Multivariate data whose norm equals unity is termed directional data. The literature describes plenty of circular distributions, with the oldest being the von Mises von Mises (1918) and studied by (Mardia, 1972, Mardia and Jupp, 2000). This distribution arises as the conditional distribution of a bivariate normal random vector with some mean vector and identity covariance matrix, given that the vector lies on the unit sphere. Various generalizations of this distribution have been proposed over the years (Dietrich and Richter, 2017, Gatto and Jammalamadaka, 2007, Kim and SenGupta, 2013), and more circular distributions have been proposed by Pewsey (2000), Jones and Pewsey (2005), Abe and Pewsey (2011), and Jones and Pewsey (2012). On the contrary, the wrapped distributions are another type of distributions that arises from wrapping a univariate random vector on the circle. These include the wrapped t family of distributions (Pewsey et al., 2007), the wrapped stable family (Pewsey, 2008), the wrapped normal, and wrapped Cauchy (WC) distributions (Mardia and Jupp, 2000) and extensions (Kato and Jones, 2010, 2013). Finally, another type of distribution somewhat less studied is the so-called projected distributions that arise as the distribution of a multivariate random vector projected onto a circle, where the projected normal (Mardia and Jupp, 2000, Presnell et al., 1998, Watson, 1983) is probably the unique distribution of this type.
In an attempt to offer increased flexibility or better capture of skewed data, the list of the aforementioned distributions may include distributions that entail more than two parameters. However, the majority suffer from the problem induced by assuming a diagonal scatter matrix. The projected normal is an exception, which has addressed this issue by adopting the Bayesian stance (Nunez-Antonio and Gutierrez-Pena, 2005, Wang and Gelfand, 2013).
Spherical (and hyper-spherical) distributions have been proposed over the years, with the oldest being the von Mises-Fisher distribution (Fisher, 1953), followed by the projected normal, also called offset normal (Mardia, 1972) or displaced normal (Kendall, 1974). This distribution was studied by Watson (1983) who showed its close proximity to the von Mises-Fisher distribution. The drawback of these two distributions is that they assume rotational symmetry, that is they are obtained via a multivariate normal (via the aforementioned manners) with an identity covariance matrix. It was this drawback that motivated the work of Kent (1982), who proposed the first elliptical symmetric distribution, a special case of the Fisher-Bingham distribution (Mardia, 1972, 1975). Paine et al. (2018) proposed a special case of the projected normal distribution, yielding the second elliptical symmetric distribution. Finally, Scealy and Wood (2019) proposed the Scaled von Mises-Fisher family of distributions that entail the same symmetry properties as the two previous distributions.
Moving along the lines of the projected types of distributions, we first propose the circular projected Cauchy distribution. To the best of our knowledge, this is the second projection type of distribution, apart from the normal distribution. We first derive the general circular projected Cauchy distribution that arises by projecting the bivariate Cauchy distribution (with some scatter matrix) onto the circle. To avoid over-identifiability issues we take two special cases; first, we consider the case of independent components, i.e., a diagonal scatter matrix, and we show its equivalence to the WC distribution. Since the diagonal scatter matrix is a rather strict assumption we relax this assumption by imposing one restriction, moving along the lines of Paine et al. (2018), a mean-constrained scatter matrix.
We then move on to the spherical case, and propose the spherical projected Cauchy distribution. As in the circular case, the scatter matrix may be equal to the identity matrix, yielding rotational symmetry. Imposing the same conditions as in Paine et al. (2018), yields an elliptically symmetric distribution. This feature is highly important with regard to spherical data, given that the literature describes only a few distributions with this property.
A remark that can be made about either circular or spherical projected Cauchy distributions is that they have a closed form normalizing constant, a feature that is not the norm with directional distributions. Another advantage is that simulation from the projected Cauchy family is straightforward.
