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

Percy S. Zhai, Ping-Shou Zhong, Wei Biao Wu

arXiv:2609.30565v1stat.MEmath.STstat.MLstat.TH

TL;DR

Testing high-dimensional Gaussian graphical models for time series is understudied under long memory. The paper proposes a local-inverse test with Gaussian-approximation and block-bootstrap guarantees, plus a correction for over-specified graphs; it also extends testing to two-sample comparisons demonstrated on Sherlock fMRI data.

  • Problem

    High-dimensional time-series graph testing is understudied, particularly for long-memory data, while much existing work focuses on estimation or short-memory settings.

    Source: §1 · pp. 1–2

  • Method

    A local-inverse statistic and data-adaptive block bootstrap test graph structure, with a consistency-empowered correction for hypothesized graphs containing false edges.

    Source: §1 · p. 2

  • Results

    The proposed tests have finite-sample Gaussian-approximation and bootstrap-validity results, including ultra-high-dimensional applicability, and the correction preserves size while guaranteeing power.

    Source: §1 · p. 2, §3.1 · p. 5, §3.3 · p. 6

  • Takeaways & Limitations

    A two-sample extension compares graphical structures using the first group's estimated graph as the second group's null support, and is demonstrated on Sherlock fMRI movie and recall data.

    Source: §1 · p. 2, §5 · p. 9

    The theory assumes β > 3/4 and requires a nondegenerate long-run variance for the score process.

    Source: §2 · p. 3, §3.1 · p. 5

Abstract

from arXiv · show

Many real-world high-dimensional time series exhibit long-memory, but Gaussian graphical model testing in this regime remains understudied. We develop a direct, data-adaptive test statistic for assessing conditional independence in the graph structure of stationary Gaussian time series. We establish a finite-sample, Berry--Esseen type Gaussian approximation bound for the statistic, which applies to both short-memory and long-memory time series. The testing procedure is fully data-adaptive using block bootstrap method, on which we provide a finite-sample validity result including in the ultra-high-dimensional scenario, and can be extended to comparing graphical structures in two-sample tests. We also develop a consistency-empowered correction to the statistic and show that such tests attain asymptotic consistency in both size and power. Our proposed method is applied to a real-world fMRI data to understand functional connectivities within brain in different periods.

1 Introduction

High-dimensional graph testing for time series remains underexplored in the long-memory regime. This paper develops a direct test for stationary Gaussian series and extends it to two-sample graph comparisons, demonstrated on Sherlock fMRI data.

1 Introduction

  • Long-memory high-dimensional time-series graph testing remains largely unexplored, while much existing time-series graph work addresses estimation or short-memory inference.Earlier i.i.d. structural tests require independence or exchangeability, and short-memory inference methods do not easily extend to long memory.Source: §1 · pp. 1–2
  • The proposed local-inverse statistic receives finite-sample Gaussian-approximation and block-bootstrap validity guarantees, including for ultra-high-dimensional testing.Its first-order behavior is governed by sample covariance error, avoiding estimation of the full precision matrix.Source: §1 · p. 2
  • The consistency-empowered correction targets hypothesized graphs containing edges absent from the true graph while preserving test size and guaranteeing power.It screens estimated precision entries near zero on the null support and artificially amplifies them.Source: §1 · p. 2
  • The two-sample extension tests whether graph structures match across samples and is demonstrated by comparing movie-watching and verbal-recall periods in Sherlock fMRI data.The first group's estimated graph serves as the null support for testing the second group's data.Source: §1 · p. 2

2 Methodology

The methodology tests a hypothesized graph using a local-inverse statistic, calibrates it with a block bootstrap, and adds a correction for over-specified null graphs.

2 Methodology

  • The setup tests whether a hypothesized graph matches supp(Ω) for a stationary Gaussian linear process, allowing p much larger than n and dependence regimes indexed by β.The stated condition requires β > 3/4; β ≤ 1 describes the long-memory regime, where autocovariances are not summable.Source: §2 · p. 3

2.1 Direct Statistic

  • The direct statistic is the root-scale maximum of sample-covariance residuals over index pairs outside the hypothesized graph's support.It uses a local-inverse estimator built from the sample covariance and support sets specified by the hypothesized graph.Source: §2.1 · p. 3

2.2 Block Bootstrap

  • The block bootstrap forms plug-in score vectors, sums them over overlapping blocks, and uses their empirical distribution to choose the direct test's critical value.The direct test rejects when its statistic exceeds that bootstrap critical value.Source: §2.2 · pp. 3–4

2.3 Consistency-empowered Correction

  • The consistency-empowered correction inflates near-zero estimated precision entries on the hypothesized support, addressing alternatives where that graph contains false edges.With calibrated tuning parameters, the corrected statistic uses the same bootstrap critical value and retains sensitivity to missing edges while restoring power against over-specified alternatives.Source: §2.3 · p. 4

3 Theoretical Guarantee for the Proposed Statistics

The direct statistic reduces graph testing to a linearized covariance-error problem, enabling Gaussian approximation and data-adaptive block-bootstrap calibration. The guarantees extend to two-sample graph tests and a consistency-empowered correction.

