Change Point Analysis in Piecewise Polynomial Signals Using Trend Filtering

Abstract

Change point analysis, an active area of research in many fields, including statistics, has attracted a lot of attention in recent years. The focus of this thesis is change point detection, where the purpose is to estimate the number and locations of changes in the structure of a data sequence. Despite the recent attention, few papers have addressed change point analysis for piecewise polynomial signals. To address this gap, the work focuses on the mean change point problem for such signals. We approach this problem by applying trend filtering and introduce a method called Pattern Recovery Using Trend Filtering (PRUTF) to estimate change point locations.

We develop an extension of the trend filtering algorithm in order to construct a dual solution path that allows us to detect change points. We demonstrate that the dual solution path constitutes a Gaussian bridge process, which enables us to derive an exact and efficient stopping rule to terminate the search algorithm. Finally, we prove that the estimates produced by this algorithm are asymptotically consistent for piecewise polynomial signals. This result holds even in the presence of staircase patterns (consecutive change points of the same sign) in the signal, which to the best of our knowledge, previous works have been unable to address.

Additionally, we employ the post-selection framework to make statistical inferences for change points once selected by the PRUTF method. The key development has been to represent the set of estimated change points and their signs as a polyhedron in the sample space. This representation gives us tools for exact statistical inference such as the construction of confidence intervals and testing the significance of the estimated change points. We also provide some truncation techniques in order to improve the confidence intervals and enhance the power of the tests.