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High-dimensional Gaussian Graphical Model Testing for Long-Memory Time Series

NeurIPS 2026 · Time series › Causal and generative time series · approach: Probabilistic and conformal forecasting · 0 citations (OpenAlex)

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Proposes a data-adaptive block-bootstrap test for Gaussian graphical model structure (precision-matrix support) in high-dimensional long-memory stationary time series, with a consistency-empowered correction and a two-sample extension, applied to Sherlock fMRI.

Problem

Testing whether the conditional-independence graph of a high-dimensional stationary Gaussian time series equals a hypothesized graph, when the series may exhibit long-range dependence (long memory) and dimension may far exceed sample size.

Approach

A direct local-inverse-based statistic built from the sample covariance (off-support residuals of Sigma*Omega0 = I) whose first-order behavior is governed by sample covariance error, so Gaussian approximation and block bootstrap for covariance error can be imported without estimating the full precision matrix; adds a screening-and-pumping correction to gain power against over-specified graphs.

Key results

• Null QQ and distance tables show reasonable Gaussian approximation and bootstrap calibration for beta=0.9 (long memory) with n in {200,400,800}, p up to 8000; beta=0.55 fails as expected.
• Under random off-band alternatives with (n,p)=(800,2000), beta=2, m=2000, kappa=0.15 the rejection rate is 1.000, while beta=0.9 power rises more slowly (e.g. 0.205 at kappa=0.18, m=2000).
• On Sherlock fMRI, all 17 subjects' recall-phase tests reject the movie-phase graph after Holm correction at alpha=0.05 (max adjusted p=0.0145).

Surprising findings

• Under long memory (beta=0.9) the consistency-empowered correction gives much smaller power gains over the direct test than in short memory, and power grows slowly with dimension.
• Bootstrap calibration is poor for some long-memory high-dimensional settings (e.g. beta=0.9, n=200 Kolmogorov distance 0.84 for bootstrap in the table of distances).

Limitations

• Theory requires temporal dependence beta>3/4; ultra-long memory (1/2<beta<=3/4) with Rosenblatt-type limits is out of scope.
• One-sample test requires the hypothesized graph to be prespecified; entrywise inference not provided.
• Consistency-empowered correction requires signal conditions (nonzero precision entries bounded away from zero) under the null.

Experimental setup

Datasets: Sherlock fMRI

Benchmarks: simulated Gaussian linear processes Sherlock fMRI

Metrics: rejection rate Kolmogorov distance Wasserstein-1 distance Holm-adjusted p-value

Baselines: direct test T_n (ablation)

Builds on: Gaussian approximation for high-dimensional time series (Chang et al.) consistency-empowered graph test (Liu et al.) block bootstrap for high-dimensional covariance under long memory

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