Beyond unimodal regression: modelling multimodality with piecewise unimodal, mixture or additive regression

Abstract

Research in the field of nonparametric shape constrained regression has been extensive and there is need for such methods in various application are as, since shape constraints can reflect prior knowledge about the underlying relationship. It is, for example, often natural that some intensity first increases and then decreases over time, which can be described by a unimodal shape constraint. But the prior knowledge in different applications is also of increasing complexity and data shapes may vary fro m few to plenty of modes and from piecewise unimodal to superpositions of unimodal function courses. Thus, we go beyond unimodal regression in this report and capture multimodality by employing piecewise unimodal regression, mixture regression or additive regression models. We give an overview of the statistical methods, namely the unimodal spline regression approach by and its aforementioned extensions for use with multimodal data. The usefulness of the methods is demonstrated by applying them to data sets from three different application areas: breath gas analysis, marine biology and astroparticle physics. Though the three application areas are quite different, the propose d extensions of unimodal regression yield very helpful results in each of it. This encourages using the methodologies proposed here in many other areas of application as well.