[Paper Review] Valid Post-Detection Inference for Change Points Identified Using Trend Filtering

Reza Valiollahi Mehrizi, Shojaeddin Chenouri|arXiv (Cornell University)|Apr 24, 2021

Statistical Methods and Inference50 references4 citations

TL;DR

This paper develops valid post-detection inference for change points identified via the PRUTF algorithm, which uses trend filtering to estimate piecewise polynomial signals. It introduces global and local post-detection inference strategies to construct accurate confidence intervals and p-values, significantly reducing the length of conventional post-selection intervals while maintaining valid coverage under known and unknown error variance.

ABSTRACT

There are many research works and methods about change point detection in the literature. However, there are only a few that provide inference for such change points after being estimated. This work mainly focuses on a statistical analysis of change points estimated by the PRUTF algorithm, which incorporates trend filtering to determine change points in piecewise polynomial signals. This paper develops a methodology to perform statistical inference, such as computing p-values and constructing confidence intervals in the newly developed post-selection inference framework. Our work concerns both cases of known and unknown error variance. As pointed out in the post-selection inference literature, the length of such confidence intervals are undesirably long. To resolve this shortcoming, we also provide two novel strategies, global post-detection, and local post-detection which are based on the intrinsic properties of change points. We run our proposed methods on real as well as simulated data to evaluate their performances.

Motivation & Objective

• To address the lack of statistical inference methods for change points detected via data-driven procedures, particularly in piecewise polynomial models.
• To develop a post-selection inference framework that accounts for the data-dependent nature of change point detection, avoiding invalid inferences from treating detected points as fixed.
• To reduce the conservativeness of standard post-selection confidence intervals, which are often too wide due to selection bias.
• To introduce two novel strategies—global and local post-detection inference—leveraging intrinsic properties of change points to improve interval precision.
• To evaluate the performance of the proposed methods on both simulated and real-world data, demonstrating robustness and accuracy under known and unknown error variance.

Proposed method

• Adapts the post-selection inference framework to change point detection in piecewise polynomial signals estimated via the PRUTF algorithm, which uses trend filtering to identify change points.
• Derives exact sampling distributions of test statistics conditional on the selected change points, enabling valid p-values and confidence intervals.
• Constructs confidence intervals using the truncated t-distribution and normal approximation, with bounds derived from the quantiles of the selection distribution.
• Introduces global post-detection inference by pooling information across multiple change points to improve interval efficiency.
• Proposes local post-detection inference that focuses on individual change points using localized test statistics and selection events.
• Employs a truncation set based on the observed test statistics to define the conditional sampling distribution, ensuring valid inference under model selection.

Experimental results

Research questions

• RQ1 How can valid statistical inference be performed for change points detected using trend filtering, accounting for the data-driven selection process?
• RQ2 What are the consequences of ignoring selection bias when constructing confidence intervals for detected change points?
• RQ3 Can global and local post-detection inference strategies reduce the length of post-selection confidence intervals while preserving coverage?
• RQ4 Under what conditions do the proposed confidence intervals maintain valid coverage when the error variance is unknown?
• RQ5 How do the proposed methods compare in performance to standard post-selection inference in terms of interval width and empirical coverage?

Key findings

• The proposed global and local post-detection inference strategies significantly reduce the length of confidence intervals compared to standard post-selection methods, especially in high-dimensional or complex signal settings.
• Theoretical bounds on confidence interval length are derived, showing that the upper bound is proportional to the quantile of the truncated t-distribution and the difference between critical values.
• For large degrees of freedom, the truncated t-distribution behaves like the normal distribution, and the proposed bounds remain valid, ensuring robustness across different error distributions.
• The function $ h(x) $, which governs the validity of the interval bounds, is positive for $ x \in (0, x_0] $, indicating that the bounds hold under reasonable conditions on the signal-to-noise ratio.
• Simulation and real data experiments confirm that the proposed methods maintain correct coverage levels while producing shorter intervals than conventional post-selection inference.
• The method achieves valid inference even when the error variance is unknown, by using estimated standard errors in the construction of test statistics and confidence intervals.