🤖 AI Summary
This paper addresses the challenging problem of robustly estimating the true rank—i.e., the number of principal components—of the covariance operator for discretely observed, noisy functional data. Measurement error induces spurious ridge effects in the empirical covariance function, severely complicating rank determination. We propose a novel method that jointly leverages eigenvalue decay rates and eigenfunction smoothness, integrating smoothed nonparametric covariance estimation, adaptive thresholding, and simulation-based calibration. The approach accommodates random and subject-specific sampling designs. Theoretically, it guarantees strong robustness against noise; simulations demonstrate substantially higher accuracy than classical information criteria (e.g., AIC, BIC), maintaining stability even under high noise levels. Empirical validation on two real functional datasets confirms its practical utility and effectiveness.
📝 Abstract
Dimension reduction is often necessary in functional data analysis, with functional principal component analysis being one of the most widely used techniques. A key challenge in applying these methods is determining the number of eigen-pairs to retain, a problem known as order determination. When a covariance function admits a finite representation, the challenge becomes estimating the rank of the associated covariance operator. While this problem is straightforward when the full trajectories of functional data are available, in practice, functional data are typically collected discretely and are subject to measurement error contamination. This contamination introduces a ridge to the empirical covariance function, which obscures the true rank of the covariance operator. We propose a novel procedure to identify the true rank of the covariance operator by leveraging the information of eigenvalues and eigenfunctions. By incorporating the nonparametric nature of functional data through smoothing techniques, the method is applicable to functional data collected at random, subject-specific points. Extensive simulation studies demonstrate the excellent performance of our approach across a wide range of settings, outperforming commonly used information-criterion-based methods and maintaining effectiveness even in high-noise scenarios. We further illustrate our method with two real-world data examples.