Spectral Analysis of High-dimensional Time Series

TL;DR

This paper provides the first non-asymptotic theoretical guarantees for high-dimensional spectral density matrix estimation in time series, enabling more reliable analysis of large-scale multivariate data.

Contribution

It introduces non-asymptotic bounds for spectral density matrix estimation, justifies high-dimensional models, and applies these results to convergence of estimators and precision matrix analysis.

Findings

Non-asymptotic bounds for spectral density matrix estimation

Convergence results for smoothed periodogram estimators

Validation through simulations and real data analysis

Abstract

A useful approach for analysing multiple time series is via characterising their spectral density matrix as the frequency domain analog of the covariance matrix. When the dimension of the time series is large compared to their length, regularisation based methods can overcome the curse of dimensionality, but the existing ones lack theoretical justification. This paper develops the first non-asymptotic result for characterising the difference between the sample and population versions of the spectral density matrix, allowing one to justify a range of high-dimensional models for analysing time series. As a concrete example, we apply this result to establish the convergence of the smoothed periodogram estimators and sparse estimators of the inverse of spectral density matrices, namely precision matrices. These results, novel in the frequency domain time series analysis, are corroborated by simulations and an analysis of the Google Flu Trends data.