Optimal nonparametric change point analysis

Abstract

We study change point detection and localization for univariate data in fully nonparametric settings, in which at each time point, we acquire an independent and identically distributed sample from an unknown distribution that is piecewise constant. The magnitude of the distributional changes at the change points is quantified using the Kolmogorov-Smirnov distance. Our framework allows all the relevant parameters, namely the minimal spacing between two consecutive change points, the minimal magnitude of the changes in the Kolmogorov-Smirnov distance, and the number of sample points collected at each time point, to change with the length of the time series. We propose a novel change point detection algorithm based on the Kolmogorov-Smirnov statistic and show that it is nearly minimax rate optimal. Our theory demonstrates a phase transition in the space of model parameters. The phase transition separates parameter combinations for which consistent localization is possible from the ones for which this task is statistically infeasible. We provide extensive numerical experiments to support our theory.

Random Forests for Change Point Detection

“…Nonparametric change point detection methods try to use measures that do not rely on parametric forms of the distribution or the nature of change. Proposals for univariate nonparametric change point detection methods include Pettitt (1979), Carlstein (1988), Dümbgen (1991), and, more recently, Zou et al (2014) and Madrid-Padilla et al (2021a). Multivariate setups are challenging even in parametric scenarios.…”

A Contrastive Approach to Online Change Point Detection

“…This problem is called online (also referred to as sequential or quickest) change point detection. Such a setup is quite different from another major research direction, offline change point detection [11,55,35,8,4,28,15,1,34], where the statistician has an access to the whole time series at once, and, instead of taking decisions on the fly, he is mostly interested in a retrospective analysis and change point localization.…”