Abstract

One way of defining probability distributions for circular variables (directions in two dimensions) is to radially project probability distributions, originally defined on ${\mathbb{R}}^2$, to the unit circle. Projected distributions have proved to be useful in the study of circular and directional data. Although any bivariate distribution can be used to produce a projected circular model, these distributions are typically parametric. In this article, we consider a bivariate Pólya tree on ${\mathbb{R}}^2$ and project it to the unit circle to define a new Bayesian nonparametric model for circular data. We study the properties of the proposed model, obtain its posterior characterization and show its performance with simulated and real datasets. Supplemental materials for this article are available online.

摘要

为圆形变量（二维方向）定义概率分布的一种方法是径向投影概率分布，最初定义在 ${\mathbb{电阻}}^2$, 到单位圆。已证明投影分布在研究圆形和定向数据中很有用。尽管可以使用任何二元分布来生成投影圆形模型，但这些分布通常是参数化的。在本文中，我们考虑一个二元 Pólya 树${\mathbb{电阻}}^2$并将其投影到单位圆以定义圆形数据的新贝叶斯非参数模型。我们研究了所提出模型的特性，获得了它的后验特征，并用模拟和真实数据集展示了它的性能。本文的补充材料可在线获取。