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An explicit link between Gaussian fields and Gaussian Markov random fields: the stochastic partial differential equation approach. (English) Zbl 1274.62360
Summary: Continuously indexed Gaussian fields (GFs) are the most important ingredient in spatial statistical modelling and geostatistics. The specification through the covariance function gives an intuitive interpretation of the field properties. On the computational side, GFs are hampered with the big n problem, since the cost of factorizing dense matrices is cubic in the dimension. Although computational power today is at an all time high, this fact seems still to be a computational bottleneck in many applications. Along with GFs, there is the class of Gaussian Markov random fields (GMRFs) which are discretely indexed. The Markov property makes the precision matrix involved sparse, which enables the use of numerical algorithms for sparse matrices, that for fields in $$\mathbb R^2$$ only use the square root 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 parameterization. We show that, using an approximate stochastic weak solution to (linear) stochastic partial differential equations, we can, for some GFs in the Matérn class, provide an explicit link, for any triangulation of $$\mathbb R^d$$, between GFs and GMRFs, formulated as a basis function representation. The consequence is that we can take the best from the two worlds and do the modelling using GFs but do the computations by using GMRFs. Perhaps more importantly, our approach generalizes to other covariance functions generated by SPDEs, including oscillating and non-stationary GFs, as well as GFs on manifolds. We illustrate our approach by analysing global temperature data with a non-stationary model defined on a sphere.

##### MSC:
 62H11 Directional data; spatial statistics 62M40 Random fields; image analysis 62H30 Classification and discrimination; cluster analysis (statistical aspects) 86A32 Geostatistics 60H15 Stochastic partial differential equations (aspects of stochastic analysis)
##### Software:
GMRFLib; METIS; symrcm; spBayes; CSparse
Full Text:
##### References:
  Adler, The Geometry of Random Fields (2010) · Zbl 1182.60017  Adler, Random Fields and Geometry (2007)  Allcroft, A latent Gaussian Markov random-field model for spatiotemporal rainfall disaggregation, Appl. Statist. 52 pp 487– (2003) · Zbl 1111.62362  Arjas, Bayesian inference of survival probabilities, under stochastic ordering constraints, J. Am. Statist. Ass. 91 pp 1101– (1996) · Zbl 0880.62024  Auslander, Introduction to Differentiable Manifolds (1977)  Banerjee, Hierarchical Modeling and Analysis for Spatial Data (2004) · Zbl 1053.62105  Banerjee, Gaussian predictive process models for large spatial data sets, J. R. Statist. Soc. B 70 pp 825– (2008) · Zbl 05563371  Bansal, Statistical analyses of brain surfaces using Gaussian random fields on 2-D manifolds, IEEE Trans. Med. Imgng 26 pp 46– (2007)  Besag, Spatial interaction and the statistical analysis of lattice systems (with discussion), J. R. Statist. Soc. B 36 pp 192– (1974) · Zbl 0327.60067  Besag, Statistical analysis of non-lattice data, Statistician 24 pp 179– (1975)  Besag, On a system of two-dimensional recurrence equations, J. R. Statist. Soc. B 43 pp 302– (1981) · Zbl 0487.60045  Besag, On conditional and intrinsic autoregressions, Biometrika 82 pp 733– (1995) · Zbl 0899.62123  Besag, First-order intrinsic autoregressions and the de Wijs process, Biometrika 92 pp 909– (2005) · Zbl 1151.62068  Besag, Bayesian image restoration with two applications in spatial statistics (with discussion), Ann. Inst. Statist. Math. 43 pp 1– (1991) · Zbl 0760.62029  Bolin, Mathematical Sciences Preprint 2009:13. (2009)  Bolin, Spatial models generated by nested stochastic partial differential equations, with an application to global ozone mapping, Ann. Appl. Statist. 