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An Analysis of Winsorized Weighted Means

  • Bonifacio LlamazaresEmail author
Article
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Abstract

The Winsorized mean is a well-known robust estimator of the population mean. It can also be seen as a symmetric aggregation function (in fact, it is an ordered weighted averaging operator), which means that the information sources (for instance, criteria or experts’ opinions) have the same importance. However, in many practical applications (for instance, in many multiattribute decision making problems) it is necessary to consider that the information sources have different importance. For this reason, in this paper we propose a natural generalization of the Winsorized means so that the sources of information can be weighted differently. The new functions, which we will call Winsorized weighted means, are a specific case of the Choquet integral and they are analyzed through several indices for which we give closed-form expressions: the orness degree, k-conjunctiveness and k-disjunctiveness indices, veto and favor indices, Shapley values and interaction indices. We also provide a closed-form expression for the Möbius transform and we show how we can aggregate data so that each information source has the desired weighting and outliers have no influence in the aggregated value.

Keywords

Winsorized weighted means Winsorized means Choquet integral Shapley values SUOWA operators 

Notes

Acknowledgements

The author is grateful to two anonymous referees for valuable suggestions and comments. This work is partially supported by the Spanish Ministry of Economy and Competitiveness (Project ECO2016-77900-P) and ERDF.

