References
- Bachl, F. E., Lindgren, F., Borchers, D. L., and Illian, J. B. (2019), “Inlabru: An R Package for Bayesian Spatial Modelling from Ecological Survey Data,” Methods in Ecology and Evolution, 10, 760–766. DOI: 10.1111/2041-210X.13168.
- Baker, G. A., Jr., and Graves-Morris, P. (1996), Padé Approximants (2nd ed.), Volume 59 of Encyclopedia of Mathematics and Its Applications. Cambridge Cambridge University Press.
- Banerjee, S., Carlin, B. P., and Gelfand, A. E. (2015), Hierarchical Modeling and Analysis for Spatial Data. (2nd ed.), Volume 135 of Monographs on Statistics and Applied Probability. Boca Raton, FL CRC Press.
- Bolin, D., and Kirchner, K. (2020), “The Rational SPDE Approach for Gaussian Random Fields with General Smoothness,” Journal of Computational and Graphical Statistics, 29, 274–285. DOI: 10.1080/10618600.2019.1665537.
- Bolin, D., and Kirchner, K. (2023), “Equivalence of Measures and Asymptotically Optimal Linear Prediction for Gaussian Random Fields with Fractional-Order Covariance Operators,” Bernoulli, 29, 1476–1504. DOI: 10.3150/22-BEJ1507.
- Bolin, D., Kirchner, K., and Kovács, M. (2018), “Weak Convergence of Galerkin Approximations for Fractional Elliptic Stochastic PDEs with Spatial White Noise,” BIT Numerical Mathematics, 58, 881–906. DOI: 10.1007/s10543-018-0719-8.
- Bolin, D., Kirchner, K., and Kovács, M. (2020), “Numerical Solution of Fractional Elliptic Stochastic PDEs with Spatial White Noise,” IMA Journal of Numerical Analysis, 40, 1051–1073. DOI: 10.1093/imanum/dry091.
- Bolin, D., and Lindgren, F. (2011), “Spatial Models Generated by Nested Stochastic Partial Differential Equations, with an Application to Global Ozone Mapping,” Annals of Applied Statistics, 5, 523–550.
- Bolin, D., and Simas, A. B. (2023), “rSPDE: Rational Approximations of Fractional Stochastic Partial Differential Equations,” R Package Version 2.2.0
- Chang, W., Cheng, J., Allaire, J., Sievert, C., Schloerke, B., Xie, Y., Allen, J., McPherson, J., Dipert, A., and Borges, B. (2021), shiny: Web Application Framework for R R package version 1.6.0.
- Cox, S. G., and Kirchner, K. (2020), “Regularity and Convergence Analysis in Sobolev and Hölder Spaces for Generalized Whittle-Matérn Fields,” Numerische Mathematik, 146, 819–873. DOI: 10.1007/s00211-020-01151-x.
- Davies, E. B. (1995), Spectral Theory and Differential Operators, Cambridge Studies in Advanced Mathematics., Cambridge Cambridge University Press.
- Evans, L. C., and Gariepy, R. F. (2015), “Measure Theory and Fine Properties of Functions,” Textbooks in Mathematics. (Revised ed.), Boca Raton, FL CRC Press.
- Fedosov, B. (1963), “Asymptotic Formulas for the Eigenvalues of the Laplacian in the Case of a Polygonal Region,” Soviet Mathematics - Doklady, 4, 1092–1096.
- Fedosov, B. (1964), “Asymptotic Formulas for the Eigenvalues of the Laplace Operator in the Case of a Polyhedron,” Soviet Mathematics - Doklady, 5, 988–990.
- Fuglstad, G.-A., Simpson, D., Lindgren, F., and Rue, H. (2015), “Does Non-Stationary Spatial Data Always Require Non-Stationary Random Fields?,” Spatial Statistics, 14, 505–531. DOI: 10.1016/j.spasta.2015.10.001.
- Fuglstad, G.-A., Simpson, D., Lindgren, F., and Rue, H. (2019), “Constructing Priors That Penalize the Complexity of Gaussian Random Fields,” Journal of the American Statistical Association, 114, 445–452. DOI: 10.1080/01621459.2017.1415907.
- Good, I. J. (1952), “Rational Decisions,” Journal of the Royal Statistical Society, Series B, 14, 107–114. DOI: 10.1111/j.2517-6161.1952.tb00104.x.
- Grisvard, P. (2011), Elliptic Problems in Nonsmooth Domains., Volume 69 of Classics in Applied Mathematics. Philadelphia, PA Society for Industrial and Applied Mathematics (SIAM.).
- Heaton, M. J., Datta, A., Finley, A. O., Furrer, R., Guinness, J., Guhaniyogi, R., Gerber, F., Gramacy, R. B., Hammerling, D., Katzfuss, M., Lindgren, F., Nychka, D. W., Sun, F., and Zammit-Mangion, A. (2019), “A Case Study Competition among Methods for Analyzing Large Spatial Data,” Journal of Agricultural, Biological, and Environmental Statistics, 24, 398–425. DOI: 10.1007/s13253-018-00348-w.