5 pp 523– (2011) · Zbl 1235.60075  Brenner, The Mathematical Theory of Finite Element Methods (2007)  Brohan, Uncertainty estimates in regional and global observed temperature changes: a new dataset from 1850, J. Geophys. Res. pp 111– (2006)  Chen, The lumped mass finite element method for a parabolic problem, J. Aust. Math. Soc. B 26 pp 329– (1985) · Zbl 0576.65110  Chilés, Geostatistics: Modeling Spatial Uncertainty (1999)  Ciarlet, The Finite Element Method for Elliptic Problems (1978)  Cressie, Statistics for Spatial Data (1993)  Cressie, Classes of nonseparable, spatio-temporal stationary covariance functions, J. Am. Statist. Ass. 94 pp 1330– (1999) · Zbl 0999.62073  Cressie, Fixed rank kriging for very large spatial data sets, J. R. Statist. Soc. B 70 pp 209– (2008) · Zbl 05563351  Cressie, Conditional-mean least-squares fitting of Gaussian Markov random fields to Gaussian fields, Computnl Statist. Data Anal. 52 pp 2794– (2008) · Zbl 1452.62707  Dahlhaus, Edge effects and efficient parameter estimation for stationary random fields, Biometrika 74 pp 877– (1987) · Zbl 0633.62094  Das , B. 2000 Global covariance modeling: a deformation approach to anisotropy PhD Thesis Department of Statistics, University of Washington  Davis, Direct Methods for Sparse Linear Systems (2006) · Zbl 1119.65021  Diggle, Model-based Geostatistics (2006)  Duff, Direct Methods for Sparse Matrices (1989)  Edelsbrunner, Geometry and Topology for Mesh Generation (2001)  Eidsvik, Technical Report 9 (2010)  Federer, Hausdorff measure and Lebesgue area, Proc. Natn. Acad. Sci. USA 37 pp 90– (1951) · Zbl 0042.28402  Federer, Colloquium lectures on geometric measure theory, Bull. Am. Math. Soc. 84 pp 291– (1978) · Zbl 0392.49021  Fuentes, High frequency kriging for nonstationary environmental processes, Environmetrics 12 pp 469– (2001)  Fuentes, Approximate likelihood for large irregular spaced spatial data, J. Am. Statist. Ass. 102 pp 321– (2008) · Zbl 1284.62589  Furrer, Covariance tapering for interpolation of large spatial datasets, J. Computnl Graph. Statist. 15 pp 502– (2006)  George, Computer Solution of Large Sparse Positive Definite Systems (1981) · Zbl 0516.65010  Gneiting, Simple tests for the validity of correlation function models on the circle, Statist. Probab. Lett. 39 pp 119– (1998) · Zbl 0910.60022  Gneiting, Nonseparable, stationary covariance functions for space-time data, J. Am. Statist. Ass. 97 pp 590– (2002) · Zbl 1073.62593  Gneiting, Matérn cross-covariance functions for multivariate random fields, J. Am. Statist. Ass. 105 pp 1167– (2010) · Zbl 1390.62194  Gschlößl, Modelling count data with overdispersion and spatial effects, Statist. Pap. 49 pp 531– (2007) · Zbl 1310.62083  Guttorp, Studies in the history of probability and statistics XLIX: on the Matérn correlation family, Biometrika 93 pp 989– (2006) · Zbl 1436.62013  Guyon, Parameter estimation for a stationary process on a d-dimensional lattice, Biometrika 69 pp 95– (1982) · Zbl 0485.62107  Hansen, GISS analysis of surface temperature change, J. Geophys. Res. 104 pp 30997– (1999)  Hansen, A closer look at United States and global surface temperature change, J. Geophys. Res. 106 pp 23947– (2001)  Hartman, Fast kriging of large data sets with Gaussian Markov random fields, Computnl Statist. Data Anal. 52 pp 2331– (2008) · Zbl 1452.62708  Heine, Models for two-dimensional stationary stochastic processes, Biometrika 42 pp 170– (1955) · Zbl 0067.36504  Henderson, Modelling spatial variation in leukemia survival data, J. Am. Statist. Ass. 97 pp 965– (2002) · Zbl 1048.62102  Higdon, A process-convolution approach to modelling temperatures in the North Atlantic Ocean, Environ. Ecol. Statist. 