On the frequency domain detection of high dimensional time series

TL;DR

This paper develops a frequency domain detection method for high-dimensional time series using spectral coherence matrices and random matrix theory, providing asymptotic analysis and numerical validation.

Contribution

It introduces a new test based on the largest eigenvalue of the spectral coherence matrix for high-dimensional time series detection, with rigorous asymptotic analysis.

Findings

Spectral coherence matrix approximated by a Wishart matrix in high dimensions.

Asymptotic behavior of eigenvalues characterized by random matrix theory.

Numerical results demonstrate the test's statistical performance.

Abstract

In this paper, we address the problem of detection, in the frequency domain, of a M-dimensional time series modeled as the output of a M x K MIMO filter driven by a K-dimensional Gaussian white noise, and disturbed by an additive M-dimensional Gaussian colored noise. We consider the study of test statistics based of the Spectral Coherence Matrix (SCM) obtained as renormalization of the smoothed periodogram matrix of the observed time series over N samples, and with smoothing span B. To that purpose, we consider the asymptotic regime in which M, B, N all converge to infinity at certain specific rates, while K remains fixed. We prove that the SCM may be approximated in operator norm by a correlated Wishart matrix, for which Random Matrix Theory (RMT) provides a precise description of the asymptotic behaviour of the eigenvalues. These results are then exploited to study the consistency of a test based on the largest eigenvalue of the SCM, and provide some numerical illustrations to evaluate the statistical performance of such a test.