In this paper we consider change-points in multiple sequences with the objective of minimizing the estimation error of a sequence by making use of information from other sequences. This is in contrast to recent interest on change-points in multiple sequences where the focus is on detection of common change-points. We start with the canonical case of a single sequence with constant change-point intensities. We consider two measures of a change-point algorithm. The first is the probability of estimating the change-point with no error. The second is the expected distance between the true and estimated change-points. We provide a theoretical upper bound for the no error probability, and a lower bound for the expected distance, that must be satisfied by all algorithms. We propose a scan-CUSUM algorithm that achieves the no error upper bound and come close to the distance lower bound. We next consider the case of non-constant intensities and establish sharp conditions under which estimation error can go to zero. We propose an extension of the scan-CUSUM algorithm for a non-constant intensity function, and show that it achieves asymptotically zero error at the boundary of the zero-error regime. We illustrate an application of the scan-CUSUM algorithm on multiple sequences sharing an unknown, non-constant intensity function. We estimate the intensity function from the change-point profile likelihoods of all sequences and apply scan-CUSUM on the estimated intensity function.
翻译:在本文中,我们考虑多个序列的变化点,目的是通过使用其他序列的信息来尽量减少序列的估计误差。这与最近对多个序列的变化点的兴趣形成对照,这些变化点的重点是检测共同的变更点。我们从单一序列的卡通案例开始,以恒定的变更点强度开始。我们考虑两种变化点算法的度量。第一个是无误估算变化点的概率。第二个是真实和估计变化点之间的预期距离。我们为无误概率提供了理论上限,而预期距离的界限较低,这必须由所有算法来满足。我们建议扫描- CUSUM算法,以无误上限,并接近于更低的界限。我们接下来考虑非一致的强度的两个情况,并设定了估算误差为零的精确度。我们提议在非一致强度函数上扩展扫描-CUUUM算法的扫描值上限,并显示它从不连续的轨道序列中实现对轨道的运行量度的不精确度。我们提议,我们从不连续的对轨道的深度进行精确度估算。