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82
Implementing approximate Bayesian inference for latent Gaussian models using integrated nested Laplace approximations: A manual for the inlaprogram
, 2008
"... Structured additive regression models are perhaps the most commonly used class of models in statistical applications. It includes, among others, (generalised) linear models, (generalised) additive models, smoothingspline models, statespace models, semiparametric regression, spatial and spatiotemp ..."
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Cited by 79 (16 self)
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Structured additive regression models are perhaps the most commonly used class of models in statistical applications. It includes, among others, (generalised) linear models, (generalised) additive models, smoothingspline models, statespace models, semiparametric regression, spatial and spatiotemporal models, logGaussian Coxprocesses, geostatistical and geoadditive models. In this paper we consider approximate Bayesian inference in a popular subset of structured additive regression models, latent Gaussian models, where the latent field is Gaussian, controlled by a few hyperparameters and with nonGaussian response variables. The posterior marginals are not available in closed form due to the nonGaussian response variables. For such models, Markov chain Monte Carlo methods can be implemented, but they are not without problems, both in terms of convergence and computational time. In some practical applications, the extent of these problems is such that Markov chain Monte Carlo is simply not an appropriate tool for routine analysis. We show that, by using an integrated nested Laplace approximation and its simplified version, we can directly compute very accurate approximations to the posterior marginals. The main benefit of these approximations
Spatial Econometrics
 PALGRAVE HANDBOOK OF ECONOMETRICS: VOLUME 1, ECONOMETRIC THEORY
, 2001
"... Spatial econometric methods deal with the incorporation of spatial interaction and spatial structure into regression analysis. The field has seen a recent and rapid growth spurred both by theoretical concerns as well as by the need to be able to apply econometric models to emerging large geocoded da ..."
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Cited by 64 (5 self)
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Spatial econometric methods deal with the incorporation of spatial interaction and spatial structure into regression analysis. The field has seen a recent and rapid growth spurred both by theoretical concerns as well as by the need to be able to apply econometric models to emerging large geocoded data bases. The review presented in this chapter outlines the basic terminology and discusses in some detail the specification of spatial effects, estimation of spatial regression models, and specification tests for spatial effects.
Multivariable geostatistics in S: the gstat package
 Computers & Geosciences
, 2004
"... This paper discusses advantages and shortcomings of the Senvironment for multivariable geostatistics, in particular when extended with the gstat package, an extension package for the Senvironments (R, SPlus). The gstat S package provides multivariable geostatistical modelling, prediction and simula ..."
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Cited by 32 (6 self)
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This paper discusses advantages and shortcomings of the Senvironment for multivariable geostatistics, in particular when extended with the gstat package, an extension package for the Senvironments (R, SPlus). The gstat S package provides multivariable geostatistical modelling, prediction and simulation, as well as several visualisation functions. In particular, it makes the calculation, simultaneous fitting, and visualisation of a large number of direct and cross (residual) variograms very easy. Gstat was started 10 years ago and was released under the GPL in 1996; gstat.org was started in 1998. Gstat was not initially written for teaching purposes, but for research purposes, emphasising flexibility, scalability and portability. It can deal with a large number of practical issues in geostatistics, including change of support (block kriging), simple/ordinary/universal (co)kriging, fast local neighbourhood selection, flexible trend modelling, variables with different sampling configurations, and efficient simulation of large spatially correlated random fields, indicator kriging and simulation, and (directional) variogram and cross variogram modelling. The formula/models interface of the Slanguage is used to define multivariable geostatistical models. This paper introduces the gstat Spackage, and discusses a number of design and implementation issues. It also draws attention to a number of papers on integration of spatial statistics software, GISand the Senvironment that were presented on the spatial statistics workshop and sessions during the conference Distributed Statistical
A tutorial on Bayesian optimization of expensive cost functions, withapplicationtoactiveusermodeling andhierarchical reinforcement learning
, 2009
"... We present a tutorial on Bayesian optimization, a method of finding the maximum of expensive cost functions. Bayesian optimization employs the Bayesian technique of setting a prior over the objective function and combining it with evidence to get a posterior function. This permits a utilitybased se ..."
