Abstract
Different approaches to robustly measure the location of data associated with a random experiment have been proposed in the literature, with the aim of avoiding the high sensitivity to outliers or data changes typical for the mean. In particular, M-estimators and trimmed means have been studied in general spaces, and can be used to handle Hilbert-valued data. Both alternatives are of interest due to their success in the classical framework. Since fuzzy set-valued data can be identified with a convex cone of a separable Hilbert space, the previous concepts have been recently applied to the one-dimensional fuzzy case. The aim of this paper is to extend M-estimators and trimmed means to p-dimensional fuzzy set-valued data, and to theoretically prove that they inherit robustness from the real settings. Some of such theoretical results are more general and directly apply to Hilbert-valued estimators and, in consequence, to functional data. A real-life example will also be included to illustrate the computation and behaviour of these estimators under contamination.
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References
Alfons A, Croux C, Gelper S (2013) Sparse least trimmed squares regression for analyzing high-dimensional large data sets. Ann Appl Stat 7(1):226–248
Aneiros G, Cao R, Fraiman R, Genest C, Vieu P (2019) Recent advances in functional data analysis and high-dimensional statistics. J Multivar Anal 170:3–9
Bobylev VN (1985) Support function of a fuzzy set and its characteristic properties. Math Notes (USSR) 37(4):281–285
Castaing C, Valadier M (1977) Convex analysis and measurable multifunctions, vol 580. Lecture notes in mathematics. Springer, Berlin
Celmiņš A (1987) Least squares model fitting to fuzzy vector data. Fuzzy Sets Syst 22:245–269
Colubi A, González-Rodríguez G (2015) Fuzziness in data analysis: towards accuracy and robustness. Fuzzy Sets Syst 281:260–271
Cuesta-Albertos JA, Fraiman R (2006) Impartial trimmed means for functional data. In: Liu RY, Serfling R, Souvaine DL (eds) Data depth: robust multivariate statistical analysis, computational geometry and applications, vol 72. DIMACS Series. American Mathematical Society, Providence, pp 121–145
Cuesta-Albertos JA, Fraiman R (2007) Impartial trimmed k-means for functional data. Comput Stat Data Anal 51(10):4864–4877
Cuesta-Albertos JA, Gordaliza A, Matrán C (1997) Trimmed \(k\)-means: an attempt to robustify quantizers. Ann Stat 25(2):553–576
Cuevas A, Febrero M, Fraiman R (2007) Robust estimation and classification for functional data via projection-based depth notions. Comput Stat 22(3):481–496
de la Rosa de Sáa S, Lubiano MA, Sinova B, Filzmoser P (2017) Robust scale estimators for fuzzy data. Adv Data Anal Classif 11(4):731–758
Donoho DL, Huber PJ (1983) The notion of breakdown point. In: Bickel PJ, Doksum K Jr, Hodges JL (eds) A Festschrift for Eric L. Wadsworth, Lehmann, pp 157–184
Fréchet M (1948) Les éléments aléatoires de nature quelconque dans un espace distancié. Ann I H Poincaré 10:215–310
García-Escudero LA, Gordaliza A, Mayo-Iscar A, Martín RS (2010) Robust clusterwise linear regression through trimming. Comput Stat Data Anal 54:3057–3069
Gil MA, Colubi A, Terán P (2013) Random fuzzy sets: why, when, how. BEIO 30(1):5–29
Hampel FR (1974) The influence curve and its role in robust estimation. J Am Stat Assoc 69:383–393
Hesketh T, Pryor R, Hesketh B (1988) An application of a computerized fuzzy graphic rating scale to the psychological measurement of individual differences. Int J Man Mach Stud 29:21–35
Huber PJ (1964) Robust estimation of a location parameter. Ann Math Stat 35:73–101
Huber PJ (1981) Robust statistics. Wiley, Hoboken
Hubert M, Rousseeuw P, Segaert P (2017) Multivariate and functional classification using depth and distance. Adv Data Anal Classif 11:445–466
Kim JS, Scott CD (2012) Robust kernel density estimation. J Mach Learn Res 13:2529–2565
Klement EP, Puri ML, Ralescu DA (1986) Limit theorems for fuzzy random variables. Proc R Soc Lond Ser A Math Phys Eng Sci 407:171–182
López-Pintado S, Romo J (2009) On the concept of depth for functional data. J Am Stat Assoc 104(486):718–734
Lubiano MA, Montenegro M, Sinova B, de la Rosa de Sáa S, Gil MA (2016) Hypothesis testing for means in connection with fuzzy rating scale-based data: algorithms and applications. Eur J Oper Res 251:918–929
Lubiano MA, Salas A, Gil MA (2017) A hypothesis testing-based discussion on the sensitivity of means of fuzzy data with respect to data shape. Fuzzy Sets Syst 328:54–69
Minkowski H (1903) Volumen und oberfläche. Math Ann 57:447–495
Puri ML, Ralescu DA (1985) The concept of normality for fuzzy random variables. Ann Probab 13:1373–1379
Puri ML, Ralescu DA (1986) Fuzzy random variables. J Math Anal Appl 114:409–422
Rivera-García D, García-Escudero LA, Mayo-Iscar A, Ortega J (2019) Robust clustering for functional data based on trimming and constraints. Adv Data Anal Classif 13:201–225
Salski A (2007) Fuzzy clustering of fuzzy ecological data. Ecol Inform 2:262–269
Sinova B, Gil MA, Van Aelst S (2016) M-estimates of location for the robust central tendency of fuzzy data. IEEE Trans Fuzzy Syst 24(4):945–956
Sinova B, González-Rodríguez G, Van Aelst S (2018) M-estimators of location for functional data. Bernoulli 24(3):2328–2357
Sugano N (2011) Fuzzy set theoretical approach to the tone triangular system. J Comput 6(11):2345–2356
Trutschnig W, González-Rodríguez G, Colubi A, Gil MA (2009) A new family of metrics for compact, convex (fuzzy) sets based on a generalized concept of mid and spread. Inf Sci 179(23):3964–3972
Valencia D, Lillo RE, Romo J (2019) A Kendall correlation coefficient between functional data. Adv Data Anal Classif 13:1083–1103
Zadeh LA (1965) Fuzzy sets. Inf Control 8(3):338–353
Zadeh LA (1975) The concept of a linguistic variable and its application to approximate reasoning—I. Inf Sci 8(3):199–249
Zadeh LA (2008) Is there a need for fuzzy logic? Inf Sci 178:2751–2779
Acknowledgements
The authors are grateful to the Editor and reviewers, as well as to their colleagues Prof. M. A. Gil and Prof. G. González-Rodríguez, for their insightful comments and suggestions. The research of Beatriz Sinova and Pedro Terán was partially supported by the Spanish Ministry of Economy and Competitiveness under Grant MTM2015-63971-P; and the Principality of Asturias/FEDER Funds under Grants GRUPIN14-101 and GRUPIN-IDI2018-000132. The research of Stefan Van Aelst was supported by Internal Funds KU Leuven (Belgium) under Grant C16/15/068. Their support is gratefully acknowledged.
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Sinova, B., Van Aelst, S. & Terán, P. M-estimators and trimmed means: from Hilbert-valued to fuzzy set-valued data. Adv Data Anal Classif 15, 267–288 (2021). https://doi.org/10.1007/s11634-020-00402-x
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DOI: https://doi.org/10.1007/s11634-020-00402-x