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 language | English |
|---|---|
| Pages (from-to) | 1-17 |
| Number of pages | 17 |
| Journal | Statistics and Computing |
| DOIs | |
| Publication status | Accepted/In press - 2014 |
| Externally published | Yes |
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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 journal › Article
}
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 -