Multi-scale wavelet coherence with its applications

The goal in this paper is to develop a novel statistical approach to characterize functional interactions between channels in a brain network. Wavelets are effective for capturing transient properties of non-stationary signals because they have compact support that can be compressed or stretched according to the dynamic properties of the signal. Wavelets give a multi-scale decomposition of signals and thus can be few for studying potential cross-scale interactions between signals. To achieve this, we develop the scale-specific sub-processes of a multivariate locally stationary wavelet stochastic process. Under this proposed framework, a novel cross-scale dependence measure is developed. This provides a measure for dependence structure of components at different scales of multivariate time series. Extensive simulation studies are conducted to demonstrate that the theoretical properties hold in practice. The proposed cross-scale analysis is applied to the electroencephalogram (EEG) data to study alterations in the functional connectivity structure in children diagnosed with attention deficit hyperactivity disorder (ADHD). Our approach identified novel interesting cross-scale interactions between channels in the brain network. The proposed framework can be applied to other signals, which can also capture the statistical association between the stocks at different time scales.