IDEAS home Printed from https://ideas.repec.org/a/taf/jnlasa/v110y2015i511p1266-1275.html
   My bibliography  Save this article

Localized Functional Principal Component Analysis

Author

Listed:
  • 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
    as

    Download full text from publisher

    File URL: http://hdl.handle.net/10.1080/01621459.2015.1016225
    Download Restriction: Access to full text is restricted to subscribers.
    File URL: https://libkey.io/10.1080/01621459.2015.1016225?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    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. 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.
    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. Zhang, Xiaoke & Zhong, Qixian & Wang, Jane-Ling, 2020. "A new approach to varying-coefficient additive models with longitudinal covariates," Computational Statistics & Data Analysis, Elsevier, vol. 145(C).
    2. 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.
    3. Guochang Wang, 2017. "Dimension reduction in functional regression with categorical predictor," Computational Statistics, Springer, vol. 32(2), pages 585-609, June.
    4. Haixu Wang & Jiguo Cao, 2023. "Nonlinear prediction of functional time series," Environmetrics, John Wiley & Sons, Ltd., vol. 34(5), August.
    5. 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).
    6. Nie, Yunlong & Cao, Jiguo, 2020. "Sparse functional principal component analysis in a new regression framework," Computational Statistics & Data Analysis, Elsevier, vol. 152(C).
    7. 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.
    8. 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.
    9. Guochang Wang & Xinyuan Song, 2018. "Functional Sufficient Dimension Reduction for Functional Data Classification," Journal of Classification, Springer;The Classification Society, vol. 35(2), pages 250-272, July.
    10. Peijun Sang & Liangliang Wang & Jiguo Cao, 2017. "Parametric functional principal component analysis," Biometrics, The International Biometric Society, vol. 73(3), pages 802-810, September.
    11. 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.
    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.

    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. Han Shang, 2014. "A survey of functional principal component analysis," AStA Advances in Statistical Analysis, Springer;German Statistical Society, vol. 98(2), pages 121-142, April.
    2. 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.
    3. Han Lin Shang & Yang Yang, 2021. "Forecasting Australian subnational age-specific mortality rates," Journal of Population Research, Springer, vol. 38(1), pages 1-24, March.
    4. Shang, Han Lin, 2013. "Bayesian bandwidth estimation for a nonparametric functional regression model with unknown error density," Computational Statistics & Data Analysis, Elsevier, vol. 67(C), pages 185-198.
    5. Shang Han Lin, 2020. "A Comparison of Hurst Exponent Estimators in Long-range Dependent Curve Time Series," Journal of Time Series Econometrics, De Gruyter, vol. 12(1), pages 1-39, January.
    6. Bastian Schäfer, 2021. "Bandwidth selection for the Local Polynomial Double Conditional Smoothing under Spatial ARMA Errors," Working Papers CIE 146, Paderborn University, CIE Center for International Economics.
    7. Bali, Juan Lucas & Boente, Graciela, 2014. "Consistency of a numerical approximation to the first principal component projection pursuit estimator," Statistics & Probability Letters, Elsevier, vol. 94(C), pages 181-191.
    8. Jaap Spreeuw & Iqbal Owadally & Muhammad Kashif, 2022. "Projecting Mortality Rates Using a Markov Chain," Mathematics, MDPI, vol. 10(7), pages 1-18, April.
    9. Shang, Han Lin & Hyndman, Rob.J., 2011. "Nonparametric time series forecasting with dynamic updating," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 81(7), pages 1310-1324.
    10. Kehui Chen & Xiaoke Zhang & Alexander Petersen & Hans-Georg Müller, 2017. "Quantifying Infinite-Dimensional Data: Functional Data Analysis in Action," Statistics in Biosciences, Springer;International Chinese Statistical Association, vol. 9(2), pages 582-604, December.
    11. Carfora, M.F. & Cutillo, L. & Orlando, A., 2017. "A quantitative comparison of stochastic mortality models on Italian population data," Computational Statistics & Data Analysis, Elsevier, vol. 112(C), pages 198-214.
    12. Santiago Gall n & Jorge Barrientos, 2021. "Forecasting the Colombian Electricity Spot Price under a Functional Approach," International Journal of Energy Economics and Policy, Econjournals, vol. 11(2), pages 67-74.
    13. Fraiman, Ricardo & Pateiro-López, Beatriz, 2012. "Quantiles for finite and infinite dimensional data," Journal of Multivariate Analysis, Elsevier, vol. 108(C), pages 1-14.
    14. Han Lin Shang & Rob J Hyndman, 2016. "Grouped functional time series forecasting: An application to age-specific mortality rates," Monash Econometrics and Business Statistics Working Papers 4/16, Monash University, Department of Econometrics and Business Statistics.
    15. Gao, Yuan & Shang, Han Lin & Yang, Yanrong, 2019. "High-dimensional functional time series forecasting: An application to age-specific mortality rates," Journal of Multivariate Analysis, Elsevier, vol. 170(C), pages 232-243.
    16. Kehui Chen & Pedro Delicado & Hans-Georg Müller, 2017. "Modelling function-valued stochastic processes, with applications to fertility dynamics," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 79(1), pages 177-196, January.
    17. 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.
    18. Azizur Rahman & Depeng Jiang, 2023. "Forecasting Canadian Age-Specific Mortality Rates: Application of Functional Time Series Analysis," Mathematics, MDPI, vol. 11(18), pages 1-14, September.
    19. repec:eca:wpaper:2013/131191 is not listed on IDEAS
    20. Ana-Maria Staicu & Yingxing Li & Ciprian M. Crainiceanu & David Ruppert, 2014. "Likelihood Ratio Tests for Dependent Data with Applications to Longitudinal and Functional Data Analysis," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 41(4), pages 932-949, December.
    21. Booth, Heather, 2006. "Demographic forecasting: 1980 to 2005 in review," International Journal of Forecasting, Elsevier, vol. 22(3), pages 547-581.

    More about this item

    Statistics

    Access and download statistics

    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:taf:jnlasa:v:110:y:2015:i:511:p:1266-1275. 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: Chris Longhurst (email available below). General contact details of provider: http://www.tandfonline.com/UASA20 .

    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.