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Functional Principal Components Analysis by Choice of Norm

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  • Ocaña, F. A.
  • Aguilera, A. M.
  • Valderrama, M. J.

Abstract

The functional principal components analysis (PCA) involves new considerations on the mechanism of measuring distances (the norm). Some properties arising in functional framework (e.g., smoothing) could be taken into account through an inner product in the data space. But this proposed inner product could make, for example, interpretational or (and) computational abilities worse. The results obtained in this paper establish equivalences between the PCA with the proposed inner product and certain PCA with a given well-suited inner product. These results have been proved in the theoretical framework given by Hilbert valued random variables, in which multivariate and functional PCAs appear jointly as particular cases.

Suggested Citation

  • Ocaña, F. A. & Aguilera, A. M. & Valderrama, M. J., 1999. "Functional Principal Components Analysis by Choice of Norm," Journal of Multivariate Analysis, Elsevier, vol. 71(2), pages 262-276, November.
    • Handle: RePEc:eee:jmvana:v:71:y:1999:i:2:p:262-276
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    References listed on IDEAS

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    1. J. O. Ramsay & X. Wang & R. Flanagan, 1995. "A Functional Data Analysis of the Pinch Force of Human Fingers," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 44(1), pages 17-30, March.
    2. Dauxois, J. & Pousse, A. & Romain, Y., 1982. "Asymptotic theory for the principal component analysis of a vector random function: Some applications to statistical inference," Journal of Multivariate Analysis, Elsevier, vol. 12(1), pages 136-154, March.
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    Cited by:

    1. van der Linde, Angelika, 2008. "Variational Bayesian functional PCA," Computational Statistics & Data Analysis, Elsevier, vol. 53(2), pages 517-533, December.
    2. Michael Greenacre & Patrick J. F Groenen & Trevor Hastie & Alfonso Iodice d’Enza & Angelos Markos & Elena Tuzhilina, 2023. "Principal component analysis," Economics Working Papers 1856, Department of Economics and Business, Universitat Pompeu Fabra.
    3. Lakraj, Gamage Pemantha & Ruymgaart, Frits, 2017. "Some asymptotic theory for Silverman’s smoothed functional principal components in an abstract Hilbert space," Journal of Multivariate Analysis, Elsevier, vol. 155(C), pages 122-132.
    4. Torokhti, Anatoli & Howlett, Phil, 2003. "Constructing fixed rank optimal estimators with method of best recurrent approximations," Journal of Multivariate Analysis, Elsevier, vol. 86(2), pages 293-309, August.
    5. Christian Acal & Manuel Escabias & Ana M. Aguilera & Mariano J. Valderrama, 2021. "COVID-19 Data Imputation by Multiple Function-on-Function Principal Component Regression," Mathematics, MDPI, vol. 9(11), pages 1-23, May.
    6. Torokhti, Anatoli & Friedland, Shmuel, 2009. "Towards theory of generic Principal Component Analysis," Journal of Multivariate Analysis, Elsevier, vol. 100(4), pages 661-669, April.
    7. Paula R. Bouzas & Ana M. Aguilera & Nuria Ruiz-Fuentes, 2012. "Functional Estimation of the Random Rate of a Cox Process," Methodology and Computing in Applied Probability, Springer, vol. 14(1), pages 57-69, March.
    8. Ana M. Aguilera, 2016. "Comments on: Probability enhanced effective dimension reduction for classifying sparse functional data," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 25(1), pages 23-26, March.
    9. Jolliffe, Ian, 2022. "A 50-year personal journey through time with principal component analysis," Journal of Multivariate Analysis, Elsevier, vol. 188(C).
    10. Christian Acal & Ana M. Aguilera & Manuel Escabias, 2020. "New Modeling Approaches Based on Varimax Rotation of Functional Principal Components," Mathematics, MDPI, vol. 8(11), pages 1-15, November.
    11. Ana Aguilera, 2016. "Comments on: Probability enhanced effective dimension reduction for classifying sparse functional data," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 25(1), pages 23-26, March.
    12. Marc Vidal & Mattia Rosso & Ana M. Aguilera, 2021. "Bi-Smoothed Functional Independent Component Analysis for EEG Artifact Removal," Mathematics, MDPI, vol. 9(11), pages 1-17, May.
    13. Michio Yamamoto & Heungsun Hwang, 2017. "Dimension-Reduced Clustering of Functional Data via Subspace Separation," Journal of Classification, Springer;The Classification Society, vol. 34(2), pages 294-326, July.
    14. Aguilera, Ana M. & Escabias, Manuel & Valderrama, Mariano J., 2008. "Discussion of different logistic models with functional data. Application to Systemic Lupus Erythematosus," Computational Statistics & Data Analysis, Elsevier, vol. 53(1), pages 151-163, September.

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