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Localized Functional Principal Component Analysis

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  • Kehui Chen
  • Jing Lei

Abstract

We propose localized functional principal component analysis (LFPCA), looking for orthogonal basis functions with localized support regions that explain most of the variability of a random process. The LFPCA is formulated as a convex optimization problem through a novel deflated Fantope localization method and is implemented through an efficient algorithm to obtain the global optimum. We prove that the proposed LFPCA converges to the original functional principal component analysis (FPCA) when the tuning parameters are chosen appropriately. Simulation shows that the proposed LFPCA with tuning parameters chosen by cross-validation can almost perfectly recover the true eigenfunctions and significantly improve the estimation accuracy when the eigenfunctions are truly supported on some subdomains. In the scenario that the original eigenfunctions are not localized, the proposed LFPCA also serves as a nice tool in finding orthogonal basis functions that balance between interpretability and the capability of explaining variability of the data. The analyses of a country mortality data reveal interesting features that cannot be found by standard FPCA methods. Supplementary materials for this article are available online.

Suggested Citation

  • Kehui Chen & Jing Lei, 2015. "Localized Functional Principal Component Analysis," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 110(511), pages 1266-1275, September.
    • Handle: RePEc:taf:jnlasa:v:110:y:2015:i:511:p:1266-1275
    DOI: 10.1080/01621459.2015.1016225
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    References listed on IDEAS

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    1. 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.
    2. Kehui Chen & Hans-Georg Müller, 2012. "Modeling Repeated Functional Observations," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 107(500), pages 1599-1609, December.
    3. 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.
    4. Peter Hall & Mohammad Hosseini‐Nasab, 2006. "On properties of functional principal components analysis," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 68(1), pages 109-126, February.
    5. Chiou, Jeng-Min & Müller, Hans-Georg, 2009. "Modeling Hazard Rates as Functional Data for the Analysis of Cohort Lifetables and Mortality Forecasting," Journal of the American Statistical Association, American Statistical Association, vol. 104(486), pages 572-585.
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    Cited by:

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    3. Peijun Sang & Liangliang Wang & Jiguo Cao, 2017. "Parametric functional principal component analysis," Biometrics, The International Biometric Society, vol. 73(3), pages 802-810, September.
    4. Yixuan Qiu & Jing Lei & Kathryn Roeder, 2023. "Gradient-based sparse principal component analysis with extensions to online learning," Biometrika, Biometrika Trust, vol. 110(2), pages 339-360.
    5. Guochang Wang, 2017. "Dimension reduction in functional regression with categorical predictor," Computational Statistics, Springer, vol. 32(2), pages 585-609, June.
    6. Haixu Wang & Jiguo Cao, 2023. "Nonlinear prediction of functional time series," Environmetrics, John Wiley & Sons, Ltd., vol. 34(5), August.
    7. Sun, Xuxue & Cai, Wenjun & Li, Mingyang, 2021. "A hierarchical modeling approach for degradation data with mixed-type covariates and latent heterogeneity," Reliability Engineering and System Safety, Elsevier, vol. 216(C).
    8. Nie, Yunlong & Cao, Jiguo, 2020. "Sparse functional principal component analysis in a new regression framework," Computational Statistics & Data Analysis, Elsevier, vol. 152(C).
    9. Robert T. Krafty & Haoyi Fu & Jessica L. Graves & Scott A. Bruce & Martica H. Hall & Stephen F. Smagula, 2019. "Measuring Variability in Rest-Activity Rhythms from Actigraphy with Application to Characterizing Symptoms of Depression," Statistics in Biosciences, Springer;International Chinese Statistical Association, vol. 11(2), pages 314-333, July.
    10. Xiongtao Dai & Zhenhua Lin & Hans‐Georg Müller, 2021. "Modeling sparse longitudinal data on Riemannian manifolds," Biometrics, The International Biometric Society, vol. 77(4), pages 1328-1341, December.
    11. Christophe Biernacki & Matthieu Marbac & Vincent Vandewalle, 2021. "Gaussian-Based Visualization of Gaussian and Non-Gaussian-Based Clustering," Journal of Classification, Springer;The Classification Society, vol. 38(1), pages 129-157, April.
    12. Ruonan Li & Luo Xiao, 2023. "Latent factor model for multivariate functional data," Biometrics, The International Biometric Society, vol. 79(4), pages 3307-3318, December.

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