Spherical Regression Models Using Projective Linear Transformations

This article studies the problem of modeling relationship between two spherical (or directional) random variables in a regression setup. Here the predictor and the response variables are constrained to be on a unit sphere and, due to this nonlinear condition, the standard Euclidean regression models do not apply. Several past papers have studied this problem, termed spherical regression, by modeling the response variable with a von Mises-Fisher (VMF) density with the mean given by a rotation of the predictor variable. The few papers that go beyond rigid rotations are limited to one- or two-dimensional spheres. This article extends the mean transformations to a larger group—the projective linear group of transformations—on unit spheres of arbitrary dimensions, while keeping the VMF density to model the noise. It develops a Newton–Raphson algorithm on the special linear group for estimating the MLE of regression parameter and establishes its asymptotic properties when the sample-size becomes large. Through a variety of experiments, using data taken from projective shape analysis, cloud tracking, etc., and some simulations, this article demonstrates improvements in the prediction and modeling performance of the proposed framework over previously used models. Supplementary materials for this article are available online.本文研究了回归框架下两个球面（spherical，或方向）随机变量之间的关系建模问题。在此问题中，预测变量与响应变量均被约束于单位球面（unit sphere）之上，受此非线性约束的影响，标准欧几里得回归模型（Euclidean regression models）无法适用。过往已有多篇文献针对该问题——即球面回归（spherical regression）——展开研究，其将响应变量建模为以预测变量经旋转变换所得结果为均值的冯·米塞斯-费舍尔（von Mises-Fisher, VMF）密度。少数突破刚性旋转（rigid rotations）框架的现有研究，仅局限于一维或二维球面场景。本文将均值变换拓展至任意维度单位球面上的更大变换群——射影线性变换群（projective linear group of transformations）——，同时保留VMF密度以刻画噪声分布。本文针对特殊线性群（special linear group）设计了牛顿-拉夫逊算法（Newton–Raphson algorithm），以估计回归参数的最大似然估计（Maximum Likelihood Estimation, MLE），并推导了样本量增大时该估计量的渐近性质。本文通过多组实验——采用来自射影形状分析（projective shape analysis）、云跟踪（cloud tracking）等领域的真实数据与仿真数据——验证了所提框架相较于现有模型在预测与建模性能上的提升。本文的补充材料可在线获取。