This paper presents a novel semi-parametric approach for local function approximation under limited global information about the underlying data generating structure.
The related technique, named hierarchical local regression (HLR), is based on an aggregate of local models with random mean parameters. The structure is hierarchical in that the local random mean parameters assume probability distributions that are defined by common globally parameterized mean-variance functions. While the global mean function expresses global information about the expected response, the variance function quantifies the uncertainty associated with the global information. On the one hand, this formulation accounts for the parsimonious nature of the global mean function. On the other hand, the information provided by estimated global parameters is combined with locally weighted data to achieve robust local adaptation in data sparse regions which occur frequently in high-dimensional situations (curse of dimensionality).
We suggest a criterion for estimation of the parameters and derive an empirical Bayes prediction formula. We present two numerical studies to illustrate different aspects of the method. One example involves prediction of power production in windmill farms based on real data.
Hierarchical local regression, Function approximation, Curse of dimensionality, Empirical Bayes, BLUP, Global information.