Assumption-Lean Inference for Spectral Differential Network Analysis of High-Dimensional Time Series

Network analysis for multivariate time series is popular in many fields, from neuroscience to seismology. The inverse spectral density is a common choice for time series network analysis due to its representation of the frequency domain correlation between two variables after removing the best linear predictor of all other variables. In many applications, the goal is to study how these networks change across different conditions. For example, in neuroscience, one might be interested in how the brain connectivity network changes before and after stimulation. Towards this goal, we develop an inference framework based on a direct estimate of the difference in two high-dimensional inverse spectral densities. We develop a new Gaussian approximation error bound for any de-biased D-trace estimation procedure which is then leveraged to both inform optimal window sizes of Welch's estimators of the spectral density and establish asymptotic normality of our de-biased D-trace estimator. Moreover, we develop an efficient algorithm based on a generalized D-trace estimation procedure to overcome the computational complexity of high-dimensional inference. The method is illustrated on synthetic data experiments and on experiments with electroencephalography data.