Sci. 241, 43–57 (2013)CrossRefGoogle Scholar
  12. 12.
    Dai, J., Xu, Q.: Approximations and uncertainty measures in incomplete information systems. Inf. Sci. 198, 62–80 (2012)zbMATHMathSciNetCrossRefGoogle Scholar
  13. 13.
    Dai, J., Xu, Q., Wang, W.: A comparative study on strategies of rule induction for incomplete data based on rough set approach. Int. J. Adv. Comput. Technol. 3, 176–183 (2011)Google Scholar
  14. 14.
    Dardzinska, A., Ras, Z.W.: Chasing unknown values in incomplete information systems. In: Workshop Notes, Foundations and New Directions of Data Mining, in Conjunction with the 3-rd International Conference on Data Mining, pp. 24–30 (2003)Google Scholar
  15. 15.
    Dardzinska, A., Ras, Z.W.: On rule discovery from incomplete information systems. In: Workshop Notes, Foundations and New Directions of Data Mining, in Conjunction with the 3-rd International Conference on Data Mining, pp. 24–30 (2003)Google Scholar
  16. 16.
    Greco, S., Matarazzo, B., Slowinski, R.: Dealing with missing data in rough set analysis of multi-attribute and multi-criteria decision problems. In: Zanakis, H., Doukidis, G., Zopounidis, Z. (eds.) Decision Making: Recent developments and Worldwide Applications, pp. 295–316. Kluwer Academic Publishers, Dordrecht (2000)CrossRefGoogle Scholar
  17. 17.
    Grzymala-Busse, J.W.: On the unknown attribute values in learning from examples. In: Raś, Z.W., Zemankova, M. (eds.) ISMIS 1991. LNCS, vol. 542, pp. 368–377. Springer, Heidelberg (1991) CrossRefGoogle Scholar
  18. 18.
    Grzymala-Busse, J.W.: LERS–a system for learning from examples based on rough sets. In: Slowinski, R. (ed.) Intelligent Decision Support, pp. 3–18. Handbook of Applications and Advances of the Rough Set Theory. Kluwer Academic Publishers, Dordrecht (1992)CrossRefGoogle Scholar
  19. 19.
    Grzymala-Busse, J.W.: A new version of the rule induction system LERS. Fundamenta Informaticae 31, 27–39 (1997)zbMATHGoogle Scholar
  20. 20.
    Grzymala-Busse, J.W.: MLEM2: A new algorithm for rule induction from imperfect data. In: Proceedings of the 9th International Conference on Information Processing and Management of Uncertainty in Knowledge-Based Systems, pp. 243–250 (2002)Google Scholar
  21. 21.
    Grzymala-Busse, J.W.: Rough set strategies to data with missing attribute values. In: Notes of the Workshop on Foundations and New Directions of Data Mining, in Conjunction with the Third International Conference on Data Mining, pp. 56–63 (2003)Google Scholar
  22. 22.
    Grzymała-Busse, J.W.: Characteristic relations for incomplete data: a generalization of the indiscernibility relation. In: Tsumoto, S., Słowiński, R., Komorowski, J., Grzymała-Busse, J.W. (eds.) RSCTC 2004. LNCS (LNAI), vol. 3066, pp. 244–253. Springer, Heidelberg (2004) CrossRefGoogle Scholar
  23. 23.
    Grzymala-Busse, J.W.: Data with missing attribute values: generalization of indiscernibility relation and rule induction. Trans. Rough Sets 1, 78–95 (2004)Google Scholar
  24. 24.
    Grzymala-Busse, J.W.: Three approaches to missing attribute values–a rough set perspective. In: Proceedings of the Workshop on Foundation of Data Mining, in Conjunction with the Fourth IEEE International Conference on Data Mining, pp. 55–62 (2004)Google Scholar
  25. 25.
    Grzymała-Busse, J.W.: Incomplete data and generalization of indiscernibility relation, definability, and approximations. In: Slezak, D., Wang, G., Szczuka, M.S., Düntsch, I., Yao, Y. (eds.) RSFDGrC 2005. LNCS (LNAI), vol. 3641, pp. 244–253. Springer, Heidelberg (2005) CrossRefGoogle Scholar
  26. 26.
