Statistical Interpretations of Three-Way Decisions

  • Yiyu Yao
  • Cong GaoEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9436)


In an evaluation based model of three-way decisions, one constructs three regions, namely, the left, middle, and right regions based on an evaluation function and a pair of thresholds. This paper examines statistical interpretations for the construction of three regions. Such interpretations rely on an understanding that the middle region consists of normal or typical instances in a population, while two side regions consist of, abnormal or untypical instances. By using statistical information such as median, mean, percentile, and standard deviation, two interpretations are discussed. One is based on non-numeric values and the other is based on numeric values. For non-numeric values, median and percentile are used to construct three pair-wise disjoint regions. For numeric values, mean and standard deviation are used. The interpretations provide a solid statistical basis of three-way decisions for applications.


Statistical interpretations Three-way decisions 



This work is partially supported by a Discovery Grant from NSERC, Canada and Sampson J. Goodfellow Scholarship.


  1. 1.
    Azam, N., Yao, J.T.: Analyzing uncertainties of probabilistic rough set regions with game-theoretic rough sets. Int. J. Approximate Reasoning 55, 142–155 (2014)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Baram, Y.: Partial classification: the benefit of deferred decision. IEEE Trans. Pattern Anal. Mach. Intell. 20, 769–776 (1998)CrossRefGoogle Scholar
  3. 3.
    Czitrom, V., Spagon, P.D.: Statistical Case Studies for Industrial Process Improvement. SIAM, Philadelphia (1997)CrossRefGoogle Scholar
  4. 4.
    Deng, X.F.: Three-Way Classification Models. Ph.D. Dissertation, Department of Computer Science, University of Regina (2015)Google Scholar
  5. 5.
    Deng, X.F., Yao, Y.Y.: A multifaceted analysis of probabilistic three-way decisions. Fundam. Informaticae 132, 291–313 (2014)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Deng, X.F., Yao, Y.Y.: Decision-theoretic three-way approximations of fuzzy sets. Inf. Sci. 279, 702–715 (2014)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Goudey, R.: Do statistical inferences allowing three alternative decision give better feedback for environmentally precautionary decision-making? J. Environ. Manage. 85, 338–344 (2007)CrossRefGoogle Scholar
  8. 8.
    Grzymala-Busse, J.W., Clarka, P.G., Kuehnhausena, M.: Generalized probabilistic approximations of incomplete data. Int. J. Approximate Reasoning 50, 180–196 (2014)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Hu, B.Q.: Three-way decisions space and three-way decisions. Inf. Sci. 281, 21–52 (2014)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Iserson, K.V., Moskop, J.C.: Triage in medicine, part I: concept, history, and types. Ann. Emerg. Med. 49, 275–281 (2007)CrossRefGoogle Scholar
  11. 11.
    Jia, X.Y., Shang, L., Zhou, X.Z., Liang, J.Y., Miao, D.Q., Wang, G.Y., Li, T.R., Zhang, Y.P. (eds.): Theory of Three-way Decisions and Applications (in Chinese). Nanjing University Press, Nanjing (2012)Google Scholar
  12. 12.
    Jia, X.Y., Tang, Z.M., Liao, W.H., Shang, L.: On an optimization representation of decision-theoretic rough set model. Int. J. Approximate Reasoning 55, 156–166 (2014)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Li, H.X., Zhang, L.B., Huang, B., Zhou, X.Z.: Sequential three-way decision and granulation for cost-sensitive face recognition, Knowledge-Based Systems (2015).
