Narrowest Significance Pursuit: inference for multiple change-points in linear models

Abstract

We propose Narrowest Significance Pursuit (NSP), a general and flexible methodology for automatically detecting localised regions in data sequences which each must contain a change-point, at a prescribed global significance level. Here, change-points are understood as abrupt changes in the parameters of an underlying linear model. NSP works by fitting the postulated linear model over many regions of the data, using a certain multiresolution sup-norm loss, and identifying the shortest interval on which the linearity is significantly violated. The procedure then continues recursively to the left and to the right until no further intervals of significance can be found. The use of the multiresolution sup-norm loss is a key feature of NSP, as it enables the transfer of significance considerations to the domain of the unobserved true residuals, a substantial simplification. It also guarantees important stochastic bounds which directly yield exact desired coverage probabilities, regardless of the form or number of the regressors.NSP works with a wide range of distributional assumptions on the errors, including Gaussian with known or unknown variance, some light-tailed distributions, and some heavy-tailed, possibly heterogeneous distributions via self-normalisation. It also works in the presence of autoregression. The mathematics of NSP is, by construction, uncomplicated, and its key computational component uses simple linear programming. In contrast to the widely studied "post-selection inference" approach, NSP enables the opposite viewpoint and paves the way for the concept of "post-inference selection". Pre-CRAN R code implementing NSP is available at https://github.com/pfryz/nsp.

Bootstrap confidence intervals for multiple change points based on moving sum procedures

“…We briefly compare the proposed bootstrap procedure with SMUCE (Frick et al, 2014), which returns a confidence set for f t at a given level 1 − α and thus confidence bands around change points. Several authors noted that the smaller α is set, the constraint imposed by SMUCE on the estimated residuals becomes more lenient and thus it tends to under-estimate the number of change points (Chen et al, 2014;Fryzlewicz, 2020). This is demonstrated in Table 2 comparing the MoLP (Cho and Kirch (2021), described in Section 4.2.2) and SMUCE with varying α ∈ {0.1, 0.2, 0.45} in terms of their detection accuracy.…”

Improving Power by Conditioning on Less in Post-selection Inference for Changepoints

“…Fearnhead, 2006) that sample from a posterior over the number and location of the changepoints naturally give measures of uncertainty, assessing uncertainty for non-Bayesian methods is more challenging. Current work in this area includes the SMUCE method (Frick et al, 2014;Li et al, 2016;Pein et al, 2017), and global methods that try to give regions that produce sets of intervals, all of which must include a change at a pre-specified significance level (Fryzlewicz, 2020(Fryzlewicz, , 2021.…”

Localising change points in piecewise polynomials of general degrees

“…This is studied thoroughly in [6], [16], [12], [13], [24], [22] and [39], among others. [14], [4], [8], [1], [18], [25] and [9] studied change point analysis in piecewise linear signals. Our work in this paper can be seen as a generalisation of the aforementioned results, allowing for polynomials of arbitrary degrees, and the magnitudes of coefficients changes to vanish as the sample size grows unbounded, although some of the aforementioned work may contain more general assumptions on the noise structure.…”