    Grzymala-Busse, J.W.: A comparison of traditional and rough set approaches to missing attribute values in data mining. In: Proceedings of the 10-th International Conference on Data Mining, Detection, Protection and Security, Royal Mare Village, Crete, pp. 155–163 (2009)Google Scholar
  27. 27.
    Grzymala-Busse, J.W.: Mining data with missing attribute values: A comparison of probabilistic and rough set approaches. In: Proceedings of the 4-th International Conference on Intelligent Systems and Knowledge Engineering, pp. 153–158 (2009)Google Scholar
  28. 28.
    Grzymala-Busse, J.W.: Rough set and CART approaches to mining incomplete data. In: Proceedings of the International Conference on Soft Computing and Pattern Recognition, IEEE Computer Society, pp. 214–219 (2010)Google Scholar
  29. 29.
    Grzymala-Busse, J.W.: A comparison of some rough set approaches to mining symbolic data with missing attribute values. In: Kryszkiewicz, M., Rybinski, H., Skowron, A., Raś, Z.W. (eds.) ISMIS 2011. LNCS, vol. 6804, pp. 52–61. Springer, Heidelberg (2011) CrossRefGoogle Scholar
  30. 30.
    Grzymała-Busse, J.W.: Generalized parameterized approximations. In: Yao, J.T., Ramanna, S., Wang, G., Suraj, Z. (eds.) RSKT 2011. LNCS, vol. 6954, pp. 136–145. Springer, Heidelberg (2011) CrossRefGoogle Scholar
  31. 31.
    Grzymala-Busse, J.W., Clark, P.G., Kuehnhausen, M.: Generalized probabilistic approximations of incomplete data. Int. J. Approximate Reasoning 132, 180–196 (2014)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Grzymala-Busse, J.W., Grzymala-Busse, W.J.: Handling missing attribute values. In: Maimon, O., Rokach, L. (eds.) Data Mining and Knowledge Discovery Handbook, pp. 37–57. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  33. 33.
    Grzymala-Busse, J.W., Grzymala-Busse, W.J.: An experimental comparison of three rough set approaches to missing attribute values. Trans. Rough Sets 6, 31–50 (2007)Google Scholar
  34. 34.
    Grzymala-Busse, J.W., Grzymala-Busse, W.J.: Improving quality of rule sets by increasing incompleteness of data sets. In: Proceedings of the Third International Conference on Software and Data Technologies, pp. 241–248 (2008)Google Scholar
  35. 35.
    Grzymala-Busse, J.W., Grzymala-Busse, W.J.: Inducing better rule sets by adding missing attribute values. In: Chan, C.-C., Grzymala-Busse, J.W., Ziarko, W.P. (eds.) RSCTC 2008. LNCS (LNAI), vol. 5306, pp. 160–169. Springer, Heidelberg (2008) CrossRefGoogle Scholar
  36. 36.
    Grzymala-Busse, J.W., Grzymala-Busse, W.J.: Handling missing attribute values. In: Maimon, O., Rokach, L. (eds.) Data Mining and Knowledge Discovery Handbook, 2nd edn, pp. 33–51. Springer, Heidelberg (2010)Google Scholar
  37. 37.
    Grzymala-Busse, J.W., Grzymala-Busse, W.J., Goodwin, L.K.: A comparison of three closest fit approaches to missing attribute values in preterm birth data. Int. J. Intell. Syst. 17(2), 125–134 (2002)zbMATHCrossRefGoogle Scholar
  38. 38.
    Grzymala-Busse, J.W., Grzymala-Busse, W.J., Hippe, Z.S., Rzasa, W.: An improved comparison of three rough set approaches to missing attribute values. In: Proceedings of the 16-th International Conference on Intelligent Information Systems, pp. 141–150 (2008)Google Scholar
  39. 39.
    Grzymala-Busse, J.W., Hippe, Z.S.: Mining data with numerical attributes and missing attribute values–a rough set approach. In: Proceedings of the IEEE International Conference on Granular Computing, pp. 144–149 (2011)Google Scholar
  40. 40.
