, Covariance properties of generalized Gaussian fields
, Generalized random fields and Karhunen-Loève expansion
, Generalized random fields on a compact domain of R d equipped with a metric, p.156
, Discretization of generalized Gaussian fields 157
, Ritz-Galerkin discretization of functions of the Laplacian
, Ritz-Galerkin discretization of GeGFs, p.161
162 7.4.1 Comparison with the Karhunen-Loève expansion ,
, Accounting for local anisotropies, p.163
, Link to stochastic partial differential equation approach
186 9.1.1 Simulation of stationary Matérn models . . 187 9.1.2 Simulation of general covariance models ,
, 195 9.2.1 Kriging estimate of non-stationary fields . . 195 9.2.2 Filtering of non-stationary fields
, Inference of non-stationary fields, p.203
Manifolds, tensor analysis, and applications, vol.75, 2012. ,
Random fields and geometry, 2009. ,
URL : https://hal.archives-ouvertes.fr/hal-01975050
Estimating deformations of isotropic gaussian random fields on the plane, The Annals of Statistics, vol.36, issue.2, pp.719-741, 2008. ,
Local likelihood estimation for nonstationary random fields, Journal of Multivariate Analysis, vol.102, issue.3, pp.506-520, 2011. ,
An introduction to numerical analysis, 1989. ,
Iterative numerical methods for sampling from high dimensional gaussian distributions, Statistics and Computing, vol.23, issue.4, pp.501-521, 2013. ,
Randomized algorithms for estimating the trace of an implicit symmetric positive semi-definite matrix, Journal of the ACM (JACM), vol.58, issue.2, p.8, 2011. ,
Spectral geometry: direct and inverse problems, vol.1207, 2006. ,
Nonlinear programming, Journal of the Operational Research Society, vol.48, issue.3, pp.334-334, 1997. ,
Numerical solution of fractional elliptic stochastic pdes with spatial white noise, IMA Journal of Numerical Analysis, 2018. ,
Graph theory with applications, 1976. ,
An introduction to pseudo-differential operators, Lecture Notes, 2012. ,
Statistics for experimenters, Wiley Series in Probability and Statistics, 2005. ,
The mathematical theory of finite element methods, vol.15, 2007. ,
On the lebesgue function for polynomial interpolation, SIAM Journal on Numerical Analysis, vol.15, issue.4, pp.694-704, 1978. ,
Analysis on manifolds via the laplacian, Lecture Notes, 2013. ,
Development of geostatistical models using Stochastic Partial Differential Equations, 2018. ,
URL : https://hal.archives-ouvertes.fr/tel-02126057
A general framework for spde-based stationary random fields, 2018. ,
URL : https://hal.archives-ouvertes.fr/hal-02075383
The lumped mass finite element method for a parabolic problem, The ANZIAM Journal, vol.26, issue.3, pp.329-354, 1985. ,
A fast computational algorithm for the discrete cosine transform, IEEE Transactions on communications, vol.25, issue.9, pp.1004-1009, 1977. ,
, Geostatistics : Modeling Spatial uncertainty. 2nd Edition. Wiley Series In Probability and Statistics, 2012.
Meshlab: an open-source mesh processing tool. The Eurographics Association, 2008. ,
An algorithm for the machine calculation of complex fourier series, Mathematics of computation, vol.19, issue.90, pp.297-301, 1965. ,
Old and new aspects in spectral geometry, vol.534, 2013. ,
Keenan's 3d model repository, 2019. ,
Krylov subspace methods for solving linear systems, 2015. ,
Performance and accuracy of lapack's symmetric tridiagonal eigensolvers, SIAM Journal on Scientific Computing, vol.30, issue.3, pp.1508-1526, 2008. ,
Maximum likelihood from incomplete data via the em algorithm, Journal of the Royal Statistical Society: Series B (Methodological), vol.39, issue.1, pp.1-22, 1977. ,
Efficient estimation of eigenvalue counts in an interval, Numerical Linear Algebra with Applications, vol.23, issue.4, pp.674-692, 2016. ,
Model-based geostatistics, Journal of the Royal Statistical Society: Series C (Applied Statistics), vol.47, issue.3, pp.299-350, 1998. ,
Auswertung der normen von interpolationsoperatoren, Mathematische Annalen, vol.164, issue.2, pp.105-112, 1966. ,
Simulating isotropic vector-valued gaussian random fields on the sphere through finite harmonics approximations. Stochastic Environmental Research and Risk Assessment, 2019. ,
