Random fields of bounded variation and computation of their variation intensity

AbstractThe main purpose of this paper is to define and characterize random fields of bounded variation, that is, random fields with sample paths in the space of functions of bounded variation, and to study their mean total variation. Simple formulas are obtained for the mean total directional variation of random fields, based on known formulas for the directional variation of deterministic functions. It is also shown that the mean variation of random fields with stationary increments is proportional to the Lebesgue measure, and an expression of the constant of proportionality, called thevariation intensity, is established. This expression shows, in particular, that the variation intensity depends only on the family of two-dimensional distributions of the stationary increment random field. When restricting to random sets, the obtained results give generalizations of well-known formulas from stochastic geometry and mathematical morphology. The interest of these general results is illustrated by computing the variation intensities of several classical stationary random field and random set models, namely Gaussian random fields and excursion sets, Poisson shot noises, Boolean models, dead leaves models, and random tessellations.

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Identification and isotropy characterization of deformed random fields through excursion sets

Advances in Applied Probability ◽

10.1017/apr.2018.32 ◽

2018 ◽

Vol 50 (3) ◽

pp. 706-725

Author(s):

Julie Fournier

Keyword(s):

Random Field ◽

Euler Characteristic ◽

Random Fields ◽

Weak Form ◽

Invariance Property ◽

Basic Domain ◽

Excursion Sets ◽

The Mean ◽

Isotropic Random

Abstract A deterministic application θ:ℝ2→ℝ2 deforms bijectively and regularly the plane and allows the construction of a deformed random field X∘θ:ℝ2→ℝ from a regular, stationary, and isotropic random field X:ℝ2→ℝ. The deformed field X∘θ is, in general, not isotropic (and not even stationary), however, we provide an explicit characterization of the deformations θ that preserve the isotropy. Further assuming that X is Gaussian, we introduce a weak form of isotropy of the field X∘θ, defined by an invariance property of the mean Euler characteristic of some of its excursion sets. We prove that deformed fields satisfying this property are strictly isotropic. In addition, we are able to identify θ, assuming that the mean Euler characteristic of excursion sets of X∘θ over some basic domain is known.

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MILES FORMULAE FOR BOOLEAN MODELS OBSERVED ON LATTICES

Image Analysis & Stereology ◽

10.5566/ias.v28.p77-92 ◽

2011 ◽

Vol 28 (2) ◽

pp. 77 ◽

Cited By ~ 19

Author(s):

Joachim Ohser ◽

Werner Nagel ◽

Katja Schladitz

Keyword(s):

Mean Curvature ◽

Surface Density ◽

Euler Number ◽

Volume Density ◽

Random Sets ◽

Boolean Models ◽

Intrinsic Volumes ◽

Binary Images ◽

Density Surface ◽

The Mean

The densities of the intrinsic volumes – in 3D the volume density, surface density, the density of the integral of the mean curvature and the density of the Euler number – are a very useful collection of geometric characteristics of random sets. Combining integral and digital geometry we develop a method for efficient and simultaneous calculation of the intrinsic volumes of random sets observed in binary images in arbitrary dimensions. We consider isotropic and reflection invariant Boolean models sampled on homogeneous lattices and compute the expectations of the estimators of the intrinsic volumes. It turns out that the estimator for the surface density is proved to be asymptotically unbiased and thusmultigrid convergent for Boolean models with convex grains. The asymptotic bias of the estimators for the densities of the integral of the mean curvature and of the Euler number is assessed for Boolean models of balls of random diameters. Miles formulae with corresponding correction terms are derived for the 3D case.

