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Some asymptotic theory for Silverman’s smoothed functional principal components in an abstract Hilbert space. (English) Zbl 1381.62147

Summary: Unlike classical principal component analysis (PCA) for multivariate data, one needs to smooth or regularize when estimating functional principal components. Silverman’s method for smoothed functional principal components has nice theoretical and practical properties. Some theoretical properties of Silverman’s method were obtained using tools in the \(L^2\) and the Sobolev spaces. This paper proposes an approach, in a general manner, to study the asymptotic properties of Silverman’s method in an abstract Hilbert space. This is achieved by exploiting the perturbation results of the eigenvalues and the corresponding eigenvectors of a covariance operator. Consistency and asymptotic distributions of the estimators are derived under mild conditions. First we restrict our attention to the first smoothed functional principal component and then extend the same method for the first \(K\) smoothed functional principal components.

MSC:

62H25 Factor analysis and principal components; correspondence analysis
62G05 Nonparametric estimation

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[1] Aguilera, A.; Aguilera-Morillo, M., Penalized pca approaches for B-spline expansions of smooth functional data, Appl. Math. Comput., 219, 7805-7819 (2013) · Zbl 1288.62098
[2] Aneiros-Pérez, G.; Vieu, P., Nonparametric time series prediction: A semi-functional partial linear modeling, J. Multivariate Anal., 99, 834-857 (2008) · Zbl 1133.62075
[3] Bali, J. L.; Boente, G.; Tyler, D. E.; Wang, J. L., Robust functional principal components: A projection-pursuit approach, Ann. Statist., 39, 2852-2882 (2011) · Zbl 1246.62145
[4] Besse, P.; Ramsay, J. O., Principal components analysis of sampled functions, Psychometrika, 51, 285-311 (1986) · Zbl 0623.62048
[5] Boente, G.; Fraiman, R., Kernel-based functional principal components, Statist. Probab. Lett., 48, 335-345 (2000) · Zbl 0997.62024
[6] Cardot, H.; Ferraty, F.; Sarda, P., Functional linear model, Statist. Probab. Lett., 45, 11-22 (1999) · Zbl 0962.62081
[7] Castro, P.; Lawton, W.; Sylvestre, E., Principal modes of variation for processes with continuous sample curves, Technometrics, 28, 329-337 (1986) · Zbl 0615.62074
[8] Cuevas, A., A partial overview of the theory of statistics with functional data, J. Statist. Plann. Inference, 147, 1-23 (2014) · Zbl 1278.62012
[9] Dauxois, J.; Pousse, A.; Romain, Y., Asymptotic theory for the principal component analysis of a vector random function: some applications to statistical inference, J. Multivariate Anal., 12, 1, 136-154 (1982) · Zbl 0539.62064
[10] Deville, J.-C., Méthodes statistiques et numériques de l’analyse harmonique, Ann. INSEE, 15, 3-101 (1974)
[11] Dunford, N.; Schwartz, J. T., Linear Operators. Part 2: Spectral Theory. Self Adjoint Operators in Hilbert Space (1963), Interscience Publishers · Zbl 0128.34803
[12] Ferraty, F.; Vieu, P., Nonparametric Functional Data Analysis: Theory and Practice (2006), Springer: Springer Berlin · Zbl 1119.62046
[13] Gilliam, D. S.; Hohage, T.; Ji, X.; Ruymgaart, F., The Fréchet derivative of an analytic function of a bounded operator with some applications, Int. J. Math. Math. Sci. (2009) · Zbl 1185.47013
[14] Gohberg, I.; Goldberg, S.; Kaashoek, M., (Classes of Linear Operators. Classes of Linear Operators, Operator Theory: Advances and Applications, vol. 1 (2013), Birkhäuser: Birkhäuser Basel)