References

  1. Aggarwal CC (2017) Outlier analysis, 2nd edn. Springer, ChamGoogle Scholar
  2. Bai C, Zhang R, Song C, Wu Y (2017) A new ordered weighted averaging operator to obtain the associated weights based on the principle of least mean square errors. Int J Intell Syst 32(3):213–226Google Scholar
  3. Barnett V, Lewis T (1994) Outliers in statistical data, 3rd edn. Wiley, ChichesterGoogle Scholar
  4. Beliakov G (2018) Comparing apples and oranges: the weighted OWA function. Int J Intell Syst 33(5):1089–1108Google Scholar
  5. Beliakov G, Dujmović J (2016) Extension of bivariate means to weighted means of several arguments by using binary trees. Inf Sci 331:137–147Google Scholar
  6. Choquet G (1953) Theory of capacities. Ann Inst Fourier 5:131–295Google Scholar
  7. Denneberg D (1994) Non-additive measures and integral. Kluwer Academic Publisher, DordrechtGoogle Scholar
  8. Dixon WJ (1960) Simplified estimation from censored normal samples. Ann Math Stat 31(2):385–391Google Scholar
  9. Dubois D, Koning JL (1991) Social choice axioms for fuzzy set aggregation. Fuzzy Sets Syst 43(3):257–274Google Scholar
  10. Grabisch M (1995) Fuzzy integral in multicriteria decision making. Fuzzy Sets Syst 69(3):279–298Google Scholar
  11. Grabisch M (1997) \(k\)-order additive discrete fuzzy measures and their representation. Fuzzy Sets Syst 92(2):167–189Google Scholar
  12. Grabisch M (2016) Set functions, games and capacities in decision making, theory and decision library, series C, vol 46. Springer, BerlinGoogle Scholar
  13. Grabisch M, Labreuche C (2010) A decade of application of the Choquet and Sugeno integrals in multi-criteria decision aid. Ann Oper Res 175(1):247–286Google Scholar
  14. Grabisch M, Labreuche C (2016) Fuzzy measures and integrals in MCDA. In: Greco S, Ehrgott M, Figueira RJ (eds) Multiple criteria decision analysis: state of the art surveys, international series in operations research and management science, vol 233, 2nd edn. Springer, New York, pp 553–603Google Scholar
  15. Grabisch M, Marichal J, Mesiar R, Pap E (2009) Aggregation functions. Cambridge University Press, CambridgeGoogle Scholar
  16. Harsanyi JC (1959) A bargaining model for cooperative \(n\)-person games. In: Tucker AW, Luce RD (eds) Contributions to the theory of games. Vol. 4, Annals of Mathematics Studies, vol 40. Princeton University Press, Princeton, pp 325–355Google Scholar
  17. Heilpern S (2002) Using Choquet integral in economics. Stat Pap 43(1):53–73Google Scholar
  18. Hoitash U, Hoitash R (2009) Conflicting objectives within the board: evidence from overlapping audit and compensation committee members. Group Decis Negot 18(1):57–73Google Scholar
  19. Huber PJ, Ronchetti EM (2009) Robust statistics, 2nd edn. Wiley, HobokenGoogle Scholar
  20. Iglewicz B, Hoaglin D (1993) How to detect and handle outliers. ASQC Quality Press, MilwaukeeGoogle Scholar
  21. Komorníková M, Mesiar R (2011) Aggregation functions on bounded partially ordered sets and their classification. Fuzzy Sets Syst 175(1):48–56Google Scholar
  22. Lenormand M (2018) Generating OWA weights using truncated distributions. Int J Intell Syst 33(4):791–801Google Scholar
  23. Leys C, Ley C, Klein O, Bernard P, Licata L (2013) Detecting outliers: do not use standard deviation around the mean, use absolute deviation around the median. J Exp Soc Psychol 49(4):764–766Google Scholar
  24. Liu X (2011) A review of the OWA determination methods: classification and some extensions. In: Yager RR, Kacprzyk J, Beliakov G (eds) Recent developments in the ordered weighted averaging operators: theory and practice. Springer, Berlin, pp 49–90Google Scholar
  25. Llamazares B (2007) Choosing OWA operator weights in the field of social choice. Inf Sci 177(21):4745–4756Google Scholar
  26. Llamazares B (2013) An analysis of some functions that generalizes weighted means and OWA operators. Int J Intell Syst 28(4):380–393Google Scholar
  27. Llamazares B (2015a) Constructing Choquet integral-based operators that generalize weighted means and OWA operators. Inf Fus 23:131–138Google Scholar
  28. Llamazares B (2015b) A study of SUOWA operators in two dimensions. Math Probl Eng 2015: Article ID 271,491Google Scholar
  29. Llamazares B (2016a) A behavioral analysis of WOWA and SUOWA operators. Int J Intell Syst 31(8):827–851Google Scholar
  30. Llamazares B (2016b) SUOWA operators: constructing semi-uninorms and analyzing specific cases. Fuzzy Sets Syst 287:119–136Google Scholar
  31. Llamazares B (2018a) Closed-form expressions for some indices of SUOWA operators. Inf Fus 41:80–90Google Scholar
  32. Llamazares B (2018b) Construction of Choquet integrals through unimodal weighting vectors. Int J Intell Syst 33(4):771–790Google Scholar
  33. Llamazares B (2019a) SUOWA operators: a review of the state of the art. Int J Intell Syst 34(5):790–818Google Scholar
  34. Llamazares B (2019b) SUOWA operators: an analysis of their conjunctive/disjunctive character. Fuzzy Sets Syst 357:117–134Google Scholar
  35. Marichal JL (1998) Aggregation operators for multicriteria decision aid. Ph.D. thesis, University of LiègeGoogle Scholar
  36. Marichal JL (2004) Tolerant or intolerant character of interacting criteria in aggregation by the Choquet integral. Eur J Oper Res 155(3):771–791Google Scholar
  37. Marichal JL (2007) \(k\)-intolerant capacities and Choquet integrals. Eur J Oper Res 177(3):1453–1468Google Scholar
  38. Murofushi T, Soneda S (1993) Techniques for reading fuzzy measures (iii): interaction index. In: Proceedings of the 9th fuzzy systems symposium, Sapporo (Japan), pp 693–696Google Scholar
  39. Murofushi T, Sugeno M (1991) A theory of fuzzy measures. Representation, the Choquet integral and null sets. J Math Anal Appl 159(2):532–549Google Scholar
  40. Murofushi T, Sugeno M (1993) Some quantities represented by the Choquet integral. Fuzzy Sets Syst 56(2):229–235Google Scholar
  41. Owen G (1972) Multilinear extensions of games. Manag Sci 18(5–part–2):64–79Google Scholar
  42. Riordan J (1968) Combinatorial identities. Wiley, New YorkGoogle Scholar
  43. Rota GC (1964) On the foundations of combinatorial theory I. Theory of Möbius functions. Z Wahrscheinlichkeitstheorie Verwandte Geb 2(4):340–368Google Scholar
  44. Seo S (2006) A review and comparison of methods for detecting outliers in univariate data sets. Master’s thesis, University of PittsburghGoogle Scholar
  45. Shapley LS (1953) A value for \(n\)-person games. In: Kuhn H, Tucker AW (eds) Contributions to the theory of games, vol 2. Princeton University Press, Princeton, pp 307–317Google Scholar
  46. Sugeno M (1974) Theory of fuzzy integrals and its applications. Ph.D. thesis, Tokyo Institute of TechnologyGoogle Scholar
  47. Torra V (1997) The weighted OWA operator. Int J Intell Syst 12(2):153–166Google Scholar
  48. Tukey JW (1977) Exploratory data analysis. Addison-Wesley, ReadingGoogle Scholar
  49. Wainer H (1976) Robust statistics: a survey and some prescriptions. J Educ Stat 1(4):285–312Google Scholar
  50. Wilcox RR (2012) Modern statistics for the social and behavioral sciences: a practical introduction. CRC Press, Boca RatonGoogle Scholar
  51. Wilcox RR, Keselman HJ (2003) Modern robust data analysis methods: measures of central tendency. Psychol Methods 8(3):254–274Google Scholar
  52. Yager RR (1988) On ordered weighted averaging operators in multicriteria decision making. IEEE Trans Syst Man Cybern 18(1):183–190Google Scholar
  53. Zhang Z, Xu Z (2014) Analysis on aggregation function spaces. Int J Uncertain Fuzziness Knowl Based Syst 22(05):737–747Google Scholar

Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Departamento de Economía Aplicada, Instituto de Matemáticas (IMUVA)Universidad de ValladolidValladolidSpain

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