Extremes of Gaussian random fields with nonadditive dependence structure

SummaryContinuously indexed Gaussian fields (GFs) are the most important ingredient in spatial statistical modelling and geostatistics. The specification through the covariance function gives an intuitive interpretation of the field properties. On the computational side, GFs are hampered with the big n problem, since the cost of factorizing dense matrices is cubic in the dimension. Although computational power today is at an all time high, this fact seems still to be a computational bottleneck in many applications. Along with GFs, there is the class of Gaussian Markov random fields (GMRFs) which are discretely indexed. The Markov property makes the precision matrix involved sparse, which enables the use of numerical algorithms for sparse matrices, that for fields in ℝ2 only use the square root of the time required by general algorithms. The specification of a GMRF is through its full conditional distributions but its marginal properties are not transparent in such a parameterization. We show that, using an approximate stochastic weak solution to (linear) stochastic partial differential equations, we can, for some GFs in the Matérn class, provide an explicit link, for any triangulation of ℝd, between GFs and GMRFs, formulated as a basis function representation. The consequence is that we can take the best from the two worlds and do the modelling by using GFs but do the computations by using GMRFs. Perhaps more importantly, our approach generalizes to other covariance functions generated by SPDEs, including oscillating and non-stationary GFs, as well as GFs on manifolds. We illustrate our approach by analysing global temperature data with a non-stationary model defined on a sphere.

Context.In the next decade, radio telescopes, such as the Square Kilometer Array (SKA), will explore the Universe at high redshift, and particularly during the epoch of reionisation (EoR). The first structures emerged during this epoch, and their radiation reionised the previously cold and neutral gas of the Universe, creating ionised bubbles that percolate at the end of the EoR (z ∼ 6). SKA will produce 2D images of the distribution of the neutral gas at many redshifts, pushing us to develop tools and simulations to understand its properties.Aims.With this paper, we aim to measure topological statistics of the EoR in the so-called reionisation time fields from both cosmological and semi-analytical simulations. This field informs us about the time of reionisation of the gas at each position; it is used to probe the inhomogeneities of reionisation histories and can be extracted from 21 cm maps. We also compare these measurements with analytical predictions obtained within Gaussian random field (GRF) theory.Methods.The GRF theory allows us to compute many statistics of a field, namely the probability distribution functions (PDFs) of the field or its gradient, isocontour length, critical point distributions, and skeleton length. We compare these theoretical predictions to measurements made on reionisation time fields extracted from anEMMAsimulation and a21cmFASTsimulation at 1 cMpc/h resolution. We also compared our results to GRFs generated from the fitted power spectra of the simulation maps.Results.BothEMMAand21cmFASTreionisation time fields (treion(r)) are close to being Gaussian fields, in contrast with the 21 cm, density, or ionisation fraction, which have all been shown to be non-Gaussian. Only accelerating ionisation fronts at the end of the EoR seem to be the cause of small non-gaussianities intreion(r). Overall, this topological description of reionisation times provides a new quantitative and reproducible way to characterise the EoR scenario. Under the assumption of GRFs, it enables the generation of reionisation models with their propagation, percolation, or seed statistics simply from the reionisation time power spectrum. Conversely, these topological statistics provide a means to constrain the properties of the power spectrum and by extension the physics that drive the propagation of radiation.

The multi-Gaussian model is used in geostatistical applications to predict functions of a regionalized variable and to assess uncertainty by determining local (conditional to neighboring data) distributions. The model relies on the assumption that the regionalized variable can be represented by a transform of a Gaussian random field with a known mean value, which is often a strong requirement. This article presents two variations of the model to account for an uncertain mean value. In the first one, the mean of the Gaussian random field is regarded as an unknown non-random parameter. In the second model, the mean of the Gaussian field is regarded as a random variable with a very large prior variance. The properties of the proposed models are compared in the context of non-linear spatial prediction and uncertainty assessment problems. Algorithms for the conditional simulation of Gaussian random fields with an uncertain mean are also examined, and problems associated with the selection of data in a moving neighborhood are discussed.

