[Paper Review] Testing for a Change in Mean After Changepoint Detection

Sean Jewell, Paul Fearnhead+1 more|arXiv (Cornell University)|Oct 9, 2019

Metabolomics and Mass Spectrometry Studies4 citations

TL;DR

This paper proposes a selective inference framework to test for a change in mean around an estimated changepoint, conditioning only on the local neighborhood of the changepoint rather than the full set of estimated changepoints. It enables valid, high-powered hypothesis tests for changepoints detected via binary segmentation, ℓ₀ segmentation, or the fused lasso, with efficient computation and correct Type I error control even when the error variance is unknown.

ABSTRACT

While many methods are available to detect structural changes in a time series, few procedures are available to quantify the uncertainty of these estimates post-detection. In this work, we fill this gap by proposing a new framework to test the null hypothesis that there is no change in mean around an estimated changepoint. We further show that it is possible to efficiently carry out this framework in the case of changepoints estimated by binary segmentation and its variants, $\ell_{0}$ segmentation, or the fused lasso. Our setup allows us to condition on much less information than existing approaches, which yields higher powered tests. We apply our proposals in a simulation study and on a dataset of chromosomal guanine-cytosine content. These approaches are freely available in the R package ChangepointInference at https://jewellsean.github.io/changepoint-inference/.

Motivation & Objective

• To address the lack of statistical inference tools for changepoints estimated by widely used methods like binary segmentation and ℓ₀ segmentation.
• To develop a framework that tests for a change in mean around an estimated changepoint, focusing on the magnitude of the shift rather than its precise location.
• To enable valid hypothesis testing with correct Type I error control by conditioning on minimal sufficient information—specifically, the changepoint and its immediate neighbors—rather than the full set of estimated changepoints.
• To extend the framework to handle unknown error variance and to support confidence interval construction for the mean shift.
• To provide a computationally efficient implementation via algorithms that exploit the structure of common changepoint detection procedures.

Proposed method

• The method uses selective inference to test the null hypothesis H₀: νᵀμ = 0, where ν is a contrast vector capturing the mean difference around a candidate changepoint.
• It conditions on the event that the j-th estimated changepoint and its adjacent changepoints are included in the model, i.e., {τ̂_{j-1}, τ̂_j, τ̂_{j+1}} ⊆ M(y′(ϕ)), to define the selective distribution of the test statistic.
• The p-value is computed as Pr(|νᵀy| ≥ |νᵀy| | conditioning event), leveraging the fact that νᵀY and the projection onto the orthogonal complement are independent under Gaussian errors.
• Efficient algorithms are developed for binary segmentation and ℓ₀ segmentation by exploiting recursive structures and dynamic programming to compute the conditioning set and p-values.
• For the fused lasso, the method adapts the framework using the piecewise constant structure of the solution path and conditional distributions under the lasso path.
• The approach remains valid even when the error variance σ² is unknown, by using a consistent estimator (e.g., median absolute deviation of first differences) in the test statistic.

Experimental results

Research questions

• RQ1 Can we construct a valid hypothesis test for a change in mean around a changepoint estimated by common methods like binary segmentation or ℓ₀ segmentation?
• RQ2 Does conditioning only on the local changepoint neighborhood—rather than the full model selection path—yield higher-powered tests with correct Type I error control?
• RQ3 Can the framework be extended to handle unknown error variance while maintaining valid inference?
• RQ4 Is the method computationally efficient enough to be applied to large-scale time series data?
• RQ5 Can the framework be adapted to other models, such as spike detection in calcium imaging data?

Key findings

• The proposed framework achieves correct Type I error control under the null hypothesis, even when the error variance is unknown, by using a consistent estimator of σ in the test statistic.
• The method maintains high statistical power by conditioning on minimal information—specifically, the estimated changepoint and its two neighbors—thereby avoiding the conservativeness of conditioning on the entire model selection path.
• The framework is computationally efficient and scalable, with algorithms that exploit the recursive structure of binary segmentation and ℓ₀ segmentation to compute p-values in O(T) time per changepoint.
• Simulation studies show that the method maintains appropriate Type I error rates and exhibits substantial power under the alternative hypothesis, even in settings with multiple, closely spaced changepoints.
• The approach is successfully applied to a chromosomal guanine-cytosine content dataset, demonstrating practical utility in real-world genomics applications.
• The method is implemented in the R package ChangepointInference, making it accessible for widespread use in applied research.