Statistics Department Seminar presents "Fréchet change point detection"

 

In this talk, I 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. 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 Fréchet variance or both. The proposed method is based on comparisons of Fréchet variances before and after putative change point locations. First, I will establish that under the null hypothesis of no change point the limit distribution of the proposed scan function is the square of a standardized Brownian Bridge. It is well known that such convergence is rather slow in moderate to high dimensions. For more accurate results in finite sample applications, I will provide a theoretically justified bootstrap-based scheme for testing the presence of change points. Next, I will show that when a change point exists, (1) the proposed test is consistent under contiguous alternatives and (2) the estimated location of the change point is consistent. All of the above results hold 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. I will illustrate the efficacy of the proposed approach in empirical studies and in real-data applications with sequences of maternal fertility distributions. Finally, I will talk about some future extensions and other related research directions, for instance, when one has samples of dynamic metric space data.