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.

翻译：本文旨在发展一种新颖的统计方法，用于刻画脑网络中通道间的功能交互作用。小波因具有紧支撑特性，可根据信号的动态特性进行压缩或拉伸，从而有效捕捉非平稳信号的瞬态特征。小波能够提供信号的多尺度分解，因此可用于研究信号间潜在的跨尺度交互作用。为实现这一目标，我们发展了多变量局部平稳小波随机过程的尺度特异性子过程。在该框架下，提出了一种新颖的跨尺度依赖度量，用于衡量多变量时间序列中不同尺度分量间的依赖结构。通过大量模拟研究验证了理论性质在实际中的有效性。所提出的跨尺度分析方法应用于脑电图（EEG）数据，以研究注意缺陷多动障碍（ADHD）儿童脑功能连接结构的变化。我们的方法识别出脑网络中通道间新颖有趣的跨尺度交互作用。该框架还可应用于其他信号，例如捕捉不同时间尺度下股票间的统计关联性。