We study the problem of detecting and locating change points in high-dimensional Vector Autoregressive (VAR) models, whose transition matrices exhibit low rank plus sparse structure. We first address the problem of detecting a single change point using an exhaustive search algorithm and establish a finite sample error bound for its accuracy. Next, we extend the results to the case of multiple change points that can grow as a function of the sample size. Their detection is based on a two-step algorithm, wherein the first step, an exhaustive search for a candidate change point is employed for overlapping windows, and subsequently, a backward elimination procedure is used to screen out redundant candidates. The two-step strategy yields consistent estimates of the number and the locations of the change points. To reduce computation cost, we also investigate conditions under which a surrogate VAR model with a weakly sparse transition matrix can accurately estimate the change points and their locations for data generated by the original model. This work also addresses and resolves a number of novel technical challenges posed by the nature of the VAR models under consideration. The effectiveness of the proposed algorithms and methodology is illustrated on both synthetic and two real data sets.

翻译：我们研究在高维矢量自动递减模式中发现和定位变化点的问题,这些变化点的过渡矩阵显示低级加上零散结构;我们首先处理利用详尽的搜索算法发现单一变化点的问题,并确立一个有限的抽样错误,以确定其准确性;然后,我们将结果扩大到多个变化点的情况,这些变化点可随着样本大小的函数而增长;其检测基于一种两步算法,第一步,即对重叠窗口采用对候选人变化点的彻底搜索,随后,采用后退消除程序来筛选多余的候选人;两步战略得出对变化点的数量和位置的一致估计;为降低计算成本,我们还调查使用微小的过渡矩阵的替代VAR模型能够准确估计变化点及其在原始模型生成数据的位置的条件;这项工作还处理和解决了所考虑的VAR模型性质带来的一些新的技术挑战;拟议的算法和方法的有效性在合成数据集和两个真实数据集上都作了说明。