Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Surojit Biswas, Buddhananda Banerjee, Arnab Kumar Laha

Abstract

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.

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

Abstract

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.
Paper Structure (22 sections, 3 theorems, 78 equations, 12 figures, 10 tables, 2 algorithms)

This paper contains 22 sections, 3 theorems, 78 equations, 12 figures, 10 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Intrinsic geometry of torus and sphere
  3. Intrinsic geometry of torus
  4. Area Decomposition of Curved Torus
  5. Intrinsic geometry of sphere
  6. Area Decomposition of Sphere
  7. Curved dispersion matrix
  8. Curved co-variance
  9. Change in mean direction of toroidal data
  10. Single changepoint
  11. Multiple Changepoint
  12. Theoretical guarantees
  13. Change in mean direction of spherical data
  14. Numerical Studies
  15. Measures of performance
  16. ...and 7 more sections

Key Result

lemma 1

Let, $a_{11}=CVar(\Phi), a_{22}=CVar(\Theta)$, and $a_{12}=a_{21}=ACov(\Phi,\Theta)$, then the curve dispersion (CD) matrix $\Sigma_{A}$ defined by is a symmetric and positive semi-definite matrix.

Figures (12)

  • Figure 1: Area between $(\phi_1, \theta_1)$, and $(\phi_2, \theta_2)$ (a) flat torus, (b) curved torus.
  • Figure 2: Area between $(\phi_1, \theta_1)$, and $(\phi_2, \theta_2)$ (a) flat sphere, (b) sphere.
  • Figure 3: The density plots of the test statistic, $\mathcal{M}_n$ under $H_{0t}$ with a sample of size $n=1000$ from von Mises sine model.
  • Figure 4: (a),(b), (c) are the contour plots; (d), (e), (f) are the corresponding surface plots of power under $H_{1t}$ when the location of the changepoint is considered at $k^{*}=\frac{n}{2},$ for von mises sine model with zero association, positive association, and negative association, respectively.
  • Figure 5: Empirical power curves of the proposed test statistic $\mathcal{M}_n$ as a function of signal length for different change point locations under a single change point scenario. Data are generated from the bivariate von Mises sine model with fixed concentration parameters and sample size $n = 600$. The first segment has mean direction $(0, 0)$, and the second segment has a shifted mean direction of $(\delta, \delta)$, where $\delta \in \{\pi/9, \pi/8, \pi/7, \pi/6\}$. Change points are placed at nine different locations ranging from early to late in the sequence: 100, 150, 200, 250, 300, 350, 400, 450, and 500. Panels (a) and (b) correspond to dependency parameters $\kappa_3 = 0$ and $\kappa_3 = -1$, respectively. Power is estimated over 1000 simulations at a nominal level of 0.05. The results show that power increases with signal strength and is higher when the change occurs closer to the center of the sequence.
  • ...and 7 more figures

Theorems & Definitions (14)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Definition 5
  • lemma 1
  • proof
  • lemma 2
  • proof
  • proof
  • ...and 4 more