We study the problem of change point (CP) detection with high dimensional time series, within the framework of frequency domain. The overarching goal is to locate all change points and for each change point, delineate which series are activated by the change, over which set of frequencies. The working assumption is that only a few series are activated per change and frequency. We solve the problem by computing a CUSUM tensor based on spectra estimated from blocks of the observed time series. A frequency-specific projection approach is applied to the CUSUM tensor for dimension reduction. The projection direction is estimated by a proposed sparse tensor decomposition algorithm. Finally, the projected CUSUM vectors across frequencies are aggregated by a sparsified wild binary segmentation for change point detection. We provide theoretical guarantees on the number of estimated change points and the convergence rate of their locations. We derive error bounds for the estimated projection direction for identifying the frequency-specific series that are activated in a change. We provide data-driven rules for the choice of parameters. We illustrate the efficacy of the proposed method by simulation and a stock returns application.

翻译：我们研究频域框架下高维时间序列的变点检测问题。总体目标在于定位所有变点，并为每个变点确定哪些序列在该变点处被激活，以及这些激活对应的频率集合。工作假设是每次变点仅激活少数序列且仅涉及特定频率。我们通过计算基于观测时间序列分块谱估计的CUSUM张量来解决该问题。采用频率特异性投影方法对CUSUM张量进行降维，投影方向通过提出的稀疏张量分解算法估计。最后，通过稀疏化野生二元分割聚合各频率的投影CUSUM向量以实现变点检测。我们提供了估计变点数量及其位置收敛速度的理论保证，推导了用于识别变点激活频率特异性序列的投影方向估计误差界，并提出了参数选择的数据驱动准则。通过模拟实验和股票收益率应用案例验证了所提方法的有效性。