Additive models with shape constraints

  • Natalya Pya

Student thesis: Doctoral ThesisPhD


In many practical situations when analyzing a dependence of one or more explanatory variables on a response variable it is essential to assume that the relationship of interest obeys certain shape constraints, such as monotonicity or monotonicity and convexity/concavity. In this thesis a new approach to shape preserving smoothing within generalized additive models has been developed. In contrast with previous quadratic programming based methods, the project develops intermediate rank penalized smoothers with shape constrained restrictions based on re-parameterized B-splines and penalties based on the P-spline ideas of Eilers and Marx (1996). Smoothing under monotonicity constraints and monotonicity together with convexity/concavity for univariate smooths; and smoothing of bivariate functions with monotonicity restrictions on both covariates and on only one of them are considered. The proposed shape constrained smoothing has been incorporated into generalized additive models with a mixture of unconstrained and shape restricted smooth terms (mono-GAM). A fitting procedure for mono-GAM is developed. Since a major challenge of any flexible regression method is its implementation in a computationally efficient and stable manner, issues such as convergence, rank deficiency of the working model matrix, initialization, and others have been thoroughly dealt with. A question about the limiting posterior distribution of the model parameters is solved, which allows us to construct Bayesian confidence intervals of the mono-GAM smooth terms by means of the delta method. The performance of these confidence intervals is examined by assessing realized coverage probabilities using simulation studies. The proposed modelling approach has been implemented in an R package monogam. The model setup is the same as in mgcv(gam) with the addition of shape constrained smooths. In order to be consistent with the unconstrained GAM, the package provides key functions similar to those associated with mgcv(gam). Performance and timing comparisons of mono-GAM with other alternative methods has been undertaken. The simulation studies show that the new method has practical advantages over the alternatives considered. Applications of mono-GAM to various data sets are presented which demonstrate its ability to model many practical situations.
Date of Award1 Aug 2010
Original languageEnglish
Awarding Institution
  • University of Bath
SupervisorSimon Wood (Supervisor)


  • monotone smoothings
  • generalized additive models

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