Smoothing Parameter and Model Selection for General Smooth Models

By Simon N. Wood, Natalya Pya, Benjamin Säfken. 2015

1,287 citations recorded by the source corpus.

ABSTRACT 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.

Subfields: Finance, Statistics and Probability

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Citations in the corpus

glmmTMB Balances Speed and Flexibility Among Packages for Zero-inflated Generalized Linear Mixed Modeling (2017, methodology): “Several R packages are available for ﬁtting zero-inﬂated models: pscl , INLA , MCMCglmm , glmmADMB , mgcv , brms , and gamlss (Table 1; Zeileis et al., 2008; Rue et al., 2009; Hadﬁeld, 2010; Skaug et al., 2012; Wood et al., 2016; Bürkner, 2017; Stasinopoulos et al., 2017).”
Trajectory-based differential expression analysis for single-cell sequencing data (2019, methodology): “We build upon recent developments that allow the joint estimation of the NB regression parameters in μgi and of the dispersion parameter g [Wood et al., 2016].”
Generalized Additive Models for Exceedances of High Thresholds With an Application to Return Level Estimation for U.S. Wind Gusts (2019, methodology): “Furthermore, Wood, Pya, and Säfken (2016) outlines how smoothing parameter uncertainty can be propagated to smooth terms.”
Simple ways to interpret effects in modeling ordinal categorical data (2018, methodology): “The measures discussed in this paper extend also to more general ordinal‐response models than those having linear predictors, such as generalized additive models for ordinal responses and their extensions with random effects, see, for example, Yee and Wild (1996) and Wood et al. (2016).”
Modelling Palaeoecological Time Series Using Generalised Additive Models (2018, methodology): “In practice, the size of the basis is an upper limit on the expected complexity of the trend, and a simple test can be used to check if the basis used was sufficiently large (Pya and Wood, 2016).”
Hierarchical generalized additive models in ecology: an introduction with mgcv (2019, positive): “There is well-developed theory of how to include effects of penalization and smoothing parameter uncertainty when estimating the model complexity penalty for AIC ( Wood, Pya & Säfken, 2016 ).”
Fast Calibrated Additive Quantile Regression (2017, methodology): “Here we do this using a Gaussian GAM, where the mean and variance are estimated jointly using the methods of Wood, Pya, and Säfken (2016).”