High-dimensional Change-point Detection Using Generalized Homogeneity Metrics

Change-point detection is a classical problem in statistics. We address the problem of detecting abrupt changes in the data-generating distributions of a sequence of high-dimensional observations beyond the first two moments. This problem remains less explored, especially in the high-dimensional context, compared to detecting changes in the mean or the covariance structure. To the best of our knowledge, this is one of the first attempts to detect and localize general types of distributional changes in the high-dimensional regime. We develop a distance-based method to (i) test for the existence of a change-point, and (ii) identify the change-point locations in an independent sequence of high-dimensional observations. Our approach rests upon recent distance-based tests for the homogeneity of two high-dimensional distributions. We construct a single change-point test statistic based on a cumulative sum process in an embedded Hilbert space and rigorously derive its limiting null distribution and prove asymptotic consistency under the high-dimensional medium sample size (HDMSS) framework. Subsequently, we combine our statistics with the Narrowest-Over-Threshold (NOT) strategy to recursively estimate and test for multiple change-point locations. We also study a componentwise monotone-invariant, rank-based extension; because its pseudo-observations are pooled empirical mid-ranks and are therefore dependent, we present this version as a practically useful heuristic extension supported by simulation evidence rather than as a fully proved analogue of the original statistic. The superior performance of our methodology compared to existing procedures is illustrated via extensive simulation studies and an application to U.S. stock return data during the global financial crisis. The proposed method is implemented in the R package KDist, available at https://github.com/zhangxiany-tamu/KDist.