5 pp 173– (1998)  Higdon, Bayesian Statistics 6 pp 761– (1999) · Zbl 0951.62091  Hjelle, Triangulations and Applications (2006)  Hrafnkelsson, Hierarchical modeling of count data with application to nuclear fall-out, Environ. Ecol. Statist. 10 pp 179– (2003)  Hughes-Oliver, Parametric nonstationary correlation models, Statist. Probab. Lett. 40 pp 267– (1998) · Zbl 0959.62123  Ilić, A numerical solution using an adaptively preconditioned Lanczos method for a class of linear systems related with the fractional Poisson equation, J. Appl. Math. Stoch. Anal. pp 104525– (2008) · Zbl 1162.65015  Jones, Stochastic processes on a sphere, Ann. Math. Statist. 34 pp 213– (1963) · Zbl 0202.46702  Jun, Nonstationary covariance models for global data, Ann. Appl. Statist. 2 pp 1271– (2008) · Zbl 1168.62381  Karypis, A fast and high quality multilevel scheme for partitioning irregular graphs, SIAM J. Scient. Comput. 20 pp 359– (1999) · Zbl 0915.68129  Kneib, A mixed model approach for geoadditive hazard regression, Scand. J. Statist. 34 pp 207– (2007) · Zbl 1142.62073  Krantz, Geometric Integration Theory (2008) · Zbl 1149.28001  Lindgren, A note on the second order random walk model for irregular locations, Scand. J. Statist. 35 pp 691– (2008) · Zbl 1199.60276  McCullagh, Generalized Linear Models (1989) · Zbl 0588.62104  Paciorek, Spatial modelling using a new class of nonstationary covariance functions, Environmetrics 17 pp 483– (2006)  Peterson, An overview of the Global Historical Climatology Network temperature database, Bull. Am. Meteorol. Soc. 78 pp 2837– (1997)  Pettitt, A conditional autoregressive Gaussian process for irregularly spaced multivariate data with application to modelling large sets of binary data, Statist. Comput. 12 pp 353– (2002)  Quarteroni, Numerical Approximation of Partial Differential Equations (2008)  Rozanov, Markov Random Fields (1982)  Rue, Fast sampling of Gaussian Markov random fields, J. R. Statist. Soc. B 63 pp 325– (2001) · Zbl 0979.62075  Rue, Gaussian Markov Random Fields: Theory and Applications (2005)  Rue, Approximate Bayesian inference for latent Gaussian models by using integrated nested Laplace approximations (with discussion), J. R. Statist. Soc. B 71 pp 319– (2009) · Zbl 1248.62156  Rue, Fitting Gaussian Markov random fields to Gaussian fields, Scand. J. Statist. 29 pp 31– (2002) · Zbl 1017.62088  Samko, Fractional Integrals and Derivatives: Theory and Applications (1992)  Sampson, Nonparametric estimation of nonstationary spatial covariance structure, J. Am. Statist. Ass. 87 pp 108– (1992)  Smith, Change of variables in Laplace’s and other second-order differential equations, Proc. Phys. Soc. 46 pp 344– (1934) · Zbl 0009.11003  Song, A compariative study of Gaussian geostatistical models and Gaussian Markov random field models, J. Multiv. Anal. 99 pp 1681– (2008) · Zbl 1142.86309  Stein, Space-time covariance functions, J. Am. Statist. Ass. 100 pp 310– (2005) · Zbl 1117.62431  Stein, Interpolation of Spatial Data: Some Theory for Kriging (1999) · Zbl 0924.62100  Stein, Approximating likelihoods for large spatial data sets, J. R. Statist. Soc. B 66 pp 275– (2004) · Zbl 1062.62094  Vecchia, Estimation and model identification for continuous spatial processes, J. R. Statist. Soc. B 50 pp 297– (1988)  Wahba, Spline interpolation and smoothing on the sphere, SIAM J. Scient. Statist. Comput. 