Circulant matrix embedding is one of the most popular and efficient methods for the exact generation of Gaussian stationary univariate series. Although the idea of circulant matrix embedding has also been used for the generation of Gaussian stationary random fields, there are many practical covariance structures of random fields where classical embedding methods break down. In this work, we propose a novel methodology that adaptively constructs feasible circulant embeddings based on convex optimization with an objective function measuring the distance of the covariance embedding to the targeted covariance structure over the domain of interest. The optimal value of the objective function will be zero if and only if there exists a feasible embedding for the a priori chosen embedding size.

We discuss a family of random fields indexed by a parameter $s\in\mathbb{R} $ which we call the fractional Gaussian fields, given by \[\mathrm{FGF}_{s}(\mathbb{R} ^{d})=(-\Delta)^{-s/2}W, \] where $W$ is a white noise on $\mathbb{R}^{d}$ and $(-\Delta)^{-s/2}$ is the fractional Laplacian. These fields can also be parameterized by their Hurst parameter $H=s-d/2$. In one dimension, examples of $\mathrm{FGF}_{s}$ processes include Brownian motion ($s=1$) and fractional Brownian motion ($1/2<s<3/2$). Examples in arbitrary dimension include white noise ($s=0$), the Gaussian free field ($s=1$), the bi-Laplacian Gaussian field ($s=2$), the log-correlated Gaussian field ($s=d/2$), Levy’s Brownian motion ($s=d/2+1/2$), and multidimensional fractional Brownian motion ($d/2<s<d/2+1$). These fields have applications to statistical physics, early-universe cosmology, finance, quantum field theory, image processing, and other disciplines. We present an overview of fractional Gaussian fields including covariance formulas, Gibbs properties, spherical coordinate decompositions, restrictions to linear subspaces, local set theorems, and other basic results. We also define a discrete fractional Gaussian field and explain how the $\mathrm{FGF}_{s}$ with $s\in(0,1)$ can be understood as a long range Gaussian free field in which the potential theory of Brownian motion is replaced by that of an isotropic $2s$-stable Levy process.

This paper discusses the following task often encountered in building Bayesian spatial models: construct a homogeneous Gaussian Markov random field (GMRF) on a lattice with correlation properties either as present in some observed data, or consistent with prior knowledge. The Markov property is essential in designing computationally efficient Markov chain Monte Carlo algorithms to analyse such models. We argue that we can restate both tasks as that of fitting a GMRF to a prescribed stationary Gaussian field on a lattice when both local and global properties are important. We demonstrate that using the Kullback–Leibler discrepancy often fails for this task, giving severely undesirable behaviour of the correlation function for lags outside the neighbourhood. We propose a new criterion that resolves this difficulty, and demonstrate that GMRFs with small neighbourhoods can approximate Gaussian fields surprisingly well even with long correlation lengths. Finally, we discuss implications of our findings for likelihood based inference for general Markov random fields when global properties are also important.

Conditional-mean least-squares fitting of Gaussian Markov random fields to Gaussian fields

Both the Hausdorff dimension and the K‐entropy supply a measure of the irregularity of the landspace surface. The relationship between the two measures is investigated over a variety of terrains in Britain and a method of calculating the entropy is checked against an independent estimate of the dimension with reasonable agreement. The calculation of the K‐entropy requires that the landscape surface be represented by an homogenous ergodic random field. This condition is satisfied by the tendency of soil‐covered terrains to progressively approximate to a form well represented by a Gaussian field.Gaussian random fields can either be very smooth, possessing derivatives of all orders at every point or they are highly irregular and non‐differentiable everywhere. Within the regular conceptualization the Rice‐Kac theory is used to predict the numbers of crossing points and the extent of excursion sets. These predictions are tested against an example terrain from the High Weald of East Sussex with very good agreement, apart from predictions of local maxima. A worked example of the calculation of the K‐entropy is given as an appendix.The potential role of information theory in geomorphology extends beyond the use made of entropy in this investigation. In particular ergodic theory has important practical and theoretical implications.