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The mean Euler characteristic and excursion probability of Gaussian random fields with stationary increments

The Annals of Applied Probability ◽

10.1214/15-aap1101 ◽

2016 ◽

Vol 26 (2) ◽

pp. 722-759 ◽

Cited By ~ 8

Author(s):

Dan Cheng ◽

Yimin Xiao

Keyword(s):

Euler Characteristic ◽

Random Fields ◽

Gaussian Random Fields ◽

Stationary Increments ◽

The Mean ◽

Excursion Probability

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On Estimation of the Mean and Covariance Parameter for Gaussian Random Fields

Statistics ◽

10.1080/02331889808802622 ◽

1998 ◽

Vol 31 (1) ◽

pp. 1-20 ◽

Cited By ~ 14

Author(s):

P. Ibarrola ◽

R. Różański ◽

R. Vélez

Keyword(s):

Random Fields ◽

Gaussian Random Fields ◽

The Mean

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Fractional random fields associated with stochastic fractional heat equations

Advances in Applied Probability ◽

10.1017/s0001867800000069 ◽

2005 ◽

Vol 37 (01) ◽

pp. 108-133 ◽

Cited By ~ 9

Author(s):

M. Ya. Kelbert ◽

N. N. Leonenko ◽

M. D. Ruiz-Medina

Keyword(s):

Random Field ◽

Random Fields ◽

Evolution Equations ◽

Heat Equations ◽

Mean Square ◽

Fractional Evolution Equations ◽

The Mean

This paper introduces a convenient class of spatiotemporal random field models that can be interpreted as the mean-square solutions of stochastic fractional evolution equations.

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The mean number of maxima above high levels in Gaussian random fields

Journal of Applied Probability ◽

10.1017/s002190020009447x ◽

1976 ◽

Vol 13 (02) ◽

pp. 377-379 ◽

Cited By ~ 1

Author(s):

A. M. Hasofer

Keyword(s):

Asymptotic Formula ◽

Random Fields ◽

Gaussian Random Fields ◽

Gaussian Field ◽

General Proof ◽

The Mean

An asymptotic formula for the mean number of maxima above a level of an n-dimensional stationary Gaussian field has been given by Nosko without proof. In this note a short general proof of this formula is given.

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Extremes of Homogeneous Gaussian Random Fields

Journal of Applied Probability ◽

10.1239/jap/1429282606 ◽

2015 ◽

Vol 52 (1) ◽

pp. 55-67 ◽

Cited By ~ 6

Author(s):

Krzysztof Dębicki ◽

Enkelejd Hashorva ◽

Natalia Soja-Kukieła

Keyword(s):

Correlation Function ◽

Asymptotic Expansion ◽

Random Fields ◽

Survival Function ◽

Random Variable ◽

Gaussian Random Fields ◽

Extremal Index ◽

Gaussian Field ◽

Gaussian Fields ◽

Sample Paths

Let {X(s, t): s, t ≥ 0} be a centred homogeneous Gaussian field with almost surely continuous sample paths and correlation function r(s, t) = cov(X(s, t), X(0, 0)) such that r(s, t) = 1 - |s|α1 - |t|α2 + o(|s|α1 + |t|α2), s, t → 0, with α1, α2 ∈ (0, 2], and r(s, t) < 1 for (s, t) ≠ (0, 0). In this contribution we derive an asymptotic expansion (as u → ∞) of P(sup(sn1(u),tn2(u)) ∈[0,x]∙[0,y]X(s, t) ≤ u), where n1(u)n2(u) = u2/α1+2/α2Ψ(u), which holds uniformly for (x, y) ∈ [A, B]2 with A, B two positive constants and Ψ the survival function of an N(0, 1) random variable. We apply our findings to the analysis of extremes of homogeneous Gaussian fields over more complex parameter sets and a ball of random radius. Additionally, we determine the extremal index of the discretised random field determined by X(s, t).

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Independence of the increments of Gaussian random fields

Nagoya Mathematical Journal ◽

10.1017/s0027763000019759 ◽

1982 ◽

Vol 85 ◽

pp. 251-268 ◽

Cited By ~ 6

Author(s):

Kazuyuki Inoue ◽

Akio Noda

Keyword(s):

Random Field ◽

Random Fields ◽

Euclidean Distance ◽

Gaussian Random Field ◽

Gaussian Random Fields

Let be a mean zero Gaussian random field (n ⋜ 2). We call X Euclidean if the probability law of the increments X(A) − X(B) is invariant under the Euclidean motions. For such an X, the variance of X(A) − X(B) can be expressed in the form r(|A − B|) with a function r(t) on [0, ∞) and the Euclidean distance |A − B|.

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