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

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 modeling circular and directional data. The circular distribution encompasses the wrapped Cauchy distribution as a special case, featuring a more convenient parameterisation. Next, we 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, resulting in an elliptically symmetric distribution. Our projected distributions exhibit attractive properties, such as closed-form normalising constants and straightforward random value generation. The distribution parameters can be estimated using maximum likelihood and we assess their bias through numerical studies. We compare our proposed distributions to existing models with real data sets, demonstrating superior fit both with and without covariates.Comment: Preprin

On Some Circular Distributions Induced by Inverse Stereographic Projection

In earlier studies of circular data, mostly circular distributions were considered and many biological data sets were assumed to be symmetric. However, presently interest has increased for skewed circular distributions as the assumption of symmetry may not be meaningful for some data. This thesis introduces three skewed circular models based on inverse stereographic projection, introduced by Minh and Farnum (2003), by considering three different versions of skewed-t considered in the literature, namely Azzalini skewed-t, two-piece skewed-t and Jones and Faddy skewed-t. Shape properties of the resulting distributions along with estimation of parameters using maximum likelihood are discussed in this thesis. Further, three real data sets (Bruderer and Jenni, 1990; Holzmann et al., 2006; Fisher, 1993) are used to illustrate the application of the new model and its extension to finite mixture modelling. Goodness of fit of the new distributions is studied using maximum log-likelihood and Akaike information criterion and chi-square values. It is found that Azzalini and Jones-Faddy skewed-t versions are good competitors; however the Jones-Faddy version is computationally more tractable

Recent advances in directional statistics

Mainstream statistical methodology is generally applicable to data observed in Euclidean space. There are, however, numerous contexts of considerable scientific interest in which the natural supports for the data under consideration are Riemannian manifolds like the unit circle, torus, sphere and their extensions. Typically, such data can be represented using one or more directions, and directional statistics is the branch of statistics that deals with their analysis. In this paper we provide a review of the many recent developments in the field since the publication of Mardia and Jupp (1999), still the most comprehensive text on directional statistics. Many of those developments have been stimulated by interesting applications in fields as diverse as astronomy, medicine, genetics, neurology, aeronautics, acoustics, image analysis, text mining, environmetrics, and machine learning. We begin by considering developments for the exploratory analysis of directional data before progressing to distributional models, general approaches to inference, hypothesis testing, regression, nonparametric curve estimation, methods for dimension reduction, classification and clustering, and the modelling of time series, spatial and spatio-temporal data. An overview of currently available software for analysing directional data is also provided, and potential future developments discussed.Comment: 61 page