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Nonlinear and additive principal component analysis for functional data

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  • Song, Jun
  • Li, Bing

Abstract

We introduce a nonlinear additive functional principal component analysis (NAFPCA) for vector-valued functional data. This is a generalization of functional principal component analysis and allows the relations among the random functions involved to be nonlinear. The method is constructed via two additively nested Hilbert spaces of functions, in which the first space characterizes the functional nature of the data, and the second space captures the nonlinear dependence. In the meantime, additivity is imposed so that we can avoid high-dimensional kernels in the functional space, which causes the curse of dimensionality. Along with the NAFPCA, we also develop a method of selection of the number of principal components and the tuning parameters that determines the degree of nonlinearity, as well as the asymptotic results for both the fully observed and the incompletely observed functional data. Simulation results show that the new method performs better than functional principal component analysis when the relations among random functions are nonlinear. We apply the new method to online handwritten digits and electroencephalogram (EEG) data sets.

Suggested Citation

  • Song, Jun & Li, Bing, 2021. "Nonlinear and additive principal component analysis for functional data," Journal of Multivariate Analysis, Elsevier, vol. 181(C).
    • Handle: RePEc:eee:jmvana:v:181:y:2021:i:c:s0047259x20302566
    DOI: 10.1016/j.jmva.2020.104675
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    References listed on IDEAS

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    1. Bing Li & Hyonho Chun & Hongyu Zhao, 2012. "Sparse Estimation of Conditional Graphical Models With Application to Gene Networks," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 107(497), pages 152-167, March.
    2. Yehua Li & Yongtao Guan, 2014. "Functional Principal Component Analysis of Spatiotemporal Point Processes With Applications in Disease Surveillance," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 109(507), pages 1205-1215, September.
    3. Bing Li & Hyonho Chun & Hongyu Zhao, 2014. "On an Additive Semigraphoid Model for Statistical Networks With Application to Pathway Analysis," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 109(507), pages 1188-1204, September.
    4. Jacques, Julien & Preda, Cristian, 2014. "Model-based clustering for multivariate functional data," Computational Statistics & Data Analysis, Elsevier, vol. 71(C), pages 92-106.
    5. Clara Happ & Sonja Greven, 2018. "Multivariate Functional Principal Component Analysis for Data Observed on Different (Dimensional) Domains," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 113(522), pages 649-659, April.
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    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. Ali Mahzarnia & Jun Song, 2022. "Multivariate functional group sparse regression: Functional predictor selection," PLOS ONE, Public Library of Science, vol. 17(4), pages 1-22, April.
    3. Jolliffe, Ian, 2022. "A 50-year personal journey through time with principal component analysis," Journal of Multivariate Analysis, Elsevier, vol. 188(C).
    4. Li, Yehua & Qiu, Yumou & Xu, Yuhang, 2022. "From multivariate to functional data analysis: Fundamentals, recent developments, and emerging areas," Journal of Multivariate Analysis, Elsevier, vol. 188(C).
    5. Yao, Binhong & Li, Peixing, 2023. "Covariance estimation error of incomplete functional data under RKHS framework," Applied Mathematics and Computation, Elsevier, vol. 443(C).

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