3.1 Gaussian Approximation

  • Under the stated conditions, the statistic is approximated by the maximum absolute value of a centered Gaussian vector, with a finite-sample bound that can cover ultra-high dimensions.The bound can converge when log p = o(n^α) for some constant α.Source: §3.1 · pp. 4–5
  • The direct statistic uses a local inverse and is dominated by a linear term in covariance error, avoiding the need for precision-matrix inference theory.This reduction links goodness-of-fit testing to Gaussian approximation of the sample covariance error.Source: §3.1 · p. 4

3.2 Block Bootstrap

  • The unknown Gaussian target law is calibrated with an overlapping block bootstrap, whose finite-sample validity result also allows ultra-high-dimensional settings.Asymptotic validity follows when Rn,l(ϵ) tends to zero.Source: §3.2 · pp. 5–6
  • Bootstrap validity additionally requires controlling the gap between the plug-in score process and its oracle analogue before reducing to the oracle block-bootstrap problem.The plug-in error is controlled under the event specified in Proposition 2.Source: §3.2 · pp. 5–6

3.3 Theoretical Extensions

  • For a two-sample test using an estimated first-group graph as the second-group null, the guarantees persist with an extra error term for graph-estimation failure.With a consistent estimator, the null rejection probability tends to α when Rn,l(ϵ) tends to zero.Source: §3.3 · p. 6
  • The consistency-empowered statistic preserves null size and can attain power against fixed alternatives, including when the hypothesized graph contains false edges.Under the null it equals the direct statistic with probability tending to one; the stated correction can provide power without affecting size.Source: §3.3 · p. 6

4 Quantitative Study

Simulations examine null calibration across dependence regimes and power under off-band alternatives, including a comparison of the consistency-empowered and direct statistics. Calibration is empirically reasonable in the covered long-memory regime, while power trends depend on signal strength, memory, and dimension.

4.1 Null Test Validity

  • Null calibration was empirically reasonable when p greatly exceeded n and temporal dependence was long-range, but failed as expected for β = 0.55 outside the theory’s range.The study considered n ∈ {200, 400, 800}, p ∈ {2000, 4000, 8000}, and β ∈ {2, 0.9, 0.55}.Source: §4.1 · p. 7

4.2 Alternative Test Power

  • Rejection rates generally rose with edge strength κ or the number of added edges m, while weak-signal rates stayed near nominal and short-memory power converged to one.Convergence was slower with longer memory or higher dimension.Source: Table 1 · p. 8, §4.2 · p. 8
  • The consistency-empowered test had higher rejection rates than the direct test, especially in short-memory, low-dimensional settings; its advantage was less pronounced under long memory.For short-memory, low-dimensional cases, rejection rates rapidly approached one as η and χ increased.Source: Table 2 · p. 9, §4.2 · p. 9

5 Sherlock fMRI Data Analysis

The Sherlock fMRI analysis tests whether each subject’s recall-phase connectivity graph contains dependencies absent from their movie-phase graph. All 17 recall tests reject after Holm correction, while all selected movie graphs pass the self-diagnostic.

5 Sherlock fMRI Data Analysis

  • The analysis compares movie and recall scans from 17 subjects across 268 cortical regions, with recall series ranging from 437 to 1824 time points.Each movie-phase series has 1976 time points.Source: §5 · p. 9
  • For each subject, the movie-phase graph is selected from a threshold path using the largest threshold whose movie self-test p-value is at least 0.05.The selected graph is the sparsest on that path to pass the self-diagnostic.Source: §5 · p. 9
  • Recall-phase p-values come from the block bootstrap and are adjusted across subjects with the Holm procedure; rejection signals off-support recall dependencies.The paper reports average column degree and maximum sparsity for each selected graph in Table 5 in Appendix C.3.Source: §5 · p. 9
  • All 17 recall tests reject after Holm correction, indicating dependencies outside the corresponding movie-phase graph for every subject.All 17 selected movie graphs pass the self-diagnostic at α = 0.05.Source: §5 · p. 9

6 Concluding Remark

The conclusion presents a local-inverse, sample-covariance-based framework for data-adaptive graph testing, while identifying dependence and graph-knowledge boundaries. The appendix also records notation and summarizes the two statistic algorithms.