Problems and results on the theory of interpolation. ii, Acta Mathematica Hungarica, vol.12, issue.1-2, pp.235-244, 1961. ,
Champs et processus gaussiens indexés par des graphes, estimation et prédiction, 2011. ,
Covariance functions on spheres cross time: Beyond spatial isotropy and temporal stationarity, Statistics & Probability Letters, vol.151, pp.1-7, 2019. ,
URL : https://hal.archives-ouvertes.fr/hal-01417668
Lebesguessche konstanten und divergente fourierreihen, Journal für die reine und angewandte Mathematik, vol.138, pp.22-53, 1910. ,
An introduction to probability theory and its applications, vol.2, 1971. ,
Geometry of finite deformations, linearization, and incremental deformations under initial stress/strain, Proceedings of International Conference Engineering Mechanics, vol.20, 2008. ,
Second-order non-stationary modeling approaches for univariate geostatistical data. Stochastic environmental research and risk assessment, vol.31, pp.1887-1906, 2017. ,
Estimation of space deformation model for non-stationary random functions, Spatial Statistics, vol.13, pp.45-61, 2015. ,
URL : https://hal.archives-ouvertes.fr/hal-01090350
Linear Algebra. Featured Titles for Linear Algebra (Advanced) Series, 2003. ,
Matrix functions, Model order reduction: theory, research aspects and applications, pp.275-303, 2008. ,
Exploring a new class of non-stationary spatial gaussian random fields with varying local anisotropy, Statistica Sinica, pp.115-133, 2015. ,
Covariance tapering for interpolation of large spatial datasets, Journal of Computational and Graphical Statistics, vol.15, issue.3, pp.502-523, 2006. ,
The resolution of the gibbs phenomenon for fourier spectral methods. Advances in The Gibbs Phenomenon, 2007. ,
applications of harmonic analysis, Generalized functions, vol.4, 1964. ,
, Computational statistics, vol.308, 2009.
Uber die abgrenzung der eigenwerte einer matrix, lzv. Akad. Nauk. USSR. Otd. Fiz-Mat. Nauk, vol.7, pp.749-754, 1931. ,
Signal processing on graphs-contributions to an emerging field, 2015. ,
URL : https://hal.archives-ouvertes.fr/tel-01256044
Stationary graph signals using an isometric graph translation, 23rd European Signal Processing Conference (EUSIPCO), pp.1516-1520, 2015. ,
URL : https://hal.archives-ouvertes.fr/hal-01155902
Compactly supported correlation functions, Journal of Multivariate Analysis, vol.83, issue.2, pp.493-508, 2002. ,
Strictly and non-strictly positive definite functions on spheres, Bernoulli, vol.19, issue.4, pp.1327-1349, 2013. ,
Matrix computations, 1996. ,
Matrix computations, pp.374-426, 1996. ,
Geometrical structure of laplacian eigenfunctions, SIAM Review, vol.55, issue.4, pp.601-667, 2013. ,
Wavelets on graphs via spectral graph theory, Applied and Computational Harmonic Analysis, vol.30, issue.2, pp.129-150, 2011. ,
URL : https://hal.archives-ouvertes.fr/inria-00541855
Non-stationary spatial modeling, Bayesian statistics, vol.6, issue.1, pp.761-768, 1999. ,
Functions of matrices: theory and computation, vol.104, 2008. ,
On the use of geostatistical filtering techniques in seismic processing, SEG Technical Program Expanded Abstracts, pp.2024-2027, 2003. ,
On the validity of commonly used covariance and variogram functions on the sphere, Mathematical Geosciences, vol.43, issue.6, pp.721-733, 2011. ,
Convergence study of the truncated karhunen-loeve expansion for simulation of stochastic processes, International journal for numerical methods in engineering, vol.52, issue.9, pp.1029-1043, 2001. ,
A stochastic estimator of the trace of the influence matrix for laplacian smoothing splines, Communications in Statistics-Simulation and Computation, vol.18, issue.3, pp.1059-1076, 1989. ,
Stochastic processes on a sphere. The Annals of mathematical statistics, vol.34, pp.213-218, 1963. ,
Riemannian geometry and geometric analysis, p.42005, 2008. ,
Covariance tapering for likelihoodbased estimation in large spatial data sets, Journal of the American Statistical Association, vol.103, issue.484, pp.1545-1555, 2008. ,
Multidimensional scaling by optimizing goodness of fit to a nonmetric hypothesis, Psychometrika, vol.29, issue.1, pp.1-27, 1964. ,
, Spectral Theory in Riemannian Geometry. EMS textbooks in mathematics, 2015.
