Title: Spectral Filtering for Spatio-temporal Dynamics and Multivariate Forecasts
Description: Due to the increasing availability of massive spatio-temporal data sets, modeling high dimensional data becomes quite challenging. A large number of research questions are rooted in identifying underlying dynamics in such spatio-temporal data. For many applications, the science suggests that the intrinsic dynamics be smooth and of low dimension. To reduce the variance of estimates and increase the computational tractability, dimension reduction is also quite necessary in the modeling procedure. In this dissertation, we propose a spectral filtering approach for dimension reduction and forecast amelioration, and apply it to multiple applications. We show the effectiveness of dimension reduction via our method and also illustrate its power for prediction in both simulation and real data examples. The resultant lower dimensional principal component series has a diagonal spectral density at each frequency whose diagonal elements are in descending order, which is not well motivated can be hard to interpret.

Therefore we propose a phase-based filtering method to create principal component series with interpretable dynamics in the time domain. Our method is based on an approach of structural decomposition and phase-aligned construction in the frequency domain, identifying lower-rank dynamics and its components embedded in a high dimensional spatio-temporal system. In both our simulated examples and real data applications, we illustrate that the proposed method is able to separate and identify meaningful lower-rank movements. Benefiting from the zero-coherence property of the principal component series, we subsequently develop a predictive model for high-dimensional forecasting via lower-rank dynamics. Our modeling approach reduces multivariate modeling task to multiple univariate modeling and is flexible in combining with regularization techniques to obtain more stable estimates and improve interpretability. The simulation results and real data analysis show that our model achieves superior forecast performance compared to the class of autoregressive models.