Collective Estimation of Spectral Density Functions for Multivariate Time Series

This dissertation presents novel nonparametric methodologies for the collective estimation and analysis of spectral density functions in multiple multivariate time series (MTS). Spectral analysis is crucial for uncovering frequency-domain characteristics of time series data, revealing patterns and interdependencies that are often difficult to detect with conventional time-domain methods. While classical spectral estimation techniques have been extensively explored, they frequently encounter issues regarding the positive definiteness, stability, and interpretability of the estimated spectral matrices. To address these limitations, we develop advanced estimation techniques based on penalized Whittle likelihood and collective spectral estimation frameworks, aiming to produce consistent, stable, and interpretable estimators. Initially, we provide a comprehensive overview of classical methods for spectral analysis, emphasizing their limitations in accurately estimating spectral density matrices. Subsequently, we propose the Nonparametric Multivariate Spectral Density Estimation (NMSDE) method, which leverages a Cholesky-based penalized Whittle likelihood approach and basis expansions to guarantee positive-definite spectral density matrices. Building upon this, we introduce the Nonparametric Multivariate Collective Spectral Density Estimation (NMCSDE) method, which simultaneously estimates multiple spectral density matrices by utilizing shared spectral features across multiple time series. By exploiting this collective estimation approach, our method significantly enhances accuracy, robustness, and interpretability compared to approaches that estimate each series individually, thereby improving clustering and classification performance based on frequency-domain characteristics. Through extensive simulation studies, we demonstrate the superiority of the proposed methods compared to traditional nonparametric estimators, achieving lower canonical angle (CAN) values and higher adjusted Rand Index (ARI) scores in clustering scenarios. A practical application to electroencephalogram (EEG) data illustrates the utility of the developed methodologies in real-world settings, particularly in identifying common spectral signatures across different subjects or different channels. Overall, this research contributes robust statistical tools for spectral analysis, providing deeper insights into multivariate temporal dependencies and interactions. Directions for future research are suggested, such as extending the methods to handle nonstationary data and improving computational efficiency for large-scale applications.