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Frequency Domain Statistical Inference for High-Dimensional Time Series

Jonas Krampe, Efstathios Paparoditis

TL;DR

The paper develops a comprehensive framework for frequency-domain inference in high-dimensional time series by combining nonparametric spectral density estimation with regularized inversion and a de-biased construction for partial coherence. It delivers asymptotically Gaussian inference for coherence and de-biased partial coherence, and introduces max-type tests with FDR control for both single and multiple hypotheses across frequency bands. The approach is validated in simulations and applied to EEG brain connectivity, illustrating accurate control of false discoveries and discovery of direct functional connections. Overall, the methods enable scalable, principled inference of complex cross-sectional and conditional dependencies in large multivariate time series, with practical impact in neuroscience and related fields.

Abstract

Analyzing time series in the frequency domain enables the development of powerful tools for investigating the second-order characteristics of multivariate processes. Parameters like the spectral density matrix and its inverse, the coherence or the partial coherence, encode comprehensively the complex linear relations between the component processes of the multivariate system. In this paper, we develop inference procedures for such parameters in a high-dimensional, time series setup. Towards this goal, we first focus on the derivation of consistent estimators of the coherence and, more importantly, of the partial coherence which possess manageable limiting distributions that are suitable for testing purposes. Statistical tests of the hypothesis that the maximum over frequencies of the coherence, respectively, of the partial coherence, do not exceed a prespecified threshold value are developed. Our approach allows for testing hypotheses for individual coherences and/or partial coherences as well as for multiple testing of large sets of such parameters. In the latter case, a consistent procedure to control the false discovery rate is developed. The finite sample performance of the inference procedures introduced is investigated by means of simulations and applications to the construction of graphical interaction models for brain connectivity based on EEG data are presented.

Frequency Domain Statistical Inference for High-Dimensional Time Series

TL;DR

The paper develops a comprehensive framework for frequency-domain inference in high-dimensional time series by combining nonparametric spectral density estimation with regularized inversion and a de-biased construction for partial coherence. It delivers asymptotically Gaussian inference for coherence and de-biased partial coherence, and introduces max-type tests with FDR control for both single and multiple hypotheses across frequency bands. The approach is validated in simulations and applied to EEG brain connectivity, illustrating accurate control of false discoveries and discovery of direct functional connections. Overall, the methods enable scalable, principled inference of complex cross-sectional and conditional dependencies in large multivariate time series, with practical impact in neuroscience and related fields.

Abstract

Analyzing time series in the frequency domain enables the development of powerful tools for investigating the second-order characteristics of multivariate processes. Parameters like the spectral density matrix and its inverse, the coherence or the partial coherence, encode comprehensively the complex linear relations between the component processes of the multivariate system. In this paper, we develop inference procedures for such parameters in a high-dimensional, time series setup. Towards this goal, we first focus on the derivation of consistent estimators of the coherence and, more importantly, of the partial coherence which possess manageable limiting distributions that are suitable for testing purposes. Statistical tests of the hypothesis that the maximum over frequencies of the coherence, respectively, of the partial coherence, do not exceed a prespecified threshold value are developed. Our approach allows for testing hypotheses for individual coherences and/or partial coherences as well as for multiple testing of large sets of such parameters. In the latter case, a consistent procedure to control the false discovery rate is developed. The finite sample performance of the inference procedures introduced is investigated by means of simulations and applications to the construction of graphical interaction models for brain connectivity based on EEG data are presented.
Paper Structure (21 sections, 21 theorems, 164 equations, 3 figures, 2 tables)

This paper contains 21 sections, 21 theorems, 164 equations, 3 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Estimation of Frequency Domain Parameters
  3. Estimation of Coherence and Partial Coherence
  4. Testing
  5. Testing Single Hypothesis
  6. Testing Multiple Hypotheses
  7. Implementation Issues and Simulations
  8. Implementation Issues
  9. Simulations
  10. A Graphical Model for Brain Connectivity
  11. Asymptotic Considerations
  12. Auxiliary Results
  13. Error bounds in the construction of de-biased partial coherences
  14. Error bounds for Gaussian approximations
  15. Covariance structure of lag-window estimators
  16. ...and 6 more sections

Key Result

Theorem 1

Suppose that $H_0^{(u,v)}$ is true. Then, under the conditions of Theorem thm.partial.coherence of Section sec.2.properties, the test eq_test satisfies $\lim_{n\rightarrow\infty} P(T_n^{(u,v)}\geq \mathcal{G}(1-\alpha))\leq \alpha$.

Figures (3)

  • Figure 1: Results for detecting non-zero partial coherences for the $50$-dimensional VARMA$(1,1)$ process with parameters as in (a) and for the sample size $n=512$ as well as for several choices of the tuning parameter $\lambda$ used in the implementation of graphical lasso. The solid lines refer to FDR and the dashed lines to power. The behavior of Testing is described by the solid and the dashed black lines while that of Regularizing by the solid and the dashed red lines. The horizontal line in each figure denotes the target FDR-level $\alpha$ (which is $0.05$ in the top figure and $0.2$ in the bottom figure).
  • Figure 2: Graphical model representing the brain connectivity for students 19 and 20. The vertex labels denote the channel labels of the EEG recording and the patient's nose is located at the top. The figures in the top row representing the brain connectivity for student 19 in the state of eyes open. In this row, the left figure displays the unconditional connectivity based on coherence and the right figure, the conditional connectivity based on partial coherence. The two figures in the second row show the conditional brain connectivity of student 20 in the two states considered, eyes open and eyes closed.
  • Figure 3: Detection of non-zero partial coherences for the VARMA$(1,1)$ process with parameters as in (a) and the VMA$(5)$ process with parameters as in (d). The upper triangular part of every plot presents results for $n=512$ and the lower triangular part for $n=4096$. A bluish dot represents a not discovery of a non-zero partial coherence, where the darker the blue color is the lower is the correct detection rate. A reddish dot represents a false discovery of a zero partial coherence, where the darker the red color is the higher is the false detection rate. The left column of plots present results for the testing procedure while the right column of plots present results of the regularizing procedure.

Theorems & Definitions (43)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Corollary 4
  • Theorem 5
  • Lemma 6
  • proof : Proof
  • Lemma 7
  • proof : Proof
  • Lemma 8
  • ...and 33 more