2 pp 5– (1981) · Zbl 0537.65008  Wall, A close look at the spatial structure implied by the CAR and SAR models, J. Statist. Planng Inf. 121 pp 311– (2004) · Zbl 1036.62097  Weir, Binary probability maps using a hidden conditional autoregressive Gaussian process with an application to Finnish common toad data, Appl. Statist. 49 pp 473– (2000) · Zbl 0965.62099  Whittle, On stationary processes in the plane, Biometrika 41 pp 434– (1954) · Zbl 0058.35601  Whittle, Stochastic processes in several dimensions, Bull. Inst. Int. Statist. 40 pp 974– (1963) · Zbl 0129.10603  Yue, Nonstationary spatial Gaussian Markov random fields, J. Computnl Graph. Statist. 19 pp 96– (2010)  Åberg, A class of non-Gaussian second order random fields, Extremes pp 1– (2010)  Åberg, Fatigue damage assessment for a spectral model of non-Gaussian random loads, Probab. Engng Mech. 24 pp 608– (2009)  Handbook of Mathematical Functions (1965)  Ainsworth, Pure and Applied Mathematics (2000)  Anderes, Local likelihood estimation for nonstationary random fields, J. Multiv. Anal. 102 pp 505– (2011) · Zbl 1207.62177  Balgovind, A stochastic-dynamic model for the spatial structure of forecast error statistics, Mnthly Weath. Rev. 111 pp 701– (1983)  Banerjee, Gaussian predictive process models for large spatial data sets, J. R. Statist. Soc. B 70 pp 825– · Zbl 05563371  Barndorff-Nielsen, Infinite divisibility of the hyperbolic and generalized inverse gaussian distributions, Probab. Theor. Reltd Flds 38 pp 309– (1977) · Zbl 0403.60026  Berg, The Dagum family of isotropic correlation functions, Bernoulli 14 pp 1134– (2008) · Zbl 1158.60350  Berger, Objective Bayesian analysis of spatially correlated data, J. Am. Statist. Ass. 96 pp 1361– (2001) · Zbl 1051.62095  Bermúdez, Perfectly Matched Layers for time-harmonic second order elliptic problems, Arch. Computnl Meth. Engng 17 pp 77– (2010) · Zbl 1359.76217  Besag, Spatial interaction and the statistical analysis of lattice systems (with discussion), J. R. Statist. Soc. B 36 pp 192– · Zbl 0327.60067  Besag, On a system of two-dimentional recurrence equations, J. R. Statist. Soc. B 43 pp 302– (1981) · Zbl 0487.60045  Besag, First-order intrinsic autoregressions and the de Wijs process, Biometrika 92 pp 909– · Zbl 1151.62068  Bhattacharya, The Hurst effect under trends, J. Appl. Probab. 20 pp 649– (1983) · Zbl 0526.60027  Bolin, Computationally efficient methods in spatial statistics, applications in environmental modeling, Licentiate Thesis (2009)  Bolin , D. Lindgren , F. Wavelet Markov models as efficient alternatives to tapering and convolution fields Mathematical Sciences Preprint 2009:13 Lund University  Bolin , D. Lindgren , F. 2011a Spatial wavelet Markov models are more efficient than covariance tapering and process convolutions  Bolin, Spatial models generated by nested stochastic partial differential equations, with an application to global ozone mapping, Ann. Appl. Statist. 5 pp 523– (2011b) · Zbl 1235.60075  Bookstein, Principal warps: thin-plate splines and the decomposition of deformations, IEEE Trans. Pattn Anal. Mach. Intell. 11 pp 567– (1989) · Zbl 0691.65002  Cameletti, Comparing air quality statistical models, Working Paper (2011)  Carr, Stochastic volatility for Lévy processess, Math. Finan. 13 pp 345– (2003) · Zbl 1092.91022  Challenor, The Oxford Handbook of Applied Bayesian Analysis pp 403– (2010)  Christensen, Deformable templates using large deformation kinematics, IEEE Trans. Im. Process. 