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Spatial Functional Principal Component Analysis with Applications to Brain Image Data


  • Yingxing Li
  • Chen Huang
  • Wolfgang Karl Härdle


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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    References listed on IDEAS

    1. I. D. Currie & M. Durban & P. H. C. Eilers, 2006. "Generalized linear array models with applications to multidimensional smoothing," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 68(2), pages 259-280, April.
    2. 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.
    3. Fang Yao & Hans-Georg Müller & Andrew J. Clifford & Steven R. Dueker & Jennifer Follett & Yumei Lin & Bruce A. Buchholz & John S. Vogel, 2003. "Shrinkage Estimation for Functional Principal Component Scores with Application to the Population Kinetics of Plasma Folate," Biometrics, The International Biometric Society, vol. 59(3), pages 676-685, September.
    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. Piotr Majer & Peter N. C. Mohr & Hauke R. Heekeren & Wolfgang K. Härdle, 2016. "Portfolio Decisions and Brain Reactions via the CEAD method," Psychometrika, Springer;The Psychometric Society, vol. 81(3), pages 881-903, September.
    6. 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.
    7. repec:wyi:journl:002174 is not listed on IDEAS
    8. Yingxing Li & David Ruppert, 2008. "On the asymptotics of penalized splines," Biometrika, Biometrika Trust, vol. 95(2), pages 415-436.
    9. 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.
    10. 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.
    11. 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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    1. repec:eee:stapro:v:136:y:2018:i:c:p:126-129 is not listed on IDEAS

    More about this item


    Principal Component Analysis; Penalized Smoothing; Asymptotics; functional Magnetic Resonance Imaging fMRI;

    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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