Wavelet-Based Modeling of Dependence Structures in Nonstationary Multivariate Time Series

Understanding the time-varying dependence structures of multivariate signals is essential for analyzing complex dynamic systems such as brain networks, financial markets, and environmental processes. However, conventional approaches typically assume stationarity and summarize dependence over the entire observation period, so they often fail to capture the transient and multiscale nature of these interactions. Building on the wavelet transform, this dissertation develops a unified statistical framework that constructs a progression of methods for modeling increasingly complex, yet practically important, dependence structures in nonstationary multivariate time series.

The first part introduces dynamic cross-scale wavelet coherence, which quantifies how fluctuations at different scales are coherently coupled over time. Moving beyond same-scale measures, it captures cross-scale dependence between pairs of signals while remaining localized in time and scale, and a nonparametric inference procedure is developed to detect these temporally evolving coupling patterns, with applications demonstrating its utility for analyzing complex neural signals. Building on this foundation, the second part proposes wavelet canonical coherence (WaveCanCoh), a framework for characterizing dynamic relationships between two sets of multivariate signals. Within the multivariate locally stationary wavelet model, WaveCanCoh identifies the dominant modes of cross-set dependence that evolve jointly across scales and time, providing a principled alternative to traditional Fourier-based canonical correlation analysis. The final part advances the methodology to partial wavelet canonical coherence (Partial Wave CanCoh), which infers direct dependencies by removing the influence of confounding processes. To accommodate high-dimensional settings, regularization and shrinkage strategies are incorporated to ensure stable estimation and interpretable results.

Together, these developments establish a rigorous and versatile statistical foundation for wavelet-based dependence analysis, progressing from pairwise crossscale coherence to canonical and partial canonical coherence between sets of signals. The proposed methodologies are validated through comprehensive simulations and applications to both large-scale neuroscience data and financial time series, revealing temporally localized, cross-scale patterns of dependence that remain inaccessible to conventional stationary approaches.