5 pp 1435– (1996)  Clark, A subordinated stochastic process model with finite variance, Econometrica 41 pp 135– (1973) · Zbl 0308.90011  Coifman, Diffusion wavelets, Appl. Computnl Harm. Anal. 21 pp 53– (2006) · Zbl 1095.94007  Conti, Bayesian emulation of complex multi-output and dynamic computer models, J. Statist. Planng Inf. 140 pp 640– (2010) · Zbl 1177.62033  Crainiceanu, Bivariate binomial spatial modeling of loa loa prevalence in tropical Africa, J. Am. Statist. Ass. 103 pp 21– (2008) · Zbl 1469.86015  Cressie, Fixed rank kriging for very large spatial data sets, J. R. Statist. Soc. B 70 pp 209– · Zbl 05563351  Cressie, Statistics for Spatio-temporal Data (2011)  Das, PhD Dissertation (2000)  Diggle, Geostatistical inference under preferential sampling (with discussion), Appl. Statist. 59 pp 191– (2010)  Diggle, Model-based geostatistics (with discussion), Appl. Statist. 47 pp 299– (1998) · Zbl 0904.62119  Eidsvik, Technical Report 9 (2010)  Eliazar, Spatial gliding, temporal trapping, and anomalous transport, Physica 187 pp 30– (2004) · Zbl 1054.82026  Eliazar, Lévy Ornstein-Uhlenbeck, and subordination: spectral vs. jump description, J. Statist. Phys. 119 pp 165– (2005) · Zbl 1125.82026  Fotopoulos, Exact asymptotic distribution of change-point mle for changes in the mean of Gaussian sequences, Ann. Appl. Statist. 4 pp 1081– (2010) · Zbl 1194.62016  Furrer, Technical Note NCAR/TN476+STR (2008)  Furrer, A framework to understand the asymptotic properties of Kriging and splines, J. Kor. Statist. Soc. 36 pp 57– (2007) · Zbl 1115.62321  Furrer , E. M. Nychka , D. W. Piret , G. Furrer , R. 2011 An asymptotic framework under the Matérn covariance model  Gelfand, Handbook of Spatial Statistics pp 495– (2010)  Giraldo, Ordinary kriging for function-valued spatial data, Environ. Ecol. Statist. (2011)  Givoli, Recent advances in the DtN FE method, Arch. Computnl Meth. Engng 6 pp 71– (1999)  Gneiting, Nonseparable, stationary covariance functions for space-time data, J. Am. Statist. Ass. 97 pp 590– (2002a) · Zbl 1073.62593  Gneiting, Compactly supported correlation functions, J. Multiv. Anal. 83 pp 493– (2002b) · Zbl 1011.60015  Gneiting, Matérn cross-covariance functions for multivariate random fields, J. Am. Statist. Ass. 105 pp 1167–  Gneiting, Stochastic models that separate fractal dimension and the Hurst effect, SIAM Rev. 46 pp 269– (2004) · Zbl 1062.60053  Godwin, PhD Dissertation (2000)  Green, Reversible jump Markov chain Monte Carlo computation and Bayesian model determination, Biometrika 82 pp 711– (1995) · Zbl 0861.62023  Griebel, A finite method for density estimation with Gaussian process priors, SIAM J. Numer. Anal. 47 pp 4759– (2010) · Zbl 1211.65007  Gumprecht, Designs for detecting spatial dependence, Geograph. Anal. 41 pp 127– (2009)  Guttorp, Studies in the history of probability and statistics XLIX: On the Matérn correlation family, Biometrika 93 pp 989– (2006) · Zbl 1436.62013  Hegland, Approximate maximum a posteriori with Gaussian process priors, Constr. Approximn 26 pp 205– (2007) · Zbl 1127.65039  Hesthaven, Nodal Discontinuous Galerkin Methods (2008) · Zbl 1134.65068  Higdon, A process-convolution approach to modelling temperatures in the North Atlantic Ocean, Environ. Ecol. Statist. 5 pp 173–  Higdon, Bayesian Statistics 6 pp 761– (1999) · Zbl 0951.62091  Höhle, Additive-multiplicative regression models for spatio-temporal epidemics, Biometr. J. 51 pp 961– (2009)  Hooten, A Hierarchical Bayesian non-linear spatio-temporal model for the spread of invasive species with application to the Eurasian Collared-Dove, Environ. Ecol. Statist. 15 pp 59– (2007)  Illian, Gibbs point processes with mixed effects, Environmetrics 21 pp 341– (2010)  Illian, Technical Report (2010)  Illian , J. B. Sørbye , S. H. Rue , H. Hendrichsen , D. K. 2011 Fitting a log gaussian cox process with temporally varying effects-a case study  Irvine, Spatial designs and properties of spatial correlation: effects on covariance estimation, J. Agric. Biol. Environ. Statist. 