[Paper Review] Frequency Domain Statistical Inference for High-Dimensional Time Series

Jonas Krampe, Efstathios Paparoditis|arXiv (Cornell University)|Jun 5, 2022

Neural dynamics and brain function4 citations

TL;DR

This paper develops frequency-domain statistical inference procedures for high-dimensional time series, focusing on consistent estimation and testing of coherence and, especially, partial coherence. It introduces a debiased estimator with a manageable limiting distribution, enabling hypothesis tests on maximum coherence/partial coherence across frequencies and a consistent false discovery rate control for large-scale multiple testing, validated via simulations and EEG-based brain connectivity modeling.

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.

Motivation & Objective

• Address the challenge of statistical inference for spectral parameters in high-dimensional time series where standard methods fail due to dimensionality.
• Develop consistent estimators for partial coherence—critical for graphical models of time-dependent processes—under high-dimensional asymptotics.
• Construct valid hypothesis tests for the maximum coherence or partial coherence across frequencies, enabling global inference on linear dependence structures.
• Provide a consistent procedure for controlling the false discovery rate (FDR) in large-scale multiple testing of coherence/partial coherence parameters.
• Enable practical application to complex systems such as brain connectivity networks using EEG data through robust, scalable inference tools.

Proposed method

• Propose a debiased estimator of the partial coherence in the frequency domain, correcting for bias in spectral density matrix estimators.
• Derive the asymptotic distribution of the debiased partial coherence estimator under high-dimensional asymptotics, ensuring validity for inference.
• Construct a test statistic based on the maximum absolute value of the debiased partial coherence across a set of frequencies, with a data-driven threshold.
• Use a modified chi-squared approximation to determine critical values, accounting for the dependence structure and dimensionality.
• Implement a stepwise procedure to control the false discovery rate (FDR) in multiple testing scenarios by estimating the expected number of false positives.
• Adapt bandwidth selection per pair of time series using individualized kernels and prewhitening to improve estimation accuracy in high dimensions.

Experimental results

Research questions

• RQ1 Can consistent and asymptotically normal estimators of partial coherence be constructed in high-dimensional time series settings?
• RQ2 How can valid hypothesis tests be developed for the maximum coherence or partial coherence across frequencies in high-dimensional systems?
• RQ3 What is the finite-sample performance of the proposed inference procedures in realistic high-dimensional scenarios?
• RQ4 Can a consistent FDR control procedure be developed for large-scale multiple testing of coherence and partial coherence parameters?
• RQ5 How well do the proposed methods perform in detecting true linear dependencies in brain connectivity networks using EEG data?

Key findings

• The proposed debiased estimator of partial coherence achieves a consistent limiting distribution, enabling valid asymptotic inference even in high-dimensional settings.
• The test based on the maximum partial coherence across frequencies maintains empirical size close to nominal levels and exhibits high power, especially as sample size increases.
• For δ = 0.2, the empirical FDR is consistently 0% across all simulation scenarios, indicating conservative but valid FDR control due to the structure of the test statistic.
• The power of the proposed testing procedure consistently exceeds that of regularization-based alternatives, even under conservative FDR control.
• In EEG data applications, the method successfully identifies meaningful brain connectivity patterns, demonstrating practical utility in neuroscience.
• The method remains robust under various data-generating processes, including VAR(1), VMA(5), and high-dimensional sparse structures, with strong performance across different sample sizes.