The Matérn model: a journey through statistics, numerical analysis and machine learning

Chris J. Oates, Robert Schaback, Moreno Bevilacqua, Emilio Porcu

Publication date: 25 July 2024

Published in: Statistical Science (Search for Journal in Brave)

kriging Sobolev spaces spatial statistics reproducing kernel Hilbert spaces maximum likelihood approximation theory covariance compact support machine learning Kernel

Mathematics Subject Classification ID

Statistics (62-XX)

Cites Work

An Explicit Link between Gaussian Fields and Gaussian Markov Random Fields: The Stochastic Partial Differential Equation Approach
The Rational SPDE Approach for Gaussian Random Fields With General Smoothness
Comparing composite likelihood methods based on pairs for spatial Gaussian random fields
Unifying compactly supported and Matérn covariance functions in spatial statistics
Modelling Point Referenced Spatial Count Data: A Poisson Process Approach
Cross-covariance functions for multivariate geostatistics
Strictly and non-strictly positive definite functions on spheres
Closed form representations for a class of compactly supported radial basis functions
2010 Rietz lecture: When does the screening effect hold?
Nonstationary modeling for multivariate spatial processes
The missing Wendland functions
Characterization theorems for the Gneiting class of space-time covariances
Fractional elliptic, hyperbolic and parabolic random fields
Hyperpriors for Matérn fields with applications in Bayesian inversion
Spatiotemporal random fields associated with stochastic fractional Helmholtz and heat equations
Sobolev error estimates and a Bernstein inequality for scattered data interpolation via radial basis functions
Positive definite radial functions and splines
Angular spectra for non-Gaussian isotropic fields
Isotropic covariance functions on spheres: some properties and modeling considerations
A class of non-Gaussian second order random fields
Tractability of multivariate problems. Volume II: Standard information for functionals.
The Dagum family of isotropic correlation functions
Asymptotically efficient prediction of a random field with a missspecified covariance function
Interpolation of spatial data. Some theory for kriging
Spatio-temporal modeling of particulate matter concentration through the SPDE approach
Matérn class tensor-valued random fields and beyond
Estimation and prediction using generalized Wendland covariance functions under fixed domain asymptotics
Anisotropy models for spatial data
Continuity properties for random fields
The screening effect in kriging
Compactly supported correlation functions
Equivalence of Gaussian measures for some nonstationary random fields
Piecewise polynomial, positive definite and compactly supported radial functions of minimal degree
Modeling temporal gradients in regionally aggregated California asthma hospitalization data
Composite likelihood estimation for a Gaussian process under fixed domain asymptotics
On fixed-domain asymptotics, parameter estimation and isotropic Gaussian random fields with Matérn covariance functions
The SPDE approach to Matérn fields: graph representations
Bayesian fixed-domain asymptotics for covariance parameters in a Gaussian process model
Criteria and characterizations for spatially isotropic and temporally symmetric matrix-valued covariance functions
Stein hypothesis and screening effect for covariances with compact support
Isotropic covariance functions on graphs and their edges
Closed form representations and properties of the generalised Wendland functions
A case study competition among methods for analyzing large spatial data
Zastavnyi operators and positive definite radial functions
A computational tool for comparing all linear PDE solvers
On positive definiteness of some radial functions
On the consistent separation of scale and variance for Gaussian random fields
Nonseparable dynamic nearest neighbor Gaussian process models for large spatio-temporal data with an application to particulate matter analysis
Duality between matrix variate \(t\) and matrix variate V.G. distributions
Metric spaces and completely monotone functions.
Regularity of the sample paths of a general second order random field
Positive definite functions on spheres
New validity conditions for the multivariate Matérn coregionalization model, with an application to exploration geochemistry
A Riemann-Stein kernel method
Asymptotically equivalent prediction in multivariate geostatistics