12 pp 450– (2007) · Zbl 1306.62296  Jandhyala, Estimation of an unknown change-point occurring in the mean and covariance matrix of a multivariate Gaussian process with application to annual mean radiosonde temperature deviations at south and north polar zones, J. Clim (2011)  Jun, An approach to producing spacetime covariance functions on spheres, Technometrics 49 pp 468– (2007)  Jun, Nonstationary covariance models for global data, Ann. Appl. Statist. 2 pp 1271– · Zbl 1168.62381  Keller, Exact nonreflecting boundary conditions, J. Computnl Phys. 82 pp 172– (1989) · Zbl 0671.65094  Kennedy, Bayesian calibration of computer models (with discussion), J. R. Statist. Soc. B 63 pp 425– (2001) · Zbl 1007.62021  Kent, Probability, Statistics and Optimisation pp 325– (1994)  Kongsgård , H. W. 2011 The genetics of conflict: low level interaction between conflict events http://ssrn.com/abstract=1768198  Kumar, Fractional normal inverse Gaussian diffusion, Statist. Probab. Lett. 81 pp 146– (2011) · Zbl 1210.60040  Le, Statistical Analysis of Environmental Space-time Processes (2006) · Zbl 1102.62126  LeVeque, Finite Volume Methods for Hyperbolic Problems (2002) · Zbl 1010.65040  Li, The value of multi-proxy reconstruction of past climate (with discussions and rejoinder), J. Am. Statist. Ass. 105 pp 883– (2010) · Zbl 1390.62190  Lindgren, Non-traditional stochastic models for ocean wave-Lagrange models and nested SPDE models, Eur. Phys. J. Specl Top. 185 pp 209– (2010)  Loeppky, Choosing the sample size of a computer experiment: a practical guide, Technometrics 51 pp 366– (2009)  Majumdar, Spatio-temporal change-point modeling, J. Statist. Planng Inf. 130 pp 149– (2005) · Zbl 1085.62103  Mardia, Proc. Conf. Geomathematics and GIS Analysis of Resources, Environment and Hazards, Beijing pp 4– (2007)  Mardia, Maximum likelihood estimation using composite likelihoods for closed exponential families, Biometrika 96 pp 975– (2010) · Zbl 1178.62059  Martin, Exact Gaussian maximum likelihood and simulation for regularly-spaced observations with Gaussian correlations, Biometrika 87 pp 727– (2000) · Zbl 0956.62084  Martin, Approximations to the covariance properties of processes averaged over irregular spatial regions, Communs Statist. Theor. Meth. 23 pp 913– (1994) · Zbl 0825.62111  Mateu, On a class of non-stationary, compactly supported spatial covariance functions, Stochast. Environ. Res. Risk Assessmnt (2011)  Møller, Log Gaussian Cox processes, Scand. J. Statist. 25 pp 451– (1998) · Zbl 0931.60038  Møller, Statistical Inference and Simulation for Spatial Point Processes (2004) · Zbl 1044.62101  Møller, Modern statistics for spatial point processes (with discussion), Scand. J. Statist. 34 pp 643– (2007) · Zbl 1157.62067  Montegranario, A regularization approach for surface reconstruction from point clouds, Appl. Math. Computn 188 pp 583– (2007) · Zbl 1114.65306  Müller, Compound optimal spatial designs, Environmetrics 21 pp 354– (2010)  Narcowich, Generalized Hermite interpolation and positive definite kernels on a Riemannian manifold, J. Math. Anal. Applic. 190 pp 165– (1995) · Zbl 0859.58032  North, Correlation models for temperature fields, J. Clim (2011)  Nychka, Splines as local smoothers, Ann. Statist. 23 pp 1175– (1995) · Zbl 0842.62025  Obukhov, Statistically homogenous random fields on the globe, Usp. Mat. Nauk. 2 pp 196– (1947)  Pardo-lgúzquiza, AMLE3D: a computer program for the statistical inference of covariance parameters by approximate maximum likelihood estimation, Comput. Geosci. 7 pp 793– (1997)  Pardo-lgúzquiza, MLMATERN: a computer program for maximum likelihood inference with the spatial Matern covariance model, Comput. Geosci. 