## Sampled forms of functional PCA in reproducing kernel Hilbert spaces.(English)Zbl 1373.62289

Summary: We consider the sampling problem for functional PCA (fPCA), where the simplest example is the case of taking time samples of the underlying functional components. More generally, we model the sampling operation as a continuous linear map from $$\mathcal{H}$$ to $$\mathbb{R}^{m}$$, where the functional components to lie in some Hilbert subspace $$\mathcal{H}$$ of $$L^{2}$$, such as a reproducing kernel Hilbert space of smooth functions. This model includes time and frequency sampling as special cases. In contrast to classical approach in fPCA in which access to entire functions is assumed, having a limited number $$m$$ of functional samples places limitations on the performance of statistical procedures. We study these effects by analyzing the rate of convergence of an $$M$$-estimator for the subspace spanned by the leading components in a multi-spiked covariance model. The estimator takes the form of regularized PCA, and hence is computationally attractive. We analyze the behavior of this estimator within a nonasymptotic framework, and provide bounds that hold with high probability as a function of the number of statistical samples $$n$$ and the number of functional samples $$m$$. We also derive lower bounds showing that the rates obtained are minimax optimal.

### MSC:

 62H25 Factor analysis and principal components; correspondence analysis 62G07 Density estimation 62G08 Nonparametric regression and quantile regression

gss; fda (R)
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### References:

 [1] Amini, A. A. and Wainwright, M. J. (2009). High-dimensional analysis of semidefinite relaxations for sparse principal components. Ann. Statist. 37 2877-2921. · Zbl 1173.62049 [2] Amini, A. A. and Wainwright, M. J. (2012). Approximation properties of certain operator-induced norms on Hilbert spaces. J. Approx. Theory 164 320-345. · Zbl 1262.41015 [3] Amini, A. A. and Wainwright, M. J. (2012). Supplement to “Sampled forms of functional PCA in reproducing kernel Hilbert spaces.” . · Zbl 1373.62289 [4] Berlinet, A. and Thomas-Agnan, C. (2004). Reproducing Kernel Hilbert Spaces in Probability and Statistics . Kluwer Academic, Boston, MA. · Zbl 1145.62002 [5] Besse, P. and Ramsay, J. O. (1986). Principal components analysis of sampled functions. Psychometrika 51 285-311. · Zbl 0623.62048 [6] Bhatia, R. (1996). Matrix Analysis . Springer, New York. · Zbl 0863.15001 [7] Boente, G. and Fraiman, R. (2000). Kernel-based functional principal components. Statist. Probab. Lett. 48 335-345. · Zbl 0997.62024 [8] Bosq, D. (2000). Linear Processes in Function Spaces : Theory and Applications. Lecture Notes in Statistics 149 . Springer, New York. · Zbl 0962.60004 [9] Cai, T. T. and Yuan, M. (2010). Nonparametric covariance function estimation for functional and longitudinal data. Technical report, Georgia Institute of Technology. [10] Cardot, H. (2000). Nonparametric estimation of smoothed principal components analysis of sampled noisy functions. J. Nonparametr. Stat. 12 503-538. · Zbl 0951.62030 [11] Dauxois, J., Pousse, A. and Romain, Y. (1982). Asymptotic theory for the principal component analysis of a vector random function: Some applications to statistical inference. J. Multivariate Anal. 12 136-154. · Zbl 0539.62064 [12] Davidson, K. R. and Szarek, S. J. (2001). Local operator theory, random matrices and Banach spaces. In Handbook of the Geometry of Banach Spaces , Vol. I 317-366. North-Holland, Amsterdam. · Zbl 1067.46008 [13] Diggle, P. J., Heagerty, P. J., Liang, K.-Y. and Zeger, S. L. (2002). Analysis of Longitudinal Data , 2nd ed. Oxford Statistical Science Series 25 . Oxford Univ. Press, Oxford. · Zbl 1031.62002 [14] Gu, C. (2002). Smoothing Spline ANOVA Models . Springer, New York. · Zbl 1051.62034 [15] 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. · Zbl 1141.62048 [16] Hall, P., Müller, H.-G. and Wang, J.-L. (2006). Properties of principal component methods for functional and longitudinal data analysis. Ann. Statist. 34 1493-1517. · Zbl 1113.62073 [17] Huang, J. Z., Shen, H. and Buja, A. (2008). Functional principal components analysis via penalized rank one approximation. Electron. J. Stat. 2 678-695. · Zbl 1320.62097 [18] Johnstone, I. M. (2001). On the distribution of the largest eigenvalue in principal components analysis. Ann. Statist. 29 295-327. · Zbl 1016.62078 [19] Johnstone, I. M. and Lu, A. Y. (2009). On consistency and sparsity for principal components analysis in high dimensions. J. Amer. Statist. Assoc. 104 682-693. · Zbl 1388.62174 [20] Ledoux, M. (2001). The Concentration of Measure Phenomenon. Mathematical Surveys and Monographs 89 . Amer. Math. Soc., Providence, RI. · Zbl 0995.60002 [21] Li, Y. and Hsing, T. (2010). Uniform convergence rates for nonparametric regression and principal component analysis in functional/longitudinal data. Ann. Statist. 38 3321-3351. · Zbl 1204.62067 [22] Mendelson, S. (2002). Geometric parameters of kernel machines. In Computational Learning Theory ( Sydney , 2002). Lecture Notes in Computer Science 2375 29-43. Springer, Berlin. · Zbl 1050.68070 [23] Paul, D. and Johnstone, I. (2008). Augmented sparse principal component analysis for high-dimensional data. Available at . [24] Pezzulli, S. and Silverman, B. W. (1993). Some properties of smoothed principal components analysis for functional data. Comput. Statist. 8 1-16. · Zbl 0775.62146 [25] Qi, X. and Zhao, H. (2010). Functional principal component analysis for discretely observed functional data. Unpublished manuscript. [26] Ramsay, J. O. and Silverman, B. W. (2002). Applied Functional Data Analysis : Methods and Case Studies . Springer, New York. · Zbl 1011.62002 [27] Ramsay, J. O. and Silverman, B. W. (2005). Functional Data Analysis , 2nd ed. Springer, New York. · Zbl 1079.62006 [28] Rice, J. A. and Silverman, B. W. (1991). Estimating the mean and covariance structure nonparametrically when the data are curves. J. R. Stat. Soc. Ser. B Stat. Methodol. 53 233-243. · Zbl 0800.62214 [29] Silverman, B. W. (1996). Smoothed functional principal components analysis by choice of norm. Ann. Statist. 24 1-24. · Zbl 0853.62044 [30] van de Geer, S. A. (2009). Empirical Processes in M-Estimation . Cambridge Univ. Press, Cambridge. · Zbl 1179.62073 [31] Wahba, G. (1990). Spline Models for Observational Data. CBMS-NSF Regional Conference Series in Applied Mathematics 59 . SIAM, Philadelphia, PA. · Zbl 0813.62001 [32] Yao, F., Müller, H.-G. and Wang, J.-L. (2005). Functional data analysis for sparse longitudinal data. J. Amer. Statist. Assoc. 100 577-590. · Zbl 1117.62451
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