Annals of Applied Statistics

Smooth Principal Component Analysis over two-dimensional manifolds with an application to neuroimaging

Eardi Lila, John A. D. Aston, and Laura M. Sangalli

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Motivated by the analysis of high-dimensional neuroimaging signals located over the cortical surface, we introduce a novel Principal Component Analysis technique that can handle functional data located over a two-dimensional manifold. For this purpose a regularization approach is adopted, introducing a smoothing penalty coherent with the geodesic distance over the manifold. The model introduced can be applied to any manifold topology, and can naturally handle missing data and functional samples evaluated in different grids of points. We approach the discretization task by means of finite element analysis, and propose an efficient iterative algorithm for its resolution. We compare the performances of the proposed algorithm with other approaches classically adopted in literature. We finally apply the proposed method to resting state functional magnetic resonance imaging data from the Human Connectome Project, where the method shows substantial differential variations between brain regions that were not apparent with other approaches.

Article information

Ann. Appl. Stat., Volume 10, Number 4 (2016), 1854-1879.

Received: December 2015
Revised: August 2016
First available in Project Euclid: 5 January 2017

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Functional data analysis principal component analysis differential regularization functional magnetic resonance imaging


Lila, Eardi; Aston, John A. D.; Sangalli, Laura M. Smooth Principal Component Analysis over two-dimensional manifolds with an application to neuroimaging. Ann. Appl. Stat. 10 (2016), no. 4, 1854--1879. doi:10.1214/16-AOAS975.