Equivalence of measures and asymptotically optimal linear prediction for Gaussian random fields with fractional-order covariance operators
Studies in the history of probability and statistics XLIX: On the Matérn correlation family
Interpolation of spatial data -- a stochastic or a deterministic problem?
Spatial Matérn Fields Driven by Non-Gaussian Noise
Geostatistical Modelling Using Non-Gaussian Matérn Fields
Localization in Matrix Computations: Theory and Applications
ON SOME MATÉRN COVARIANCE FUNCTIONS FOR SPATIO-TEMPORAL RANDOM FIELDS
Geostatistics
Maximum likelihood estimation of models for residual covariance in spatial regression
Kernel techniques: From machine learning to meshless methods
Bayesian Wombling
On some properties of Buhmann functions
Towards reconciling two asymptotic frameworks in spatial statistics
Generalized Whittle–Matérn random field as a model of correlated fluctuations
Fixed Rank Kriging for Very Large Spatial Data Sets
Gaussian Predictive Process Models for Large Spatial Data Sets
Nonseparable, Stationary Covariance Functions for Space–Time Data
Directional Rates of Change Under Spatial Process Models
A new class of radial basis functions with compact support
Error Analysis of Nodal Meshless Methods
Stochastic Models That Separate Fractal Dimension and the Hurst Effect
Approximating Likelihoods for Large Spatial Data Sets
Fractional random fields associated with stochastic fractional heat equations
A Valid Matérn Class of Cross-Covariance Functions for Multivariate Random Fields With Any Number of Components
Gneiting Class, Semi-Metric Spaces and Isometric Embeddings
Non‐Gaussian geostatistical modeling using (skew) t processes
Semivariogram methods for modeling Whittle–Matérn priors in Bayesian inverse problems
On spectral theory of random fields in the ball
Spatiotemporal covariance functions for Laplacian ARMA fields in higher dimensions
Finite Element Representations of Gaussian Processes: Balancing Numerical and Statistical Accuracy
Surface Finite Element Approximation of Spherical Whittle--Matérn Gaussian Random Fields
Probabilistic Numerics
Semi-exact control functionals from Sard’s method
Inverses of Matérn covariances on grids
On Identifiability and Consistency of The Nugget in Gaussian Spatial Process Models
Space-Time Estimation and Prediction under Infill Asymptotics with Compactly Supported Covariance Functions
Optimal stencils in Sobolev spaces
Multivariate Type G Matérn Stochastic Partial Differential Equation Random Fields
On the Improved Rates of Convergence for Matérn-Type Kernel Ridge Regression with Application to Calibration of Computer Models
Analysis of Boundary Effects on PDE-Based Sampling of Whittle--Matérn Random Fields
Bayesian Probabilistic Numerical Methods
Matérn Cross-Covariance Functions for Multivariate Random Fields
The role of the range parameter for estimation and prediction in geostatistics
Inconsistent Estimation and Asymptotically Equal Interpolations in Model-Based Geostatistics
Statistical Methods for Regular Monitoring Data
Control Functionals for Monte Carlo Integration
Non-Gaussian Bayesian Geostatistical Modeling
Graphical Gaussian process models for highly multivariate spatial data
Scattered Data Approximation
The origins of kriging
Estimators of fractal dimension: assessing the roughness of time series and spatial data
Fractional Gaussian fields: a survey
A general framework for Vecchia approximations of Gaussian processes
Beyond Matérn: On A Class of Interpretable Confluent Hypergeometric Covariance Functions
Galerkin-Chebyshev approximation of Gaussian random fields on compact Riemannian manifolds
Modeling Temporally Evolving and Spatially Globally Dependent Data
Estimating the parameters of some common Gaussian random fields with nugget under fixed-domain asymptotics
Space-time covariance models on networks
Gaussian Whittle-Matérn fields on metric graphs
30 years of space-time covariance functions
Nearest-neighbor sparse Cholesky matrices in spatial statistics
Permutation and Grouping Methods for Sharpening Gaussian Process Approximations
Gaussian likelihood inference on data from trans-Gaussian random fields with Matérn covariance function
Title not available (Why is that?)
Title not available (Why is that?)
Title not available (Why is that?)
Title not available (Why is that?)
Title not available (Why is that?)
Title not available (Why is that?)
Title not available (Why is that?)
Title not available (Why is that?)
Title not available (Why is that?)
Title not available (Why is that?)
Title not available (Why is that?)