Skip to main content
SpringerLink
Log in

M-estimators and trimmed means: from Hilbert-valued to fuzzy set-valued data

  • Regular Article
  • Published:
Advances in Data Analysis and Classification Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2

Similar content being viewed by others

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

    Article  MathSciNet  Google Scholar 

  • 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

    Article  MathSciNet  Google Scholar 

  • Bobylev VN (1985) Support function of a fuzzy set and its characteristic properties. Math Notes (USSR) 37(4):281–285

    Article  MathSciNet  Google Scholar 

  • Castaing C, Valadier M (1977) Convex analysis and measurable multifunctions, vol 580. Lecture notes in mathematics. Springer, Berlin

    Book  Google Scholar 

  • Celmiņš A (1987) Least squares model fitting to fuzzy vector data. Fuzzy Sets Syst 22:245–269

    Article  MathSciNet  Google Scholar 

  • Colubi A, González-Rodríguez G (2015) Fuzziness in data analysis: towards accuracy and robustness. Fuzzy Sets Syst 281:260–271

    Article  MathSciNet  Google Scholar 

  • 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

    Google Scholar 

  • Cuesta-Albertos JA, Fraiman R (2007) Impartial trimmed k-means for functional data. Comput Stat Data Anal 51(10):4864–4877

    Article  MathSciNet  Google Scholar 

  • Cuesta-Albertos JA, Gordaliza A, Matrán C (1997) Trimmed \(k\)-means: an attempt to robustify quantizers. Ann Stat 25(2):553–576

    Article  MathSciNet  Google Scholar 

  • 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

    Article  MathSciNet  Google Scholar 

  • 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

    Article  MathSciNet  Google Scholar 

  • 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

    Google Scholar 

  • Fréchet M (1948) Les éléments aléatoires de nature quelconque dans un espace distancié. Ann I H Poincaré 10:215–310

    MATH  Google Scholar 

  • García-Escudero LA, Gordaliza A, Mayo-Iscar A, Martín RS (2010) Robust clusterwise linear regression through trimming. Comput Stat Data Anal 54:3057–3069

    Article  MathSciNet  Google Scholar 

  • Gil MA, Colubi A, Terán P (2013) Random fuzzy sets: why, when, how. BEIO 30(1):5–29

    Google Scholar 

  • Hampel FR (1974) The influence curve and its role in robust estimation. J Am Stat Assoc 69:383–393

    Article  MathSciNet  Google Scholar 

  • 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

    Article  Google Scholar 

  • Huber PJ (1964) Robust estimation of a location parameter. Ann Math Stat 35:73–101

    Article  MathSciNet  Google Scholar 

  • Huber PJ (1981) Robust statistics. Wiley, Hoboken

    Book  Google Scholar 

  • Hubert M, Rousseeuw P, Segaert P (2017) Multivariate and functional classification using depth and distance. Adv Data Anal Classif 11:445–466

    Article  MathSciNet  Google Scholar 

  • Kim JS, Scott CD (2012) Robust kernel density estimation. J Mach Learn Res 13:2529–2565

    MathSciNet  MATH  Google Scholar 

  • 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

    MathSciNet  MATH  Google Scholar 

  • López-Pintado S, Romo J (2009) On the concept of depth for functional data. J Am Stat Assoc 104(486):718–734

    Article  MathSciNet  Google Scholar 

  • 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

    Article  MathSciNet  Google Scholar 

  • 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

    Article  MathSciNet  Google Scholar 

  • Minkowski H (1903) Volumen und oberfläche. Math Ann 57:447–495

    Article  MathSciNet  Google Scholar 

  • Puri ML, Ralescu DA (1985) The concept of normality for fuzzy random variables. Ann Probab 13:1373–1379

    Article  MathSciNet  Google Scholar 

  • Puri ML, Ralescu DA (1986) Fuzzy random variables. J Math Anal Appl 114:409–422

    Article  MathSciNet  Google Scholar 

  • 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

    Article  MathSciNet  Google Scholar 

  • Salski A (2007) Fuzzy clustering of fuzzy ecological data. Ecol Inform 2:262–269

    Article  Google Scholar 

  • 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

    Article  Google Scholar 

  • Sinova B, González-Rodríguez G, Van Aelst S (2018) M-estimators of location for functional data. Bernoulli 24(3):2328–2357

    Article  MathSciNet  Google Scholar 

  • Sugano N (2011) Fuzzy set theoretical approach to the tone triangular system. J Comput 6(11):2345–2356

    Article  Google Scholar 

  • 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

    Article  MathSciNet  Google Scholar 

  • Valencia D, Lillo RE, Romo J (2019) A Kendall correlation coefficient between functional data. Adv Data Anal Classif 13:1083–1103

    Article  MathSciNet  Google Scholar 

  • Zadeh LA (1965) Fuzzy sets. Inf Control 8(3):338–353

    Article  Google Scholar 

  • Zadeh LA (1975) The concept of a linguistic variable and its application to approximate reasoning—I. Inf Sci 8(3):199–249

    Article  MathSciNet  Google Scholar 

  • Zadeh LA (2008) Is there a need for fuzzy logic? Inf Sci 178:2751–2779

    Article  MathSciNet  Google Scholar 

Download references

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.

Author information

Authors and Affiliations

  1. Department of Statistics and O.R. and D.M., University of Oviedo, 33007, Oviedo, Spain

    Beatriz Sinova & Pedro Terán

  2. Department of Mathematics, KU Leuven, 3001, Leuven, Belgium

    Stefan Van Aelst

Authors
  1. Beatriz Sinova
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Stefan Van Aelst
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Pedro Terán
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Beatriz Sinova.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Electronic supplementary material

Below is the link to the electronic supplementary material.

Supplementary material 1 (pdf 122 KB)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

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

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11634-020-00402-x

Keywords

Mathematics Subject Classification

Search

Navigation