Circular and Spherical Projected Cauchy Distributions: A Novel Framework for Circular and Directional Data Modeling

TL;DR

This paper introduces a new family of projected Cauchy distributions for circular and spherical data, offering flexible, well-behaved models with practical estimation and superior fitting capabilities compared to existing models.

Contribution

The paper proposes circular and spherical projected Cauchy distributions with enhanced parameterization and properties, including a generalized form and conditions for elliptic symmetry, improving data modeling.

Findings

Closed-form normalising constant facilitates computation

Distribution parameters can be efficiently estimated via maximum likelihood

Proposed models outperform existing alternatives in real data fitting

Abstract

We introduce a novel family of projected distributions on the circle and the sphere, namely the circular and spherical projected Cauchy distributions, as promising alternatives for modelling circular and spherical data. The circular distribution encompasses the wrapped Cauchy distribution as a special case, while featuring a more convenient parameterisation. We also propose a generalised wrapped Cauchy distribution that includes an extra parameter, enhancing the fit of the distribution. In the spherical context, we impose two conditions on the scatter matrix of the Cauchy distribution, resulting in an elliptically symmetric distribution. Our projected distributions exhibit attractive properties, such as a closed-form normalising constant and straightforward random value generation. The distribution parameters can be estimated using maximum likelihood, and we assess their bias through numerical studies. Further, we compare our proposed distributions to existing models with real datasets, demonstrating equal or superior fitting both with and without covariates.