Skip to main content
SpringerLink
Log in

Estimators for the common principal components model based on reweighting: influence functions and Monte Carlo study

  • Published:
Metrika Aims and scope Submit manuscript

Abstract

The common principal components model for several groups of multivariate observations is a useful parsimonious model for the scatter structure which assumes equal principal axes but different variances along those axes for each group. Due to the lack of resistance of the classical maximum likelihood estimators for the parameters of this model, several robust estimators have been proposed in the literature: plug-in estimators and projection-pursuit (PP) type estimators. In this paper, we show that it is possible to improve the low efficiency of the projection-pursuit estimators by applying a reweighting step. More precisely, we consider plug-in estimators obtained by plugging a reweighted estimator of the scatter matrices into the maximum likelihood equations defining the principal axes. The weights considered penalize observations with large values of the influence measures defined by Boente et al. (2002). The new estimators are studied in terms of theoretical properties (influence functions and asymptotic variances) and are compared with other existing estimators in a simulation study.

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

Similar content being viewed by others

References

  • Adrover J (1998) Minimax bias-robust estimation of the dispersion matrix of a multivariate distribution. Ann Stat 26:2301–2320

    Article  MATH  MathSciNet  Google Scholar 

  • Boente G, Orellana L (2001) A robust approach to common principal components. In: Fernholz L, Morgenthaler S, Stahel W (eds) Statistics in genetics and in the environmental sciences. Birkhauser, Basel, pp 117–147

    Google Scholar 

  • Boente G, Pires AM, Rodrigues IM (2002) Influence functions and outlier detection under the common principal components model: a robust approach. Biometrika 89:861–875

    Article  MATH  MathSciNet  Google Scholar 

  • Boente G, Critchley F, Orellana L (2004) Influence functions for robust estimators under proportional scatter matrices. Available on http://www.ic.fcen.uba.ar/preprints/boecriore.pdf

  • Boente G, Pires AM, Rodrigues IM (2005) Reweighted estimators for the common principal components model: influence functions and Monte Carlo study. Publicaciones Previas del Instituto de Cálculo. Available on http://www.ic.fcen.uba.ar/preprints/boentepiresrodrigues.pdf

  • Boente G, Pires AM, Rodrigues IM (2006) General projection-pursuit estimators for the common principal components model: influence functions and Monte Carlo study. J Multivariate Anal 97:124–147

    Article  MATH  MathSciNet  Google Scholar 

  • Critchley F (1985) Influence in principal components analysis. Biometrika 72:627–636

    Article  MATH  MathSciNet  Google Scholar 

  • Croux C, Haesbroeck G (1999) Influence function and efficiency of the minimum covariance determinant scatter matrix estimator. J Multivariate Anal 71:161–190

    Article  MATH  MathSciNet  Google Scholar 

  • Croux C, Haesbroeck G (2000) Principal component analysis based on robust estimators of the covariance or correlation matrix: influence functions and efficiencies. Biometrika 87:603–618

    Article  MATH  MathSciNet  Google Scholar 

  • Croux C, Ruiz-Gazen A (2005) High breakdown estimators for principal components: the projection-pursuit approach revisited. J Multivariate Anal 95:206–226

    Article  MATH  MathSciNet  Google Scholar 

  • Davies PL (1987) Asymptotic behavior of S−estimates of multivariate location parameters and dispersion matrices. Ann Stat 15:1269–1292

    Article  MATH  Google Scholar 

  • Donoho DL (1982) Breakdown properties of multivariate location estimators, Qualifying paper. Harvard University, Boston

    Google Scholar 

  • Filzmoser P (2004) A multivariate outlier detection method. In: Aivazian S, Filzmoser P, Kharin Y (eds) Proceedings of the Seventh international conference on computer data analysis and modelling, vol 1. Belarusian State University, Minsk, pp. 18–22

    Google Scholar 

  • Filzmoser P, Reimann C, Garrett RG (2005) Multivariate outlier detection in exploration geochemistry. Comput Geosci 31:579–587

    Article  Google Scholar 

  • Flury BK (1984) Common principal components in k groups. J Am Stat Assoc 79:892–898

    Article  MathSciNet  Google Scholar 

  • Flury BK (1988) Common principal components and related multivariate models. Wiley, New York

