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：我们考虑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阶数的分段多项式。即我们寻求互不相交的区间，每个区间在给定置信水平下均匀地必须包含一个变点位置。我们提出了一种基于稀疏网格上多尺度多位置局部检验的方法，该方法能自适应网格选择，即通过选择更稀疏的网格可明确降低多重检验的代价。该方法计算复杂度始终为$\mathcal{O}(n \log (n))$阶（$n$为数据长度），因此具有快速性；同时具有最优性，即在某些温和条件下，每个变点均能以高概率被检测到，且返回的区间宽度在忽略对数因子后匹配相关变点问题的极小极大定位速率。详细的仿真研究表明，我们的方法与同类问题的最先进算法相比具有竞争力。该方法已在R包ChangePointInference中实现，可通过https://github.com/gaviosha/ChangePointInference获取。