Sequential change point detection in high dimensional time series

Abstract

Change point detection in high dimensional data has found considerable interest in recent years. Most of the literature designs methodology for a retrospective analysis, where the whole sample is already available when the statistical inference begins. This paper takes a different point of view and develops monitoring schemes for the online scenario, where high dimensional data arrives steadily and the goal is to detect changes as fast as possible controlling at the same time the probability of a type I error of a false alarm. We develop sequential procedures capable of detecting changes in the mean vector of a successively observed high dimensional time series with spatial and temporal dependence. The statistical properties of the methods are analyzed in the case where both, the sample size and dimension converge to infinity. In this scenario it is shown that the new monitoring schemes have asymptotic level alpha under the null hypothesis of no change and are consistent under the alternative of a change in at least one component of the high dimensional mean vector. Moreover, we also prove that the new detection scheme identifies all components affected by a change. The finite sample properties of the new methodology are illustrated by means of a simulation study and in the analysis of a data example. Our approach is based on a new type of monitoring scheme for one-dimensional data which turns out to be often more powerful than the usually used CUSUM and Page- CUSUM methods, and the component-wise statistics are aggregated by the maximum statistic. From a mathematical point of view we use Gaussian approximations for high dimensional time series to prove our main results and derive extreme value convergence for the maximum of the maximal increment of dependent Brownian motions. In particular we show that the range of a Brownian motion on a given interval is in the domain of attraction of the Gumbel distribution.