Generalized structured additive regression based on Bayesian P-splines

By Andreas Brezger, Stefan Lang. Computational Statistics & Data Analysis, 2006

402 citations recorded by the source corpus.

Generalized additive models (GAM) for modeling nonlinear effects of continuous covariates are now well established tools for the applied statistician. A Bayesian version of GAM's and extensions to generalized structured additive regression (STAR) are developed. One or two dimensional P-splines are used as the main building block. Inference relies on Markov chain Monte Carlo (MCMC) simulation techniques, and is either based on iteratively weighted least squares (IWLS) proposals or on latent utility representations of (multi)categorical regression models. The approach covers the most common univariate response distributions, e.g., the binomial, Poisson or gamma distribution, as well as multicategorical responses. For the first time, Bayesian semiparametric inference for the widely used multinomial logit model is presented. Two applications on the forest health status of trees and a space-time analysis of health insurance data demonstrate the potential of the approach for realistic modeling of complex problems. Software for the methodology is provided within the public domain package BayesX.

Subfields: Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/8f9579f2e4654326dc4bf81e6d8c190b95efeb90

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Citations in the corpus

Twenty years of P-splines (2015, methodology): “Eilers and Marx (2002) proposed GLASS (generalized linear additive smooth structures), while Brezger and Lang (2006), referring to Fahrmeir et al. (2004), proposed STAR (structured additive regression).”
A Bayesian Semiparametric Latent Variable Model for Mixed Responses (2007, positive): “The semiparametrically structured additive predictor used in our LVM is described in Fahrmeir, Kneib and Lang (2004), and Brezger and Lang (2006) in the context of semiparametric generalized regression for univariate responses.”
A spatially explicit height–diameter model for Scots pine in Estonia (2011, background): “This type of data can be also characterized as having a unit, cluster and occasion-specific heterogeneity (Brezger and Lang 2006), whereas in forest management a unit may refer to an ecoregion or forest district, and a cluster may refer to a sample plot.”
Fast stable restricted maximum likelihood and marginal likelihood estimation of semiparametric generalized linear models (2011, methodology): “Lang and Brezger (2004), Brezger and Lang (2006) and Fahrmeir and Lang (2001)) is likely to be more efficient than the method that is proposed here, albeit applicable to a more restricted range of penalized GLMs, because of restrictions on the Sj that are required to maintain computational efficienc”
Some asymptotic results on generalized penalized spline smoothing (2009, background): “In the non-Gaussian case, σ2" is fixed, whereas the full conditional for θ has no analytic form and sampling can be done through Metropolis–Hastings steps; see Brezger and Lang (2006) for details.”
High dimensional structured additive regression models: Bayesian regularization, smoothing and predictive performance (2011, methodology): “…smoothing is based on Bayesian penalized splines with Gaussian smoothness priors, corresponding to difference penalties for basis function coefficients, and on Gaussian Markov random-field priors for spatial effects as in Fahrmeir et al. (2004), Lang and Brezger (2004) or Brezger and Lang (2006).”
Bayesian structured additive distributional regression for multivariate responses (2015, methodology): “To approximate potentially non-linear effects, we use Bayesian P-splines; see Eilers and Marx (1996) and Brezger and Lang (2006) for detailed explanations.”
Geoadditive Survival Models (2006, methodology): “For updating the parameter vectors β j, which correspond to the time-independent functions fj(xj), as well as spatial effects βspat, fixed effects γ , and random effects b, we use a slightly modified version of an MH algorithm based on iteratively weighted least squares (IWLS) proposals, developed f”
Monotonic Regression Based on Bayesian P–Splines (2008, methodology): “Further possible extensions are varying coefficient terms and interactions of covariates (see Brezger and Lang 2003; Lang and Brezger 2004).”
Scale-Dependent Priors for Variance Parameters in Structured Additive Distributional Regression (2016, methodology): “This is based on the idea of iteratively weighted least squares (IWLS, Gamerman, 1997) proposals originally developed for updating the vectors of regression coefficients, see, e.g. Brezger and Lang (2006) and Gamerman (1997).”
BayesX: Analyzing Bayesian Structural Additive Regression Models (2005, methodology): “The functions f1 and f2 are the main effects of the two covariates and f1|2 is a two-dimensional interaction surface which can be modeled e.g. by twodimensional P-splines (Lang and Brezger 2004; Brezger and Lang 2005).”
Simultaneous Confidence Bands for Penalized Spline Estimators (2010, methodology): “Inferences in the fully Bayesian approach are then typically based on Markov chain Monte Carlo (MCMC) simulation techniques (see Brezger and Lang 2006 for details).”
Bayesian Structured Additive Distributional Regression (2013, methodology): “To construct suitable proposal densities, we follow the idea of iteratively weighted least squares proposals [Gamerman, 1997, Brezger and Lang, 2006] and construct local quadratic approximations to the full conditionals.”