Intrinsic Geometry-Based Angular Covariance: A Novel Framework for Nonparametric Changepoint Detection in Meteorological Data

In many temporal datasets, the parameters of the underlying distribution may change abruptly at unknown times. Detecting such changepoints is crucial for numerous applications. Although such a problem has been extensively studied for linear data, there has been notably less research on bivariate angular data. To the best of our knowledge, this paper presents the first attempt to address the changepoint detection problem for the mean direction of toroidal and spherical data. By defining the square of an angle'' through intrinsic geometry, we construct a curved dispersion matrix for bivariate angular data, analogous to the linear dispersion matrix in Euclidean space. Using the analogous measure of the Mahalanobis distance,’’ we develop two new non-parametric tests to identify changes in the mean direction parameters for toroidal and spherical distributions. The pivotal distributions of the test statistics are shown to follow the Kolmogorov distribution under the null hypothesis. Under the alternative hypothesis, we establish the consistency of the proposed tests. We also apply the proposed methods to detect changes in mean direction for hourly wind-wave direction (toroidal) measurements and the path (spherical) of the cyclonic storm ``Biporjoy,’’ which occurred between 6th and 19th June 2023 over the Arabian Sea, western coast of India.

💡 Research Summary

The paper introduces a novel non‑parametric framework for detecting change‑points in the mean direction of bivariate angular data that lie on a torus (S¹×S¹) or a sphere (S²). Recognizing that traditional change‑point methods are designed for Euclidean data, the authors develop intrinsic‑geometric tools that respect the curvature and periodicity of directional data.

First, they define the “square of an angle” as the proportionate area between the origin (0,0) and a point (θ, θ) on the curved manifold. For the torus, the surface element dA = r(R + r cosθ) dθ dϕ is derived from the first fundamental form, and the torus is partitioned into four mutually exclusive regions; the smallest region’s area, normalized by the total torus area 4π²rR, yields the proportionate area. An analogous construction is performed for the sphere, where dA = r² sinθ dθ dϕ and the minimum of four region areas, divided by 4πr², defines the proportionate area. This proportionate area serves as the intrinsic analogue of the Euclidean squared distance.

Using these squared‑angle measures, the authors construct a “curved dispersion matrix” Σ, a 2 × 2 matrix that captures variance and covariance of the two angular components on the manifold. The matrix is built from sample averages of the squared‑angle quantities, thereby inheriting the geometry of the underlying space.

With Σ in hand, they define a Mahalanobis‑type distance between the sample mean directions of two segments of the data split at a candidate change‑point t:

Dₜ = √{ ( \bar{X}{1:t} − \bar{X}{t+1:n} )ᵀ Σ⁻¹ ( \bar{X}{1:t} − \bar{X}{t+1:n} ) }.

The distance is accumulated across time to form a CUSUM‑type statistic. Two versions of the statistic are presented: one for toroidal data, another for spherical data.

Theoretical results show that under the null hypothesis (no change in mean direction) the maximum of the CUSUM process converges to the Kolmogorov distribution, the same limiting law as classic CUSUM tests for linear data. This is proved by demonstrating that the intrinsic distance process satisfies the same functional central limit theorem conditions after appropriate scaling. Under the alternative hypothesis (a genuine change in mean direction), the statistic diverges to infinity, establishing consistency of the tests.

A comprehensive simulation study evaluates the finite‑sample performance. Torus data are generated from the von Mises‑sine distribution, while sphere data follow the Fisher distribution. Across varying sample sizes, change‑point locations, and concentration parameters, the proposed tests maintain nominal size and exhibit higher power than existing linear‑based or parametric angular methods.

The methodology is applied to real meteorological data from Cyclone “Biporjoy” (June 6–19, 2023) over the Arabian Sea. Hourly wind direction and wave direction constitute toroidal observations, whereas the cyclone’s latitude–longitude trajectory forms spherical observations. The tests successfully identify several change‑points coinciding with known dynamical transitions: abrupt shifts in wind–wave alignment during intensification, and directional turning points in the cyclone track associated with interaction with atmospheric pressure systems. These findings demonstrate the practical utility of the approach for climate monitoring and disaster preparedness.

In summary, the paper makes five key contributions: (1) a unified intrinsic definition of the square of an angle for torus and sphere, (2) the introduction of a curved dispersion matrix as a natural covariance analogue on manifolds, (3) Mahalanobis‑type CUSUM statistics tailored to angular data, (4) rigorous asymptotic theory linking the test statistics to the Kolmogorov distribution and proving consistency, and (5) empirical validation on both simulated and real-world cyclone data. By bridging differential geometry and change‑point analysis, the work opens a new avenue for robust, non‑parametric inference on directional data that lie on curved spaces.