We study the problems arising from modeling high-dimensional tensor-valued time series under a Tucker decomposition-based factor model with multiple structural change points. First, we propose an algorithm for detecting the multiple change points, which utilizes the low-rank structure of the data for statistical and computational efficiency. Also, the multi-dimensional array setting poses unique challenges, as some changes are associated with a subset of the modes, and the changes in different modes may interact with one another. Recognizing these, we investigate the problem of identifying each change with the tensor modes post-segmentation. To this end, we formalize the mode-identifiability of each change and propose an algorithm for detecting the modes at which the data are undergoing 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. Additionally we analyze the datasets on New York City taxi usage and Fama--French portfolio returns using the proposed suite of methods.

翻译：我们研究了在高维张量值时间序列建模中，基于Tucker分解因子模型且存在多个结构变点时产生的问题。首先，我们提出了一种检测多个变点的算法，该算法利用数据的低秩结构提升统计与计算效率。同时，多维数组的设置带来了独特的挑战——部分变点仅与某些模态相关，且不同模态的变点可能相互影响。针对这些问题，我们进一步研究了在分割后识别每个变点所属张量模态的任务。为此，我们形式化地定义了每个变点的模态可识别性，并提出了一种算法，用于检测数据发生模态可识别变化时的模态。我们在弱矩条件下证明了变点检测与模态识别方法的一致性，并在模拟数据集上展示了其优异性能——特别地，模态识别步骤能改善分割后对模态载荷空间的估计。此外，我们利用所提出的方法套件分析了纽约市出租车使用量数据集和Fama–French投资组合收益率数据。