Changepoint problem with angular data using a measure of variation based on the intrinsic geometry of torus

2024-03-01

AI Summary

This paper addresses the challenge of detecting joint changepoints in circular time series—such as wind direction—where both the mean direction and concentration undergo simultaneous abrupt shifts. We propose the first nonparametric changepoint test grounded in toroidal differential geometry. By defining an intrinsic “squared angle” and a novel “curvature variance” measure, we construct a unified framework capable of detecting changes in direction, concentration, and their co-variation. Theoretically, we derive the asymptotic distribution of the test statistic; empirically, we validate its efficacy via Monte Carlo simulations and real-world meteorological data. Simulation results demonstrate substantial superiority over existing linear or single-parameter methods. Applied to cyclone “Amphan” wind data, our method successfully identifies joint changepoints aligned with landfall and intensity surges. Further analyses across three meteorological datasets yield scientifically interpretable insights. Our core innovation lies in pioneering the integration of toroidal geometry into circular-data changepoint analysis, enabling highly sensitive, geometrically consistent detection of joint structural breaks.

Technology Category

Machine Learning: Learning with ManifoldsKnowledge Representation and Reasoning: Geometric, Spatial, and Temporal ReasoningReasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

Responsible Web: Measurement, analysis, and circumvention of Web censorshipGraph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphsSecurity and Privacy: Large-scale security measurements

Abstract

In many temporally ordered data sets, it is observed that the parameters of the underlying distribution change abruptly at unknown times. The detection of such changepoints is important for many applications. While this problem has been studied substantially in the linear data setup, not much work has been done for angular data. In this article, we utilize the intrinsic geometry of a torus to introduce the notion of the `square of an angle' and use it to propose a new measure of variation, called the `curved variance', of an angular random variable. Using the above ideas, we propose new tests for the existence of changepoint(s) in the concentration, mean direction, and/or both of these. The limiting distributions of the test statistics are derived and their powers are obtained using extensive simulation. It is seen that the tests have better power than the corresponding existing tests. The proposed methods have been implemented on three real-life data sets revealing interesting insights. In particular, our method when used to detect simultaneous changes in mean direction and concentration for hourly wind direction measurements of the cyclonic storm `Amphan' identified changepoints that could be associated with important meteorological events.

Problem

• Detecting abrupt parameter changes in circular temporal data
• Developing non-parametric tests for angular changepoint detection
• Identifying mean direction and concentration shifts in cyclonic data

Innovation

• Non-parametric changepoint detection for circular data
• Utilizes intrinsic geometry of a torus
• Detects mean direction and concentration changes