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Robust functional principal components for irregularly spaced longitudinal data

  • Ricardo A. MaronnaEmail author
Regular Article
  • 4 Downloads

Abstract

Consider longitudinal data \(x_{ij},\) with \(i=1,...,n\) and \(j=1,...,p,\) where \(x_{ij}\) is the observation of the smooth random function \(X_{i}\left( .\right) \) at time \(t_{j}.\) The goal of this paper is to develop a parsimonious representation of the data by a linear combination of a set of \(q<p\) smooth functions \(H_{k}\left( .\right) \) (\(k=1,..,q)\) in the sense that \(x_{ij}\approx \mu _{j}+\sum _{k=1}^{q}\beta _{ki}H_{k}\left( t_{j}\right) .\) This representation should be resistant to atypical \(X_{i}\)’s (“case contamination”), resistant to isolated gross errors at some cells (ij) (”cell contamination”), and applicable when some of the \(x_{ij}\) are missing (”irregularly spaced—or ’incomplete’—data”). Two approaches will be proposed for this problem. One deals with the three requirements stated above, and is based on ideas similar to MM-estimation (Yohai in Ann Stat 15:642–656, 1987). The other is a simple and fast estimator which can be applied to complete data with case- and cellwise contamination, and is based on applying a standard robust principal components estimate and smoothing the principal directions. Experiments with real and simulated data suggest that with complete data the simple estimator outperforms its competitors, while the MM estimator is competitive for incomplete data.

Keywords

MM-estimator B-splines Sparse data 

Notes

Acknowledgements

This research was partially supported by Grant 20020170100022BA from the University of Buenos Aires, Argentina. The author thanks the two anonymous reviewers for their insightful comments that much helped to improve the paper’s coherence.

References

  1. Bali JL, Boente G, Tyler DE, Wang J-L (2011) Robust functional principal components: a projection-pursuit approach. Ann Stat 39:2852–2882MathSciNetCrossRefGoogle Scholar
  2. Bay SD (1999) The UCI KDD Archive [http://kdd.ics.uci.edu], University of California, Irvine, Department of Information and Computer Science
  3. Boente G, Salibian-Barrera M (2015) S-estimators for functional principal component analysis. JASA 110:1100–1111MathSciNetCrossRefGoogle Scholar
  4. Cevallos Valdiviezo H (2016) On methods for prediction based on complex data with missing values and robust principal component analysis, PhD thesis, Ghent University (supervisors Van Aelst S. and Van den Poel, D.)Google Scholar
  5. Cleveland WS (1979) Robust locally weighted regression and smoothing scatterplots. JASA 74:829–836MathSciNetCrossRefGoogle Scholar
  6. Gnanadesikan R, Kettenring JR (1972) Robust estimates, residuals, and outlier detection with multiresponse data. Biometrics 28:81–124CrossRefGoogle Scholar
  7. Grecki T, Krzyko M, Waszak L, Woyski W (2018) Selected statistical methods of data analysis for multivariate functional data. Stat Pap 59:153–182MathSciNetCrossRefGoogle Scholar
  8. James G, Hastie TG, Sugar CA (2001) Principal component models for sparse functional data. Biometrika 87:587–602MathSciNetCrossRefGoogle Scholar
  9. Lee S, Shin H, Billor N (2013) M-type smoothing spline estimators for principal functions. Comput Stat Data Anal 66:89–100MathSciNetCrossRefGoogle Scholar
  10. Locantore N, Marron JS, Simpson DG, Tripoli N, Zhang JT, Cohen KL (1999) Robust principal components for functional data. Test 8:1–28MathSciNetCrossRefGoogle Scholar
  11. Maronna R (2005) Principal components and orthogonal regression based on robust scales. Technometrics 47:264–273MathSciNetCrossRefGoogle Scholar
  12. Maronna RA, Martin RD, Yohai VJ, Salibian-Barrera M (2019) Robust statistics: theory and methods (with R), 2nd edn. Wiley, ChichesterzbMATHGoogle Scholar
  13. Rousseeuw PJ, Croux C (1993) Alternatives to the median absolute deviation. JASA 88:1273–1283MathSciNetCrossRefGoogle Scholar
  14. Yao F, Müller H-G, Wang J-L (2005) Functional data analysis for sparse longitudinal data. JASA 100:577–590MathSciNetCrossRefGoogle Scholar
  15. Yohai VJ (1987) High breakdown-point and high efficiency robust estimates for regression. Ann Stat 15:642–656MathSciNetCrossRefGoogle Scholar
  16. Yohai VJ, Zamar RH (1988) High breakdown-point estimates of regression by means of the minimization of an efficient scale. J Am Stat Assoc 83:406–413MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Mathematics Department, Faculty of Exact SciencesUniversity of La PlataLa PlataArgentina

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