Nonparametric Rotations for Sphere-Sphere Regression

ABSTRACT

Regression of data represented as points on a hypersphere has traditionally been treated using parametric families of transformations that include the simple rigid rotation as an important, special case. On the other hand, nonparametric methods have generally focused on modeling a scalar response through a spherical predictor by representing the regression function as a polynomial, leading to component-wise estimation of a spherical response. We propose a very flexible, simple regression model where for each location of the manifold a specific rotation matrix is to be estimated. To make this approach tractable, we assume continuity of the regression function that, in turn, allows for approximations of rotation matrices based on a series expansion. It is seen that the nonrigidity of our technique motivates an iterative estimation within a Newton–Raphson learning scheme, which exhibits bias reduction properties. Extensions to general shape matching are also outlined. Both simulations and real data are used to illustrate the results. Supplementary materials for this article are available online.

KEYWORDS:

• Fisher's method of scoring
• Local smoothing
• Singular value decomposition
• Skew-symmetric matrices
• Spherical kernels
• Wahba's problem

Acknowledgments

The authors are grateful to David Ruppert, an Associate Editor and two anonymous referees for their many helpful suggestions, which have greatly helped to improve the manuscript. The authors are also indebted to Louis-Paul Rivest for suggesting the motivating example, as well for drawing their attention to the importance of some graphical representations. Finally, Giovanni Lafratta, Stefania Fensore, and Cinzia Nepa are thanked for their comments on previous versions of the article.