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62
Bayesian PSplines
 Journal of Computational and Graphical Statistics
, 2004
"... Psplines are an attractive approach for modelling nonlinear smooth effects of covariates within the generalized additive and varying coefficient models framework. In this paper we propose a Bayesian version for Psplines and generalize the approach for one dimensional curves to two dimensional surf ..."
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Cited by 67 (21 self)
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Psplines are an attractive approach for modelling nonlinear smooth effects of covariates within the generalized additive and varying coefficient models framework. In this paper we propose a Bayesian version for Psplines and generalize the approach for one dimensional curves to two dimensional surface fitting for modelling interactions between metrical covariates. A Bayesian approach to Psplines has the advantage of allowing for simultaneous estimation of smooth functions and smoothing parameters. Moreover, it can easily be extended to more complex formulations, for example to mixed models with random effects for serially or spatially correlated response. Additionally, the assumption of constant smoothing parameters can be replaced by allowing the smoothing parameters to be locally adaptive. This is particularly useful in situations with changing curvature of the underlying smooth function or where the function is highly oscillating. Inference is fully Bayesian and uses recent MCMC techniques for drawing random samples from the posterior. In a couple of simulation studies the performance of Bayesian Psplines is studied and compared to other approaches in the literature. We illustrate the approach by a complex application on rents for flats in Munich.
Bayesian inference for generalized additive mixed models based on markov random field priors
 C
, 2001
"... Summary. Most regression problems in practice require ¯exible semiparametric forms of the predictor for modelling the dependence of responses on covariates. Moreover, it is often necessary to add random effects accounting for overdispersion caused by unobserved heterogeneity or for correlation in lo ..."
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Cited by 63 (19 self)
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Summary. Most regression problems in practice require ¯exible semiparametric forms of the predictor for modelling the dependence of responses on covariates. Moreover, it is often necessary to add random effects accounting for overdispersion caused by unobserved heterogeneity or for correlation in longitudinal or spatial data. We present a uni®ed approach for Bayesian inference via Markov chain Monte Carlo simulation in generalized additive and semiparametric mixed models. Different types of covariates, such as the usual covariates with ®xed effects, metrical covariates with nonlinear effects, unstructured random effects, trend and seasonal components in longitudinal data and spatial covariates, are all treated within the same general framework by assigning appropriate Markov random ®eld priors with different forms and degrees of smoothness. We applied the approach in several casestudies and consulting cases, showing that the methods are also computationally feasible in problems with many covariates and large data sets. In this paper, we choose two typical applications.
On Block Updating in Markov Random Field Models For . . .
 SCANDINAVIAN JOURNAL OF STATISTICS
, 2002
"... Gaussian Markov random field (GMRF) models are commonlyufz to model spatial correlation in disease mapping applications. For Bayesian inference by MCMC, so far mainly singlesiteuinglealgorithms have been considered. However, convergence and mixing properties ofsuD algorithms can be extremely ..."
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Cited by 51 (7 self)
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Gaussian Markov random field (GMRF) models are commonlyufz to model spatial correlation in disease mapping applications. For Bayesian inference by MCMC, so far mainly singlesiteuinglealgorithms have been considered. However, convergence and mixing properties ofsuD algorithms can be extremely poordu to strong dependencies ofparameters in the posteriordistribuQ84K In this paper, we propose variou block sampling algorithms in order to improve the MCMC performance. The methodology is rather general, allows for nonstandardfu6 conditionals, and can be applied in amoduzK fashion in a large nugef of di#erent scenarios. For illu##Kzf0 n we consider three di#erent applications: twoformu8Df0z3 for spatial modelling of a single disease (with andwithou additionaluditionalfL parameters respectively), and one formu## ion for the joint analysis of two diseases. TheresuKK indicate that the largest benefits are obtained ifparameters and the corresponding hyperparameter areuefz#L jointly in one large block. Implementation ofsuQ block algorithms is relatively easy usyf methods for fast sampling ofGaungf3 Markov random fields (Rus 2001). By comparison, Monte Carlo estimates based on singlesiteungles can be rather misleading, even for very long rugfOu resuL6 may have wider relevance for efficient MCMCsimu6z8#f in hierarchical models with Markov random field components.