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Cited by 29 (2 self)
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We present a tutorial on Bayesian optimization, a method of finding the maximum of expensive cost functions. Bayesian optimization employs the Bayesian technique of setting a prior over the objective function and combining it with evidence to get a posterior function. This permits a utilitybased selection of the next observation to make on the objective function, which must take into account both exploration (sampling from areas of high uncertainty) and exploitation (sampling areas likely to offer improvement over the current best observation). We also present two detailed extensions of Bayesian optimization, with experiments—active user modelling with preferences, and hierarchical reinforcement learning— and a discussion of the pros and cons of Bayesian optimization based on our experiences. 1
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,
Geostatistical SpaceTime Models, Stationarity, Separability and Full Symmetry
"... Geostatistical approaches to modeling spatiotemporal data rely on parametric covariance models and rather stringent assumptions, such as stationarity, separability and full symmetry. This paper reviews recent advances in the literature on spacetime covariance functions in light of the aforemention ..."
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Cited by 15 (3 self)
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Geostatistical approaches to modeling spatiotemporal data rely on parametric covariance models and rather stringent assumptions, such as stationarity, separability and full symmetry. This paper reviews recent advances in the literature on spacetime covariance functions in light of the aforementioned notions, which are illustrated using wind data from Ireland. Experiments with timeforward kriging predictors suggest that the use of more complex and more realistic covariance models results in improved predictive performance.
Covariance tapering for likelihoodbased estimation in large spatial data sets
 Journal of the American Statistical Association
, 2008
"... Likelihoodbased methods such as maximum likelihood, REML, and Bayesian methods are attractive approaches to estimating covariance parameters in spatial models based on Gaussian processes. Finding such estimates can be computationally infeasible for large datasets, however, requiring O(n 3) calculat ..."
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Cited by 11 (0 self)
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Likelihoodbased methods such as maximum likelihood, REML, and Bayesian methods are attractive approaches to estimating covariance parameters in spatial models based on Gaussian processes. Finding such estimates can be computationally infeasible for large datasets, however, requiring O(n 3) calculations for each evaluation of the likelihood based on n observations. We propose the method of covariance tapering to approximate the likelihood in this setting. In this approach, covariance matrices are “tapered, ” or multiplied elementwise by a compactly supported correlation matrix. This produces matrices which can be be manipulated using more efficient sparse matrix algorithms. We present two approximations to the Gaussian likelihood using tapering. The first tapers the model covariance matrix only, whereas the second tapers both the model and sample covariance matrices. Tapering the model covariance matrix can be viewed as changing the underlying model to one in which the spatial covariance function is the direct product of the original covariance function and the tapering function. Focusing on the particular case of the Matérn class of covariance functions, we give conditions under which tapered and untapered covariance functions give equivalent (mutually absolutely continuous) measures for Gaussian processes on bounded domains. This allows us to evaluate
Approximate bayesian inference in spatial generalized linear mixed models
, 2006
"... In this paper we propose fast approximate methods for computing posterior marginals in spatial generalized linear mixed models. We consider the common geostatistical special case with a high dimensional latent spatial variable and observations at only a few known registration sites. Our methods of i ..."
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Cited by 10 (5 self)
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In this paper we propose fast approximate methods for computing posterior marginals in spatial generalized linear mixed models. We consider the common geostatistical special case with a high dimensional latent spatial variable and observations at only a few known registration sites. Our methods of inference are deterministic, using no random sampling. We present two methods of approximate inference. The first is very fast to compute and via examples we find that this approximation is ’practically sufficient’. By this expression we mean that the results obtained by this approximate method do not show any bias or dispersion effects that might affect decision making. The other approximation is an improved version of the first one, and via examples we demonstrate that the inferred posterior approximations of this improved version are ’practically exact’. By this expression we mean that one would have to run Markov chain Monte Carlo simulations for longer than is typically done to detect any indications of bias or dispersion error effects in the approximate results. The two methods of approximate inference can help to expand the scope of geostatistical models, for instance in the context of model choice, model assessment, and sampling design. The