Spatio-temporal analysis of Gaussian WSS processes via complex correlation and partial correlation screening

We propose a framework for spatio-temporal correlation analysis of jointly Gaussian Wide Sense Stationary (WSS) multivariate time series. The goal is to identify the hub time series, i.e., the ones that are highly correlated with a number of other time series. When the dimension of the multivariate time series and the number of time samples are relatively large, direct correlation analysis in the time domain could be computationally intractable. As an alternative, we apply the Discrete Fourier Transform to the time series and perform correlation analysis in the frequency domain. We extend the previous theory of hub screening to the complex domain to accommodate complex-valued Fourier transforms. The theory allows p-values to be assigned to time series for being a hub under the null hypothesis that the time series are independent of each other. It also specifies thresholds for which thresholded sample (partial) correlation matrices can be used to identify hubs. We then use an independence property of Gaussian WSS time series in the frequency domain to perform multiple inference for detecting hub time series. Experimental results on both synthetic data and real financial data illustrate the accuracy of our theoretical results and the usefulness of the proposed framework.