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.