Improved monotone polynomial fitting with applications and variable selection

Abstract

We investigate existing and new isotonic parameterisations for monotone polynomials, the latter which have been previously unconsidered in the statistical literature. We show that this new parameterisation is faster and more flexible than its alternatives enabling polynomials to be constrained to be monotone over either a compact interval or a semi-compact interval of the form [a;∞), in addition to over the whole real line. Due to the speed and efficiency of algorithms based on our new parameterisation the use of standard bootstrap methodology becomes feasible. We investigate the use of the bootstrap under monotonicity constraints to obtain confidence and prediction bands for the fitted curves and show that an adjustment by using either the ‘m out of n’ bootstrap or a post hoc symmetrisation of the confidence bands is necessary to achieve more uniform coverage probabilities. However, the same such adjustments appear unwarranted for prediction bands. Furthermore, we examine the model selection problem, not only for monotone polynomials, but also in a general sense, with a focus on graphical methods. Specifically, we describe how to visualize measures of description loss and of model complexity to facilitate the model selection problem. We advocate the use of the bootstrap to assess the stability of selected models and to enhance our graphical tools and demonstrate which variables are important using variable inclusion plots, showing that these can be invaluable plots for the model building process. We also describe methods for using the ‘m out of n’ bootstrap to select the degree of the fitted monotone polynomial and demonstrate it’s effectiveness in the specific constrained regression scenario. We demonstrate the effectiveness of all of these methods using numerous case studies, which highlight the necessity and usefulness of our techniques. All algorithms discussed in this thesis are available in the R package MonoPoly (version 0.3-6 or later).