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Projected distributions have proved to be useful in the study of circular anddirectional data. Although any multivariate distribution can be used to producea projected model, these distributions are typically parametric. In thisarticle we consider a multivariate P\'olya tree on $R^k$ and project it to theunit hypersphere $S^k$ to define a new Bayesian nonparametric model fordirectional data. We study the properties of the proposed model and inparticular, concentrate on the implied conditional distributions of somedirections given the others to define a directional-directional regressionmodel. We also define a multivariate linear regression model with P\'olya treeerror and project it to define a linear-directional regression model. We obtainthe posterior characterisation of all models and show their performance withsimulated and real datasets.

Multivariate and regression models for directional data based on projected P\'olya trees