From Symmetry to Asymmetry on the Disc Manifold: Modeling of Marion Island Data

Abstract

The joint modeling of angular and linear observations is crucial as data of this nature are prevalent in multiple disciplines, for example the joint modeling of wind direction and another climatological variable such as wind speed or air temperature, the direction an animal moves and the distance moved, or wave direction and wave height. Hence, there is a need for developing flexible distributions on the hyper-disc, which has support of the interior of the hyper-sphere, as it allows for modeling the combination of angular and linear observations. This paper addresses this need by developing flexible distributions for the disc that have the ability to capture any inherent bimodality present in the data. A new class of bivariate distributions is proposed which has support on the unit disc in two dimensions that includes, as a special case, the existing Möbius distribution on the disc. This class is obtained by expressing the density function in a general form using a measurable function termed as generator. Special cases of this generator are considered to demonstrate the flexibility. By applying a conformal mapping to the generator function a new Möbius distribution class emanates. This class of bivariate distributions on the disc is the first to account for bimodality and skewness present in the data. The flexible behavior of the proposed models in terms of bimodality and skewness is graphically demonstrated. Preliminary evidential analysis of the wind data observed at Marion Island reveals the absence of unimodality in the data. The fit of the proposed models, which account for bimodality, to the Marion Island wind data were evaluated analytically and visually.