Spectral Density Shrinkage for High-Dimensional Time Series

• Activity Number: 215
• Type: Invited
• Date/Time: Monday, August 5, 2013 : 2:00 PM to 3:50 PM
• Sponsor: Section on Statistics in Imaging
• Author(s): Mark Fiecas*+ and Rainer von Sachs
• Companies: University of California, San Diego and Universite Catholique de Louvain
• Keywords: High-dimensional time series ; Shrinkage estimation ; Neuroimaging data ; fMRI ; Spectral analysis ; Bootstrap

Abstract

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 difficult or sometimes impossible to compute statistics that are based on the spectral density matrix because these matrices are numerically unstable. In this talk, we discuss the importance of regularization for analyzing high-dimensional time series. We discuss the shrinkage framework for estimating high-dimensional spectral density matrices. In this framework, the shrinkage estimator is derived from a penalized log-likelihood. The optimal penalty parameter has a closed-form solution, and can be estimated using the bootstrap. We have developed a new bootstrap procedure for multivariate time series, and can show that it is theoretically valid. We show via simulations and an empirical fMRI data set that failure to regularize the estimates of the variance-covariance matrix or the spectral density matrix can yield very unstable statistics, and that this can be alleviated by adopting the shrinkage framework.