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M-type smoothing spline estimators for principal functions

Author

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  • Lee, Seokho
  • Shin, Hyejin
  • Billor, Nedret

Abstract

We propose a robust method for estimating principal functions based on MM estimation. Specifically, we formulate functional principal component analysis into alternating penalized M-regression with a bounded loss function. The resulting principal functions are given as M-type smoothing spline estimators. Using the properties of a natural cubic spline, we develop a fast computation algorithm even for long and dense functional data. The proposed method is efficient in that the maximal information from whole observed curve is retained since it partly downweighs abnormally observed individual measurements in a single curve rather than removing or downweighing a whole curve. We demonstrate the performance of the proposed method on simulated and real data and compare it with the conventional functional principal component analysis and other robust functional principal component analysis techniques.

Suggested Citation

  • Lee, Seokho & Shin, Hyejin & Billor, Nedret, 2013. "M-type smoothing spline estimators for principal functions," Computational Statistics & Data Analysis, Elsevier, vol. 66(C), pages 89-100.
    • Handle: RePEc:eee:csdana:v:66:y:2013:i:c:p:89-100
    DOI: 10.1016/j.csda.2013.03.022
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    References listed on IDEAS

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    1. Shen, Haipeng & Huang, Jianhua Z., 2008. "Sparse principal component analysis via regularized low rank matrix approximation," Journal of Multivariate Analysis, Elsevier, vol. 99(6), pages 1015-1034, July.
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    6. Pallavi Sawant & Nedret Billor & Hyejin Shin, 2012. "Functional outlier detection with robust functional principal component analysis," Computational Statistics, Springer, vol. 27(1), pages 83-102, March.
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    8. Hyndman, Rob J. & Shahid Ullah, Md., 2007. "Robust forecasting of mortality and fertility rates: A functional data approach," Computational Statistics & Data Analysis, Elsevier, vol. 51(10), pages 4942-4956, June.
    9. Croux, Christophe & Ruiz-Gazen, Anne, 2005. "High breakdown estimators for principal components: the projection-pursuit approach revisited," Journal of Multivariate Analysis, Elsevier, vol. 95(1), pages 206-226, July.
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    Cited by:

    1. Boente, Graciela & Rodriguez, Daniela & Sued, Mariela, 2019. "The spatial sign covariance operator: Asymptotic results and applications," Journal of Multivariate Analysis, Elsevier, vol. 170(C), pages 115-128.
    2. Italo R. Lima & Guanqun Cao & Nedret Billor, 2019. "Robust simultaneous inference for the mean function of functional data," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 28(3), pages 785-803, September.
    3. Boente, Graciela & Parada, Daniela, 2023. "Robust estimation for functional quadratic regression models," Computational Statistics & Data Analysis, Elsevier, vol. 187(C).
    4. Bali, Juan Lucas & Boente, Graciela, 2017. "Robust estimators under a functional common principal components model," Computational Statistics & Data Analysis, Elsevier, vol. 113(C), pages 424-440.
    5. Graciela Boente & Matías Salibián-Barrera, 2021. "Robust functional principal components for sparse longitudinal data," METRON, Springer;Sapienza Università di Roma, vol. 79(2), pages 159-188, August.
    6. Italo R. Lima & Guanqun Cao & Nedret Billor, 2019. "M-based simultaneous inference for the mean function of functional data," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 71(3), pages 577-598, June.
    7. Ricardo A. Maronna, 2021. "Robust functional principal components for irregularly spaced longitudinal data," Statistical Papers, Springer, vol. 62(4), pages 1563-1582, August.
    8. Martínez-Hernández, Israel & Genton, Marc G. & González-Farías, Graciela, 2019. "Robust depth-based estimation of the functional autoregressive model," Computational Statistics & Data Analysis, Elsevier, vol. 131(C), pages 66-79.
    9. Haolun Shi & Jiguo Cao, 2022. "Robust Functional Principal Component Analysis Based on a New Regression Framework," Journal of Agricultural, Biological and Environmental Statistics, Springer;The International Biometric Society;American Statistical Association, vol. 27(3), pages 523-543, September.

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