Non-parametric regression on the hyper-sphere with uniform design

This paper deals with the estimation of a function f defined on the sphere S of R from a sample of noisy observation points. We introduce an estimation procedure based on waveletlike functions on the sphere called needlets and study two estimators f and f respectively made adaptive through the use of a stochastic and deterministic needlet-shrinkage method. We show hereafter that these estimators are nearly-optimal in the minimax framework, explain why f outperforms f and run finite sample simulations with f to demonstrate that our estimation procedure is easy to implement and fares well in practice. We are motivated by applications in geophysical and atmospheric sciences.