AI 中文总结
该研究将默比乌斯变换与球面最优传输关联，推广出广义默比乌斯变换，构建的相关分布在古地磁方向和短周期彗星轨道建模中性能优于现有模型。
AI 中文摘要
从均匀分布生成球面柯西分布的默比乌斯变换，是一维最优传输的切-法提升：即余弦关于位置轴的单调重排。这明确了默比乌斯变换的概率本质，并提出推广：将柯西目标替换为任意旋转对称分布（如泊松核或球面心脏线），可得到广义默比乌斯变换。对冯·米塞斯-费希尔（von Mises-Fisher）基分布进行默比乌斯传输，可得到球面上易处理的各向异性分布，其具有闭式密度，继承基分布的归一化常数且可直接模拟。默比乌斯-冯·米塞斯-费希尔分布及各向异性缩放冯·米塞斯-费希尔分布（后者也源自默比乌斯传输），在古地磁方向和短周期彗星轨道上得到验证，其性能优于经典及近期提出的替代模型。
英文摘要
The Möbius transformation that generates the spherical Cauchy distribution from the uniform is the tangent-normal lift of a one-dimensional optimal transport: the monotone rearrangement of the cosine about the location axis. This identifies the probabilistic nature of the Möbius transformation and suggests a generalization: replacing the Cauchy target by any rotationally symmetric law, for instance the Poisson kernel or spherical cardioid, yields a generalized Möbius transformation. Möbius transport of a von Mises-Fisher base gives tractable anisotropic distributions on the sphere, with closed-form densities that inherit the base normalizing constant and allow immediate simulation. The Möbius-von Mises-Fisher and isotropic scaled von Mises-Fisher distributions, the latter also arising from a Möbius transport, are illustrated on paleomagnetic directions and short-period comet orbits, where they outperform classical and recently proposed alternatives.
Comments9 pages, 3 figures, 2 tables
