Nonparametric Tests for Homogeneity Based on Non-Bipartite Matching

Abstract

Given a sequence of observations, has a change occurred in the underlying probability distribution with respect to observation order? This problem of detecting change points arises in a variety of applications including health prognostics for mechanical systems, syndromic disease surveillance in geographically dispersed populations, anomaly detection in information networks, and multivariate process control in general. Detecting change points in high-dimensional settings is challenging, and most change-point methods for multidimensional problems rely upon distributional assumptions or the use of observation history to model probability distributions. We present three new nonparametric statistical tests for heterogeneity based on the combinatorial properties of minimum non-bipartite matching (MNBM). The key idea underlying each of these tests is that if a sequence of independent random observations undergoes a change in distribution—either an abrupt “shift” or a gradual “drift”—a MNBM based on inter-point distances tends to produce pairings that are closer in the sequence labeling than would be the case if the observations were drawn from the same distribution. Our tests follow on the work of Rosenbaum (2005) who used MNBM to derive a simple cross-match test statistic for the two-sample problem based on this idea. Similar ideas are present in the minimum spanning tree (MST) test derived by Friedman and Rafsky (1979, 1981). We extend these approaches by utilizing ensembles of orthogonal MNBMs which greatly increase information extraction from the data, leading to tests that compare favorably to parametric procedures while maintaining level and good power properties across distributions.