Large Spectral Density Matrix Estimation by Thresholding

TL;DR

This paper introduces a new thresholding method for estimating high-dimensional spectral density matrices of multivariate time series, with theoretical guarantees and practical advantages in neuroscience data analysis.

Contribution

It develops a non-asymptotic theory for regularized spectral density estimation using thresholded periodograms, allowing consistent estimation under sparsity in high dimensions.

Findings

Consistent estimation under high-dimensional sparsity regime.

Automatic edge selection for coherence network learning.

Validated effectiveness through simulations and fMRI data.

Abstract

Spectral density matrix estimation of multivariate time series is a classical problem in time series and signal processing. In modern neuroscience, spectral density based metrics are commonly used for analyzing functional connectivity among brain regions. In this paper, we develop a non-asymptotic theory for regularized estimation of high-dimensional spectral density matrices of Gaussian and linear processes using thresholded versions of averaged periodograms. Our theoretical analysis ensures that consistent estimation of spectral density matrix of a $p$-dimensional time series using $n$ samples is possible under high-dimensional regime $\log p / n \rightarrow 0$ as long as the true spectral density is approximately sparse. A key technical component of our analysis is a new concentration inequality of average periodogram around its expectation, which is of independent interest. Our estimation consistency results complement existing results for shrinkage based estimators of multivariate spectral density, which require no assumption on sparsity but only ensure consistent estimation in a regime $p^2/n \rightarrow 0$. In addition, our proposed thresholding based estimators perform consistent and automatic edge selection when learning coherence networks among the components of a multivariate time series. We demonstrate the advantage of our estimators using simulation studies and a real data application on functional connectivity analysis with fMRI data.