    Grzymała-Busse, J.W., Hu, M.: A comparison of several approaches to missing attribute values in data mining. In: Ziarko, W.P., Yao, Y. (eds.) RSCTC 2000. LNCS (LNAI), vol. 2005, p. 378. Springer, Heidelberg (2001) CrossRefGoogle Scholar
  41. 41.
    Grzymala-Busse, J.W., Rzasa, W.: Local and global approximations for incomplete data. In: Greco, S., Hata, Y., Hirano, S., Inuiguchi, M., Miyamoto, S., Nguyen, H.S., Słowiński, R. (eds.) RSCTC 2006. LNCS (LNAI), vol. 4259, pp. 244–253. Springer, Heidelberg (2006) CrossRefGoogle Scholar
  42. 42.
    Grzymala-Busse, J.W., Rzasa, W.: Local and global approximations for incomplete data. Trans. Rough Sets 8, 21–34 (2008)MathSciNetGoogle Scholar
  43. 43.
    Grzymala-Busse, J.W., Rzasa, W.: A local version of the MLEM2 algorithm for rule induction. Fundamenta Informaticae 100, 99–116 (2010)zbMATHMathSciNetGoogle Scholar
  44. 44.
    Grzymala-Busse, J.W., Wang, A.Y.: Modified algorithms LEM1 and LEM2 for rule induction from data with missing attribute values. In: Proceedings of the 5-th International Workshop on Rough Sets and Soft Computing in conjunction with the Third Joint Conference on Information Sciences, pp. 69–72 (1997)Google Scholar
  45. 45.
    Grzymala-Busse, J.W., Yao, Y.: Probabilistic rule induction with the LERS data mining system. Int. J. Intell. Syst. 26, 518–539 (2011)CrossRefGoogle Scholar
  46. 46.
    Grzymala-Busse, J.W., Ziarko, W.: Data mining based on rough sets. In: Wang, J. (ed.) Data Mining: Opportunities and Challenges, pp. 142–173. Idea Group Publ, Hershey (2003)CrossRefGoogle Scholar
  47. 47.
    Guan, L., Wang, G.: Generalized approximations defined by non-equivalence relations. Inf. Sci. 193, 163–179 (2012)zbMATHMathSciNetCrossRefGoogle Scholar
  48. 48.
    Hong, T.P., Tseng, L.H., Chien, B.C.: Learning coverage rules from incomplete data based on rough sets. In: Proceedings of the IEEE International Conference on Systems, Man and Cybernetics, pp. 3226–3231 (2004)Google Scholar
  49. 49.
    Hong, T.P., Tseng, L.H., Wang, S.L.: Learning rules from incomplete training examples by rough sets. Expert Syst. Appl. 22, 285–293 (2002)CrossRefGoogle Scholar
  50. 50.
    Kryszkiewicz, M.: Rough set approach to incomplete information systems. In: Proceedings of the Second Annual Joint Conference on Information Sciences, pp. 194–197 (1995)Google Scholar
  51. 51.
    Kryszkiewicz, M.: Rules in incomplete information systems. Inf. Sci. 113(3–4), 271–292 (1999)zbMATHMathSciNetCrossRefGoogle Scholar
  52. 52.
    Latkowski, R.: On decomposition for incomplete data. Fundamenta Informaticae 54, 1–16 (2003)zbMATHMathSciNetGoogle Scholar
  53. 53.
    Latkowski, R., Mikołajczyk, M.: Data decomposition and decision rule joining for classification of data with missing values. In: Tsumoto, S., Słowiński, R., Komorowski, J., Grzymała-Busse, J.W. (eds.) RSCTC 2004. LNCS (LNAI), vol. 3066, pp. 254–263. Springer, Heidelberg (2004) CrossRefGoogle Scholar
  54. 54.
    Leung, Y., Wu, W., Zhang, W.: Knowledge acquisition in incomplete information systems: a rough set approach. Eur. J. Ope. Res. 168, 164–180 (2006)zbMATHMathSciNetCrossRefGoogle Scholar
  55. 55.
