Multiple Change Point Analysis: Fast Implementation and Strong Consistency

Abstract

One of the main challenges in identifying structural changes in stochastic processes is to carry out analysis for time series with dependency structure in a computationally tractable way. Another challenge is that the number of true change points is usually unknown, requiring a suitable model selection criterion to arrive at informative conclusions. To address the first challenge, we model the data generating process as a segment-wise autoregression, which is composed of several segments (time epochs), each of which modeled by an autoregressive model. We propose a multi-window method that is both effective and efficient for discovering the structural changes. The proposed approach was motivated by transforming a segment-wise autoregression into a multivariate time series that is asymptotically segment-wise independent and identically distributed. To address the second challenge, we derive theoretical guarantees for (almost surely) selecting the true number of change points of segment-wise independent multivariate time series. Specifically, under mild assumptions, we show that a Bayesian Information Criterion (BIC)-like criterion gives a strongly consistent selection of the optimal number of change points, while an Akaike Information Criterion (AIC)-like criterion cannot. Finally, we demonstrate the theory and strength of the proposed algorithms by experiments on both synthetic and real-world data, including the Eastern US temperature data and the El Nino data from 1854 to 2015. The experiment leads to some interesting discoveries about temporal variability of the summer-time temperature over the Eastern US, and about the most dominant factor of ocean influence on climate.