The fitting of multifunctions : an approach to nonparametric multimodal regression.

Abstract

In the last decades a lot of research has been devoted to smoothing in the sense of nonparametric regression. However, this work has nearly exclusively concentrated on fitting regression functions. When the conditional distribution of y|x is multimodal, the assumption of a functional relationship y = m(x) + noise might be too restrictive. We introduce a nonparametric approach to fit multifunctions, allowing to assign a set of output values to a given x. The concept is based on conditional mean shift, which is an easily implemented tool to detect the local maxima of a conditional density function. The methodology is illustrated by environmental data examples.