6 Concluding Remark

  • The framework reduces graph-specification testing to simultaneous covariance inference via a local inverse, avoiding full precision-matrix estimation; its theory assumes β > 3/4.For 1/2 < β ≤ 3/4, the asymptotic distribution is described as Rosenblatt-type and is outside the paper’s scope; the one-sample test also requires a predetermined graph.Source: §6 · p. 10

A.1 Notation

  • The notation section defines vector and matrix norms, support, cardinality, and asymptotic-order conventions used throughout.It specifies |·|∞ as the maximum absolute entry and |·|1 as the matrix ℓ1 norm.Source: §A.1 · p. 13

A.2 Algorithms

  • The appendix distinguishes the direct statistic ˆTn from the consistency-empowered corrected statistic ˜Tn and outlines block-bootstrap threshold computation for both.Algorithms 1 and 2 compute their respective statistics using the same bootstrap rejection threshold.Source: §A.2 · p. 13

B.1 Detailed Discussion on Two-sample Extension

The two-sample extension is directional: it uses a graph estimated from the first sample as the null graph for testing the second. Its corollary gives a Gaussian approximation bound under conditions that include consistency of the first-sample graph estimator.

B.1 Detailed Discussion on Two-sample Extension

  • The extension tests whether the second sample’s graph differs from the first sample’s graph estimate, making the comparison directional.The first-sample estimate supplies the second sample’s null graph.Source: §B.1 · p. 14
  • The second-sample bootstrap uses the estimated first-sample null graph to form its directional CDF and critical value.The critical value is defined from the directional bootstrap CDF.Source: §B.1 · p. 14
  • Corollary 5 states a Gaussian approximation bound when the first-sample graph-estimation error vanishes and the second sample satisfies the stated theorem assumptions.The result also specifies conditions involving the approximation error and continuity at the Gaussian maximum’s (1 − α)-quantile.Source: §B.1 · pp. 14–15

B.2 Detailed Discussion on Consistency-empowered Estimator

The consistency-empowered correction uses long-run variability to screen local-inverse coordinates and is designed to retain null calibration while achieving power under fixed alternatives. The appendix also describes FFT-based generation of approximate simulation copies.

B.2 Detailed Discussion on Consistency-empowered Estimator

  • The correction screens inside-support local-inverse coordinates using a standard error based on long-run variance, estimated with overlapping blocks.The estimator is stated to be uniformly consistent under the lemma’s conditions; the variance accounts for temporal dependence.Source: §B.2 · p. 15
  • The screening result assumes nonzero inside-support precision coordinates are uniformly bounded away from zero; shrinking coordinates would require an explicit beta-min condition.The stated signal condition prevents true nonzero coordinates from being mistakenly pumped under the null.Source: §B.2 · p. 16
  • Under the stated conditions, the corrected statistic retains the direct test’s Gaussian approximation and bootstrap validity under the null, and its power tends to one under fixed alternatives.The null condition includes nonzero precision coordinates bounded away from zero; fixed alternatives do not approach the null as n grows.Source: §B.2 · p. 16

C.1 Generating Copies of Time Series using Fast Fourier Transformation

  • For simulations, the FFT construction generates approximate length-n copies by splitting a long circular convolution, with N = n^2 used for the truncation length.The resulting blocks are weakly dependent, and the construction keeps the high-dimensional simulation regimes computationally feasible.Source: §C.1 · pp. 16–17

C.2 Additional Null Test Results

The section describes null-test approximation diagnostics under ultra-long memory and defines the distance measures used to assess Gaussian approximation and block-bootstrap proximity.

C.2 Additional Null Test Results

  • Figure 3 plots the null-test statistic against Gaussian and block-bootstrap approximations under ultra-long memory, with proximity to y=x indicating closer agreement.The setting uses β=0.55, beyond Condition 1's range.Source: Figure 3 · p. 17
  • Table 3 compares the null-test statistic with its finite-sample Gaussian and overlapping-block-bootstrap approximations using Kolmogorov distance.Smaller distances indicate closer approximation; the distance ranges from 0 to 1.Source: Table 3 · p. 18
  • Table 4 compares the null-test statistic with its finite-sample Gaussian and overlapping-block-bootstrap approximations using Wasserstein-1 distance.Smaller distances indicate closer approximation; the distance ranges from 0 to infinity.Source: Table 4 · p. 18

C.3 Sherlock fMRI Experiment Result

The section reports the Sherlock movie-phase graph and recall-containment test setup, alongside Hurst-exponent and autocorrelation evidence of long memory in movie-phase time series.

