Unimodal spline regression and its use in various applications with single or multiple modes

Abstract

Research in the field of non-parametric shape constrained regression has been extensive and there is need for such methods in various application areas, since shape constraints can reflect prior knowledge about the underlying relationship. This thesis develops semi-parametric spline regression approaches to unimodal regression. However, the prior knowledge in different applications is also of increasing complexity and data shapes may vary from few to plenty of modes and from piecewise unimodal to accumulations of identically or diversely shaped unimodal functions. Thus, we also go beyond unimodal regression in this thesis and propose to capture multimodality by employing piecewise unimodal regression or deconvolution models based on unimodal peak shapes. More explicitly, this thesis proposes unimodal spline regression methods that make use of Bernstein-Schoenberg-splines and their shape preservation property. To achieve unimodal and smooth solutions we use penalized splines, and extend the penalized spline approach towards penalizing against general parametric functions, instead of using just difference penalties. For tuning parameter selection under a unimodality constraint a restricted maximum likelihood and an alternative Bayesian approach for unimodal regression are developed. We compare the proposed methodologies to other common approaches in a simulation study and apply it to a dose-response data set. All results suggest that the unimodality constraint or the combination of unimodality and a penalty can substantially improve estimation of the functional relationship. A common feature of the approaches to multimodal regression is that the response variable is modelled using several unimodal spline regressions. This thesis examines mixture models of unimodal regressions, piecewise unimodal regression and deconvolution models with identical or diverse unimodal peak shapes. The usefulness of these extensions of unimodal regression is demonstrated by applying them to data sets from three different application areas: marine biology, astroparticle physics and breath gas analysis. The proposed methodologies are implemented in the statistical software environment R and the implementations and their usage are explained in this thesis as well.