Detection and Mode-Identification of Multiple Change Points in Tensor Factor Models

We study the problem of modeling high-dimensional tensor-valued time series under a Tucker decomposition-based factor model while permitting the presence of multiple structural change points. Such data call for a method that (i) handles high dimensionality and complex dependence while achieving consistency in multiple change points and further, (ii) addresses a unique challenge posed by the multi-dimensional array setting, of identifying the modes associated with each change while the shifts in different modes interact with one another. To this end, we propose a change point detection procedure utilizing the low-rank structure of the data for statistical and computational efficiency. In addition, we formalize the $\textit{mode-identifiability}$ of each change and propose an algorithm for detecting the modes at which the data undergo a mode-identifiable shift. We establish the consistency of both change point detection and mode-identification methods under a weak moment condition, and demonstrate their good performance on simulated datasets where, in particular, it is shown that the mode-identification step can improve the post-segmentation estimation of the mode-wise loading space. We also analyze the datasets on New York City taxi usage and Fama--French portfolio returns using the proposed suite of methods.