arXiv subjects

Christian Lantuéjoul

Publications and source records attributed to Christian Lantuéjoul.

5 recordsLinked to original sources

arXiv2026

On the compatibility between the spatial moments and the codomain of a real random field

While any symmetric and positive semidefinite mapping can be the non-centered covariance of a Gaussian random field, it is known that these conditions are no longer sufficient when the random field is valued in a two-point set. The question therefore arises of what are the necessary and sufficient conditions for a mapping $ρ: \X \times \X \to \R$ to be the non-centered covariance of a random field with values in a subset ${\cE}$ of $\R$. Such conditions are presented in the general case when ${\cE}$ is a closed subset of the real line, then examined for some specific cases. In particular, if ${\cE}=\R$ or $\Z$, it is shown that the conditions reduce to $ρ$ being symmetric and positive semidefinite. If ${\cE}$ is a closed interval or a two-point set, the necessary and sufficient conditions are more restrictive: the symmetry, positive semidefiniteness, upper and lower boundedness of $ρ$ are no longer enough to guarantee the existence of a random field valued in ${\cE}$ and having $ρ$ as its non-centered covariance. Similar characterizations are obtained for semivariograms and higher-order spatial moments, as well as for multivariate random fields.

math.PR↗

arXiv2026

Towards a complete characterization of indicator variograms and madograms

Indicator variograms and madograms are structural tools used in many disciplines of the natural sciences and engineering to describe random sets and random fields. To date, several necessary conditions are known for a function to be a valid indicator variogram but, except for intractable corner-positive inequalities, a complete characterization of indicator variograms is missing. Likewise, only partial characterizations of madograms are known. This paper provides novel necessary and sufficient conditions for a given function to be the variogram of an indicator random field with constant mean value or to be the madogram of a random field, and establishes under which conditions these two families of functions coincide. Our results apply to any set of points where the random field is defined and rely on distance geometry and Gaussian random field theory.

math.PR↗

arXiv2020

The Turning Arcs: a Computationally Efficient Algorithm to Simulate Isotropic Vector-Valued Gaussian Random Fields on the $d$-Sphere

Random fields on the sphere play a fundamental role in the natural sciences. This paper presents a simulation algorithm parenthetical to the spectral turning bands method used in Euclidean spaces, for simulating scalar- or vector-valued Gaussian random fields on the $d$-dimensional unit sphere. The simulated random field is obtained by a sum of Gegenbauer waves, each of which is variable along a randomly oriented arc and constant along the parallels orthogonal to the arc. Convergence criteria based on the Berry-Esséen inequality are proposed to choose suitable parameters for the implementation of the algorithm, which is illustrated through numerical experiments. A by-product of this work is a closed-form expression of the Schoenberg coefficients associated with the Chentsov and exponential covariance models on spheres of dimensions greater than or equal to 2.

math.ST↗

arXiv2019

Simulating space-time random fields with nonseparable Gneiting-type covariance functions

Two algorithms are proposed to simulate space-time Gaussian random fields with a covariance function belonging to an extended Gneiting class, the definition of which depends on a completely monotone function associated with the spatial structure and a conditionally negative definite function associated with the temporal structure. In both cases, the simulated random field is constructed as a weighted sum of cosine waves, with a Gaussian spatial frequency vector and a uniform phase. The difference lies in the way to handle the temporal component. The first algorithm relies on a spectral decomposition in order to simulate a temporal frequency conditional upon the spatial one, while in the second algorithm the temporal frequency is replaced by an intrinsic random field whose variogram is proportional to the conditionally negative definite function associated with the temporal structure. Both algorithms are scalable as their computational cost is proportional to the number of space-time locations, which may be unevenly spaced in space and/or in time. They are illustrated and validated through synthetic examples.

stat.CO↗

arXiv2015

Truncation map estimation based on bivariate probabilities and validation for the truncated plurigaussian model

The truncated plurigaussian model is often used to simulate the spatial distribution of random categorical variables such as geological facies. The problems addressed in this paper are the estimation of parameters of the truncation map for the truncated plurigaussian model. Unlike standard truncation maps, in this paper a colored Voronoi tessellation with number of nodes, locations of nodes, and category associated with each node all treated as unknowns in the optimization. Parameters were adjusted to match categorical bivariate unit-lag probabilities, which were obtained from a larger pattern joint distribution estimates from the Bayesian maximum-entropy approach conditioned to the unit-lag probabilities. The distribution of categorical variables generated from the estimated truncation map was close to the target unit-lag bivariate probabilities. The validation of the predictive performance of the model is evaluated using scoring rules, and conditioning of the latent Gaussian fields to log-data is generalized for the case when the truncated bigaussian model is governed by a colored Voronoi tessellation of the truncation map.

math.ST↗