Multiple Change Point Detection in Reduced Rank 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 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 datasets. Supplementary materials for this article are available online.

KEYWORDS:

• Algorithms
• Consistency
• Time series data and their applications

Supplementary Materials

The supplementary materials contain all proofs of the theoretical results, together with auxiliary lemmas, additional details on the detection algorithms, and additional numerical experiments.

Acknowledgments

The authors would like to thank Associate Editor and three anonymous referees for many constructive comments and suggestions.

Additional information

Funding

The work of George Michailidis has been supported in part by NSF grants DMS2124507, DMS1821220, and DMS1830175.