A Kronecker-based Covariance Specification for Spatially Continuous Multivariate Data

We propose a covariance specification for modeling spatially continuous multivariate data. This model is based on a reformulation of Kronecker’s product of covariance matrices for Gaussian random fields. We illustrate the case with the Matérn function used for specifying marginal covariances. The structure holds for other choices of covariance functions with parameters varying in their usual domains, which makes the estimation process more accessible. The reduced computational time and flexible generalization for increasing number of variables, make it an attractive alternative for modelling spatially continuous data. Theoretical results for the likelihood function and the derivatives of the covariance matrix are presented. The proposed model is fitted to the literature’s soil250 dataset, and adequacy measures, forecast errors and estimation times are compared with the ones obtained based on classical models. Furthermore, the model is fitted to the classic meuse dataset to illustrate the model’s flexibility in a four-variate analysis. A simulation study is performed considering different parametric scenarios to evaluate the asymptotic properties of the maximum likelihood estimators. The satisfactory results, its simpler structure and the reduced estimation time make the proposed model a candidate approach for multivariate analysis of spatial data.

Series representations and simulations of isotropic random fields in the Euclidean space

This paper introduces the series expansion for homogeneous, isotropic and mean square continuous random fields in the Euclidean space, which involves the Bessel function and the ultraspherical polynomial, but differs from the spectral representation in terms of the ordinary spherical harmonics that has more terms at each level.The series representation provides a simple and efficient approach for simulation of isotropic (non-Gaussian) random fields.

Stochastic elliptic operators defined by non-Gaussian random fields with uncertain spectrum

This paper presents a construction and the analysis of a class of non-Gaussian positive-definite matrix-valued homogeneous random fields with uncertain spectral measure for stochastic elliptic operators. Then the stochastic elliptic boundary value problem in a bounded domain of the 3D-space is introduced and analyzed for stochastic homogenization.

Lossless, scalable implicit likelihood inference for cosmological fields

We present a comparison of simulation-based inference to full, field-based analytical inference in cosmological data analysis. To do so, we explore parameter inference for two cases where the information content is calculable analytically: Gaussian random fields whose covariance depends on parameters through the power spectrum; and correlated lognormal fields with cosmological power spectra. We compare two inference techniques: i) explicit field-level inference using the known likelihood and ii) implicit likelihood inference with maximally informative summary statistics compressed via Information Maximising Neural Networks (IMNNs). We find that a) summaries obtained from convolutional neural network compression do not lose information and therefore saturate the known field information content, both for the Gaussian covariance and the lognormal cases, b) simulation-based inference using these maximally informative nonlinear summaries recovers nearly losslessly the exact posteriors of field-level inference, bypassing the need to evaluate expensive likelihoods or invert covariance matrices, and c) even for this simple example, implicit, simulation-based likelihood incurs a much smaller computational cost than inference with an explicit likelihood. This work uses a new IMNN implementation in Jax that can take advantage of fully-differentiable simulation and inference pipeline. We also demonstrate that a single retraining of the IMNN summaries effectively achieves the theoretically maximal information, enhancing the robustness to the choice of fiducial model where the IMNN is trained.

Stochastic Downscaling to Chaotic Weather Regimes using Spatially Conditioned Gaussian Random Fields with Adaptive Covariance

Downscaling aims to link the behaviour of the atmosphere at fine scales to properties measurable at coarser scales, and has the potential to provide high resolution information at a lower computational and storage cost than numerical simulation alone. This is especially appealing for targeting convective scales, which are at the edge of what is possible to simulate operationally. Since convective scale weather has a high degree of independence from larger scales, a generative approach is essential. We here propose a statistical method for downscaling moist variables to convective scales using conditional Gaussian random fields, with an application to wet bulb potential temperature (WBPT) data over the UK. Our model uses an adaptive covariance estimation to capture the variable spatial properties at convective scales. We further propose a method for the validation, which has historically been a challenge for generative models.