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  • Alfeld, P., Neamtu, M. and Schumaker, L. L. (1996). Fitting scattered data on sphere-like surfaces using spherical splines. J. Comput. Appl. Math. 73 5–43.
  • Belkin, M. and Niyogi, P. (2001). Laplacian eigenmaps and spectral techniques for embedding and clustering. In Advances in Neural Information Processing Systems 14 (T. G. Dietterich, S. Becker and Z. Ghahramani, eds.) 585–591.
  • Buckner, R. L., Andrews-Hanna, J. R. and Schacter, D. L. (2008). The brain’s default network: Anatomy, function, and relevance to disease. Ann. N. Y. Acad. Sci. 1124 1–38.
  • Cai, D., He, X., Han, J. and Huang, T. S. (2011). Graph regularized nonnegative matrix factorization for data representation. IEEE Trans. Pattern Anal. Mach. Intell. 33 1548–1560.
  • Chung, M. K., Hanson, J. L. and Pollak, S. D. (2014). Statistical analysis on brain surfaces. Technical report, University of Wisconsin–Madison.
  • Chung, M. K., Robbins, S. M., Dalton, K. M., Davidson, R. J., Alexander, A. L. and Evans, A. C. (2005). Cortical thickness analysis in autism with heat kernel smoothing. NeuroImage 25 1256–1265.
  • Dassi, F., Ettinger, B., Perotto, S. and Sangalli, L. M. (2015). A mesh simplification strategy for a spatial regression analysis over the cortical surface of the brain. Appl. Numer. Math. 90 111–131.
  • Duchon, J. (1977). Splines minimizing rotation-invariant semi-norms in Sobolev spaces. In Constructive Theory of Functions of Several Variables (Proc. Conf., Math. Res. Inst., Oberwolfach, 1976) 85–1000. Springer, Berlin.
  • Dziuk, G. (1988). Finite elements for the Beltrami operator on arbitrary surfaces. In Partial Differential Equations and Calculus of Variations (S. Hildebrandt and R. Leis, eds.). Lecture Notes in Math. 1357 142–155. Springer, Berlin.
  • Essen, D. C. V., Ugurbil, K., Auerbach, E., Barch, D., Behrens, T. E. J., Bucholz, R., Chang, A., Chen, L., Corbetta, M., Curtiss, S. W., Penna, S. D., Feinberg, D., Glasser, M. F., Harel, N., Heath, A. C., Larson-Prior, L., Marcus, D., Michalareas, G., Moeller, S., Oostenveld, R., Petersen, S. E., Prior, F., Schlaggar, B. L., Smith, S. M., Snyder, A. Z., Xu, J. and Yacoub, E. (2012). The Human Connectome Project: A data acquisition perspective. NeuroImage 62 2222–2231.
  • Ettinger, B., Perotto, S. and Sangalli, L. M. (2016). Spatial regression models over two-dimensional manifolds. Biometrika 103 71–88.
  • Glasser, M. F., Sotiropoulos, S. N., Wilson, J. A., Coalson, T. S., Fischl, B., Andersson, J. L., Xu, J., Jbabdi, S., Webster, M., Polimeni, J. R., Essen, D. C. V. and Jenkinson, M. (2013). The minimal preprocessing pipelines for the Human Connectome Project. NeuroImage 80 105–124.
  • Gordon, E. M., Laumann, T. O., Adeyemo, B., Huckins, J. F., Kelley, W. M. and Petersen, S. E. (2014). Generation and evaluation of a cortical area parcellation from resting-state correlations. Cereb. Cortex.
  • Green, P. J. and Silverman, B. W. (1993). Nonparametric Regression and Generalized Linear Models. CRC Press, Boca Raton.
  • Hall, P. and Hosseini-Nasab, M. (2006). On properties of functional principal components analysis. J. R. Stat. Soc. Ser. B Stat. Methodol. 68 109–126.
  • Harrison, S. J., Woolrich, M. W., Robinson, E. C., Glasser, M. F., Beckmann, C. F., Jenkinson, M. and Smith, S. M. (2015). Large-scale probabilistic functional modes from resting state fMRI. NeuroImage 109 217–231.
  • Huang, J. Z., Shen, H. and Buja, A. (2008). Functional principal components analysis via penalized rank one approximation. Electron. J. Stat. 2 678–695.
  • Jolliffe, I. T., Trendafilov, N. T. and Uddin, M. (2003). A modified principal component technique based on the LASSO. J. Comput. Graph. Statist. 12 531–547.
  • Lila, E., Aston, J. A. D. and Sangalli, L. M. Supplement to “Smooth Principal Component Analysis over two-dimensional manifolds with an application to neuroimaging.” DOI:10.1214/16-AOAS975SUPP.
  • Marron, J. S., Ramsay, J. O., Sangalli, L. M. and Srivastava, A. (2015). Functional data analysis of amplitude and phase variation. Statist. Sci. 30 468–484.
  • Ogawa, S., Lee, T. M., Kay, A. R. and Tank, D. W. (1990). Brain magnetic resonance imaging with contrast dependent on blood oxygenation. Proc. Natl. Acad. Sci. USA 87 9868–9872.
  • Ramsay, T. (2002). Spline smoothing over difficult regions. J. R. Stat. Soc. Ser. B Stat. Methodol. 64 307–319.
  • Ramsay, J. O. and Silverman, B. W. (2005). Functional Data Analysis, 2nd ed. Springer, New York.
  • Rice, J. A. and Silverman, B. W. (1991). Estimating the mean and covariance structure nonparametrically when the data are curves. J. Roy. Statist. Soc. Ser. B 53 233–243.
  • Riesz, F. and Sz.-Nagy, B. (1955). Functional Analysis. Frederick Ungar Publishing Co., New York.
  • Sangalli, L. M., Ramsay, J. O. and Ramsay, T. O. (2013). Spatial spline regression models. J. R. Stat. Soc. Ser. B. Stat. Methodol. 75 681–703.
  • Shen, H. and Huang, J. Z. (2008). Sparse principal component analysis via regularized low rank matrix approximation. J. Multivariate Anal. 99 1015–1034.
  • Silverman, B. W. (1996). Smoothed functional principal components analysis by choice of norm. Ann. Statist. 24 1–24.
  • Wahba, G. (1981). Spline interpolation and smoothing on the sphere. SIAM J. Sci. Statist. Comput. 2 5–16.
  • Zhou, L. and Pan, H. (2014). Principal component analysis of two-dimensional functional data. J. Comput. Graph. Statist. 23 779–801.
  • Zou, H. and Hastie, T. (2005). Regularization and variable selection via the elastic net. J. R. Stat. Soc. Ser. B Stat. Methodol. 67 301–320.
  • Zou, H., Hastie, T. and Tibshirani, R. (2006). Sparse principal component analysis. J. Comput. Graph. Statist. 15 265–286.

Supplemental materials

  • Supplement to “Smooth Principal Component Analysis over two-dimensional manifolds with an application to neuroimaging”. The online supplementary material contains the theoretic details of the Finite Element discretization approach. Moreover, it includes further simulations on the sphere investigating both the methodology and its robustness to alignment issues.