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

Area of Science:

• Statistics
• Functional Data Analysis

Background:

• Principal Component Analysis (PCA) is crucial in functional data analysis.
• Smoothing and regularization are essential for estimating principal component curves in FPCA.
• Silverman's method offers theoretical and practical advantages for fully observed curves but lacks theoretical understanding for discrete data.

Purpose of the Study:

• To generalize smoothed functional PCA to situations with discrete, observed sample curves.
• To establish theoretical properties of Silverman's method for discrete functional data.
• To analyze the estimation errors of eigenvalues and eigenfunctions.

Main Methods:

• Establishing the existence of solutions for successive optimization problems.
• Deriving upper bounds for the bias of estimation errors.
• Proving functional central limit theorems for the variation of estimation errors.

Main Results:

• Theoretical properties of smoothed functional PCA for discrete data are established.
• Convergence rates for estimating eigenvalues and eigenfunctions are determined, depending on sample size and smoothing parameters.
• Asymptotic normality of estimations is proven under specific conditions on smoothing parameters.

Conclusions:

• The study provides a theoretical foundation for using smoothed functional PCA with discrete data.
• The findings offer insights into the accuracy and convergence of functional principal component estimations.
• This work extends the applicability of advanced functional data analysis techniques.