Spherical regression

Methods are introduced for regressing points on the surface of one sphere on points on another. Complex variables and stereographic projection are used to deal with theoretical problems of directional statistics much as they have been used historically to deal with problems in non-Euclidean geometry. The complex plane harbours the group of Möbius transformations, and stereographic projection is used as a bridge to map these Möbius transforms to regression link functions on the surface of a unit sphere. A special form for these links is introduced which employs the complex plane and stereographic projection to effect angular scale changes on the sphere. The family of special forms is closed under orthogonal transformations of the dependent variable and Möbius transformations of the independent variable, and incorporates independence and proper and improper rotations as special cases. Parameter estimation and inference are exemplified using the von Mises--Fisher spherical distribution and vectorcardiogram data. All statistical results and calculations have been formulated in the real domain. Copyright Biometrika Trust 2003, Oxford University Press.