Multiple Spherical Regression

• Activity Number: 430
• Type: Contributed
• Date/Time: Thursday, August 12, 2004 : 10:30 AM to 12:20 PM
• Sponsor: General Methodology
• Author(s): Tom Downs*+ and Kathy Downs
• Companies: Rice University and Research Statistics, Inc.
• Keywords: Moebius mapping ; polar decomposition ; stereographic projection ; vectorcardiogram

Abstract

Complex variables, Moebius mappings and stereographic projection, used historically to study non-Euclidean geometry, are used herein to construct non-Euclidean links for regressing one spatial direction on several others. Each spatial direction is a point on a unit sphere centered at the origin of an xyt rectangular coordinate system whose xy plane is treated as an extended complex plane. A link function m for regressing the mean direction of the dependent direction v on a vector u of p independent spatial directions is defined as the spherical projective image of the complex number w = a'z/c'z, where z is a (p+1)-vector of homogeneous coordinates for the complex projective images of the p spatial direction components of u, while a and c are linearly independent complex-valued parameter vectors. Properties and interpretations of the link function m are obtained from the polar decomposition of the 2x(p+1) matrix M with rows a' and c'. Simple spherical regression ensues when p=1, and has been studied by Downs (2003, Spherical Regression. Biometrika 90, 3, pp. 655-668). Parameter estimation and inference are illustrated with vectorcardiogram data.