Multivariate time series may be subject to partial structural changes over certain frequency band, for instance, in neuroscience. We study the change point detection problem with high dimensional time series, within the framework of frequency domain. The overarching goal is to locate all change points and delineate which series are activated by the change, over which frequencies. In practice, the number of activated series per change and frequency could span from a few to full participation. We solve the problem by first computing a CUSUM tensor based on spectra estimated from blocks of the time series. A frequency-specific projection approach is applied for dimension reduction. The projection direction is estimated by a proposed tensor decomposition algorithm that adjusts to the sparsity level of changes. Finally, the projected CUSUM vectors across frequencies are aggregated 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 activated in a change. We provide data-driven rules for the choice of parameters. The efficacy of the proposed method is illustrated by simulation and a stock returns application.

翻译：多元时间序列可能在特定频段发生部分结构变化，例如在神经科学领域。本研究在频域框架下探讨高维时间序列的变点检测问题，核心目标是定位所有变点，并识别每个变点激活的具体时间序列及其对应的频率范围。在实际应用中，每个变点在不同频率下激活的序列数量可能从少数几个到全体参与不等。我们首先基于时间序列分块估计的谱计算CUSUM张量，采用频率特异性投影方法进行降维处理。投影方向的估计通过提出的自适应变化稀疏水平的张量分解算法实现。最后，聚合跨频率的投影CUSUM向量进行变点检测。我们为估计变点数量及其位置收敛速度提供了理论保证，推导了识别频率特异性激活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规则。通过仿真实验和股票收益率应用验证了所提方法的有效性。