Bayesian changepoint detection via logistic regression and the topological analysis of image series

We present a Bayesian method for multivariate changepoint detection that allows for simultaneous inference on the location of a changepoint and the coefficients of a logistic regression model for distinguishing pre-changepoint data from post-changepoint data. In contrast to many methods for multivariate changepoint detection, the proposed method is applicable to data of mixed type and avoids strict assumptions regarding the distribution of the data and the nature of the change. The regression coefficients provide an interpretable description of a potentially complex change. For posterior inference, the model admits a simple Gibbs sampling algorithm based on Pólya-gamma data augmentation. We establish conditions under which the proposed method is guaranteed to recover the true underlying changepoint. As a testing ground for our method, we consider the problem of detecting topological changes in time series of images. We demonstrate that our proposed method BCLR, combined with a topological feature embedding, performs well on both simulated and real image data. The method also successfully recovers the location and nature of changes in more traditional changepoint tasks. An implementation of our method is available in the Python package bclr.

本文提出一种用于多变量变点检测（multivariate changepoint detection）的贝叶斯方法，该方法可同时完成变点位置以及用于区分变点前与变点后数据的逻辑回归模型（logistic regression model）系数的推断。与多数多变量变点检测方法相比，本文所提方法可适用于混合型数据，且无需对数据分布与变化本质施加严苛假设。该逻辑回归模型的系数可对潜在的复杂变化提供可解释性描述。在后验推断环节，该模型可依托波利亚-伽马数据增强（Pólya-gamma data augmentation）技术，采用简易的吉布斯采样（Gibbs sampling）算法。本文推导了可保证所提方法准确恢复真实隐式变点的理论条件。为验证所提方法的有效性，本文将其应用于图像时间序列的拓扑变化检测任务。实验结果表明，本文提出的BCLR方法结合拓扑特征嵌入（topological feature embedding）技术后，在模拟图像数据与真实图像数据集上均表现出色。该方法在更为经典的变点检测任务中，同样可准确恢复变化的位置与本质。本文所提方法的代码实现可通过Python包bclr获取。