Approximate Bayesian inference for large spatial datasets using predictive process models

Eidsvik, Jo; Finley, Andrew O.; Banerjee, Sudipto; Rue, Håvard

doi:10.1016/j.csda.2011.10.022

Cited by 56 publications

(38 citation statements)

References 37 publications

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“…The maximum likelihood estimator then chooses the parameter that maximizes l(θ), reasoning that under the corresponding stochastic model observing the data f becomes most likely. An extension that works for both the case of a non-trivial mean of the form (20) and the case of a generalized covariance function was proposed by Kitanidis [42]. The idea is to use the information of n−q allowable linear combinations of f only, rather than the complete data vector.…”

Section: Kernel Selection and Parameter Estimationmentioning

confidence: 99%

“…Note the To formulate the precise statement of [74], consider a random field Z on a bounded domain T with mean function of the form (20) and covariance function K. Let ZK(x 0 ) be the kriging prediction at x 0 ∈ T based on observations of Z at some set X n ⊂ T , derived under the (false) assumption thatK is the covariance function. Assume further that x 0 / ∈ X n , that the sequence (X n ) n∈N of point sets is getting dense in T and that…”

Section: Interpolation With Misspecified Kernelsmentioning

confidence: 99%

“…When all sampling locations are on a (near-)regular lattice, spectral methods to approximate the likelihood can be used and allow to reduce the computational cost to an order of O(n log(n)) [31,13,28]. These techniques cannot be applied to scattered data, but other approaches to approximating likelihoods [79,78,5,46], covariance tapering [29], or simplified Gaussian models of low rank [3,12,20] have been proposed and have been shown to be quite effective in reducing the computational effort to an order that allows the application of REML in most practical situations.…”

Section: Kernel Selection and Parameter Estimationmentioning

confidence: 99%

“…The universal kriging interpolant can be also derived within a Bayesian framework. As a prior distribution for f one assumes a Gaussian random field with covariance function K and mean function m as in (20). For a complete specification of the prior for f , a distribution assumption needs to be made for β 1 , .…”

Section: Ordinary and Universal Krigingmentioning

confidence: 99%

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Interpolation of spatial data – A stochastic or a deterministic problem?

2013

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Interpolation of spatial data is a very general mathematical problem with various applications. In geostatistics, it is assumed that the underlying structure of the data is a stochastic process which leads to an interpolation procedure known as kriging. This method is mathematically equivalent to kernel interpolation, a method used in numerical analysis for the same problem, but derived under completely different modelling assumptions. In this article we present the two approaches and discuss their modelling assumptions, notions of optimality and the different concepts to quantify the interpolation accuracy. Their relation is much closer than has been appreciated so far, and even results on convergence rates of kernel interpolants can be translated to the geostatistical framework. We sketch the different answers obtained in the two fields concerning the issue of kernel misspecification, present some methods for kernel selection, and discuss the scope of these methods with a data example from the computer experiments literature.

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Section: Kernel Selection and Parameter Estimationmentioning

confidence: 99%

Section: Interpolation With Misspecified Kernelsmentioning

confidence: 99%

Section: Kernel Selection and Parameter Estimationmentioning

confidence: 99%

Section: Ordinary and Universal Krigingmentioning

confidence: 99%

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Interpolation of spatial data – A stochastic or a deterministic problem?

2013

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“…They also provide the R-INLA software (http://www.r-inla.org) to fit a wide range of models which in most cases reduce computation time to seconds and allow for the use of larger datasets. See Eidsvik et al [5] for a discussion of the benefits of using INLA on large datasets.…”

Section: Introductionmentioning

confidence: 99%

Approximate Bayesian inference for spatial econometrics models

Bivand

Gómez‐Rubio

Rue³

2014

Spatial Statistics

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In this paper we explore the use of the Integrated Laplace Approximation (INLA) for Bayesian inference in some widely used models in Spatial Econometrics. Bayesian inference often relies on computationally intensive simulation methods, such as Markov Chain Monte Carlo. When only marginal inference is needed, INLA provides a fast and accurate estimate of the posterior marginals of the parameters in the model. Furthermore, we have compared the results provided by these models to those obtained with a more general class of Generalised Linear Models with random effects. In these models, spatial autocorrelation is modelled by means of correlated Gaussian random effects.We also discuss a procedure to extend the class of models that the R-INLA software can fit. This approach is based on conditioning on one or more parameters so that the resulting models can be fitted with R-INLA across sets of values of the fixed parameters. The posterior marginals of these parameters of interest are then obtained by combining the marginal likelihoods (which are conditioned on the values of the parameters fixed) of the fitted models and a prior on these parameters. This approach can also be used to fit even more general models.Finally, we discuss the use of all these models on two datasets based on median housing prices for census tracts in Boston and the probability of business re-opening in New Orleans in the aftermath of hurricane Katrina.

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Adaptive Gaussian predictive process models for large spatial datasets

et al. 2011

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Large point referenced datasets occur frequently in the environmental and natural sciences. Use of Bayesian hierarchical spatial models for analyzing these datasets is undermined by onerous computational burdens associated with parameter estimation. Low-rank spatial process models attempt to resolve this problem by projecting spatial effects to a lower-dimensional subspace. This subspace is determined by a judicious choice of “knots” or locations that are fixed a priori. One such representation yields a class of predictive process models (e.g., Banerjee et al., 2008) for spatial and spatial-temporal data. Our contribution here expands upon predictive process models with fixed knots to models that accommodate stochastic modeling of the knots. We view the knots as emerging from a point pattern and investigate how such adaptive specifications can yield more flexible hierarchical frameworks that lead to automated knot selection and substantial computational benefits.

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Approximate Bayesian inference for large spatial datasets using predictive process models

Cited by 56 publications

References 37 publications

Interpolation of spatial data – A stochastic or a deterministic problem?

Interpolation of spatial data – A stochastic or a deterministic problem?

Approximate Bayesian inference for spatial econometrics models

Adaptive Gaussian predictive process models for large spatial datasets

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