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The goal in this thesis is to develop a novel statistical approach to identity functional interactions between regions in a brain network. Wavelets are effective for capturing time varying 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 provide a multi-scale decomposition of signals and thus can be few for exploring potential cross-scale interactions between signals. To achieve this, we propose the scale-specific sub-processes of a multivariate locally stationary wavelet stochastic process. Under this proposed framework, a novel cross-scale dependence measurement is developed, which provides a measure for dependence structure of components at different scales of multivariate time series. Extensive simulation experiments are conducted to demonstrate that the theoretical properties hold in practice. The developed cross-scale analysis is performed on 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 extended to other signals, which can also capture the statistical association between the stocks at different time scales. |
