Partial Coherence Estimation via Spectral Matrix Shrinkage under Quadratic Loss

TL;DR

This paper introduces new quadratic loss-based shrinkage estimators for spectral and precision matrices to improve partial coherence estimation in multivariate time series, especially when true coherencies are large.

Contribution

It derives novel QL estimators for spectral and precision matrices and compares their performance to HS-based methods, demonstrating advantages in certain scenarios.

Findings

QL estimators outperform HS in large partial coherence cases

The precision matrix QL estimator is particularly robust and reliable

Simulations based on EEG data validate the estimators' effectiveness

Abstract

Partial coherence is an important quantity derived from spectral or precision matrices and is used in seismology, meteorology, oceanography, neuroscience and elsewhere. If the number of complex degrees of freedom only slightly exceeds the dimension of the multivariate stationary time series, spectral matrices are poorly conditioned and shrinkage techniques suggest themselves. When true partial coherencies are quite large then for shrinkage estimators of the diagonal weighting kind it is shown empirically that the minimization of risk using quadratic loss (QL) leads to oracle partial coherence estimators superior to those derived by minimizing risk using Hilbert-Schmidt (HS) loss. When true partial coherencies are small the methods behave similarly. We derive two new QL estimators for spectral matrices, and new QL and HS estimators for precision matrices. In addition for the full estimation (non-oracle) case where certain trace expressions must also be estimated, we examine the behaviour of three different QL estimators, the precision matrix one seeming particularly robust and reliable. For the empirical study we carry out exact simulations derived from real EEG data for two individuals, one having large, and the other small, partial coherencies. This ensures our study covers cases of real-world relevance.