    Li, D., Deogun, I., Spaulding, W., Shuart, B.: Dealing with missing data: algorithms based on fuzzy set and rough set theories. Trans. Rough Sets 4, 37–57 (2005)Google Scholar
  56. 56.
    Li, H., Yao, Y., Zhou, X., Huang, B.: Two-phase rule induction from incomplete data. In: Wang, G., Li, T., Grzymala-Busse, J.W., Miao, D., Skowron, A., Yao, Y. (eds.) RSKT 2008. LNCS (LNAI), vol. 5009, pp. 47–54. Springer, Heidelberg (2008) CrossRefGoogle Scholar
  57. 57.
    Li, T., Ruan, D., Geert, W., Song, J., Xu, Y.: A rough sets based characteristic relation approach for dynamic attribute generalization in data mining. Knowl. Based Syst. 20(5), 485–494 (2007)CrossRefGoogle Scholar
  58. 58.
    Li, T., Ruan, D., Song, J.: Dynamic maintenance of decision rules with rough set under characteristic relation. In: Proceedings of the International Conference on Wireless Communications, Networking and Mobile Computing, pp. 3713–3716 (2007)Google Scholar
  59. 59.
    Meng, Z., Shi, Z.: A fast approach to attribute reduction in incomplete decision systems with tolerance relation-based rough sets. Inf. Sci. 179, 2774–2793 (2009)zbMATHMathSciNetCrossRefGoogle Scholar
  60. 60.
    Meng, Z., Shi, Z.: Extended rough set-based attribute reduction in inconsistent incomplete decision systems. Inf. Sci. 204, 44–69 (2012)MathSciNetCrossRefGoogle Scholar
  61. 61.
    Nakata, M., Sakai, H.: Rough sets handling missing values probabilistically interpreted. In: Slezak, D., Wang, G., Szczuka, M.S., Düntsch, I., Yao, Y. (eds.) RSFDGrC 2005. LNCS (LNAI), vol. 3641, pp. 325–334. Springer, Heidelberg (2005) CrossRefGoogle Scholar
  62. 62.
    Nakata, M., Sakai, H.: Applying rough sets to information tables containing missing values. In: Proceedings of the 39-th International Symposium on Multiple-Valued Logic, pp. 286–291 (2009)Google Scholar
  63. 63.
    Pawlak, Z.: Rough sets. Int. J. Comput. Inf. Sci. 11, 341–356 (1982)zbMATHMathSciNetCrossRefGoogle Scholar
  64. 64.
    Pawlak, Z.: Rough Sets: Theoretical Aspects of Reasoning about Data. Kluwer Academic Publishers, Dordrecht (1991) zbMATHCrossRefGoogle Scholar
  65. 65.
    Pawlak, Z., Grzymala-Busse, J.W., Slowinski, R., Ziarko, W.: Rough sets. Commun. ACM 38, 89–95 (1995)CrossRefGoogle Scholar
  66. 66.
    Pawlak, Z., Skowron, A.: Rough sets: some extensions. Inf. Sci. 177, 28–40 (2007)zbMATHMathSciNetCrossRefGoogle Scholar
  67. 67.
    Pawlak, Z., Wong, S.K.M., Ziarko, W.: Rough sets: probabilistic versus deterministic approach. Int. J. Man-Mach. Stud. 29, 81–95 (1988)zbMATHCrossRefGoogle Scholar
  68. 68.
    Peng, H., Zhu, S.: Handling of incomplete data sets using ICA and SOM in data mining. Neural Comput. Appl. 16, 167–172 (2007)CrossRefGoogle Scholar
  69. 69.
    Qi, Y.S., Wei, L., Sun, H.J., Song, Y.Q., Sun, Q.S.: Characteristic relations in generalized incomplete information systems. In: International Workshop on Knowledge Discovery and Data Mining, pp. 519–523 (2008)Google Scholar
  70. 70.
    Qi, Y.S., Sun, H., Yang, X.B., Song, Y., Sun, Q.: Approach to approximate distribution reduct in incomplete ordered decision system. J. Inf. Comput. Sci. 3, 189–198 (2008)Google Scholar
  71. 71.
