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Spatial Functional Principal Component Analysis with Applications to Brain Image Data

Author

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  • Yingxing Li
  • Chen Huang
  • Wolfgang Karl Härdle

Abstract

This paper considers a fast and effective algorithm for conducting functional principle component analysis with multivariate factors. Compared with the univariate case, our approach could be more powerful in revealing spatial connections or extracting important features in images. To facilitate fast computation, we connect Singular Value Decomposition with penalized smoothing and avoid estimating a huge dimensional covariance operator. Under regularity assumptions, the results indicate that we may enjoy the optimal convergence rate by employing the smoothness assumption inherent to functional objects. We apply our method on the analysis of brain image data. Our extracted factors provide excellent recovery of the risk related regions of interests in human brain and the estimated loadings are very informative in revealing the individual risk attitude.

Suggested Citation

  • Yingxing Li & Chen Huang & Wolfgang Karl Härdle, 2017. "Spatial Functional Principal Component Analysis with Applications to Brain Image Data," SFB 649 Discussion Papers SFB649DP2017-024, Sonderforschungsbereich 649, Humboldt University, Berlin, Germany.
    • Handle: RePEc:hum:wpaper:sfb649dp2017-024
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    File URL: http://sfb649.wiwi.hu-berlin.de/papers/pdf/SFB649DP2017-024.pdf
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    References listed on IDEAS

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    4. Luo Xiao & Yingxing Li & David Ruppert, 2013. "Fast bivariate P-splines: the sandwich smoother," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 75(3), pages 577-599, June.
    5. Peter N. C. Mohr & Guido Biele & Hauke R. Heekeren, 2010. "Neural Processing of Risk," SFB 649 Discussion Papers SFB649DP2010-065, Sonderforschungsbereich 649, Humboldt University, Berlin, Germany.
    6. repec:wyi:journl:002174 is not listed on IDEAS
    7. Yingxing Li & David Ruppert, 2008. "On the asymptotics of penalized splines," Biometrika, Biometrika Trust, vol. 95(2), pages 415-436.
    8. 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.
    9. Wolfgang Karl Härdle & Lukas Borke, 2017. "GitHub API based QuantNet Mining infrastructure in R," SFB 649 Discussion Papers SFB649DP2017-008, Sonderforschungsbereich 649, Humboldt University, Berlin, Germany.
    10. Alena Bömmel & Song Song & Piotr Majer & Peter Mohr & Hauke Heekeren & Wolfgang Härdle, 2014. "Risk Patterns and Correlated Brain Activities. Multidimensional Statistical Analysis of fMRI Data in Economic Decision Making Study," Psychometrika, Springer;The Psychometric Society, vol. 79(3), pages 489-514, July.
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    Citations

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    Cited by:

    1. Giraldo, Ramón & Dabo-Niang, Sophie & Martínez, Sergio, 2018. "Statistical modeling of spatial big data: An approach from a functional data analysis perspective," Statistics & Probability Letters, Elsevier, vol. 136(C), pages 126-129.
    2. Dennis Schroers, 2024. "Robust Functional Data Analysis for Stochastic Evolution Equations in Infinite Dimensions," Papers 2401.16286, arXiv.org.
    3. Colin Lewis-Beck & Zhengyuan Zhu & Victoria Walker & Brian Hornbuckle, 2020. "Modeling Crop Phenology in the US Corn Belt Using Spatially Referenced SMOS Satellite Data," Journal of Agricultural, Biological and Environmental Statistics, Springer;The International Biometric Society;American Statistical Association, vol. 25(4), pages 657-675, December.
    4. Haozhe Zhang & Yehua Li, 2020. "Unified Principal Component Analysis for Sparse and Dense Functional Data under Spatial Dependency," Papers 2006.13489, arXiv.org, revised Jun 2021.
    5. Tomasz Górecki & Lajos Horváth & Piotr Kokoszka, 2020. "Tests of Normality of Functional Data," International Statistical Review, International Statistical Institute, vol. 88(3), pages 677-697, December.
    6. Park, Yeonjoo & Kim, Hyunsung & Lim, Yaeji, 2023. "Functional principal component analysis for partially observed elliptical process," Computational Statistics & Data Analysis, Elsevier, vol. 184(C).

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    More about this item

    Keywords

    Principal Component Analysis; Penalized Smoothing; Asymptotics; functional Magnetic Resonance Imaging fMRI;
    All these keywords.

    JEL classification:

    • C13 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Estimation: General
    • C20 - Mathematical and Quantitative Methods - - Single Equation Models; Single Variables - - - General
    • E37 - Macroeconomics and Monetary Economics - - Prices, Business Fluctuations, and Cycles - - - Forecasting and Simulation: Models and Applications

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