Projected distributions have proved to be useful in the study of circular and directional data. Although any multivariate distribution can be used to produce a projected model, these distributions are typically parametric. In this article we consider a multivariate P\'olya tree on $R^k$ and project it to the unit hypersphere $S^k$ to define a new Bayesian nonparametric model for directional data. We study the properties of the proposed model and in particular, concentrate on the implied conditional distributions of some directions given the others to define a directional-directional regression model. We also define a multivariate linear regression model with P\'olya tree error and project it to define a linear-directional regression model. We obtain the posterior characterisation of all models and show their performance with simulated and real datasets.

翻译：投影分布已被证明在圆形和方向数据研究中具有实用性。尽管任何多元分布均可用于生成投影模型，但这些分布通常是参数化的。本文考虑$R^k$上的多元Pólya树，并将其投影到单位超球面$S^k$上，从而定义一种新的方向数据贝叶斯非参数模型。我们研究了所提出模型的性质，特别关注在某些方向条件隐含下的条件分布，以定义方向-方向回归模型。此外，我们定义了具有Pólya树误差的多元线性回归模型，并将其投影以定义线性-方向回归模型。我们获得了所有模型的后验特征，并通过模拟和真实数据集展示了其性能。