## 4 Citations

### A diffusion-based spatio-temporal extension of Gaussian Mat\'ern fields

- Computer Science, Mathematics
- 2020

This work develops a spatio-temporal extension of the Gaussian Mat ´ ern ﬁelds formulated as solutions to a stochastic partial differential equation that provides a sparse representation based on a sparse element approximation that is well suited for statistical inference and which is implemented in the R-INLA software.

### Practical Hilbert space approximate Bayesian Gaussian processes for probabilistic programming

- Computer Science
- 2020

A novel approach for low-rank approximate Bayesian Gaussian processes, based on a basis function approximation via Laplace eigenfunctions for stationary covariance functions, which exhibits an attractive computational complexity due to its linear structure and is easy to implement in probabilistic programming frameworks.

### Finite Element Representations of Gaussian Processes: Balancing Numerical and Statistical Accuracy

- Computer ScienceSIAM/ASA Journal on Uncertainty Quantification
- 2022

The theory implies that, under certain smoothness assumptions, one can reduce the computation and memory cost without hindering the estimation accuracy by setting 𝑛 ≪ 𝓁 in the large 𝐁 asymptotics.

### The SPDE approach for spatio-temporal datasets with advection and diffusion

- Computer Science
- 2022

This work presents the advection-diﬀusion SPDE with ﬁrst order derivative in time to enlarge the SPDE family to the space-time context and proposes Computationally eﬃcient methods to estimate the parameters of the SPDe and to predict the spatio-temporalﬁeld by kriging.

### Realistic and Fast Modeling of Spatial Extremes over Large Geographical Domains

- Computer Science
- 2021

A more realistic Bayesian framework based on a novel Gaussian scale mixture model, where the Gaussian process component is defined by a stochastic partial differential equation that yields a sparse precision matrix, and the random scale component is modeled as a lowrank Pareto-tailed or Weibull-tailed spatial process determined by compactly supported basis functions.

### Efficient algorithms for Bayesian Inverse Problems with Whittle-Matérn Priors

- Mathematics, Computer ScienceArXiv
- 2022

An eﬃcient method for handling all admissible noninteger values of the exponent is derived and it is shown how to incorporate this prior representation into the inﬁnite-dimensional Bayesian formulation, and how to compute the maximum a posteriori estimate, and approximate the posterior variance.

### Boltzmann–Gibbs Random Fields with Mesh-free Precision Operators Based on Smoothed Particle Hydrodynamics

- Computer Science, MathematicsTheory of Probability and Mathematical Statistics
- 2022

A new Boltzmann–Gibbs model which features local interactions in the energy functional and is embodied in a spatial coupling function which uses smoothed kernel-function approximations of spatial derivatives inspired from the theory of smoothed particle hydrodynamics.

### Beyond Stationary Simulation; Modern Approaches to Stochastic Modelling

- Computer Science
- 2022

Three stochastic simulation methods using non-stationary covariance, multipoint simulation, and conditional GANs are presented: an SPDE method was used as a benchmark comparison and the application of machine learning techniques can be used to surpass previous simulation generation speeds and allows for greater parameterization.

### Parallelized integrated nested Laplace approximations for fast Bayesian inference

- Computer ScienceArXiv
- 2022

This work presents parallelization strategies for the methodology of integrated nested Laplace approximations (INLA), a popular framework for performing approximate Bayesian inference on the class of Latent Gaussian models, and demonstrates the performance of the new parallelization scheme on a number of real-world applications.

### Mitigating spatial confounding by explicitly correlating Gaussian random fields

- Environmental Science, MathematicsEnvironmetrics
- 2022

Spatial models are used in a variety of research areas, such as environmental sciences, epidemiology, or physics. A common phenomenon in such spatial regression models is spatial confounding. This…

## References

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- Computer Science, MathematicsJournal of Computational and Graphical Statistics
- 2019

This work proposes a new method, the rational SPDE approach, which in spatial dimension is applicable for any , and thus remedies the mentioned limitation, and shows that it facilitates likelihood-based inference for all model parameters including β.

### Exploring a New Class of Non-stationary Spatial Gaussian Random Fields with Varying Local Anisotropy

- Computer Science, Mathematics
- 2013

The results show that the use of an SPDE with non-constant coefficients is a promising way of creating non-stationary spatial GMRFs that allow for physical interpretability of the parameters, although there are several remaining challenges that would need to be solved before these models can be put to general practical use.

### A general framework for SPDE-based stationary random fields

- MathematicsBernoulli
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This paper presents theoretical advances in the application of the Stochastic Partial Differential Equation (SPDE) approach in geostatistics. We show a general approach to construct stationary models…

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- Computer Science
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The article studies non‐Gaussian extensions of a recently discovered link between certain Gaussian random fields, expressed as solutions to stochastic partial differential equations (SPDEs), and…

### Equivalence of measures and asymptotically optimal linear prediction for Gaussian random fields with fractional-order covariance operators

- Mathematics
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We consider Gaussian measures μ, μ̃ on a separable Hilbert space, with fractional-order covariance operators A−2β resp. Ã−2β̃ , and derive necessary and sufficient conditions on A, Ã and β, β̃ > 0…

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This volume is essential reading for statisticians working in spatial theory and its applications, as well as quantitative researchers in a wide range of science fields where spatial data analysis is important.

### Models and Methods for Random Fields in Spatial Statistics with Computational Efficiency from Markov Properties

- Computer Science, Mathematics
- 2012

The SPDE method is extended to a larger class of non-Gaussian random fields with Matern covariance functions, including certain Laplace Moving Average (LMA) models, and it is shown how the SPDE formulation can be used to obtain an efficient simulation method and an accurate parameter estimation technique for a LMA model.

### Multilevel approximation of Gaussian random fields: Covariance compression, estimation and spatial prediction

- MathematicsArXiv
- 2021

It is proved that the precision and covariance operators may be identified with bi-infinite matrices and finite sections may be diagonally preconditioned rendering the condition number independent of the dimension p of this section, and a tapering strategy by thresholding results in optimally numerically sparse approximations.