  14. 14.
    Li, H.X., Zhou, X.Z., Huang, B., Liu, D.: Cost-sensitive three-way decision: a sequential strategy. In: Lingras, P., Wolski, M., Cornelis, C., Mitra, S., Wasilewski, P. (eds.) RSKT 2013. LNCS, vol. 8171, pp. 325–337. Springer, Heidelberg (2013) CrossRefGoogle Scholar
  15. 15.
    Liang, D.C., Liu, D.: Deriving three-way decisions from intuitionistic fuzzy decision-theoretic rough sets. Inf. Sci. 300, 28–48 (2015)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Liang, D.C., Pedrycz, W., Liu, D., Hu, P.: Three-way decisions based on decision-theoretic rough sets under linguistic assessment with the aid of group decision making. Appl. Soft Comput. 29, 256–269 (2015)CrossRefGoogle Scholar
  17. 17.
    Liu, D., Li, T.R., Liang, D.C.: Incorporating logistic regression to decision-theoretic rough sets for classifications. Int. J. Approximate Reasoning 55, 197–210 (2014)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Liu, D., Li, T.R., Liang, D.C.: Three-way government decision analysis with decision-theoretic rough sets. Int. J. Uncertainty Fuzziness Knowl.-Based Syst. 20, 119–132 (2012)CrossRefGoogle Scholar
  19. 19.
    Liu, D., Li, T.R., Miao, D.Q., Wang, G.Y., Liang, J.Y. (eds.): Three-way Decisions and Granular Computing (in Chinese). Science Press, Beijing (2013) Google Scholar
  20. 20.
    Liu, D., Liang, D.C., Wang, C.C.: A novel three-way decision model based on incomplete information system. Knowledge-Based Systems (2015).
  21. 21.
    Pater, C.: The blood pressure “uncertainty range” - a pragmatic approach to overcome current diagnostic uncertainties (II). Curr. Controlled Trials Cardiovasc. Med. 6, 5 (2005)CrossRefGoogle Scholar
  22. 22.
    Pawlak, Z.: Rough sets. Int. J. Comput. Inf. Sci. 11, 341–356 (1982)CrossRefGoogle Scholar
  23. 23.
    Pedrycz, W., Skowron, A., Kreinovich, V.: Handbook of Granular Computing. Wiley, Chichester (2008) CrossRefGoogle Scholar
  24. 24.
    Peters, J.F., Ramanna, S.: Proximal three-way decisions: theory and applications in social networks. Knowledge-Based Systems (2015).
  25. 25.
    Rousseeuw, P.J., Ruts, I., Tukey, J.W.: The bagplot: a bivariate boxplot. Am. Stat. 53, 382–387 (1999)Google Scholar
  26. 26.
    Sanders, D.H., Smidt, R.K., Adatia, A., Larson, G.A.: Statistics: A First Course. McGraw-Hill Ryerson, Toronto (2001) Google Scholar
  27. 27.
    Sang, Y.L., Liang, J.Y., Qian, Y.H.: Decision-theoretic rough sets under dynamic granulation. Knowledge-Based Systems (2015).
  28. 28.
    Sattler, J.M.: Assessment of Children’s Intelligence. W.B. Saunders Company, Philadelphia (1975) Google Scholar
  29. 29.
    Schechter, C.B.: Sequential analysis in a Bayesian model of diastolic blood pressure measurement. Med. Decis. Making 8, 191–196 (1988)CrossRefGoogle Scholar
  30. 30.
    Schofield, H.: Assess. Test. Introduction. Allen & Unwin, London (1972) Google Scholar
  31. 31.
    Shakiba, A., Hooshmandasl, M.R.: S-approximation spaces: a three-way decision approach. Fundam. Informaticae 39, 307–328 (2015)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Wald, A.: Sequential Anal. Wiley, New York (1947) zbMATHGoogle Scholar
  33. 33.
    Yao, J.T., Azam, N.: Web-based medical decision support systems for three-way medical decision making with game-theoretic rough sets. IEEE Trans. Fuzzy Syst. 23, 3–15 (2014)CrossRefGoogle Scholar
  34. 34.
    Yao, J.T., Herbert, J.P.: A game-theoretic perspective on rough set analysis. J. Chongqing Univ. Posts Telecommun. 20, 291–298 (2008)Google Scholar