35 pp 1139– (2008)  Piterbarg, Asymptotic Methods in the Theory of Gaussian Processes and Fields (1996) · Zbl 0841.60024  Ramsay, Spline smoothing over difficult regions, J. R. Statist. Soc. B 64 pp 307– (2002) · Zbl 1067.62037  Roberts, Exponential convergence of Langevin diffusions and their discrete approximations, Bernoulli 2 pp 341– (1996) · Zbl 0870.60027  Rosanov, Markov random fields and stochastic partial differential equations, Math. USSR Sbor 32 pp 515– (1977) · Zbl 0396.60057  Rue, Gaussian Markov Random Fields: Theory and Applications (2005)  Rue, Approximate Bayesian inference for latent Gaussian models by using integrated nested Laplace approximations (with discussion), J. R. Statist. Soc. B 71 pp 319– · Zbl 1248.62156  Ruiz-Medina, The Dagum and auxiliary covariance families: towards reconciling two-parameter models that separate fractal dimension and the Hurst effect, Probab. Engng Mech. 26 pp 259– (2011)  Sampson, Handbook of Spatial Statistics pp 119– (2010)  Sampson, Nonparametric estimation of nonstationary spatial covariance structure, J. Am. Statist. Ass. 87 pp 108–  Santner, The Design and Analysis of Computer Experiments (2003) · Zbl 1041.62068  Schmidt, Considering covariates in the covariance structure of spatial processes, Environmetrics 22 pp 487– (2011)  Schmidt, Bayesian inference for non-stationary spatial covariance structures via spatial deformations, J. R. Statist. Soc. B 65 pp 743– (2003) · Zbl 1063.62034  Simpson, Preprint Statistics 16/2010 (2010)  Sokolov, Lévy flights from a continuous-time process, Phys. Rev. E 63 pp 011104– (2000)  Stein , M. L. Interpolation of Spatial Data: Some Theory for Kriging Springer · Zbl 0924.62100  Stein, Technical Report 21 (2005)  Sun, Advances and Challenges in Space-time Modelling of Natural Events (2011)  Vapnik, Statistical Learning Theory (1998)  Vecchia, Estimation and model identificaton for continuous spatial process, J. R. Statist. Soc. B 50 pp 297– (1988)  Wegener, Doctoral Thesis (2010)  Wendland, Scattered Data Approximation (2005)  Whittle, On stationary processes in the plane, Biometrika 41 pp 434– · Zbl 0058.35601  Wikle, Hierarchical Bayesian models for predicting the spread of ecological processes, Ecology 84 pp 1382– (2003)  Wikle, Polynomial nonlinear spatio-temporal integro-difference equation models, J. Time Ser. Anal. (2011) · Zbl 1294.62225  Wikle, Applications of Computational Statistics in the Environmental Sciences: Hierarchical Bayes and MCMC Methods pp 145– (2006)  Wikle, A general science-based framework for nonlinear spatiotemporal dynamical models, Test 19 pp 417– (2010) · Zbl 1203.37141  Wikle, Spatiotemporal hierarchical Bayesian modeling: tropical ocean surface winds, J. Am. Statist. Ass. 96 pp 382– (2001) · Zbl 1022.62117  Wiktorsson, Simulation of stochastic integrals with respect to Lévy processes of type G, Stoch. Processes Appl. 101 pp 113– (2002) · Zbl 1075.60051  Wood, Soap film smoohing, J. R. Statist. Soc. B 70 pp 931– (2008)  Wu, Closed-form valuations of basket options using a multivariate normal inverse Gaussian model, Insur. Math. Econ. 44 pp 95– (2009) · Zbl 1156.91389  Yadrenko, Spectral Theory of Random Fields (1983) · Zbl 0539.60048  Zhang, Inconsistent estimation and asymptotically equal interpolations in model-based geostatistics, J. Am. Statist. Ass. 99 pp 250– (2004) · Zbl 1089.62538  Zhu, Spatial sampling design for prediction with estimated parameters, J. Agric. Biol. Environ. Statist. 11 pp 24– (2006)  Zimmerman, Optimal network design for spatial prediction, covariance parameter estimation, and empirical prediction, Environmetrics 17 pp 635– (2006)
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