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Quantifying Infinite-Dimensional Data: Functional Data Analysis in Action

Author

Listed:
  • Kehui Chen

    () (University of Pittsburgh
    University of Pittsburgh)

  • Xiaoke Zhang

    (University of Delaware)

  • Alexander Petersen

    (University of California)

  • Hans-Georg Müller

    (University of California)

Abstract

Abstract Functional data analysis (FDA) is concerned with inherently infinite-dimensional data objects and therefore can be viewed as part of the methodology for big data. The size of functional data may vary from terabytes as encountered in functional magnetic resonance imaging (fMRI) and other applications in brain imaging to just a few kilobytes in longitudinal data with small or modest sample sizes. In this contribution, we highlight some applications of FDA methodology through various data illustrations. We briefly review some basic computational tools that can be used to accelerate implementations of FDA methodology. The analyses presented in this paper illustrate the principal analysis by conditional expectation (PACE) package for FDA, where our applications include both relatively simple and more complex functional data from the biomedical sciences. The data we discuss range from functional data that result from daily movement profile tracking and that are modeled as repeatedly observed functions per subject, to medfly longitudinal behavior profiles, where the goal is to predict remaining lifetime of individual flies. We also discuss the quantification of connectivity of fMRI signals that is of interest in brain imaging and the prediction of continuous traits from high-dimensional SNPs in genomics. The methods of FDA that we demonstrate for these analyses include functional principal component analysis, functional regression and correlation, the modeling of dependent functional data, and the stringing of high-dimensional data into functional data and can be implemented with the PACE package.

Suggested Citation

  • Kehui Chen & Xiaoke Zhang & Alexander Petersen & Hans-Georg Müller, 0. "Quantifying Infinite-Dimensional Data: Functional Data Analysis in Action," Statistics in Biosciences, Springer;International Chinese Statistical Association, vol. 0, pages 1-23.
  • Handle: RePEc:spr:stabio:v::y::i::d:10.1007_s12561-015-9137-5
    DOI: 10.1007/s12561-015-9137-5
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    References listed on IDEAS

    as
    1. Müller, Hans-Georg & Yao, Fang, 2008. "Functional Additive Models," Journal of the American Statistical Association, American Statistical Association, vol. 103(484), pages 1534-1544.
    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. Kehui Chen & Hans‐Georg Müller, 2012. "Conditional quantile analysis when covariates are functions, with application to growth data," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 74(1), pages 67-89, January.
    4. Fang Yao & Hans-Georg Müller, 2010. "Functional quadratic regression," Biometrika, Biometrika Trust, vol. 97(1), pages 49-64.
    5. Manuel Febrero-Bande & Wenceslao González-Manteiga, 2013. "Generalized additive models for functional data," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 22(2), pages 278-292, June.
    6. Yao, Fang & Muller, Hans-Georg & Wang, Jane-Ling, 2005. "Functional Data Analysis for Sparse Longitudinal Data," Journal of the American Statistical Association, American Statistical Association, vol. 100, pages 577-590, June.
    7. Peter Hall & Hans-Georg Müller & Fang Yao, 2008. "Modelling sparse generalized longitudinal observations with latent Gaussian processes," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 70(4), pages 703-723.
    8. Hongxiao Zhu & Fang Yao & Hao Helen Zhang, 2014. "Structured functional additive regression in reproducing kernel Hilbert spaces," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 76(3), pages 581-603, June.
    9. 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.
    10. Hans-Georg Müller & Yichao Wu & Fang Yao, 2013. "Continuously additive models for nonlinear functional regression," Biometrika, Biometrika Trust, vol. 100(3), pages 607-622.
    11. Chen, Kun & Chen, Kehui & Müller, Hans-Georg & Wang, Jane-Ling, 2011. "Stringing High-Dimensional Data for Functional Analysis," Journal of the American Statistical Association, American Statistical Association, vol. 106(493), pages 275-284.
    12. Kneip A. & Utikal K. J, 2001. "Inference for Density Families Using Functional Principal Component Analysis," Journal of the American Statistical Association, American Statistical Association, vol. 96, pages 519-542, June.
    13. Wenjing Yang & Hans‐Georg Müller & Ulrich Stadtmüller, 2011. "Functional singular component analysis," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 73(3), pages 303-324, June.
    14. 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.
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