    MATH  Google Scholar 

  • Huber P (1977) Robust covariances. In: Guta SS, Moore DS (eds) Statistical decision theory and related topics. Academic, New York, pp 165–191

    Google Scholar 

  • Jaupi L, Saporta G (1993) Using the influence function in robust principal components analysis. In: Morgenthaler S, Ronchetti E, Stahel W (eds) New directions in statistical data analysis and robustness. Birkhauser, Basel, pp 147–156

    Google Scholar 

  • Kent J, Tyler D (1996) Constrained M-estimation for multivariate location and scatter. Ann Stat 24:1346–1370

    Article  MATH  MathSciNet  Google Scholar 

  • Li G, Chen Z (1985) Projection-pursuit approach to robust dispersion matrices and principal components: primary theory and Monte Carlo. J Am Stat Assoc 80:759–766

    Article  MATH  Google Scholar 

  • Lopuhaä HP (1999) Asymptotics of reweighted estimators of multivariate location and scatter. Ann Stat 27:1–28

    Article  Google Scholar 

  • Maronna R (1976) Robust M-estimates of multivariate location and scatter. Ann Stat 4:51–67

    Article  MATH  MathSciNet  Google Scholar 

  • Maronna R, Yohai V (1998) Robust estimation of multivariate locaton and scatter. In: Kotz S, Johnson Nl (eds) Encyclopedia of statistical sciences, update vol 2. Wiley, New York, pp. 589–596

    Google Scholar 

  • Maronna R, Martin D, Yohai V (2006) Robust statistics: theory and methods. Wiley, New York

    Book  MATH  Google Scholar 

  • Pires AM, Branco J (2002) Partial influence functions. J Multivariate Anal 83:451–468

    Article  MATH  MathSciNet  Google Scholar 

  • Pison G, Rousseeuw PJ, Filzmoser P, Croux C (2000) A robust version of principal factor analysis. In: Bethlehem J, van der Heijden P (eds) Compstat: proceedings in computational statistics. Physica-Verlag, Heidelberg, pp 385–390

    Google Scholar 

  • Pison G, Rousseeuw PJ, Filzmoser P, Croux C (2003) Robust factor analysis. J Multivariate Anal 84:145–172

    Article  MATH  MathSciNet  Google Scholar 

  • Rodrigues IM (2003) Métodos Robustos em Análise de Componentes Principais Comuns. Unpublished PhD Thesis (in portuguese), Universidade Técnica de Lisboa. Available on http://www.math.ist.utl.pt/∼apires/phd.html

  • Rousseeuw PJ (1985) Multivariate estimation with high breakdown point. In: Grossmann W, Pflug G, Vincze I, Wertz W (eds) Mathematical statistics and applications. Reidel, Dordrecht, pp 283–297

    Google Scholar 

  • Rousseeuw PJ, van Zomeren BC (1990) Unmasking multivariate outliers and leverage points. J Am Stat Assoc 85:633–639

    Article  Google Scholar 

  • Shi L (1997) Local influence in principal components analysis. Biometrika 84:175–186

    Article  MATH  MathSciNet  Google Scholar 

  • Stahel WA (1981) Robuste schätzungen: infinitesimale optimalität und schätzungen von Kovarianzmatrizen, PhD thesis, ETH Zürich

  • Tyler D (1991) Some issues in the robust estimation of multivariate location and scatter. In: Stahel W, Weisberg S (eds) Directions in Robust statistics and diagnostics Part 2. Springer, Heidelberg, pp. 327–336

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. CONICET and Departamento de Matemática e Instituto de Cálculo, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, Ciudad Universitaria, Pabellón 2, Buenos Aires, C1428EHA, Argentina

    Graciela Boente

  2. Departamento de Matemática and CEMAT, Instituto Superior Técnico, Technical University of Lisbon (TULisbon), Avda. Rovisco Pais, 1, 1049–001, Lisboa, Portugal

    Ana M. Pires & Isabel M. Rodrigues

Authors
  1. Graciela Boente
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ana M. Pires
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Isabel M. Rodrigues
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Graciela Boente.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Boente, G., Pires, A.M. & Rodrigues, I.M. Estimators for the common principal components model based on reweighting: influence functions and Monte Carlo study. Metrika 67, 189–218 (2008). https://doi.org/10.1007/s00184-007-0129-4

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00184-007-0129-4

Keywords

Search

Navigation