Abstract
In regression problems, it is often of interest to assume that the relationship between a predictor variable and a response variable is monotone. In this thesis, we focus on general additive regression, in which each covariate effect can be described by a univariate smooth function. We additionally restrict the functions to be monotone, establishing the additive isotonic/monotone regression. Methods incorporating such restrictions in classical low dimensional settings (where the number of covariates $p$ does not exceed the number of observations $n$), mostly require us to provide monotonicity directions a priori. These directions, however, might be unknown, particularly when facing a problem with a large number of covariates. To our knowledge, there exist two methods of solving additive isotonic regression in high dimensional settings ($p>n$), namely liso and monotone splines lasso, both with their adaptive extensions. Apart from the non-adaptive liso these methods have the distinctive ability to perform automatic direction discovery, alleviating the need for prior knowledge of monotonicity directions, and possibly making the methods attractive for both low and high dimensional settings. In this thesis, we want to evaluate the effect of prior knowledge of the correct monotonicity directions in additive monotone regression. Uniquely, the adaptive liso can be used both with and without prior information on monotonicity directions. Liso without the adaptive step requires the monotonicity directions as input, while monotone splines lasso cannot exploit prior knowledge about monotonicity directions. Hence, by comparing the two possible approaches to the adaptive liso we can study the effect of providing the additional information on monotonicity directions. We perform a simulation study in which we compare the selection performance, accuracy of estimated directions, as well as estimation and prediction mean square errors. We find that providing the correct monotonicity directions generally improves the adaptive liso method in terms of all of the considered criteria, in four different low dimensional study cases. For that reason, and since in the classical setting all but one method require prior knowledge on monotonicity directions, we suggest a few methods that can be used for pre-estimating monotonicity directions. We present a simulation comparison of their performances and find that it is most beneficial to pre-estimate monotonicity directions using ridge regression, with such an approach outperforming the method relying on the sign discovery procedure of the adaptive liso in four low dimensional study cases. Inspired by this, we decided to also investigate the effect of pre-estimating monotonicity directions in a high dimensional simulation, in which case the true model is built only on a subset of all the available covariates. We find that it is still beneficial to pre-estimate monotonicity directions by ridge regression in terms of estimation errors, however, such estimated models include more noise variables in comparison to the adaptive liso with no prior information, which results in a worse prediction performance.