Observations in various applications are frequently represented as a time series of multidimensional arrays, called tensor time series, preserving the inherent multidimensional structure. In this paper, we present a factor model approach, in a form similar to tensor CP decomposition, to the analysis of high-dimensional dynamic tensor time series. As the loading vectors are uniquely defined but not necessarily orthogonal, it is significantly different from the existing tensor factor models based on Tucker-type tensor decomposition. The model structure allows for a set of uncorrelated one-dimensional latent dynamic factor processes, making it much more convenient to study the underlying dynamics of the time series. A new high order projection estimator is proposed for such a factor model, utilizing the special structure and the idea of the higher order orthogonal iteration procedures commonly used in Tucker-type tensor factor model and general tensor CP decomposition procedures. Theoretical investigation provides statistical error bounds for the proposed methods, which shows the significant advantage of utilizing the special model structure. Simulation study is conducted to further demonstrate the finite sample properties of the estimators. Real data application is used to illustrate the model and its interpretations.

翻译：各类应用中的观测数据常以多维数组的时间序列形式呈现，即张量时间序列，其保留了固有的多维结构。本文提出一种类张量CP分解形式的因子模型方法，用于分析高维动态张量时间序列。由于载荷向量具有唯一确定性但非正交性，该模型与基于Tucker型张量分解的现有张量因子模型存在显著差异。该模型结构允许一组不相关的一维潜在动态因子过程，从而极大便利了时间序列潜在动态机制的研究。针对此类因子模型，我们提出了一种新型高阶投影估计方法，该方法充分利用了Tucker型张量因子模型中常用的高阶正交迭代过程及通用张量CP分解过程的核心思想。理论分析给出了所提方法的统计误差界，证明了利用特殊模型结构的显著优势。通过仿真研究进一步展示了估计量的有限样本性质，并利用实际数据案例验证了模型及其解释能力。