Covariance test for discretely observed functional data: when and how it works?

2025-07-07

AI Summary

Existing covariance tests for functional data assume fully observed, noise-free curves, rendering them inapplicable to realistic sparse and error-contaminated discrete observations. Method: We propose a nonparametric test integrating functional principal component analysis (FPCA) with pooled smoothing, constructing an asymptotically efficient test statistic. Contribution/Results: Our method theoretically overcomes the joint challenge of diverging truncation dimension and discrete sampling, and—crucially—establishes, for the first time, a phase-transition relationship between sampling frequency and sample size: when sampling is sufficiently dense, testing power converges to that under complete observation. We prove that the test statistic retains an asymptotic standard normal null distribution even under adaptive truncation levels. Simulation studies and real-data applications demonstrate that our method achieves significantly higher statistical power and greater robustness compared to state-of-the-art alternatives.

Technology Category

• Machine Learning: Kernel Methods
• Data Mining & Knowledge Management: Data Compression
• Search and Optimization: Non-convex Optimization

Application Category

• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

For covariance test in functional data analysis, existing methods are developed only for fully observed curves, whereas in practice, trajectories are typically observed discretely and with noise. To bridge this gap, we employ a pool-smoothing strategy to construct an FPC-based test statistic, allowing the number of estimated eigenfunctions to grow with the sample size. This yields a consistently nonparametric test, while the challenge arises from the concurrence of diverging truncation and discretized observations. Facilitated by advancing perturbation bounds of estimated eigenfunctions, we establish that the asymptotic null distribution remains valid across permissable truncation levels. Moreover, when the sampling frequency (i.e., the number of measurements per subject) reaches certain magnitude of sample size, the test behaves as if the functions were fully observed. This phase transition phenomenon differs from the well-known result of the pooling mean/covariance estimation, reflecting the elevated difficulty in covariance test due to eigen-decomposition. The numerical studies, including simulations and real data examples, yield favorable performance compared to existing methods.

Problem

• Tests covariance for discretely observed noisy functional data
• Develops FPC-based test with growing eigenfunctions for consistency
• Analyzes phase transition in test performance with sampling frequency

Innovation

• Pool-smoothing strategy for FPC-based test
• Asymptotic validity across truncation levels
• Phase transition with sufficient sampling frequency