Some theoretical properties of Silverman's method for Smoothed functional principal component analysis

JOURNAL OF MULTIVARIATE ANALYSIS

卷:102

Article

Qi, Xin¹  Zhao, Hongyu¹ 

[1] Yale Univ, Dept Epidemiol & Publ Hlth, New Haven, CT 06520 USA

关键词: Functional PCA;    Smoothing methods;    Roughness penalty;    Convergence rates;    Functional central limit theorem;    Asymptotic normality;   

DOI  :  10.1016/j.jmva.2010.12.001

来源: Elsevier

【 摘 要 】

Principal component analysis (PCA) is one of the key techniques in functional data analysis. One important feature of functional PCA is that there is a need for smoothing or regularizing of the estimated principal component curves. Silverman's method for smoothed functional principal component analysis is an important approach in a situation where the sample curves are fully observed due to its theoretical and practical advantages. However, lack of knowledge about the theoretical properties of this method makes it difficult to generalize it to the situation where the sample curves are only observed at discrete time points. In this paper, we first establish the existence of the solutions of the successive optimization problems in this method. We then provide upper bounds for the bias parts of the estimation errors for both eigenvalues and eigenfunctions. We also prove functional central limit theorems for the variation parts of the estimation errors. As a corollary, we give the convergence rates of the estimations for eigenvalues and eigenfunctions, where these rates depend on both the sample size and the smoothing parameters. Under some conditions on the convergence rates of the smoothing parameters, we can prove the asymptotic normalities of the estimations. (C) 2010 Elsevier Inc. All rights reserved.

【 授权许可】

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