- Herrmann, L., Kirchner, K., and Schwab, C. (2020), “Multilevel Approximation of Gaussian Random Fields: Fast Simulation,” Mathematical Models and Methods in Applied Sciences, 30, 181–223. DOI: 10.1142/S0218202520500050.
- Hildeman, A., Bolin, D., and Rychlik, I. (2021), “Deformed SPDE Models with an Application to Spatial Modeling of Significant Wave Height,” Spatial Statistics, 42, 100449–100427. Paper NoDOI: 10.1016/j.spasta.2020.100449.
- Hofreither, C. (2021), “An Algorithm for Best Rational Approximation Based on Barycentric Rational Interpolation,” Numerical Algorithms, 88, 365–388. DOI: 10.1007/s11075-020-01042-0.
- Kaufman, C. G., Schervish, M. J., and Nychka, D. W. (2008), “Covariance Tapering for Likelihood-Based Estimation in Large Spatial Data Sets,” Journal of the American Statistical Association, 103, 1545–1555. DOI: 10.1198/016214508000000959.
- Khristenko, U., Scarabosio, L., Swierczynski, P., Ullmann, E., and Wohlmuth, B. (2019), “Analysis of Boundary Effects on PDE-Based Sampling of Whittle-Matérn Random Fields,” SIAM/ASA Journal on Uncertainty Quantification, 7, 948–974. DOI: 10.1137/18M1215700.
- Lindgren, F., Bakka, H., Bolin, D., Krainski, E., and Rue, H. (2020), “A Diffusion-based Spatio-Temporal Extension of Gaussian Matérn fields,” arXiv: 2006.04917v2.
- Lindgren, F., Bolin, D., and Rue, H. (2022), “The SPDE Approach for Gaussian and non-Gaussian Fields: 10 Years and Still Running,” Spatial Statistics, 50, 100599. Paper NoDOI: 10.1016/j.spasta.2022.100599.
- Lindgren, F., and Rue, H. (2015), “Bayesian Spatial Modelling with R-INLA,” Journal of Statistical Software, 63, 1–25. DOI: 10.18637/jss.v063.i19.
- Lindgren, F., Rue, H., and Lindström, J. (2011), “An Explicit Link between Gaussian Fields and Gaussian Markov Random Fields: The Stochastic Partial Differential Equation Approach,” Journal of the Royal Statistical Society Series B: Statistical Methodology, 73, 423–498. DOI: 10.1111/j.1467-9868.2011.00777.x.
- Liu, Z., and Rue, H. (2022), “Leave-Group-Out Cross-validation for Latent Gaussian Models,” arXiv:2210.04482.
- Lototsky, S. V., and Rozovsky, B. L. (2017), Stochastic Partial Differential Equations. Universitext, Cham Springer.
- Matérn, B. (1960), Spatial Variation: Stochastic Models and their Application to Some Problems in Forest Surveys and Other Sampling Investigations Statens Skogsforskningsinstitut, Stockholm. Meddelanden Från Statens Skogsforskningsinstitut, Band 49, Nr. 5.
- R Core Team (2022), R: A Language and Environment for Statistical Computing. Vienna, Austria R Foundation for Statistical Computing.
- Remez, E. Y. (1934), “Sur la Détermination Des Polynômes D’Approximation de Degré Donnée,” Communications of the Kharkov Mathematical Society, 10, 41–63.
- Rue, H., and Held, L. (2005), Gaussian Markov Random Fields., Volume 104 of Monographs on Statistics and Applied Probability. Boca Raton, FL Chapman & Hall/CRC. Theory and Applications.
- Rue, H., Martino, S., and Chopin, N. (2009), “Approximate Bayesian Inference for Latent Gaussian Models by Using Integrated Nested Laplace Approximations,” Journal of the Royal Statistical Society Series B: Statistical Methodology, 71, 319–392. DOI: 10.1111/j.1467-9868.2008.00700.x.
- Simpson, D., Rue, H., Riebler, A., Martins, T. G., and Sørbye, S. H. (2017), “Penalising Model Component Complexity: A Principled, Practical Approach to Constructing Priors,” Statistical Science, 32, 28. DOI: 10.1214/16-STS576.
- Stein, M. L. (1999), Interpolation of Spatial Data. Springer Series in Statistics. New York Springer-Verlag. Some Theory for Kriging.
- Stein, M. L. (2002), “The Screening Effect in Kriging,” The Annals of Statistics, 30, 298–323. DOI: 10.1214/aos/1015362194.
- Steinwart, I., and Scovel, C. (2012), “Mercer’s Theorem on General Domains: On the Interaction between Measures, Kernels, and RKHSs,” Constructive Approximation, 35, 363–417. DOI: 10.1007/s00365-012-9153-3.
- Whittle, P. (1963), “Stochastic Processes in Several Dimensions,” Bulletin of the International Statistical Institute, 40, 974–994.