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Some bivariate distributions on a discrete torus with application to wind direction datasets

Brajesh Kumar Dhakad, Jayant Jha, Debepsita Mukherjee

TL;DR

This paper develops two analytically tractable bivariate models for ordinal circular data on a discrete torus, namely the bivariate wrapped geometric (BWG) and its generalized version (BGWG). By combining wrapped symmetric geometric marginals with a cosine-based dependence, the authors derive closed-form joint pmfs, marginals, conditionals, and a circular–circular dependence measure, enabling exact likelihood-based inference. Through simulation and three wind-direction datasets from India, BGWG demonstrates competitive or superior fit relative to discretized continuous bivariate circular models, with interpretable dependence parameters ρ and δ. The approach yields efficient computation, avoids discretization errors, and has potential extensions to multivariate and time-series settings via copulas and hidden Markov models. Overall, the work provides a practical framework for analyzing bivariate ordinal circular data and paves the way for broader applications in environmental and biological directional data.

Abstract

Many datasets are observed on a finite set of equally spaced directions instead of the exact angles, such as the wind direction data. However, in the statistical literature, bivariate models are only available for continuous circular random variables. This article presents two bivariate circular distributions, namely bivariate wrapped geometric (BWG) and bivariate generalized wrapped geometric (BGWG), for analyzing bivariate discrete circular data. We consider wrapped geometric distributions and a trigonometric function to construct the models. The models are analytically tractable due to the exact closed-form expressions for the trigonometric moments. We thoroughly discuss the distributional properties of the models, including the interpretation of parameters and dependence structure. The estimation methodology based on maximizing the likelihood functions is illustrated for simulated datasets. Finally, the proposed distributions are utilized to analyze pairwise wind direction measurements obtained at different stations in India, and the interpretations for the fitted models are briefly discussed.

Some bivariate distributions on a discrete torus with application to wind direction datasets

TL;DR

This paper develops two analytically tractable bivariate models for ordinal circular data on a discrete torus, namely the bivariate wrapped geometric (BWG) and its generalized version (BGWG). By combining wrapped symmetric geometric marginals with a cosine-based dependence, the authors derive closed-form joint pmfs, marginals, conditionals, and a circular–circular dependence measure, enabling exact likelihood-based inference. Through simulation and three wind-direction datasets from India, BGWG demonstrates competitive or superior fit relative to discretized continuous bivariate circular models, with interpretable dependence parameters ρ and δ. The approach yields efficient computation, avoids discretization errors, and has potential extensions to multivariate and time-series settings via copulas and hidden Markov models. Overall, the work provides a practical framework for analyzing bivariate ordinal circular data and paves the way for broader applications in environmental and biological directional data.

Abstract

Many datasets are observed on a finite set of equally spaced directions instead of the exact angles, such as the wind direction data. However, in the statistical literature, bivariate models are only available for continuous circular random variables. This article presents two bivariate circular distributions, namely bivariate wrapped geometric (BWG) and bivariate generalized wrapped geometric (BGWG), for analyzing bivariate discrete circular data. We consider wrapped geometric distributions and a trigonometric function to construct the models. The models are analytically tractable due to the exact closed-form expressions for the trigonometric moments. We thoroughly discuss the distributional properties of the models, including the interpretation of parameters and dependence structure. The estimation methodology based on maximizing the likelihood functions is illustrated for simulated datasets. Finally, the proposed distributions are utilized to analyze pairwise wind direction measurements obtained at different stations in India, and the interpretations for the fitted models are briefly discussed.
Paper Structure (30 sections, 6 theorems, 44 equations, 6 figures, 18 tables)

This paper contains 30 sections, 6 theorems, 44 equations, 6 figures, 18 tables.

Table of Contents

  1. Introduction
  2. Bivariate model for ordinal circular data
  3. Wrapped symmetric geometric distribution
  4. Bivariate wrapped geometric distribution
  5. Properties of the model
  6. Marginal wrapped symmetric geometric distribution
  7. Conditional wrapped geometric distribution
  8. Generalized BWG distribution
  9. Measuring dependence using rho
  10. Interpretation of rho in BWG and BGWG distributions
  11. Simulation study
  12. Data analysis
  13. Dataset-I
  14. Dataset-II
  15. Dataset-III
  16. ...and 15 more sections

Key Result

Theorem 1

The closed form expression for the proposed joint pmf of $(X_1, X_2)$ is given by where, $C_1 = \frac{(1-q)(1-s)}{(1-q^{m_1})(1-s^{m_2})(1+q)(1+s)}(\frac{(q^2-2q\cos\frac{2\pi}{m_1} + 1)(s^2 -2s\cos\frac{2\pi}{m_2} + 1)}{(q^2-2q\cos\frac{2\pi}{m_1} + 1)(s^2 -2s\cos\frac{2\pi}{m_2} + 1) + \rho(1-q)^2(1-s)^2}).$

Figures (6)

  • Figure 1: Probability mass plot of BWG distribution for various values of $\rho$
  • Figure 2: Probability mass plot of CWG distribution for fixed $X_1 = 6 \pi/m_1$
  • Figure 3: Representation of the directions
  • Figure 4: Heatmap of pmf of estimated BGWG
  • Figure 5: Heatmap of pmf of estimated BGWG
  • ...and 1 more figures

Theorems & Definitions (7)

  • Theorem 1
  • Corollary 1
  • Theorem 2
  • Theorem 3
  • Corollary 2
  • Theorem 4
  • Remark 1