A Comparison of Single and Multiple Changepoint Techniques for Time Series Data

Abstract

This paper describes and compares several prominent single and multiple changepoint techniques for time series data. Due to their importance in inferential matters, changepoint research on correlated data has accelerated recently. Unfortunately, small perturbations in model assumptions can drastically alter changepoint conclusions; for example, heavy positive correlation in a time series can be misattributed to a mean shift should correlation be ignored. This paper considers both single and multiple changepoint techniques. The paper begins by examining cumulative sum (CUSUM) and likelihood ratio tests and their variants for the single changepoint problem; here, various statistics, boundary cropping scenarios, and scaling methods (e.g., scaling to an extreme value or Brownian Bridge limit) are compared. A recently developed test based on summing squared CUSUM statistics over all times is shown to have realistic Type I errors and superior detection power. The paper then turns to the multiple changepoint setting. Here, penalized likelihoods drive the discourse, with AIC, BIC, mBIC, and MDL penalties being considered. Binary and wild binary segmentation techniques are also compared. We introduce a new distance metric specifically designed to compare two multiple changepoint segmentations. Algorithmic and computational concerns are discussed and simulations are provided to support all conclusions. In the end, the multiple changepoint setting admits no clear methodological winner, performance depending on the particular scenario. Nonetheless, some practical guidance will emerge.

Changepoint Detection: An Analysis of the Central England Temperature Series

“…These two penalties were judged as "winners" in a changepoint detection context in the recent statistical comparison in [16]. AIC penalties are not considered here because they often estimate an excessive number of changepoints [16]. The BIC penalty for having 𝑚 changepoints at the times 𝜏 1 , .…”

“…Binary segmentation works with any single changepoint technique, termed an at most one change (AMOC) method. Many AMOC tests have been developed, including cumulative sums (CUSUM) [34], likelihood ratios [35], Chow tests [36], and sum of squared CUSUM tests [16]. Binary segmentation first analyzes the entire series for a changepoint.…”

“…The procedure is repeated iteratively until no subsegments are deemed to have additional changepoints. While simple and computationally convenient, binary segmentation is one of the poorer performing multiple changepoint techniques [16], often being fooled by changepoints that occur close to one another or shifts that move the series in antipodal directions. Attempts to fix binary segmentation exist -see the wild binary segmentation and related methods in [19] and [37].…”

“…It is important to allow autocorrelation and mean shifts features in tandem in models since both features can inject similar run patterns into a time series. [4,16,17] show how misleading changepoint inferences can be should one ignore autocorrelation in a series. This point will be reinforced as we proceed.…”

Autocovariance Estimation in the Presence of Changepoints

“…Changepoint techniques may incorrectly segment the data (estimate too many or too few changepoints) should non-zero autocorrelation be ignored. Two primary ways of tackling this issue have been pursued: 1) including autocorrelation in the multiple changepoint model [Li and Lund, 2012], and 2) decorrelating the time series prior to any changepoint analysis [Chakar et al, 2017, Shi et al, 2021. In either case, one needs the autocovariance function of the data.…”