Abstract

Co-authors: Daren Wang (Carnegie Mellon University), Alessandro Rinaldo (Carnegie Mellon University) 

In this paper, we study covariance change point detection problem in high dimension. Specifically, we assume that the time series  $X_i \in \mathbb{R}^{p$, $i = 1, \ldots, n$ are independent $p$-dimensional sub-Gaussian random vectors and that the corresponding covariance matrices $\{\Sigma_i\}_{i=1}}n$ are stationary within segments and only change at certain time points.  Our generic model setting allows $p$ grows with $n$ and we do not place any additional structural assumptions on the covariance matrices.  We introduce algorithms based on binary segmentation (e.g. Vostrikova, 1981) and wild binary segmentation (Fryzlewicz, 2014) and establish the consistency results under suitable conditions.  To improve the detection performance in high dimension, we propose   Wild Binary Segmentation through Independent Projection (WBSIP).  We show that WBSIP can optimally estimate the locations of the change points.  Our analysis also reveals a phase transition effect based on our generic model assumption and to the best of our knowledge, this type of results have not been established elsewhere in the change point detection literature.