Shape constrained additive models

Natalya Pya, Simon N. Wood

Research output: Contribution to journalArticle

50 Citations (Scopus)

Abstract

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)1-17
Number of pages17
JournalStatistics and Computing
DOIs
Publication statusAccepted/In press - 2014
Externally publishedYes

Fingerprint

Additive Models
P-splines
Shape Constraint
Generalized Additive Models
Refuse incinerators
Efficient Estimation
Air Pollution
Smoothing Parameter
Robust Algorithm
Coverage Probability
Component Model
Term
Second derivative
Air pollution
Splines
Proximity
Categorical or nominal
Confidence interval
Predictors
Health

Keywords

  • 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

Cite this

Shape constrained additive models. / Pya, Natalya; Wood, Simon N.

In: Statistics and Computing, 2014, p. 1-17.

Research output: Contribution to journalArticle

Pya, Natalya ; Wood, Simon N. / Shape constrained additive models. In: Statistics and Computing. 2014 ; pp. 1-17.
@article{e4d28bd55c7c4839b93f56efe7aeee92,
title = "Shape constrained additive models",
abstract = "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.",
keywords = "Convex smoothing, Generalized additive model, Monotonic smoothing, P-splines",
author = "Natalya Pya and Wood, {Simon N.}",
year = "2014",
doi = "10.1007/s11222-013-9448-7",
language = "English",
pages = "1--17",
journal = "Statistics and Computing",
issn = "0960-3174",
publisher = "Springer Netherlands",

}

TY - JOUR

T1 - Shape constrained additive models

AU - Pya, Natalya

AU - Wood, Simon N.

PY - 2014

Y1 - 2014

N2 - 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.

AB - 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.

KW - Convex smoothing

KW - Generalized additive model

KW - Monotonic smoothing

KW - P-splines

UR - http://www.scopus.com/inward/record.url?scp=84894277295&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84894277295&partnerID=8YFLogxK

U2 - 10.1007/s11222-013-9448-7

DO - 10.1007/s11222-013-9448-7

M3 - Article

SP - 1

EP - 17

JO - Statistics and Computing

JF - Statistics and Computing

SN - 0960-3174

ER -