[15] Goia, A.; Vieu, P., An introduction to recent advances in high/infinite dimensional statistics, J. Multivariate Anal., 146, 1-6 (2016) · Zbl 1384.00073
[16] Hall, P.; Hosseini-Nasab, M., On properties of functional principal components analysis, J. R. Stat. Soc. Ser. B Stat. Methodol., 68, 109-126 (2006) · Zbl 1141.62048
[17] Hall, P.; Müller, H.-G.; Wang, J. L., Properties of principal component methods for functional and longitudinal data analysis, Ann. Statist., 34, 1493-1517 (2006) · Zbl 1113.62073
[18] Horváth, L.; Kokoszka, P., Inference for Functional Data With Applications (2012), Springer: Springer Berlin · Zbl 1279.62017
[19] Hsing, T.; Eubank, R., Theoretical Foundations of Functional Data Analysis, With an Introduction to Linear Operators (2015), Wiley: Wiley New York · Zbl 1338.62009
[20] Huang, J. Z.; Shen, H.; Buja, A., Functional principal components analysis via penalized rank one approximation, Electron. J. Stat., 2, 678-695 (2008) · Zbl 1320.62097
[21] Hunter, J.; Nachtergaele, B., Applied Analysis (2001), World Scientific · Zbl 0981.46002
[22] Ingrassia, S.; Costanzo, G. D., Functional principal component analysis of financial time series, (New Developments in Classification and Data Analysis (2005), Springer), 351-358
[23] Kneip, A.; Utikal, K. J., Inference for density families using functional principal component analysis, J. Amer. Statist. Assoc., 96, 519-542 (2001) · Zbl 1019.62060
[24] Kokoszka, P.; Reimherr, M., Asymptotic normality of the principal components of functional time series, Stochastic Process. Appl., 123, 1546-1562 (2013) · Zbl 1275.62066
[25] Lax, P., Functional Analysis, Pure and Applied Mathematics (2002), Wiley: Wiley New York
[26] Luo, L.; Zhu, Y.; Xiong, M., Smoothed functional principal component analysis for testing association of the entire allelic spectrum of genetic variation, Eur. J. Hum. Genet., 21, 217-224 (2013)
[27] Ocaña, F. A.; Aguilera, A. M.; Escabias, M., Computational considerations in functional principal component analysis, Comput. Statist., 22, 449-465 (2007) · Zbl 1196.62080
[28] Ocaña, F. A.; Aguilera, A. M.; Valderrama, M. J., Functional principal components analysis by choice of norm, J. Multivariate Anal., 71, 262-276 (1999) · Zbl 0944.62059
[29] Qi, X.; Zhao, H., Some theoretical properties of Silverman’s method for smoothed functional principal component analysis, J. Multivariate Anal., 102, 741-767 (2011) · Zbl 1327.62223
[30] Ramsay, J. O.; Dalzell, C., Some tools for functional data analysis, J. R. Stat. Soc. Ser. B, 539-572 (1991) · Zbl 0800.62314
[31] Ramsay, J. O.; Silverman, B. W., Functional Data Analysis (2005), Springer: Springer Berlin · Zbl 0882.62002
[32] Reiss, P. T.; Ogden, R. T., Functional principal component regression and functional partial least squares, J. Amer. Statist. Assoc., 102, 984-996 (2007) · Zbl 1469.62237
[33] Rice, J. A.; Silverman, B. W., Estimating the mean and covariance structure nonparametrically when the data are curves, J. R. Stat. Soc. Ser. B Stat. Methodol., 53, 233-243 (1991) · Zbl 0800.62214
[35] Silverman, B. W., Smoothed functional principal components analysis by choice of norm, Ann. Statist., 24, 1-24 (1996) · Zbl 0853.62044
[36] Yao, F.; Lee, T., Penalized spline models for functional principal component analysis, J. R. Stat. Soc. Ser. B Stat. Methodol., 68, 3-25 (2006) · Zbl 1141.62050
[37] Yao, F.; Müller, H.-G., Functional quadratic regression, Biometrika, 97, 49-64 (2010) · Zbl 1183.62113
[38] Zhikov, V. V., Weighted Sobolev spaces, Sb. Math., 189, 1139-1170 (1998) · Zbl 0919.46026
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