Shape constrained additive models

Natalya Pya, Simon N. Wood

Research output: Contribution to journalArticlepeer-review

88 Citations (Scopus)


A framework is presented for generalized additive modelling under shape constraints on the component functions of the linear predictor of the GAM. We represent shape constrained model components by mildly non-linear extensions of P-splines. Models can contain multiple shape constrained and unconstrained terms as well as shape constrained multi-dimensional smooths. The constraints considered are on the sign of the first or/and the second derivatives of the smooth terms. A key advantage of the approach is that it facilitates efficient estimation of smoothing parameters as an integral part of model estimation, via GCV or AIC, and numerically robust algorithms for this are presented. We also derive simulation free approximate Bayesian confidence intervals for the smooth components, which are shown to achieve close to nominal coverage probabilities. Applications are presented using real data examples including the risk of disease in relation to proximity to municipal incinerators and the association between air pollution and health.

Original languageEnglish
Pages (from-to)543-559
Number of pages17
JournalStatistics and Computing
Issue number3
Publication statusPublished - Feb 25 2014


  • Convex smoothing
  • Generalized additive model
  • Monotonic smoothing
  • P-splines

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Statistics and Probability
  • Statistics, Probability and Uncertainty
  • Computational Theory and Mathematics

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