Change Detection with Probabilistic Models on Persistence Diagrams

Detecting structural changes in time-series data is crucial in many applications. However, the changes in the data may appear as global structural changes that cannot be detected by conventional methods. In recent years, Topological Data Analysis (TDA) has been used to detect such global structural changes. In TDA, information on connected components or holes of data is encoded into a two-dimensional plot called a persistence diagram (PD), which can be used to detect global changes in time-series. However, only a few studies on TDA conducted change detection assuming probabilistic structure on PDs. In this paper, we introduce probability structures into PD, with which we conduct change detection on the basis of the minimum description length principle. We propose the following two methods: (1) A parametric method: We employ the Gaussian mixture model for PD modeling and then detect global changes by tracking the changes in the optimal number of mixture components. (2) A non-parametric method: We employ kernel densities for PD modeling and then detect changes by tracking changes in their global complexity. These methods not only improve the detection accuracy of global structural changes but also provide the explainability of global changes. We showcase the effectiveness of the proposed methods using synthetic data and real-world financial time-series data.