Modelling beyond Regression Functions: an Application of Multimodal Regression to Speed-Flow Data

An enormous amount of publications deals with smoothing in the sense of nonparametric regression. However, nearly all of the literature treats the case where predictors and response are related in the form of a function y = m(x) + noise. In many situations this simple functional model does not capture adequately the essential relation between predictor and response. We show by means of speed-flow diagrams, that a more general setting may be required, allowing for multifunctions instead of only functions. It turns out that in this case the conditional modes are more appropriate for the estimation of the underlying relation than the commonly used mean or the median. Estimation is achieved using a conditional mean-shift procedure, which is adapted to the present situation.