IDEAS home Printed from https://ideas.repec.org/a/eee/jmvana/v71y1999i2p262-276.html
   My bibliography  Save this article

Functional Principal Components Analysis by Choice of Norm

Author

Listed:
  • 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
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0047-259X(99)91844-4
    Download Restriction: Full text for ScienceDirect subscribers only
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    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.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. 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.
    2. 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.
    3. 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.
    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. 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.
    7. Jolliffe, Ian, 2022. "A 50-year personal journey through time with principal component analysis," Journal of Multivariate Analysis, Elsevier, vol. 188(C).
    8. 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.
    9. 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.
    10. Torokhti, Anatoli & Friedland, Shmuel, 2009. "Towards theory of generic Principal Component Analysis," Journal of Multivariate Analysis, Elsevier, vol. 100(4), pages 661-669, April.
    11. 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.
    12. 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.
    13. 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.
    14. van der Linde, Angelika, 2008. "Variational Bayesian functional PCA," Computational Statistics & Data Analysis, Elsevier, vol. 53(2), pages 517-533, December.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Mingotti, Nicola & Lillo Rodríguez, Rosa Elvira & Romo, Juan, 2015. "A Random Walk Test for Functional Time Series," DES - Working Papers. Statistics and Econometrics. WS ws1506, Universidad Carlos III de Madrid. Departamento de Estadística.
    2. María Edo & Walter Sosa Escudero & Marcela Svarc, 2021. "A multidimensional approach to measuring the middle class," The Journal of Economic Inequality, Springer;Society for the Study of Economic Inequality, vol. 19(1), pages 139-162, March.
    3. Jarry, Gabriel & Delahaye, Daniel & Nicol, Florence & Feron, Eric, 2020. "Aircraft atypical approach detection using functional principal component analysis," Journal of Air Transport Management, Elsevier, vol. 84(C).
    4. Guangxing Wang & Sisheng Liu & Fang Han & Chong‐Zhi Di, 2023. "Robust functional principal component analysis via a functional pairwise spatial sign operator," Biometrics, The International Biometric Society, vol. 79(2), pages 1239-1253, June.
    5. Michal Benko & Wolfgang Härdle & Alois Kneip, 2006. "Common Functional Principal Components," SFB 649 Discussion Papers SFB649DP2006-010, Sonderforschungsbereich 649, Humboldt University, Berlin, Germany.
    6. van Delft, Anne, 2020. "A note on quadratic forms of stationary functional time series under mild conditions," Stochastic Processes and their Applications, Elsevier, vol. 130(7), pages 4206-4251.
    7. Bali, Juan Lucas & Boente, Graciela, 2015. "Influence function of projection-pursuit principal components for functional data," Journal of Multivariate Analysis, Elsevier, vol. 133(C), pages 173-199.
    8. Qi, Xin & Zhao, Hongyu, 2011. "Some theoretical properties of Silverman's method for Smoothed functional principal component analysis," Journal of Multivariate Analysis, Elsevier, vol. 102(4), pages 741-767, April.
    9. Ci-Ren Jiang & John A. D. Aston & Jane-Ling Wang, 2016. "A Functional Approach to Deconvolve Dynamic Neuroimaging Data," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 111(513), pages 1-13, March.
    10. Delsol, Laurent & Ferraty, Frédéric & Vieu, Philippe, 2011. "Structural test in regression on functional variables," Journal of Multivariate Analysis, Elsevier, vol. 102(3), pages 422-447, March.
    11. Michal Benko & Alois Kneip, 2005. "Common functional component modelling," SFB 649 Discussion Papers SFB649DP2005-016, Sonderforschungsbereich 649, Humboldt University, Berlin, Germany.
    12. Beran, Jan & Liu, Haiyan, 2016. "Estimation of eigenvalues, eigenvectors and scores in FDA models with dependent errors," Journal of Multivariate Analysis, Elsevier, vol. 147(C), pages 218-233.
    13. Chung Chang & Yakuan Chen & R. Ogden, 2014. "Functional data classification: a wavelet approach," Computational Statistics, Springer, vol. 29(6), pages 1497-1513, December.
    14. Yuping Hu & Siyu Wu & Sanying Feng & Junliang Jin, 2020. "Estimation in Partial Functional Linear Spatial Autoregressive Model," Mathematics, MDPI, vol. 8(10), pages 1-12, October.
    15. Biau Gérard & Mas André, 2012. "PCA-kernel estimation," Statistics & Risk Modeling, De Gruyter, vol. 29(1), pages 19-46, March.
    16. Haixu Wang & Jiguo Cao, 2023. "Nonlinear prediction of functional time series," Environmetrics, John Wiley & Sons, Ltd., vol. 34(5), August.
    17. Andrii Babii & Eric Ghysels & Junsu Pan, 2022. "Tensor Principal Component Analysis," Papers 2212.12981, arXiv.org, revised Aug 2023.
    18. André Mas, 2002. "Testing for the Mean of Random Curves : from Penalization to Dimension Selection," Working Papers 2002-08, Center for Research in Economics and Statistics.
    19. Berkes, István & Horváth, Lajos & Rice, Gregory, 2016. "On the asymptotic normality of kernel estimators of the long run covariance of functional time series," Journal of Multivariate Analysis, Elsevier, vol. 144(C), pages 150-175.
    20. Nie, Yunlong & Cao, Jiguo, 2020. "Sparse functional principal component analysis in a new regression framework," Computational Statistics & Data Analysis, Elsevier, vol. 152(C).

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:jmvana:v:71:y:1999:i:2:p:262-276. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Catherine Liu (email available below). General contact details of provider: http://www.elsevier.com/wps/find/journaldescription.cws_home/622892/description#description .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.