循环最优运输的统计
The Statistics of Circular Optimal Transport
摘要：
经验最佳运输（OT）计划和距离提供了有效的工具，比较和统计匹配概率测度定义在一个给定的地面空间。基本上这是分布极限法，我们推导出一个中心极限定理的经验OT距离的圆形数据。我们的极限结果只需要温和的假设一般，包括突出的例子，如冯米塞斯或包装柯西家庭。最值得注意的是，当数据从概率测度中取样进行比较时，不需要任何假设，这与实际线形成严格对比。一个引导原则如下立即作为我们的证明依赖于阿达玛微分的OT功能。这为各种统计推断任务铺平了道路，并以基于渐近OT的圆形分布拟合优度检验为例。我们讨论了数值实现，一致性和调查其统计功效。对于测试均匀性，事实证明，这种方法表现得特别好的单峰替代品，几乎是强大的瑞利测试，最强大的不变测试冯米塞斯替代品。对于具有许多模式的制度，圆形OT检验的效力较低，这可以通过相应的运输计划的形状来解释。
英文摘要：
Empirical optimal transport (OT) plans and distances provide effective tools to compare and statistically match probability measures defined on a given ground space. Fundamental to this are distributional limit laws and we derive a central limit theorem for the empirical OT distance of circular data. Our limit results require only mild assumptions in general and include prominent examples such as the von Mises or wrapped Cauchy family. Most notably, no assumptions are required when data are sampled from the probability measure to be compared with, which is in strict contrast to the real line. A bootstrap principle follows immediately as our proof relies on Hadamard differentiability of the OT functional. This paves the way for a variety of statistical inference tasks and is exemplified for asymptotic OT based goodness of fit testing for circular distributions. We discuss numerical implementation, consistency and investigate its statistical power. For testing uniformity, it turns out that this approach performs particularly well for unimodal alternatives and is almost as powerful as Rayleigh's test, the most powerful invariant test for von Mises alternatives. For regimes with many modes the circular OT test is less powerful which is explained by the shape of the corresponding transport plan.
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