Riemannian Metrics refer to a mathematical framework used to measure distances and angles on a manifold, specifically within the space of symmetric positive definite matrices. This is relevant for change point detection, as it allows for the analysis of correlation matrices, which are crucial in understanding the relationships between variables in data sequences.
Riemannian geometry provides tools for defining geodesic distances, the shortest paths between points on the manifold. In this case, the geodesic distance helps quantify changes in correlation structures over time. Using these metrics, the proposed Rio-CPD framework can effectively track shifts in correlations and enhance the CUSUM (Cumulative Sum) statistic, making it more sensitive to changes.
The careful selection of Riemannian metrics ensures that the framework remains computationally efficient while maintaining accuracy in detecting change points, thereby improving performance compared to existing methods.
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The objective of change point detection is to identify abrupt changes at potentially multiple points within a data sequence. This task is particularly challenging in the online setting where various types of changes can occur, including shifts in both the marginal and joint distributions of the data. This paper tackles these challenges by sequentially tracking correlation matrices on their Riemannian geometry, where the geodesic distances accurately capture the development of correlations. We propose Rio-CPD, a non-parametric correlation-aware online change point detection framework that combines the Riemannian geometry of the manifold of symmetric positive definite matrices and the cumulative sum statistic (CUSUM) for detecting change points. Rio-CPD enhances CUSUM by computing the geodesic distance from present observations to the Frechet mean of previous observations. With careful choice of metrics equipped to the Riemannian geometry, Rio-CPD is simple and computationally efficient. Experimental results on both synthetic and real-world datasets demonstrate that Rio-CPD outperforms existing methods in detection accuracy and efficiency.
