Fréchet Change Point Detection

Abstract

We propose a method to infer the presence and location of changepoints 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 Fréchet mean or Fréchet variance or both. The proposed method is based on comparisons of Fréchet 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.

Weighted-Graph-Based Change Point Detection

“…As corollaries of previous theorems, we can get asymptotic null distribution, power, and localization consistency of S 3 . The results are similar to those in Dubey and Müller (2019) and for brevity are included in the Appendix B.8. Comparing Corollary B.3 or Theorem 3 in Dubey and Müller (2019) against Theorem 4.2, we find that when using S 3 instead of S 1 , we are capable of idenfitying both changes in location or scale, but pay a price of increasing the order of magnitude of local alternatives µ 0 − µ 1 H we can detect from n −1/2 to n −1/4 .…”

“…Their statistics, before taking max with respect to t, is equivalent to (properly normalized) 4 T 2 1 (t) + T 2 2 (t), where T 1 (t) is defined in Equation ( 26) and can be seen as a non-centered version of T 1 (t). In this sense S 1 , S 2 and that of Dubey and Müller (2019) are highly related. One difference is that we analyze the two components, T 1 (t) and T 2 (t), separately, and the combination of them follows automatically (see Section 4.1.4).…”

“…The proposed method is also closely related to that of Dubey and Müller (2019) that developed a test statistic for detecting change point in Frechet mean and/or Frechet variance in the AMOC setting. Their statistics, before taking max with respect to t, is equivalent to (properly normalized) 4 T 2 1 (t) + T 2 2 (t), where T 1 (t) is defined in Equation ( 26) and can be seen as a non-centered version of T 1 (t).…”

“…, we are free of solving the optimization problem in Dubey and Müller (2019), granting the proposed statistics greater practicality. Finally, we also generalize our statistics to the multiple change points setting and prove next its asymptotic correctness.…”

“…We investigate the power and localization accuracy for the proposed statistics in simulated datasets. We compare the proposed statistics (S 1 , S 2 , S 3 , all with higher order corrections introduced in Section 4.1.1) against Z w , M and that of Dubey and Müller (2019), which we refer to as D. Notice that S 3 is the same as D (see Definition 27), except that we are using higher order corrections introduced in Section 4.1.1. For graph-based methods, as suggested by Chen and Friedman (2017), we use 5-MST to construct the graph.…”