Conditional prior proposals in dynamic models
 SCANDINAVIAN JOURNAL OF STATISTICS
, 1999
"... Dynamic models extend state space models to nonnormal observations. This paper suggests a specific hybrid MetropolisHastings algorithm as a simple device for Bayesian inference via Markov chain Monte Carlo in dynamic models. Hastings proposals from the (conditional) prior distribution of the unk ..."
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Cited by 33 (3 self)
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Dynamic models extend state space models to nonnormal observations. This paper suggests a specific hybrid MetropolisHastings algorithm as a simple device for Bayesian inference via Markov chain Monte Carlo in dynamic models. Hastings proposals from the (conditional) prior distribution of the unknown, timevarying parameters are used to update the corresponding full conditional distributions. It is shown through simulated examples that the methodology has optimal performance in situations where the prior is relatively strong compared to the likelihood. Typical examples include smoothing priors for categorical data. A specific blocking strategy is proposed to ensure good mixing and convergence properties of the simulated Markov chain. It is also shown that the methodology is easily extended to robust transition models using mixtures of normals. The applicability is illustrated with an analysis of a binomial and a binary time series, known in the literature.
Modelling Risk from a Disease in Time and Space
 Statistics in Medicine 17
, 1997
"... This paper combines existing models for longitudinal and spatial data in a hierarchical Bayesian framework, with particular emphasis on the role of time and space varying covariate effects. Data analysis is implemented via Markov chain Monte Carlo methods. The methodology is illustrated by a ten ..."
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Cited by 31 (8 self)
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This paper combines existing models for longitudinal and spatial data in a hierarchical Bayesian framework, with particular emphasis on the role of time and space varying covariate effects. Data analysis is implemented via Markov chain Monte Carlo methods. The methodology is illustrated by a tentative reanalysis of Ohio lung cancer data 196888. Two approaches that adjust for unmeasured spatial covariates, particularly tobacco consumption, are described. The first includes random effects in the model to account for unobserved heterogeneity; the second adds a simple urbanization measure as a surrogate for smoking behaviour. The Ohio dataset has been of particular interest because of the suggestion that a nuclear facility in the southwest of the state may have caused increased levels of lung cancer there. However, we contend here that the data are inadequate for a proper investigation of this issue. Email: leo@stat.unimuenchen.de 1 Introduction Data on disease incidence or mor...
Lang S: Generalized structured additive regression based on Bayesian P splines
 Computational Statistics & Data Analysis
"... 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 Psplines as th ..."
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Cited by 26 (7 self)
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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 Psplines as the main building block. The approach extends previous work by Lang and 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. As we will demonstrate through two applications on the forest health status of trees and a spacetime 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. Key words: geoadditive models, IWLS proposals, multicategorical response, structured additive
Bayesian Prediction of Spatial Count Data Using Generalised Linear Mixed Models
, 2001
"... Introduction Site specic farming is aiming at targeting inputs of fertiliser, pesticide, and herbicide according to locally determined requirements. In connection with herbicide application on a eld, it is important to map the weed intensity so that the dose of herbicide applied at any location can ..."
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Cited by 24 (3 self)
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Introduction Site specic farming is aiming at targeting inputs of fertiliser, pesticide, and herbicide according to locally determined requirements. In connection with herbicide application on a eld, it is important to map the weed intensity so that the dose of herbicide applied at any location can be adjusted to the amount of weed present at the location. In a Danish project on precision farming (Olesen, 1997) one objective was to investigate whether observations of soil properties could be used for prediction of weed intensity. In practice the farmer or his advisor should then establish a relation between soil properties and weed occurrence from extensive observations collected one year and use this for prediction of the weed intensity in subsequent years where only a limited number of weed count observations would be 1 collected. Many soil properties are fairly constant over time so that observations of soil samples obtained the rst year can also be used in subseq
An Explicit Link between Gaussian Fields and . . .