Conditioning realizations of stationary Gaussian random fields to a set of data is traditionally based on simple kriging. In practice, this approach may be demanding as it does not account for the uncertainty in the spatial average of the random field. In this paper, an alternative model is presented, in which the Gaussian field is decomposed into a random mean, constant over space but variable over the realizations, and an independent residual. It is shown that, when the prior variance of the random mean is infinitely large (reflecting prior ignorance on the actual spatial average), the realizations of the Gaussian random field are made conditional by substituting ordinary kriging for simple kriging. The proposed approach can be extended to models with random drifts that are polynomials in the spatial coordinates, by using universal or intrinsic kriging for conditioning the realizations, and also to multivariate situations by using cokriging instead of kriging.

The analysis of the covariance structure is a crucial topic in multivariate statistics. Not only does the covariance matrix contain valuable information about the dependence structure between variables, enabling informed conclusions and optimal decision-making, but it also helps in calibrating statistical models. With the increasing use of more complex models, it has become essential to verify assumptions about the structure of covariance matrices. In this study, we review several significant covariance structures and demonstrate how it is possible to simultaneously test the presence of different structures in the diagonal blocks of a covariance matrix. We examine the distribution of the likelihood ratio test statistic and derive the expression for its h-th null moment. To ensure practical usability, we develop near-exact approximations for the likelihood ratio statistic. We provide a practical application using real data, along with numerical studies and simulations, to illustrate the applicability of the test and assess the precision of the developed near-exact approximations. Finally, we demonstrate how the proposed methodology can be extended to the study of more complex covariance structures.

Modeling asymptotically independent spatial extremes based on Laplace random fields

Gaussian and non-Gaussian random fields associated with Markov processes

Probabilistic estimation of thermal crack propagation in clays with Gaussian processes and random fields

For many models of solids, we frequently assume that the material parameters do not vary in space nor that they vary from one product realization to another. If the length scale of the application approaches the length scale of the microstructure however, spatially fluctuating parameter fields (which vary from one realization of the field to another) can be incorporated to make the model capture the stochasticity of the underlying micro-structure. Randomly fluctuating parameter fields are often described as Gaussian fields. Gaussian fields, however, assume that the probability density function of a material parameter at a given location is a univariate Gaussian distribution. This entails for instance that negative parameter values can be realized, whereas most material parameters have physical bounds (e.g., Young’s modulus cannot be negative). In this contribution, randomly fluctuating parameter fields are therefore described using the copula theorem and Gaussian fields, which allow different types of univariate marginal distributions to be incorporated but with the same correlation structure as Gaussian fields. It is convenient to keep the Gaussian correlation structure, as it allows us to draw samples from Gaussian fields and transform them into the new random fields. The benefit of this approach is that any type of univariate marginal distribution can be incorporated. If the selected univariate marginal distribution has bounds, unphysical material parameter values will never be realized. We then use Bayesian inference to identify the distribution parameters (which govern the random field). Bayesian inference regards the parameters that are to be identified as random variables and requires a user-defined prior distribution of the parameters to which the observations are inferred. For homogenized Young’s modulus of a columnar polycrystalline material of interest in this study, the results show that with a relatively wide prior (i.e., a prior distribution without strong assumptions), a single specimen is sufficient to accurately recover the distribution parameter values.

Gaussian fields are considered as Gibbsian fields. Thermodynamic functions are calculated for them and the variational principle is proved. As an application we get an approximation of log likelihood and an information theoretic interpretation of the asymptotic behaviour of the maximum likelihood estimator for Gaussian Markov fields.