  35. 35.
    Yao, Y.Y.: An outline of a theory of three-way decisions. In: Yao, J.T., Yang, Y., Słowiński, R., Greco, S., Li, H., Mitra, S., Polkowski, L. (eds.) RSCTC 2012. LNCS, vol. 7413, pp. 1–17. Springer, Heidelberg (2012) CrossRefGoogle Scholar
  36. 36.
    Yao, Y.Y.: Granular computing and sequential three-way decisions. In: Lingras, P., Wolski, M., Cornelis, C., Mitra, S., Wasilewski, P. (eds.) RSKT 2013. LNCS, vol. 8171, pp. 16–27. Springer, Heidelberg (2013) CrossRefGoogle Scholar
  37. 37.
    Yao, Y.Y.: Interval-set algebra for qualitative knowledge representation. In: Proceedings of the 5th International Conference on Computing and Information (ICCI), pp. 370–374 (1993)Google Scholar
  38. 38.
    Yao, Y.Y.: Perspectives of granular computing. In: Proceedings of 2005 IEEE International Conference on Granular Computing, vol. 1, pp. 85–90 (2005)Google Scholar
  39. 39.
    Yao, Y.Y.: Rough sets and three-way decisions. In: Ciucci, D., Wang, G.Y., Mitra, S., Wu, W.Z. (eds.) RSKT 2015. LNCS (LNAI), vol. 9436, pp. 62–73. Springer International Publishing, Switzerland (2015)CrossRefGoogle Scholar
  40. 40.
    Yao, Y.Y., Wong, S.K.M., Lingras, P.: A decision-theoretic rough set model. In: Proceedings of the 5th International Symposium on Methodologies for Intelligent Systems, pp. 17–25 (1990)Google Scholar
  41. 41.
    Yao, Y.Y., Yu, H.: An introduction of three-way decisions. In: Yu, H., Wang, G.Y., Li, T.R., Liang, J.Y., Miao, D.Q., Yao, Y.Y. (eds.) Three-Way Decisions: Methods and Practices for Complex Problem Solving, pp. 1–19. Science Press, Beijing (2015) (in Chinese)Google Scholar
  42. 42.
    Yu, H., Liu, Z.G., Wang, G.Y.: An automatic method to determine the number of clusters using decision-theoretic rough set. Int. J. Approximate Reasoning 55, 101–115 (2014)MathSciNetCrossRefGoogle Scholar
  43. 43.
    Yu, H., Su, T., Zeng, X.H.: A three-way decisions clustering algorithm for incomplete data. In: Miao, D., Pedrycz, W., Slezak, D., Peters, G., Hu, Q., Wang, R. (eds.) RSKT 2014. LNCS, vol. 8818, pp. 765–776. Springer, Heidelberg (2014) Google Scholar
  44. 44.
    Yu, H., Wang, G.Y., Li, T.R., Liang, J.Y., Miao, D.Q., Yao, Y.Y. (eds.): Three-Way Decisions: Methods and Practices for Complex Problem Solving. Science Press, Beijing (2015) (in Chinese)Google Scholar
  45. 45.
    Yu, H., Zhang, C., Wang, G.Y.: A tree-based incremental overlapping clustering method using the three-way decision theory. Knowledge-Based Systems (2015).
  46. 46.
    Zhang, H.R., Min, F.: Three-way recommender systems based on random forests. Knowledge-Based Systems (2015).
  47. 47.
    Zhang, H.Y., Yang, S.Y., Ma, J.M.: Ranking interval sets based on inclusion measures and applications to three-way decisions. Knowledge-Based Systems (2015).
  48. 48.
    Zhang, Y.: Optimizing Gini coefficient of probabilistic rough set regions using game-theoretic rough sets. In: 26th Canadian Conference of Electrical And Computer Engineering (CCECE), pp. 1–4 (2013)Google Scholar
  49. 49.
    Zhou, B.: Multi-class decision-theoretic rough sets. Int. J. Approximate Reasoning 55, 211–224 (2014)MathSciNetCrossRefGoogle Scholar
  50. 50.
    Zhou, B., Yao, Y.Y., Luo, J.G.: Cost-sensitive three-way email spam filtering. J. Intell. Inf. Syst. 42, 19–45 (2014)CrossRefGoogle Scholar

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Authors and Affiliations

  1. 1.Department of Computer ScienceUniversity of ReginaReginaCanada

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