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

1. Problem Statement

Functional Data Analysis (FDA) often deals with data observed discretely and with noise (e.g., medical imaging, meteorology), rather than as fully observed continuous curves. While hypothesis testing for mean functions in FDA is well-established, testing for covariance operators (second-order structures) remains challenging, particularly for discretely observed data.

• The Gap: Existing covariance tests are primarily designed for fully observed curves or rely on fixed truncation (using a constant number of eigenfunctions). Fixed truncation violates the nonparametric nature of the problem, as it may miss differences in high-order components.
• The Challenge: Bridging the gap between discrete observations and continuous theory requires handling:
  1. Noise and Sparsity: Data is observed at $N$ discrete time points with measurement error.
  2. Diverging Truncation: To be truly nonparametric, the number of eigenfunctions ($K$) used in the test must grow with the sample size ($n$).
  3. Perturbation Bounds: Theoretical analysis requires sharp bounds on the estimation error of eigenfunctions derived from pooled, noisy data, which is significantly more complex than for fully observed data.

2. Methodology

The authors propose a nonparametric covariance test based on Functional Principal Components (FPC) that accommodates discretely observed data through a pool-smoothing strategy.

A. Data Model

• Two groups of processes, $X(s)$ and $Y(t)$, are observed at discrete time points $s_{ip}, t_{iq}$ with additive noise: $X_{ip} = X_i(s_{ip}) + \epsilon_{ip}$.
• The sampling frequency $N$ (number of points per subject) can vary from "sparse" ($N \ll n$) to "dense" ($N \gg n$).

B. Estimation Strategy

1. Pool-Smoothing: Instead of smoothing individual curves (which fails in sparse regimes), the authors pool all observations from both groups to estimate the population covariance functions ($\hat{C}_X, \hat{C}_Y$) and the pooled covariance operator ($\hat{C}_P$).
2. Eigen-decomposition: Eigenfunctions ($\hat{e}_j$) and eigenvalues are extracted from $\hat{C}_P$.
3. Sample Splitting: To handle the dependence between the estimated eigenfunctions and the data used to calculate test statistics, the authors employ a sample-splitting technique:
   • Split data into two disjoint sets ($Z$ and $Z'$).
   • Estimate eigenfunctions on $Z$.
   • Calculate FPC scores and test statistics on $Z'$ using the eigenfunctions from $Z$.
4. Diagonal Elimination: To correct for bias introduced by measurement noise in the estimation of second moments (variances of scores), the authors remove diagonal terms ($p=p'$) in the Monte Carlo integration approximation.

C. Test Statistics

1. Non-standardized Statistic ($\hat{T}_{K, Z'}$): A theoretical benchmark statistic constructed from the squared differences of projected covariance operators. It converges to a quadratic form of a Gaussian measure in infinite-dimensional space.
2. Normalized Statistic ($\hat{D}_{K, Z'}$): A practical test statistic where the non-standardized statistic is normalized by estimated variances of the scores. This allows for a Chi-squared ($\chi^2$) test with degrees of freedom $K(K+1)/2$.

3. Key Theoretical Contributions

The paper establishes rigorous asymptotic theory for the proposed test under a diverging truncation regime ($K \to \infty$ as $n \to \infty$).

A. Perturbation Bounds for Diverging Eigenfunctions

The authors derive new perturbation bounds for estimated eigenfunctions based on pool-smoothing of discretely observed data. This extends previous work (e.g., Hall and Horowitz, 2007) which was limited to fully observed data or fixed $K$.

• The bounds depend on the sample size $n$, sampling frequency $N$, bandwidth $h$, and the decay rate of eigenvalues ($\alpha$).

B. Phase Transition Phenomenon

A central finding is the existence of a phase transition regarding the relationship between sampling frequency ($N$) and sample size ($n$). The behavior of the test changes depending on whether $N$ is "sparse" or "dense":

1. Non-standardized Statistic (Theoretical Benchmark):

   • Dense Regime: If $N \gtrsim n^{1/(\alpha+1)}$, the test behaves as if data were fully observed. The allowable truncation level $K$ can grow as $O(n^{1/(2\alpha+2)})$.
   • Sparse Regime: If $N$ is smaller, $K$ must be restricted to $O(N^{1/2})$ (roughly) to maintain validity.

2. Normalized Statistic (Practical Implementation):

   • Dense Regime: If $N \gtrsim n^{(\alpha+1)/(\alpha+3)}$, the test achieves a valid asymptotic $\chi^2$ distribution with $K$ growing up to $O(n^{1/(2\alpha+6)})$.
   • Sparse Regime: If $N$ is below this threshold, the allowable $K$ shrinks to $O(N^{1/(2\alpha+2)})$.

Significance of Phase Transition:

• The transition threshold for covariance testing ($N \sim n^{(\alpha+1)/(\alpha+3)}$) is higher than that for mean/covariance estimation ($N \sim n^{1/4}$).
• This reflects the elevated difficulty of covariance testing due to the need to estimate a diverging series of eigenfunctions and their variances, which introduces additional instability compared to simple estimation.

4. Results

A. Simulation Studies

• Setup: Compared the proposed method ($T_{pool}$) against existing methods ($T_{PK}$ from Panaretos et al., 2010; $T_{FH}$ from Fremdt et al., 2013) under various signal strengths and sparsity levels ($N=6, 15, 30$).
• Findings:
  • Size Control: $T_{pool}$ maintains the nominal significance level (0.05) even as $K$ increases, provided $N$ and $n$ are sufficient. Competitors ($T_{FH}, T_{PK}$) suffer from severe size inflation (false positives) as $K$ grows, especially in sparse designs.
  • Power: $T_{pool}$ demonstrates superior power in detecting differences in high-order components, particularly in sparse regimes where competitors fail.
  • Robustness: The method is robust to the choice of bandwidth $h$.

B. Real Data Application

• Dataset: Longitudinal Diffusion Tensor Imaging (DTI) data from the Alzheimer's Disease Neuroimaging Initiative (ADNI), comparing Cognitively Normal (CN) vs. Alzheimer's Disease (AD) groups.
• Result: The proposed test ($T_{pool}$) detected significant differences in covariance structures at higher truncation levels ($K \ge 5$), with p-values dropping to 0.004.
• Comparison: Competing methods yielded inconsistent or misleading results (e.g., $T_{PK}$ showed p-values of 0 for all $K$, indicating severe size inflation; $T_{FH}$ failed to detect differences at higher $K$).
• Validation: Permutation and bootstrap tests confirmed that $T_{pool}$ controls the Type I error rate correctly, while competitors did not.

5. Significance and Conclusion

• Theoretical Breakthrough: This is the first work to provide a consistent nonparametric covariance test for discretely observed functional data with diverging truncation. It rigorously quantifies the "phase transition" between sparse and dense sampling regimes.
• Practical Guidance: The paper provides clear guidelines on how many principal components ($K$) can be safely used based on the sampling frequency ($N$) and sample size ($n$). It advises shrinking $K$ in sparse regimes but allowing it to grow in dense regimes.
• Methodological Innovation: The combination of pool-smoothing, sample-splitting, and diagonal-elimination effectively controls bias and variance, overcoming the theoretical hurdles of estimating eigenfunctions from noisy, discrete data.
• Impact: The proposed method enables more powerful and reliable inference on the second-order structures of functional data in real-world scenarios where dense, noise-free observations are rarely available.