Change Point Detection for Random Objects with Periodic Behavior

Time-varying random objects have been increasingly encountered in modern data analysis. Moreover, in a substantial number of these applications, periodic behaviour of the random objects has been observed. We develop a novel procedure to identify and localize abrupt changes in the distribution of non-Euclidean random objects with periodic behaviour. The proposed procedure is flexible and broadly applicable, accommodating a variety of suitable change point detectors for random objects. We further construct a specific detector used in the proposed procedure which is nonparametric and effectively captures the entire distribution of these random objects. The theoretical results cover the limiting distribution of the detector under the null hypothesis of no change point, the power of the test in the presence of change points under local alternatives and the consistency in estimating the number and locations of change points, whether dealing with a single change point or multiple ones. We demonstrate that the most competitive method currently in the literature for change point detection in random objects is degraded by periodic behaviour, as periodicity leads to blurring of the changes that this procedure aims to discover. Through comprehensive simulation studies, we demonstrate the superior power and accuracy of our approach in both detecting change points and pinpointing their locations. Our main application is to weighted networks, represented through graph Laplacians. The proposed method delivers highly interpretable results, as evidenced by the identification of meaningful change points in the New York City Citi Bike sharing system that align with significant historical events.