Smooth Gaussian Fields Research Articles

Smooth Matérn Gaussian random fields: Euler characteristic, expected number and height distribution of critical points

This paper studies Gaussian random fields with Matérn covariance functions with smooth parameter ν>2. Two cases of parameter spaces, the Euclidean space and N-dimensional sphere, are considered. For such smooth Gaussian fields, we have derived the explicit formulae for the expected Euler characteristic of the excursion set, the expected number and height distribution of critical points. The results are valuable for approximating the excursion probability in family-wise error control and for computing p-values in peak inference.

Erratum: “Existence of an unbounded nodal hypersurface for smooth Gaussian fields in dimension d≥3”

Erratum: “Existence of an unbounded nodal hypersurface for smooth Gaussian fields in dimension d≥3”

Uniqueness of Unbounded Component for Level Sets of Smooth Gaussian Fields

Abstract For a large family of stationary continuous Gaussian fields $f$ on ${\mathbb {R}}^{d}$, including the Bargmann–Fock and Cauchy fields, we prove that there exists at most one unbounded connected component in the level set $\{f=\ell \}$ (as well as in the excursion set $\{f\geq \ell \}$) almost surely for every level $\ell \in {\mathbb {R}}$, thus proving a conjecture proposed by Duminil-Copin, Rivera, Rodriguez, and Vanneuville. As the fields considered are typically very rigid (e.g., analytic almost surely), there is no sort of finite energy property available and the classical approaches to prove uniqueness become difficult to implement. We bypass this difficulty using a soft shift argument based on the Cameron–Martin theorem.

Norm inflation for a non-linear heat equation with gaussian initial conditions

We consider a non-linear heat equation ∂tu=Δu+B(u,Du)+P(u)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\partial _t u = \\Delta u + B(u,Du)+P(u)$$\\end{document} posed on the d-dimensional torus, where P is a polynomial of degree at most 3 and B is a bilinear map that is not a total derivative. We show that, if the initial condition u0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$u_0$$\\end{document} is taken from a sequence of smooth Gaussian fields with a specified covariance, then u exhibits norm inflation with high probability. A consequence of this result is that there exists no Banach space of distributions which carries the Gaussian free field on the 3D torus and to which the DeTurck–Yang–Mills heat flow extends continuously, which complements recent well-posedness results of Cao–Chatterjee and the author with Chandra–Hairer–Shen. Another consequence is that the (deterministic) non-linear heat equation exhibits norm inflation, and is thus locally ill-posed, at every point in the Besov space B∞,∞-1/2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$B^{-1/2}_{\\infty ,\\infty }$$\\end{document}; the space B∞,∞-1/2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$B^{-1/2}_{\\infty ,\\infty }$$\\end{document} is an endpoint since the equation is locally well-posed for B∞,∞η\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$B^{\\eta }_{\\infty ,\\infty }$$\\end{document} for every η>-12\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\eta >-\\frac{1}{2}$$\\end{document}.

Existence of an unbounded nodal hypersurface for smooth Gaussian fields in dimension d≥3

For the Bargmann–Fock field on Rd with d≥3, we prove that the critical level ℓc(d) of the percolation model formed by the excursion sets {f≥ℓ} is strictly positive. This implies that for every ℓ sufficiently close to 0 (in particular for the nodal hypersurfaces corresponding to the case ℓ=0), {f=ℓ} contains an unbounded connected component that visits “most” of the ambient space. Our findings actually hold for a more general class of positively correlated smooth Gaussian fields with rapid decay of correlations. The results of this paper show that the behavior of nodal hypersurfaces of these Gaussian fields in Rd for d≥3 is very different from the behavior of nodal lines of their 2-dimensional analogues.

Smooth Gaussian fields and percolation

Smooth Gaussian fields and percolation

On convergence of volume of level sets of stationary smooth Gaussian fields

On convergence of volume of level sets of stationary smooth Gaussian fields

A covariance formula for topological events of smooth Gaussian fields

We derive a covariance formula for the class of ‘topological events’ of smooth Gaussian fields on manifolds; these are events that depend only on the topology of the level sets of the field, for example, (i) crossing events for level or excursion sets, (ii) events measurable with respect to the number of connected components of level or excursion sets of a given diffeomorphism class and (iii) persistence events. As an application of the covariance formula, we derive strong mixing bounds for topological events, as well as lower concentration inequalities for additive topological functionals (e.g., the number of connected components) of the level sets that satisfy a law of large numbers. The covariance formula also gives an alternate justification of the Harris criterion, which conjecturally describes the boundary of the percolation university class for level sets of stationary Gaussian fields. Our work is inspired by (Ann. Inst. Henri Poincaré Probab. Stat. 55 (2019) 1679–1711), in which a correlation inequality was derived for certain topological events on the plane, as well as by (Asymptotic Methods in the Theory of Gaussian Processes and Fields (1996) Amer. Math. Soc.), in which a similar covariance formula was established for finite-dimensional Gaussian vectors.

On the number of excursion sets of planar Gaussian fields

The Nazarov–Sodin constant describes the average number of nodal set components of smooth Gaussian fields on large scales. We generalise this to a functional describing the corresponding number of level set components for arbitrary levels. Using results from Morse theory, we express this functional as an integral over the level densities of different types of critical points, and as a result deduce the absolute continuity of the functional as the level varies. We further give upper and lower bounds showing that the functional is at least bimodal for certain isotropic fields, including the important special case of the random plane wave.

