Modular regression - a Lego system for building structured additive distributional regression models with tensor product interactions
- 188 Downloads
Semiparametric regression models offer considerable flexibility concerning the specification of additive regression predictors including effects as diverse as nonlinear effects of continuous covariates, spatial effects, random effects, or varying coefficients. Recently, such flexible model predictors have been combined with the possibility to go beyond pure mean-based analyses by specifying regression predictors on potentially all parameters of the response distribution in a distributional regression framework. In this paper, we discuss a generic concept for defining interaction effects in such semiparametric distributional regression models based on tensor products of main effects. These interactions can be assigned anisotropic penalties, i.e. different amounts of smoothness will be associated with the interacting covariates. We investigate identifiability and the decomposition of interactions into main effects and pure interaction effects (similar as in a smoothing spline analysis of variance) to facilitate a modular model building process. The decomposition is based on orthogonality in function spaces which allows for considerable flexibility in setting up the effect decomposition. Inference is based on Markov chain Monte Carlo simulations with iteratively weighted least squares proposals under constraints to ensure identifiability and effect decomposition. One important aspect is therefore to maintain sparse matrix structures of the tensor product also in identifiable, decomposed model formulations. The performance of modular regression is verified in a simulation on decomposed interaction surfaces of two continuous covariates and two applications on the construction of spatio-temporal interactions for the analysis of precipitation on the one hand and functional random effects for analysing house prices on the other hand.
KeywordsConstrained sampling Distributional regression Functional random effects Markov chain Monte Carlo simulations Penalised splines Smoothing spline analysis of variance Space–time models Tensor product interactions
Mathematics Subject Classification62G08 62J12 62H11
We thank the referees and the associate editor for many valuable comments that lead to a significant improvement in our paper upon the original submission. We are grateful to Jim Hodges for pointing us to the alternative representation of the tensor product precision matrix based on eigen decompositions. Financial support by the German Research Foundation (DFG), Grant KN 922/9-1 is gratefully acknowledged.
- Adler D, Kneib T, Lang S, Umlauf N, Zeileis A (2012) BayesXsrc: R Package Distribution of the BayesX C++ Sources. R package version 3.0-0. https://CRAN.R-project.org/package=BayesXsrc. Accessed 29 Jan 2019
- Belitz C, Brezger A, Klein N, Kneib T, Lang S, Umlauf N (2015) BayesX—Software for Bayesian inference in structured additive regression models. Version 3.0.2. http://www.bayesx.org. Accessed 29 Jan 2019
- Klein N (2018) sdPrior: scale-dependent hyperpriors in structured additive distributional regression. R package version 1.0Google Scholar
- R Core Team (2017) R: a Language and Environment for Statistical Computing, R Foundation for Statistical Computing, Vienna, Austria. ISBN 3-900051-07-0. https://www.R-project.org/. Accessed 29 Jan 2019
- Stauffer R, Mayr GJ, Messner JW, Umlauf N, Zeileis A (2016) Spatio-temporal precipitation climatology over complex terrain using a censored additive regression model. Int J Climatol 15:3264Google Scholar
- Umlauf N, Klein N, Zeileis A, Köhler M (2018) bamlss : Bayesian additive models for location scale and shape (and Beyond). R package version 1.0-0. http://CRAN.R-project.org/package=bamlss. Accessed 29 Jan 2019
- Wood S (2015) mgcv: Mixed GAM computation vehicle with GCV/AIC/REML smoothness estimations. R package version 1.8-5Google Scholar