We consider the problem of uncertainty quantification in change point regressions, where the signal can be piecewise polynomial of arbitrary but fixed degree. That is we seek disjoint intervals which, uniformly at a given confidence level, must each contain a change point location. We propose a procedure based on performing local tests at a number of scales and locations on a sparse grid, which adapts to the choice of grid in the sense that by choosing a sparser grid one explicitly pays a lower price for multiple testing. The procedure is fast as its computational complexity is always of the order $\mathcal{O} (n \log (n))$ where $n$ is the length of the data, and optimal in the sense that under certain mild conditions every change point is detected with high probability and the widths of the intervals returned match the mini-max localisation rates for the associated change point problem up to log factors. A detailed simulation study shows our procedure is competitive against state of the art algorithms for similar problems. Our procedure is implemented in the R package ChangePointInference which is available via https://github.com/gaviosha/ChangePointInference.

翻译：我们研究变点回归中的不确定性量化问题，其中信号可以是任意固定次数的分段多项式。具体而言，我们旨在寻找互不相交的区间，这些区间在给定置信水平下必须各自包含一个变点位置。我们提出一种基于稀疏网格上多尺度与多位置局部检验的方法，该方法能自适应网格选择——通过选择更稀疏的网格可显式降低多重检验代价。该算法具有高效性，其计算复杂度始终保持在 $\mathcal{O} (n \log (n))$ 量级（其中 $n$ 为数据长度），同时具备最优性：在特定温和条件下，每个变点都能以高概率被检测到，且返回的区间宽度在对数因子范围内匹配相应变点问题的极小极大定位速率。详细的仿真研究表明，本算法在处理同类问题时与前沿算法相比具有竞争力。该算法已实现于 R 软件包 ChangePointInference 中，可通过 https://github.com/gaviosha/ChangePointInference 获取。