Multi-scale wavelet coherence

TL;DR

The paper tackles nonstationarity in brain signals by introducing a cross-scale, multivariate locally stationary wavelet model (CS-MvLSW) that explicitly models scale-specific subprocesses and cross-scale interactions via a cross-scale spectral matrix $\boldsymbol{S}_{jj'}(u)$ and cross-scale covariance $\boldsymbol{Q}_{jj'}(u)$. It defines dual-scale coherence $\boldsymbol{\rho}_{jj'}(u)$ and provides two estimation paths—subprocess-based and process-based—with bias-corrected, smoothed periodograms to obtain consistent estimates of cross-scale spectra and coherence. Through simulations, the authors show that the CS-MvLSW framework robustly captures cross-scale dependencies, outperforming existing multivariate LSW methods, especially when cross-scale structure is present. In EEG analysis of ADHD versus control children, the approach uncovers novel cross-scale interactions (e.g., beta–gamma coupling) and reveals group differences in both single-scale and cross-scale connectivity, offering insights into multiscale neural coordination. Overall, the CS-MvLSW framework provides a principled, scalable tool for studying how long-term dynamics in one brain region relate to short-term activity in others, with broad applicability to other multiscale, nonstationary time series.

Abstract

This paper develops a novel statistical approach to characterize temporally localised cross-oscillatory interactions between channels in a functional brain network. Brain signals are generally nonstationary and the proposed framework uses wavelets as an effective tool for capturing (i) single-scale channel transient features, due to their adaptiveness to the dynamic signal properties, and (ii) cross-scale channel interactions, due to their multi-scale nature. Our approach formalises scale-specific subprocesses and cross-scale (CS) dependencies for a new class of multivariate locally stationary (MvLSW) wavelet processes that we refer to as CS-MvLSW. Under this model, we develop a novel spectral domain time-varying cross-scale dependence measure and its appropriate estimation. Extensive simulation studies demonstrate that the theoretically established properties hold in practice. The proposed CS-MvLSW framework remains accurate under pronounced cross-scale dependence, whereas existing MvLSW modelling can deteriorate even for single-scale coherence when such complex structure is present in the process. The proposed cross-scale analysis is applied to electroencephalogram (EEG) data to study alterations in the functional connectivity structure in children diagnosed with attention deficit hyperactivity disorder (ADHD). Our approach identified novel, clinically pertinent cross-scale interactions in the functional brain network, differentiating brain connectivity between control and ADHD groups.

Figures (13)

• Figure 1: EEG signals recorded for 10s from 6 channels of a particular participant diagnosed with ADHD.
• Figure 2: (a) The Haar wavelet at different scales and locations; (b) two nonstationary signals and cross-scale coherence among their components at multiple different scales.
• Figure 3: Single-scale coherence estimates at $j=1$ (top) and $j=2$ (bottom) for $T=1024$ (Scenario 1), averaged over 1,000 replicates. Black: truth; coloured: mvLSW, subprocess-based (approximated), and process-based estimates.
• Figure 4: Estimated cross-scale coherence (Scenario 2) for three pairs of scales $(j_1, j_2)$, averaged over 1,000 replicates. Black: truth; coloured: process-based estimator.
• Figure 5: Relationships between the scale and frequency band of components at every scale in terms of the sampling rate of the original signal.

Theorems & Definitions (21)

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