Applied analysis. Courier Corporation, 1988. ,
Simulation of stochastic partial differential equations and stochastic active contours, 2007. ,
Fast simulation of gaussian random fields, Monte Carlo Methods and Applications, vol.17, issue.3, pp.195-214, 2011. ,
Isotropic gaussian random fields on the sphere: regularity, fast simulation and stochastic partial differential equations, The Annals of Applied Probability, vol.25, issue.6, pp.3047-3094, 2015. ,
Fundamentals of differential geometry, vol.191, 2012. ,
Geostatistical simulation: models and algorithms, 2013. ,
Spectral simulation of isotropic gaussian random fields on a sphere, Mathematical Geosciences, 2019. ,
Learning riemannian metrics, Proceedings of the Nineteenth conference on Uncertainty in Artificial Intelligence, pp.362-369, 2002. ,
Introduction to Smooth Manifolds, vol.218, 2012. ,
A class of non-stationary covariance functions with compact support. Stochastic environmental research and risk assessment, vol.30, pp.973-987, 2016. ,
An explicit link between gaussian fields and gaussian markov random fields: the spde approach (with discussion), JR 671 Stat Soc, Series B, vol.73, pp.423-498, 2011. ,
Stationary stochastic processes: theory and applications, 2012. ,
On the minimum flops problem in the sparse cholesky factorization, SIAM Journal on Matrix Analysis and Applications, vol.35, issue.1, pp.1-21, 2014. ,
M-factorial kriging for seismic data noise attenuation, 11th International Congress of the Brazilian Geophysical Society & EXPOGEF, pp.1651-1654, 2009. ,
A fast cosine transform in one and two dimensions, IEEE Transactions on Acoustics Speech and Signal Processing, vol.28, issue.1, pp.27-34, 1980. ,
Random fields on the sphere: representation, limit theorems and cosmological applications, vol.389, 2011. ,
Stationary graph processes and spectral estimation, IEEE Transactions on Signal Processing, vol.65, issue.22, pp.5911-5926, 2017. ,
Chebyshev polynomials, 2002. ,
, The theory of regionalized variables and its applications, vol.5, p.211, 1971.
Pour une analyse krigeante des données régionalisées. Centre de Géostatistique, 1982. ,
mize: Unconstrained Numerical Optimization Algorithms, 2019. ,
, Stability of the Lanczos Method for Matrix Function Approximation. arXiv, 2017.
Networks: An Introduction, 2010. ,
Numerical optimization, 2006. ,
Discrete-time signal processing, 2001. ,
A theorem on fourier transforms of radial functions, Journal of Mathematical Analysis and Applications, vol.69, issue.2, pp.559-562, 1979. ,
Graph signal processing: Overview, challenges, and applications, Proceedings of the IEEE, vol.106, pp.808-828, 2018. ,
Multidimensional fourier series, Nonlinear Ocean Waves and the Inverse Scattering Transform, vol.97, pp.115-145, 2010. ,
Spatial modelling using a new class of nonstationary covariance functions, Environmetrics: The official journal of the International Environmetrics Society, vol.17, issue.5, pp.483-506, 2006. ,
Stochastic processes, SIAM, vol.24, 1999. ,
Improved learning of riemannian metrics for exploratory analysis, Neural Networks, vol.17, issue.8-9, pp.1087-1100, 2004. ,
Stationary signal processing on graphs, IEEE Transactions on Signal Processing, vol.65, issue.13, pp.3462-3477, 2017. ,
Nonstationarity in R n is second-order stationarity in R 2n, Journal of applied probability, vol.40, issue.3, pp.815-820, 2003. ,
Modelling of non-stationary spatial structure using parametric radial basis deformations, geoENV II-Geostatistics for Environmental Applications, pp.175-186, 1999. ,
Reducing non-stationary random fields to stationarity and isotropy using a space deformation, Statistics & probability letters, vol.48, issue.1, pp.23-32, 2000. ,
The matrix cookbook, vol.7, p.510, 2008. ,
Signals, systems, and transforms, 2003. ,
M-factorial kriging-an efficient aid to noisy seismic data interpretation, Petroleum Geostatistics, 2015. ,
Spatially adaptive non-stationary covariance functions via spa-tially adaptive spectra, 2004. ,
Spatio-temporal covariance and crosscovariance functions of the great circle distance on a sphere, Journal of the American Statistical Association, vol.111, issue.514, pp.888-898, 2016. ,
Nonseparable stationary anisotropic space-time covariance functions, Stochastic Environmental Research and Risk Assessment, vol.21, issue.2, pp.113-122, 2006. ,
Covariance functions that are stationary or nonstationary in space and stationary in time, Statistica Neerlandica, vol.61, issue.3, pp.358-382, 2007. ,
Quasi-arithmetic means of covariance functions with potential applications to space-time data, Journal of Multivariate Analysis, vol.100, issue.8, pp.1830-1844, 2009. ,
On the non-reducibility of non-stationary correlation functions to stationary ones under a class of mean-operator transformations, Stochastic environmental research and risk assessment, vol.24, issue.5, pp.599-610, 2010. ,
Numerical recipes 3rd edition: The art of scientific computing, 2007. ,
Comparison of level set models in image segmentation, IET Image Processing, vol.12, issue.12, pp.2212-2221, 2018. ,
Introduction à l'analyse numérique des équations aux dérivées partielles, vol.2, 1998. ,
An introduction to the approximation of functions, 1969. ,
Gaussian Markov random fields: theory and applications, 2005. ,
Iterative methods for sparse linear systems, vol.82, 2003. ,
Nonparametric estimation of nonstationary spatial covariance structure, Journal of the American Statistical Association, vol.87, issue.417, pp.108-119, 1992. ,
A simple expression for multivariate lagrange interpolation, 2008. ,
Fehlerfortpflanzung bei interpolation, Numerische Mathematik, vol.3, issue.1, pp.62-71, 1961. ,
The emerging field of signal processing on graphs: Extending high-dimensional data analysis to networks and other irregular domains, IEEE Signal Processing Magazine, vol.3, issue.30, pp.83-98, 2013. ,
Stress tensors, riemannian metrics and the alternative descriptions in elasticity, Trends and Applications of Pure Mathematics to Mechanics, pp.369-383, 1984. ,
Fast sampling from a gaussian markov random field using krylov subspace approaches, 2008. ,
, Statistical methods. Eight Edition, 1989.