C.3 Sherlock fMRI Experiment Result

  • Table 5 reports selected movie-graph degree summaries and movie self-diagnostic, raw recall-containment, and Holm-adjusted recall p-values at α=0.05.The Holm adjustment controls family-wise error across the 17 subjects.Source: Table 5 · p. 18

C.4 Long Memory of Sherlock fMRI Time Series

  • 99.8% of movie-phase subject-ROI series have Hurst exponent H>0.5, and the median Hurst exponent is 0.719.The passage identifies H>0.5 as consistent with long-memory.Source: §C.4 · p. 19, Figure 4 · p. 19
  • The selected 50th-, 75th-, 90th-, and 95th-percentile Hurst series have significant autocorrelation lags near 100, with especially slow decay in the latter two.The passage treats this pattern together with H>0.5 as evidence of long-memory.Source: §C.4 · p. 19

D.1 Proof of Proposition 1

The proof sections bound local-inverse estimation error, establish Gaussian approximation through selected score coordinates, and control the plug-in estimator's deviation from its oracle counterpart.

D.1 Proof of Proposition 1

  • On the stated event, the local-inverse column estimates have maximum ℓ1 error bounded by C1s0^2|Δ|∞^2.The proof first establishes invertibility of the estimated local matrices on that event.Source: §D.1 · pp. 19–21

D.2 Proof of Theorem 1

  • Theorem 1's proof applies Gaussian approximation to selected score coordinates represented through a derived Gaussian linear process, then transfers the approximation to the statistic.The argument invokes a nondegeneracy condition for the selected coordinates.Source: §D.2 · pp. 21–23

D.3 Proof of Proposition 2

  • The plug-in local-inverse columns differ from their oracle counterparts by at most C′1s0|Δ|∞ in maximum ℓ1 norm.The proof also bounds the maximum ℓ1 norms of the oracle and plug-in columns.Source: §D.3 · p. 24

D.4 Proof of Theorem 2

The proof of Theorem 2 applies block-bootstrap approximation to oracle scores, then controls plug-in errors to transfer the result to the proposed statistic.

D.4 Proof of Theorem 2

  • The proof constructs an oracle score process as a derived Gaussian linear process and applies a selected-coordinate block-bootstrap theorem to its overlapping block sums.The selected coordinate count is at most p^2.Source: §D.4 · pp. 24–25
  • The proof compares the plug-in block CDF with the oracle CDF and bounds the plug-in block-score error before transferring bootstrap validity.The comparison controls the block CDF uniformly over its argument.Source: §D.4 · pp. 25–26
  • The argument controls covariance-estimation and score-process errors, then combines those bounds to obtain the theorem's probability bound.It also bounds the maximum of the score process using a Gaussian tail bound and a union bound.Source: §D.4 · p. 26

E.1 Proof of Corollary 5

These proofs establish asymptotic size and power for the corrected statistic, derive supporting plug-in control and screening behavior, and transfer direct-statistic bounds to a random null.

E.1 Proof of Corollary 5

  • The random-null proof reduces to a fixed null graph, where Theorems 1 and 2 give Gaussian-approximation and bootstrap bounds; continuity at the limiting quantile yields exact asymptotic level.On the event that the estimated null graph equals the first-sample graph, the statistic is the direct statistic computed from the second sample with that fixed null.Source: §E.1 · pp. 27–28

E.2 Proof of Lemma 1

  • For local-inverse scores, the proof compares plug-in and oracle block estimators; on the stated event, the maximum coordinate error in vhat0,ij is at most Cs0|Σhat_n−Σ|∞.The variance-estimator comparison also controls deterministic bias and stochastic fluctuation of overlapping block averages.Source: §E.2 · pp. 28, 30

E.3 Proof of Lemma 2

  • Under the null, screening sets every Δhat_j to zero with probability tending to one; for a fixed strictly included alternative, at least one is nonzero with probability tending to one.The alternative argument uses a coordinate whose restricted population inverse is zero.Source: §E.3 · p. 31

E.4 Proof of Theorem 6

  • The corrected statistic has asymptotic size under the null and rejection probability tending to one for every fixed alternative E ≠ E0.Under the null, it agrees with the direct statistic with probability tending to one, so Gaussian approximation and bootstrap validity follow from Theorems 1 and 2.Source: §E.4 · pp. 32, 34
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