RIO-CPD: A Riemannian Geometric Method for Correlation-aware Online Change Point Detection

TL;DR

This paper introduces Rio-CPD, a non-parametric, correlation-aware online change point detection framework that integrates the Riemannian geometry of the manifold of symmetric positive definite matrices with the cumulative sum (CUSUM) statistic for detecting change points.

Abstract

Change point detection aims to identify abrupt shifts occurring at multiple points within a data sequence. This task becomes particularly challenging in the online setting, where different types of changes can occur, including shifts in both the marginal and joint distributions of the data. In this paper, we address these challenges by tracking the Riemannian geometry of correlation matrices, allowing Riemannian metrics to compute the geodesic distance as an accurate measure of correlation dynamics. We introduce Rio-CPD, a non-parametric, correlation-aware online change point detection framework that integrates the Riemannian geometry of the manifold of symmetric positive definite matrices with the cumulative sum (CUSUM) statistic for detecting change points. Rio-CPD employs a novel CUSUM design by computing the geodesic distance between current observations and the Fréchet mean of prior observations. With appropriate choices of Riemannian metrics, Rio-CPD offers a simple yet effective and computationally efficient algorithm. Experimental results on both synthetic and real-world datasets demonstrate that Rio-CPD outperforms existing methods on detection accuracy, average detection delay and efficiency.

Figures (4)

• Figure 1: Comparison between $\textnormal{Rio-CPD}$ and subspace model methods. $B_i$ represents the $i$-th batch of time series data, and $d_i$ denotes the distances between neighboring batches within the subspace.
• Figure 2: The overview of the proposed $\textnormal{Rio-CPD}$ framework. First, a correlation matrix is constructed for each batch of data within the sliding window. Next, the distance between the current observation and the centroid of previous ones is calculated. A change point is likely to belong to a different cluster, resulting in a larger distance from the centroid. Finally, we compute the CUSUM statistics using this distance and perform a hypothesis test to detect the change point.
• Figure 3: Detected change points by $\textnormal{Rio-CPD}$ and MSSA-CPD on the Microservice dataset, false alarms omitted for MSSA-CPD.
• Figure 4: Running time comparison on three datasets.

Theorems & Definitions (6)

• Definition 2.1: Riemannian Manifold
• Definition 2.2: Exponential and Logarithm Map of Matrices
• Definition 2.3: Fréchet mean
• Definition 3.1: CUSUM Statistic
• Remark 3.2
• Remark 3.3