Fr\'echet Change Point Detection

TL;DR

This paper introduces a change-point detection method in metric spaces based on Fréchet variance comparisons, providing theoretical guarantees and demonstrating effectiveness through simulations and real data examples.

Contribution

It presents a novel, tuning-parameter-free approach for detecting change-points in complex metric space data with proven consistency and bootstrap inference methods.

Findings

Method accurately detects change-points in simulations.

Theoretical guarantees ensure consistency under broad conditions.

Effective in real-world applications like fertility data and network analysis.

Abstract

We propose a method to infer the presence and location of change-points in the distribution of a sequence of independent data taking values in a general metric space, where change-points are viewed as locations at which the distribution of the data sequence changes abruptly in terms of either its Fr\'echet mean or Fr\'echet variance or both. The proposed method is based on comparisons of Fr\'echet variances before and after putative change-point locations and does not require a tuning parameter except for the specification of cut-off intervals near the endpoints where change-points are assumed not to occur. Our results include theoretical guarantees for consistency of the test under contiguous alternatives when a change-point exists and also for consistency of the estimated location of the change-point if it exists, where under the null hypothesis of no change-point the limit distribution of the proposed scan function is the square of a standardized Brownian Bridge. These consistency results are applicable for a broad class of metric spaces under mild entropy conditions. Examples include the space of univariate probability distributions and the space of graph Laplacians for networks. Simulation studies demonstrate the effectiveness of the proposed methods, both for inferring the presence of a change-point and estimating its location. We also develop theory that justifies bootstrap-based inference and illustrate the new approach with sequences of maternal fertility distributions and communication networks.