Hilbert space methods for reduced-rank gaussian process regression, 2014. ,
Introduction to fourier analysis on euclidean spaces, p.32, 1971. ,
Nonstationary spatial covariance functions, 2005. ,
Interpolation of spatial data: some theory for kriging, 2012. ,
A note on the volume of a simplex, The American Mathematical Monthly, vol.73, issue.3, pp.299-301, 1966. ,
Probability, Markov chains, queues, and simulation: the mathematical basis of performance modeling, 2009. ,
An analysis of the finite element method, vol.212, 1973. ,
Optimization theory and methods: nonlinear programming, vol.1, 2006. ,
Geostatistics for large datasets, Advances and challenges in space-time modelling of natural events, pp.55-77, 2012. ,
, Accelerated filtering on graphs using lanczos method, 2015.
, Numerical partial differential equations: finite difference methods, vol.22, 2013.
Central limit theorems for hilbert-space valued random fields satisfying a strong mixing condition, ALEA, vol.8, pp.77-94, 2011. ,
The multivariate normal distribution, 2012. ,
Approximation theory and approximation practice, vol.128, 2013. ,
Two results on polynomial interpolation in equally spaced points, Journal of Approximation Theory, vol.65, issue.3, pp.247-260, 1991. ,
The bounding of polynomials prescribed at equally distributed points, Proc. Pedag. Inst. Vitebsk, vol.3, pp.117-127, 1940. ,
Multivariate geostatistics: an introduction with applications, 2013. ,
A monte carlo implementation of the em algorithm and the poor man's data augmentation algorithms, Journal of the American statistical Association, vol.85, issue.411, pp.699-704, 1990. ,
On stationary processes in the plane, Biometrika, pp.434-449, 1954. ,
Advanced calculus, 2002. ,
nloptr: R Interface to NLopt, 2018. ,
En particulier, le phénomène en question est décrit par un champ aléatoire (généralement gaussien) et les données observées sont considérées comme résultant d'une réalisation particulière de ce champ aléatoire. Afin de faciliter la modélisation et les traitements géostatistiques qui en découlent, il est d'usage de supposer ce champ comme stationnaire et donc de supposer que la structuration spatiale des données se ,
, En effet, comment définir cette notion de stationnarité lorsque les données sont indexées sur des domaines non euclidiens (comme des sphères ou autres surfaces lisses)? Quid également du cas où les données présentent structuration spatiale qui change manifestement d'un endroit à l'autre du domaine d'étude? En outre, opter pour des modèles plus complexes, lorsque cela est possible, s'accompagne en général d'une augmentation drastique des coûts opérationnels (calcul et mémoire), Cependant, lorsqu'on travaille avec des jeux de données spatialisées complexes, cette hypothèse devient inadaptée
nous proposons une solution à ces problèmes s'appuyant sur la définition de champs aléatoires généralisés sur des variétés riemanniennes. D'une part, travailler avec des champs aléatoires généralisés permet d'étendre naturellement des travaux récents s'attachant à tirer parti d'une caractérisation des champs aléatoires utilisés en géostatistique comme des solutions d'équations aux dérivées partielles stochastiques. D'autre part, travailler sur des variétés riemanniennes permet à la fois de définir des champs sur des domaines qui ne sont que localement euclidiens, et sur des domaines vus comme déformés localement ,
, Ces champs généralisés sont ensuite discrétisés en utilisant une approche par éléments finis, et nous en donnons une formule analytique pour une large classe de champs généralisés englobant les champs généralement utilisés dans les applications. Enfin, afin de résoudre le problème du passage à l'échelle pour les grands jeux de données, nous proposons des algorithmes inspirés du traitement du signal sur graphe permettant la simulation