UC Santa Cruz

Spectral analysis has been widely used to characterize the properties of one or more time series in the frequency domain. Accurate inference of spectral density matrices is critical for understanding the structure underlying the components of a given multivariate temporal process, and for revealing potential relationships across its components. However, inference of spectral density matrices suffers from the curse of dimensionality. This dissertation first develops methods to estimate the spectral density matrix and functions of this matrix for high-dimensional stationary time series under a Bayesian framework. We consider a Whittle likelihood-based spectral modeling approach and impose a discounted regularized horseshoe prior on the coefficients that define a spline representation of the Cholesky factorization components of the inverse spectral density matrix. Next, we extend the model to estimate the time-varying power spectrum and its functions for high-dimensional nonstationary time series. Under a locally stationary basis representation, two types of priors on basis coefficients are developed: a slow-varying double gamma shrinkage prior is used to induce power spectral estimates to evolve smoothly over time, while a piecewise linear function with a global-local shrinkage prior is proposed for cases in which the power spectral estimates are expected to display abrupt changes. Finally, we further develop methods to conduct quantile spectral analysis for multivariate stationary time series by modeling the matrix of quantile cross-spectral density kernels via its low-rank factorization. Several customized stochastic gradient variational Bayes (SGVB) approaches, supported by parallel computation and GPU accelerations, are developed to obtain fast approximate posterior inference in all the spectral modeling frameworks mentioned above. Extensive simulation studies and data analyses show that our models and methods for posterior inference are accurate and time efficient. Furthermore, our methods are superior compared to competing methods for standard and quantile spectral analysis of multivariate time series.