Lipschitz-Killing curvatures of excursion sets for two-dimensional random fields

In the present paper we study three geometrical characteristics for the excursion sets of a two-dimensional stationary isotropic random field. First, we show that these characteristics can be estimated without bias if the considered field satisfies a kinematic formula, this is for instance the case of fields given by a function of smooth Gaussian fields or of some shot noise fields. By using the proposed estimators of these geometric characteristics, we describe some inference procedures for the estimation of the parameters of the field. An extensive simulation study illustrates the performances of each estimator. Then, we use the Euler characteristic estimator to build a test to determine whether a given field is Gaussian or not, when compared to various alternatives. The test is based on a sparse information, i.e., the excursion sets for two different levels of the field to be tested. Finally, the proposed test is adapted to an applied case, synthesized 2D digital mammograms.

Small ball probabilities for smooth Gaussian fields and tensor products of compact operators

We find the logarithmic L2‐small ball asymptotics for a class of zero mean Gaussian fields with covariances having the structure of “tensor product”. The main condition imposed on marginal covariances is slow growth at the origin of counting functions of their eigenvalues. That is valid for Gaussian functions with smooth covariances. Another type of marginal functions considered as well are classical Wiener process, Brownian bridge, Ornstein–Uhlenbeck process, etc., in the case of special self‐similar measure of integration. Our results are based on a new theorem on spectral asymptotics for the tensor products of compact self‐adjoint operators in Hilbert space which is of independent interest. Thus, we continue to develop the approach proposed in the paper , where the regular behavior at infinity of marginal eigenvalues was assumed.

The Tail of the Maximum of Smooth Gaussian Fields on Fractal Sets

We study the probability distribution of the maximum M S of a smooth stationary Gaussian field defined on a fractal subset S of ℝn. Our main result is the equivalent of the asymptotic behavior of the tail of the distribution ℙ(M S >u) as u→+∞. The basic tool is the Rice formula for the moments of the number of local maxima of a random field.

On random surface area

Consider a random smooth Gaussian field G(x): $ F \to \mathbb{R} $ , where F is a compact set in $ {\mathbb{R}^d} $ . We derive a formula for the average area of a surface determined by the equation G(x) = 0 and give some applications. As an auxiliary result, we obtain an integral expression for the area of a surface determined by zeros of a nonrandom smooth field. Bibliography: 13 titles.

Adaptive Bayesian estimation using a Gaussian random field with inverse Gamma bandwidth

We consider nonparametric Bayesian estimation inference using a rescaled smooth Gaussian field as a prior for a multidimensional function. The rescaling is achieved using a Gamma variable and the procedure can be viewed as choosing an inverse Gamma bandwidth. The procedure is studied from a frequentist perspective in three statistical settings involving replicated observations (density estimation, regression and classification). We prove that the resulting posterior distribution shrinks to the distribution that generates the data at a speed which is minimax-optimal up to a logarithmic factor, whatever the regularity level of the data-generating distribution. Thus the hierachical Bayesian procedure, with a fixed prior, is shown to be fully adaptive.

A general expression for the distribution of the maximum of a Gaussian field and the approximation of the tail

We study the probability distribution F ( u ) of the maximum of smooth Gaussian fields defined on compact subsets of R d having some geometric regularity. Our main result is a general expression for the density of F . Even though this is an implicit formula, one can deduce from it explicit bounds for the density, and hence for the distribution, as well as improved expansions for 1 − F ( u ) for large values of u . The main tool is the Rice formula for the moments of the number of roots of a random system of equations over the reals. This method enables also to study second-order properties of the expected Euler characteristic approximation using only elementary arguments and to extend these kinds of results to some interesting classes of Gaussian fields. We obtain more precise results for the “direct method” to compute the distribution of the maximum, using the spectral theory of GOE random matrices.

A Gaussian kinematic formula

In this paper we consider probabilistic analogues of some classical integral geometric formulae: Weyl–Steiner tube formulae and the Chern–Federer kinematic fundamental formula. The probabilistic building blocks are smooth, real-valued random fields built up from i.i.d. copies of centered, unit-variance smooth Gaussian fields on a manifold M. Specifically, we consider random fields of the form fp=F(y1(p),…,yk(p)) for F∈C2(ℝk;ℝ) and (y1,…,yk) a vector of C2 i.i.d. centered, unit-variance Gaussian fields. The analogue of the Weyl–Steiner formula for such Gaussian-related fields involves a power series expansion for the Gaussian, rather than Lebesgue, volume of tubes: that is, power series expansions related to the marginal distribution of the field f. The formal expansions of the Gaussian volume of a tube are of independent geometric interest. As in the classical Weyl–Steiner formulae, the coefficients in these expansions show up in a kinematic formula for the expected Euler characteristic, χ, of the excursion sets M∩f−1[u,+∞)=M∩y−1(F−1[u,+∞)) of the field f. The motivation for studying the expected Euler characteristic comes from the well-known approximation $\mathbb{P}[\sup_{p\in M}f(p)\geq u]\simeq\mathbb{E}[\chi(f^{-1}[u,+\infty))]$.

Bayesian Inversion of Geoelectrical Resistivity Data

SummaryEnormous quantities of geoelectrical data are produced daily and often used for large scale reservoir modelling. To interpret these data requires reliable and efficient inversion methods which adequately incorporate prior information and use realistically complex modelling structures. We use models based on random coloured polygonal graphs as a powerful and flexible modelling framework for the layered composition of the Earth and we contrast our approach with earlier methods based on smooth Gaussian fields. We demonstrate how the reconstruction algorithm may be efficiently implemented through the use of multigrid Metropolis–coupled Markov chain Monte Carlo methods and illustrate the method on a set of field data.