    Qian, Y., Dang, C., Liang, J., Zhang, H., Ma, J.: On the evaluation of the decision performance of an incomplete decision table. Data Knowl. Eng. 65, 373–400 (2008)CrossRefGoogle Scholar
  72. 72.
    Qian, Y., Li, D., Wang, F., Ma, N.: Approximation reduction in inconsistent incomplete decision tables. Knowl. Based Syst. 23, 427–433 (2010)CrossRefGoogle Scholar
  73. 73.
    Ślȩzak, D., Ziarko, W.: The investigation of the bayesian rough set model. Int. J. Approx. Reason. 40, 81–91 (2005)CrossRefGoogle Scholar
  74. 74.
    Song, J., Li, T., Ruan, D.: A new decision tree construction using the cloud transform and rough sets. In: Wang, G., Li, T., Grzymala-Busse, J.W., Miao, D., Skowron, A., Yao, Y. (eds.) RSKT 2008. LNCS (LNAI), vol. 5009, pp. 524–531. Springer, Heidelberg (2008) CrossRefGoogle Scholar
  75. 75.
    Song, J., Li, T., Wang, Y., Qi, J.: Decision tree construction based on rough set theory under characteristic relation. In: Proceedings of the ISKE 2007, the 2-nd International Conference on Intelligent Systems and Knowledge Engineering Conference, pp. 788–792 (2007)Google Scholar
  76. 76.
    Stefanowski, J., Tsoukiàs, A.: On the extension of rough sets under incomplete information. In: Zhong, N., Skowron, A., Ohsuga, S. (eds.) RSFDGrC 1999. LNCS (LNAI), vol. 1711, pp. 73–82. Springer, Heidelberg (1999) CrossRefGoogle Scholar
  77. 77.
    Stefanowski, J., Tsoukias, A.: Incomplete information tables and rough classification. Computat. Intell. 17(3), 545–566 (2001)CrossRefGoogle Scholar
  78. 78.
    Wang, G.: Extension of rough set under incomplete information systems. In: Proceedings of the IEEE International Conference on Fuzzy Systems, pp. 1098–1103 (2002)Google Scholar
  79. 79.
    Wong, S.K.M., Ziarko, W.: INFER–an adaptive decision support system based on the probabilistic approximate classification. In: Proceedings of the 6-th International Workshop on Expert Systems and their Applications, pp. 713–726 (1986)Google Scholar
  80. 80.
    Yang, X., Yang, J.: Incomplete Information System and Rough Set Theory: Model and Attribute Reduction. Springer, Heidelberg (2012) CrossRefGoogle Scholar
  81. 81.
    Yang, X., Zhang, M., Dou, H., Yang, J.: Neighborhood systems-based rough sets in incomplete information systems. Knowl. Based Syst. 24, 858–867 (2011)CrossRefGoogle Scholar
  82. 82.
    Yao, Y.Y.: Probabilistic rough set approximations. Int. J. Approx. Reason. 49, 255–271 (2008)zbMATHCrossRefGoogle Scholar
  83. 83.
    Yao, Y.Y., Wong, S.K.M.: A decision theoretic framework for approximate concepts. Int. J. Man Mach. Stud. 37, 793–809 (1992)CrossRefGoogle Scholar
  84. 84.
    Yao, Y.Y., Wong, S.K.M., Lingras, P.: A decision-theoretic rough set model. In: Ras, Z.W., Zemankova, M., Emrich, M.L. (eds.) Methodologies for Intelligent Systems, North-Holland, pp. 388–395 (1990)Google Scholar
  85. 85.
    Ziarko, W.: Variable precision rough set model. J. Comput. Syst. Sci. 46(1), 39–59 (1993)zbMATHMathSciNetCrossRefGoogle Scholar
  86. 86.
    Ziarko, W.: Probabilistic approach to rough sets. Int. J. Approx. Reason. 49, 272–284 (2008)zbMATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Department of Electrical Engineering and Computer ScienceUniversity of KansasLawrenceUSA
  2. 2.Department of Expert Systems and Artificial IntelligenceUniversity of Information Technology and ManagementRzeszowPoland

Personalised recommendations