 PREPRINTS IN MATHEMATICAL SCIENCES
, 2010
"... Continuously indexed Gaussian fields (GFs) is the most important ingredient in spatial statistical modelling and geostatistics. The specification through the covariance function gives an intuitive interpretation of its properties. On the computational side, GFs are hampered with the bign problem, ..."
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Cited by 21 (7 self)
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Continuously indexed Gaussian fields (GFs) is the most important ingredient in spatial statistical modelling and geostatistics. The specification through the covariance function gives an intuitive interpretation of its properties. On the computational side, GFs are hampered with the bign problem, since the cost of factorising dense matrices is cubic in the dimension. Although the computational power today is alltimehigh, this fact seems still to be a computational bottleneck in applications. Along with GFs, there is the class of Gaussian Markov random fields (GMRFs) which are discretely indexed. The Markov property makes the involved precision matrix sparse which enables the use of numerical algorithms for sparse matrices, that for fields in R 2 only use the squareroot of the time required by general algorithms. The specification of a GMRF is through its full conditional distributions but its marginal properties are not transparent in such a parametrisation. In this paper, we show that using an approximate stochastic weak solution to (linear) stochastic partial differential equations (SPDEs), we can, for some GFs in the Matérn class, provide an explicit link, for any triangulation of R d, between GFs and GMRFs. The consequence is that we can take the best from the two worlds and do the modelling using GFs but do the computations using GMRFs. Perhaps more importantly,
Approximating Hidden Gaussian Markov Random Fields
 JOURNAL OF THE ROYAL STATISTICAL SOCIETY, SERIES B
, 2003
"... This paper discusses how to construct approximations to a unimodal hidden Gaussian Markov random field on a graph of dimension n when the likelihood consists of mutually independent data. We demonstrate that a class of nonGaussian approximations can be constructed for a wide range of likelihood ..."
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Cited by 19 (4 self)
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This paper discusses how to construct approximations to a unimodal hidden Gaussian Markov random field on a graph of dimension n when the likelihood consists of mutually independent data. We demonstrate that a class of nonGaussian approximations can be constructed for a wide range of likelihood models. They have the appealing properties that exact samples can be drawn from them, the normalisation constant is computable, and the computational complexity is only O(n 2 ) in the spatial case. The nonGaussian approximations are refined versions of a Gaussian approximation. The latter serves well if the likelihood is nearGaussian, but it is not sufficiently accurate when the likelihood is not nearGaussian or if n is large. The accuracy of our approximations can be tuned by intuitive parameters to near any precision. We apply
Penalized structured additive regression for spacetime data: a Bayesian perspective
 STATISTICA SINICA
, 2004
"... We propose extensions of penalized spline generalized additive models for analyzing spacetime regression data and study them from a Bayesian perspective. Nonlinear effects of continuous covariates and time trends are modelled through Bayesian versions of penalized splines, while correlated spati ..."
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Cited by 16 (11 self)
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We propose extensions of penalized spline generalized additive models for analyzing spacetime regression data and study them from a Bayesian perspective. Nonlinear effects of continuous covariates and time trends are modelled through Bayesian versions of penalized splines, while correlated spatial effects follow a Markov random field prior. This allows to treat all functions and effects within a unified general framework by assigning appropriate priors with different forms and degrees of smoothness. Inference can be performed either with full (FB) or empirical Bayes (EB) posterior analysis. FB inference using MCMC techniques is a slight extension of previous work. For EB inference, a computationally efficient solution is developed on the basis of a generalized linear mixed model representation. The second approach can be viewed as posterior mode estimation and is closely related to penalized likelihood estimation in a frequentist setting. Variance components, corresponding to inverse smoothing parameters, are then estimated by marginal likelihood. We carefully compare both inferential procedures in simulation studies and illustrate them through data applications. The methodology is available in the open domain statistical package BayesX and as an Splus/R function.