Spectral density shrinkage for high-dimensional time series

Time series data obtained from neurophysiological signals is often high-dimensional and the length of the time series is often short relative to the number of dimensions. Thus, it is dicult or sometimes impossible to compute statistics that are based on the spectral density matrix because these matrices are numerically unstable. In this work, we discuss the importance of regularization for spectral analysis of high-dimensional time series and propose shrinkage estimation for estimating high-dimensional spectral density matrices. The shrinkage estimator is derived from a penalized log-likelihood, and the optimal penalty parameter has a closed-form solution, which can be estimated using the bootstrap. We developed the multivariate Time-frequency Toggle (TFT) bootstrap procedure for multivariate time series, and showed that it is theoretically valid. We show via simulations and an fMRI data set that failure to regularize the estimates of the spectral density matrix can yield unstable statistics, and that this can be alleviated by shrinkage estimation