Abstract
This paper develops the analytical form of the degrees of freedom in functional principal components analysis. Under the framework of unbiased risk estimation, we derive an unbiased estimator with a clear analytical formula for the degrees of freedom in the one-way penalized functional principal components analysis paradigm. Specifically, a new analytical formula incorporating binary smoothing parameters is also derived based on the singular value decomposition and half-smoothed method regarding the two-way penalized functional principal components analysis framework. The performance of our procedures is demonstrated by simulation studies.
Highlights
Functional principal component analysis (FPCA) is a key method for analyzing principal component from smoothing data, such as the data observed from temperature curves with sinusoidal nature
FPCA has become a crucial research focus in many statistical fields. e author in [1] proposed the principle of the roughness penalty to deal with data curves using the smoothing spline method. e authors in [1, 2] proposed the roughness penalty to analyze functional data by decomposing variation in a two-way data table
Utilizing the rank-one approximation to the data matrix and penalizing only the right eigenvectors, the authors in [3] considered that a sample X was observed from a linear model consisting of the combination of original left singular vectors, right eigenvectors, and observed noise
Summary
Functional principal component analysis (FPCA) is a key method for analyzing principal component from smoothing data, such as the data observed from temperature curves with sinusoidal nature. Utilizing the rank-one approximation to the data matrix and penalizing only the right eigenvectors, the authors in [3] considered that a sample X was observed from a linear model consisting of the combination of original left singular vectors, right eigenvectors, and observed noise. By singular value decomposition (SVD), the authors in [4] analyzed two-way functional data by penalizing left and right singular vectors of a generated covariance matrix.
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