Multiple Change Points Detection in Low Rank and Sparse High Dimensional Vector Autoregressive Models

Abstract

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 twostep algorithm, wherein the first step, an exhaustive search for a candidate change point is employed for overlapping windows, and subsequently a backwards 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.

Collective anomaly detection in High-dimensional VAR Models

“…They use a block fused lasso penalty by assuming that the model parameters in a block is fixed, while our objective function controls only the sparsity of change in building a test statistic and search many segments to find an anomalous interval. Also, Safikhani and Shojaie (2020) and Bai et al (2020) assume that the l 2 -norm of a change in VAR parameter is bounded away from zero, whereas our assumption on the l 2 -norm of a change is related to the sparsity of change which is in line with the assumptions used in Wang et al (2019).…”

“…Among those relevant works already introduced earlier in this section, the work of Safikhani and Shojaie (2020) and Bai et al (2020) are most closely related to our work, in that they also control the change in VAR parameters with a lasso penalty in their objective functions, however their approaches are different from our method in several aspects. To obtain the initial estimate of change-points before screening, Safikhani and Shojaie (2020) use a fused lasso penalty on a full model considering all time points being a candidate for change-point.…”

“…Thus their objective function controls the sparsity of VAR parameters and the sparsity of its difference at the same time. Bai et al (2020) follows a similar procedure to Safikhani and Shojaie (2020) under the multiple change-point framework. They use a block fused lasso penalty by assuming that the model parameters in a block is fixed, while our objective function controls only the sparsity of change in building a test statistic and search many segments to find an anomalous interval.…”

Multiple Change Point Detection in Structured VAR Models: the VARDetect R Package

“…There are a number of tuning parameters in the TBSS and LSTSP algorithms. The theoretical rates for those tuning parameters are provided in relevant papers (Safikhani and Shojaie 2020;Bai et al 2020). Next, we provide guidelines for their selection.…”

“…However, in many of the applications mentioned above, the number of time series is large, thus giving rise to high dimensional VAR models. Hence, there has been recent work of developing methods for detecting change points in such high dimensional VAR models (Wang, Yu, Rinaldo, and Willett 2019;Cribben, Wager, and Lindquist 2013;Bai, Safikhani, and Michailidis 2020). Most of the work has focused on sparse VAR models .…”