Penalized Likelihood and Bayesian Function Selection in Regression Models

Challenging research in various fields has driven a wide range of methodological advances in variable selection for regression models with high-dimensional predictors. In comparison, selection of nonlinear functions in models with additive predictors has been considered only more recently. Several competing suggestions have been developed at about the same time and often do not refer to each other. This article provides a state-of-the-art review on function selection, focusing on penalized likelihood and Bayesian concepts, relating various approaches to each other in a unified framework. In an empirical comparison, also including boosting, we evaluate several methods through applications to simulated and real data, thereby providing some guidance on their performance in practice

Variable Selection and Model Choice in Structured Survival Models

In many situations, medical applications ask for flexible survival models that allow to extend the classical Cox-model via the inclusion of time-varying and nonparametric effects. These structured survival models are very flexible but additional difficulties arise when model choice and variable selection is desired. In particular, it has to be decided which covariates should be assigned time-varying effects or whether parametric modeling is sufficient for a given covariate. Component-wise boosting provides a means of likelihood-based model fitting that enables simultaneous variable selection and model choice. We introduce a component-wise likelihood-based boosting algorithm for survival data that permits the inclusion of both parametric and nonparametric time-varying effects as well as nonparametric effects of continuous covariates utilizing penalized splines as the main modeling technique. Its properties and performance are investigated in simulation studies. The new modeling approach is used to build a flexible survival model for intensive care patients suffering from severe sepsis. A software implementation is available to the interested reader

High-dimensional Structured Additive Regression Models: Bayesian Regularisation, Smoothing and Predictive Performance

Data structures in modern applications frequently combine the necessity of flexible regression techniques such as nonlinear and spatial effects with high-dimensional covariate vectors. While estimation of the former is typically achieved by supplementing the likelihood with a suitable smoothness penalty, the latter are usually assigned shrinkage penalties that enforce sparse models. In this paper, we consider a Bayesian unifying perspective, where conditionally Gaussian priors can be assigned to all types of regression effects. Suitable hyperprior assumptions on the variances of the Gaussian distributions then induce the desired smoothness or sparseness properties. As a major advantage, general Markov chain Monte Carlo simulation algorithms can be developed that allow for the joint estimation of smooth and spatial effects and regularised coefficient vectors. Two applications demonstrate the usefulness of the proposed procedure: A geoadditive regression model for data from the Munich rental guide and an additive probit model for the prediction of consumer credit defaults. In both cases, high-dimensional vectors of categorical covariates will be included in the regression models. The predictive ability of the resulting high-dimensional structure additive regression models compared to expert models will be of particular relevance and will be evaluated on cross-validation test data

Bayesian joint models with INLA exploring marine mobile predator-prey and competitor species habitat overlap

EPSRC grant Ecowatt 2050 EP/K012851/1 ACKNOWLEDGMENTS We would like to thank the associate editor and the anonymous reviewers for their useful and constructive suggestions which led to a considerable improvement of the manuscript. The authors would also like to thank the following people/organizations for making large datasets available for use in this paper: Mark Lewis (Joint Nature Conservation Committee), Philip Hammond (Scottish Oceans Institute, University of St. Andrews), Susan Lusseau (Marine Scotland Science), Darren Stevens (The Sir Alister Hardy Foundation for Ocean Science, PML), and Yuri Artioli (Plymouth Marine Laboratory). This work was supported by the Engineering and Physical Sciences Research Council (EcoWatt250; EPSRC EP/K012851/1).Peer reviewedPublisher PD

Spike-and-Slab Priors for Function Selection in Structured Additive Regression Models

Structured additive regression provides a general framework for complex Gaussian and non-Gaussian regression models, with predictors comprising arbitrary combinations of nonlinear functions and surfaces, spatial effects, varying coefficients, random effects and further regression terms. The large flexibility of structured additive regression makes function selection a challenging and important task, aiming at (1) selecting the relevant covariates, (2) choosing an appropriate and parsimonious representation of the impact of covariates on the predictor and (3) determining the required interactions. We propose a spike-and-slab prior structure for function selection that allows to include or exclude single coefficients as well as blocks of coefficients representing specific model terms. A novel multiplicative parameter expansion is required to obtain good mixing and convergence properties in a Markov chain Monte Carlo simulation approach and is shown to induce desirable shrinkage properties. In simulation studies and with (real) benchmark classification data, we investigate sensitivity to hyperparameter settings and compare performance to competitors. The flexibility and applicability of our approach are demonstrated in an additive piecewise exponential model with time-varying effects for right-censored survival times of intensive care patients with sepsis. Geoadditive and additive mixed logit model applications are discussed in an extensive appendix

Large-scale Nonlinear Variable Selection via Kernel Random Features

We propose a new method for input variable selection in nonlinear regression. The method is embedded into a kernel regression machine that can model general nonlinear functions, not being a priori limited to additive models. This is the first kernel-based variable selection method applicable to large datasets. It sidesteps the typical poor scaling properties of kernel methods by mapping the inputs into a relatively low-dimensional space of random features. The algorithm discovers the variables relevant for the regression task together with learning the prediction model through learning the appropriate nonlinear random feature maps. We demonstrate the outstanding performance of our method on a set of large-scale synthetic and real datasets.Comment: Final version for proceedings of ECML/PKDD 201

Generalized structured additive regression based on Bayesian P-splines

Generalized additive models (GAM) for modelling nonlinear effects of continuous covariates are now well established tools for the applied statistician. In this paper we develop Bayesian GAM's and extensions to generalized structured additive regression based on one or two dimensional P-splines as the main building block. The approach extends previous work by Lang und Brezger (2003) for Gaussian responses. 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. Our 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, we present Bayesian semiparametric inference for the widely used multinomial logit models. As we will demonstrate through two applications on the forest health status of trees and a space-time analysis of health insurance data, the approach allows realistic modelling of complex problems. We consider the enormous flexibility and extendability of our approach as a main advantage of Bayesian inference based on MCMC techniques compared to more traditional approaches. Software for the methodology presented in the paper is provided within the public domain package BayesX

Identification of nonlinear time-varying systems using an online sliding-window and common model structure selection (CMSS) approach with applications to EEG

The identification of nonlinear time-varying systems using linear-in-the-parameter models is investigated. A new efficient Common Model Structure Selection (CMSS) algorithm is proposed to select a common model structure. The main idea and key procedure is: First, generate K 1 data sets (the first K data sets are used for training, and theK 1 th one is used for testing) using an online sliding window method; then detect significant model terms to form a common model structure which fits over all the K training data sets using the new proposed CMSS approach. Finally, estimate and refine the time-varying parameters for the identified common-structured model using a Recursive Least Squares (RLS) parameter estimation method. The new method can effectively detect and adaptively track the transient variation of nonstationary signals. Two examples are presented to illustrate the effectiveness of the new approach including an application to an EEG data set