TY - JOUR
T1 - Smoothing Parameter and Model Selection for General Smooth Models
AU - Wood, Simon N.
AU - Pya, Natalya
AU - Säfken, Benjamin
N1 - Funding Information:
We thank the anonymous referees for a large number of very helpful comments that substantially improved the paper and Phil Reiss for spotting an embarrassing error in Supplementary Appendix A. SNW and NP were funded by EPSRC grant EP/K005251/1 and NP was also funded by EPSRC grant EP/I000917/1. BS was funded by the German Research Association (DFG) Research Training Group ?Scaling Problems in Statistics? (RTG 1644). SNW is grateful to Carsten Dorman and his research group at the University of Freiburg, where the extended GAM part of this work was carried out.
Publisher Copyright:
© 2016 The Author(s). Published with license by Taylor and Francis © Simon N. Wood, Natalya Pya, and Benjamin Säfken.
Copyright:
Copyright 2017 Elsevier B.V., All rights reserved.
PY - 2016
Y1 - 2016
N2 - This article discusses a general framework for smoothing parameter estimation for models with regular likelihoods constructed in terms of unknown smooth functions of covariates. Gaussian random effects and parametric terms may also be present. By construction the method is numerically stable and convergent, and enables smoothing parameter uncertainty to be quantified. The latter enables us to fix a well known problem with AIC for such models, thereby improving the range of model selection tools available. The smooth functions are represented by reduced rank spline like smoothers, with associated quadratic penalties measuring function smoothness. Model estimation is by penalized likelihood maximization, where the smoothing parameters controlling the extent of penalization are estimated by Laplace approximate marginal likelihood. The methods cover, for example, generalized additive models for nonexponential family responses (e.g., beta, ordered categorical, scaled t distribution, negative binomial and Tweedie distributions), generalized additive models for location scale and shape (e.g., two stage zero inflation models, and Gaussian location-scale models), Cox proportional hazards models and multivariate additive models. The framework reduces the implementation of new model classes to the coding of some standard derivatives of the log-likelihood. Supplementary materials for this article are available online.
AB - This article discusses a general framework for smoothing parameter estimation for models with regular likelihoods constructed in terms of unknown smooth functions of covariates. Gaussian random effects and parametric terms may also be present. By construction the method is numerically stable and convergent, and enables smoothing parameter uncertainty to be quantified. The latter enables us to fix a well known problem with AIC for such models, thereby improving the range of model selection tools available. The smooth functions are represented by reduced rank spline like smoothers, with associated quadratic penalties measuring function smoothness. Model estimation is by penalized likelihood maximization, where the smoothing parameters controlling the extent of penalization are estimated by Laplace approximate marginal likelihood. The methods cover, for example, generalized additive models for nonexponential family responses (e.g., beta, ordered categorical, scaled t distribution, negative binomial and Tweedie distributions), generalized additive models for location scale and shape (e.g., two stage zero inflation models, and Gaussian location-scale models), Cox proportional hazards models and multivariate additive models. The framework reduces the implementation of new model classes to the coding of some standard derivatives of the log-likelihood. Supplementary materials for this article are available online.
KW - Additive model
KW - AIC
KW - Distributional regression
KW - GAM
KW - Location scale and shape model
KW - Ordered categorical regression
KW - Penalized regression spline
KW - REML
KW - Smooth Cox model
KW - Smoothing parameter uncertainty
KW - Statistical algorithm
KW - Tweedie distribution
UR - http://www.scopus.com/inward/record.url?scp=85010676878&partnerID=8YFLogxK
U2 - 10.1080/01621459.2016.1180986
DO - 10.1080/01621459.2016.1180986
M3 - Article
AN - SCOPUS:85010676878
VL - 111
SP - 1548
EP - 1563
JO - Journal of the American Statistical Association
JF - Journal of the American Statistical Association
SN - 0162-1459
IS - 516
ER -