Spectral Change Point Estimation for High Dimensional Time Series by Sparse Tensor Decomposition

Multivariate time series may be subject to partial structural changes over certain frequency band, for instance, in neuroscience. We study the change point detection problem with high dimensional time series, within the framework of frequency domain. The overarching goal is to locate all change points and delineate which series are activated by the change, over which frequencies. In practice, the number of activated series per change and frequency could span from a few to full participation. We solve the problem by first computing a CUSUM tensor based on spectra estimated from blocks of the time series. A frequency-specific projection approach is applied for dimension reduction. The projection direction is estimated by a proposed tensor decomposition algorithm that adjusts to the sparsity level of changes. Finally, the projected CUSUM vectors across frequencies are aggregated for change point detection. We provide theoretical guarantees on the number of estimated change points and the convergence rate of their locations. We derive error bounds for the estimated projection direction for identifying the frequency-specific series activated in a change. We provide data-driven rules for the choice of parameters. The efficacy of the proposed method is illustrated by simulation and a stock returns application.