Nonparametric multivariate breakpoint detection for the means, variances, and covariances of a discrete time stochastic process

Abstract

We introduce a nonparametric breakpoint detection method for the means and covariances of a multivariate discrete time stochastic process. Breakpoints are defined as left or right endpoints of maximal intervals of local time homogeneity for the means and covariances. The breakpoint detection method is an adaptive algorithm that estimates the last maximal interval of homogeneity. Applied recursively, it allows us to find an arbitrary number of breakpoints. We then study a second breakpoint detection algorithm that makes use of a sliding window. The quality of both methods is analysed. For the adaptive algorithm, we provide the quality of the estimation of the one-step-ahead means and covariance matrix as well as upper bounds on the type I and type II errors when applying the procedure to a change-point model. Regarding the second method, the probability of correctly detecting the breakpoint of a change-point model is bounded from below. Numerical simulations assess the performance of both methods using simulated data.

Keywords:

• breakpoint detection
• time series
• multivariate data
• covariance matrix estimation
• adaptive algorithm

AMS Subject Classifications :

• 62M10
• 62G05

Acknowledgements

The author is grateful to Anatoli Juditski for helpful discussions. The author also thanks the two reviewers and the associate editor for beneficial comments and suggestions.

Notes

Since we deal with a discrete time stochastic process, the intervals I considered are discrete sets of consecutive time steps belonging to .