Cauchy Families of Directional Distributions Closed Under Location and Scale Transformations

ABSTRACT

Directional statistics deals with angular data that come from non-linear objects such as circle circumferences or toroidal surfaces. A fundamental problem in directional statistics is that arithmetic cannot be meaningfully done on angles. Naive changes of location and scale like λ' = (λ – μ)/σ for a spherical longitude λ are inappropriate and often misleading since they are not interpretable as one-one mappings from a sphere onto itself. Finding ways to obtain angular scale changes and to construct families of spherical probability distributions that are closed under such scale changes have been unsuccessful. But, such families are successfully constructed herein by indirect but historically powerful methods. Thus, a unit sphere with a uniform probability distribution on its surface is centrally rotated to a suitable position, and then stereographically projected onto an extended complex plane, a linear surface especially amenable to directional and statistical computations. A central dilation is performed on the plane, the dilated plane is projected back in effect as a rescaled sphere, and the rescaled sphere is again rotated. This process induces a family of spherical Cauchy-type probability distributions on the sphere that is closed under composition of such processes (rotate sphere, project sphere to plane, dilate plane, project dilated plane back as a rescaled sphere, and rotate again). The distributions so induced can be generalized to higher dimensional spheres that are also closed under location and scale transformations. These distributions enjoy numerous interrelationships with one another and with linear and circular Cauchy distributions.