scholarly journals Identification and isotropy characterization of deformed random fields through excursion sets

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.

scholarly journals Excursion sets of infinitely divisible random fields with convolution equivalent Lévy measure

2017 ◽  
Vol 54 (3) ◽  
pp. 833-851 ◽  
Author(s):  
Anders Rønn-Nielsen ◽  
Eva B. Vedel Jensen
Keyword(s):  
Random Field ◽  
Large Class ◽  
Random Fields ◽  
Lévy Measure ◽  
Infinitely Divisible ◽  
Excursion Sets ◽  
Excursion Set ◽  
Fixed Radius ◽  
Asymptotic Probability ◽  
The Right

Abstract We consider a continuous, infinitely divisible random field in ℝd, d = 1, 2, 3, given as an integral of a kernel function with respect to a Lévy basis with convolution equivalent Lévy measure. For a large class of such random fields, we compute the asymptotic probability that the excursion set at level x contains some rotation of an object with fixed radius as x → ∞. Our main result is that the asymptotic probability is equivalent to the right tail of the underlying Lévy measure.

scholarly journals Fractional random fields associated with stochastic fractional heat equations

2005 ◽  
Vol 37 (01) ◽  
pp. 108-133 ◽  
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.

Extrapolation Problem for Continuous Time Periodically Correlated Isotropic Random Fields

2017 ◽  
Vol 19 ◽  
pp. 1-23
Author(s):  
Iryna Golichenko ◽  
Oleksand Masyutka ◽  
Mikhail Moklyachuk
Keyword(s):  
Random Field ◽  
Spectral Characteristics ◽  
Mean Square ◽  
Spectral Densities ◽  
Optimal Linear ◽  
Extrapolation Problem ◽  
Time Argument ◽  
The Mean ◽  
Mean Square Errors ◽  
Isotropic Random

The problem of optimal linear estimation of functionals depending on the unknown values of a random fieldζ(t,x), which is mean-square continuous periodically correlated with respect to time argumenttє R and isotropic on the unit sphere Sn with respect to spatial argumentxєSn. Estimates are based on observations of the fieldζ(t,x) +Θ(t,x) at points (t,x) :t< 0;xєSn, whereΘ(t,x) is an uncorrelated withζ(t,x) random field, which is mean-square continuous periodically correlated with respect to time argumenttє R and isotropic on the sphereSnwith respect to spatial argumentxєSn. Formulas for calculating the mean square errors and the spectral characteristics of the optimal linear estimate of functionals are derived in the case of spectral certainty where the spectral densities of the fields are exactly known. Formulas that determine the least favourable spectral densities and the minimax (robust) spectral characteristics are proposed in the case where the spectral densities are not exactly known while a class of admissible spectral densities is given.

About approximation of 3-D random fields and statistical simulation

2003 ◽  
Vol 11 (3) ◽  
Author(s):  
Zoya O. Vyzhva
Keyword(s):  
Random Fields ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Statistical Simulation ◽  
Gaussian Random Fields ◽  
Statistical Characteristics ◽  
Mean Square ◽  
The Mean ◽  
Isotropic Random ◽  
Three Dimensional Space

The estimator of the mean-square approximation of 3-D homogeneous and isotropic random field is investigated. The problem of statistical simulation of realizations of random fields in threedimensional space is considered. The algorithm for the receiving of this realization has been formulated, which has been constructed on the base the mean-square approximation of random fields estimator. It has been constructed the statistical model for the Gaussian random fields in three-dimensional space, which has been given by its statistical characteristics.

scholarly journals CHARACTERIZATION OF THE FORMATION OF FILTER PAPER USING THE BARTLETT SPECTRUM OF THE FIBER STRUCTURE

2013 ◽  
Vol 32 (2) ◽  
pp. 77 ◽  
Author(s):  
Martin Lehmann ◽  
Jobst Eisengräber-Pabst ◽  
Joachim Ohser ◽  
Ali Moghiseh
Keyword(s):  
Quality Control ◽  
Filter Paper ◽  
Random Fields ◽  
Image Data ◽  
Gaussian Random Fields ◽  
Fiber Structure ◽  
Bessel Transform ◽  
The Mean ◽  
Paper Structure

The formation index of filter paper is one of the most important characteristics used in industrial quality control. Its estimation is often based on subjective comparison chart rating or, even more objective, on the power spectrum of the paper structure observed on a transmission light table. It is shown that paper formation can be modeled as Gaussian random fields with a well defined class of correlation functions, and a formation index can be derived from the density of the Bartlett spectrum estimated from image data. More precisely, the formation index is the mean of the Bessel transform of the correlation taken for wave lengths between 2 and 5 mm.

Boundary corrections for the expected Euler characteristic of excursion sets of random fields, with an application to astrophysics

1995 ◽  
Vol 27 (4) ◽  
pp. 943-959 ◽  
Author(s):  
K. J. Worsley
Keyword(s):  
Euler Characteristic ◽  
Random Fields ◽  
Background Radiation ◽  
Three Dimensions ◽  
Connected Components ◽  
Excursion Sets ◽  
Microwave Background Radiation ◽  
Excursion Set ◽  
Set Of Points ◽  
Fixed Region

Certain images arising in astrophysics and medicine are modelled as smooth random fields inside a fixed region, and experimenters are interested in the number of ‘peaks', or more generally, the topological structure of ‘hot-spots' present in such an image. This paper studies the Euler characteristic of the excursion set of a random field; the excursion set is the set of points where the image exceeds a fixed threshold, and the Euler characteristic counts the number of connected components in the excursion set minus the number of ‘holes'. For high thresholds the Euler characteristic is a measure of the number of peaks. The geometry of excursion sets has been studied by Adler (1981) who gives the expectation of two excursion set characteristics, called the DT (differential topology) and IG (integral geometry) characteristics, which equal the Euler characteristic provided the excursion set does not touch the boundary of the region. Worsley (1995) finds a boundary correction which gives the expectation of the Euler characteristic itself in two and three dimensions. The proof uses a representation of the Euler characteristic given by Hadwiger (1959). The purpose of this paper is to give a general result for any number of dimensions. The proof takes a different approach and uses a representation from Morse theory. Results are applied to the recently discovered anomalies in the cosmic microwave background radiation, thought to be the remnants of the creation of the universe.

Inference for stationary random fields given Poisson samples

1986 ◽  
Vol 18 (2) ◽  
pp. 406-422 ◽  
Author(s):  
Alan F. Karr
Keyword(s):  
Random Field ◽  
Poisson Process ◽  
Random Fields ◽  
Covariance Function ◽  
Mean Squared Error ◽  
Compact Sets ◽  
Mild Conditions ◽  
Minimum Mean Squared Error ◽  
Squared Error ◽  
The Mean

Given a d-dimensional random field and a Poisson process independent of it, suppose that it is possible to observe only the location of each point of the Poisson process and the value of the random field at that (randomly located) point. Non-parametric estimators of the mean and covariance function of the random field—based on observation over compact sets of single realizations of the Poisson samples—are constructed. Under fairly mild conditions these estimators are consistent (in various senses) as the set of observation becomes unbounded in a suitable manner. The state estimation problem of minimum mean-squared error reconstruction of unobserved values of the random field is also examined.

scholarly journals Piecewise-Multilinear Interpolation of a Random Field

2013 ◽  
Vol 45 (04) ◽  
pp. 945-959 ◽  
Author(s):  
Konrad Abramowicz ◽  
Oleg Seleznjev
Keyword(s):  
Random Field ◽  
Random Fields ◽  
Asymptotic Approximation ◽  
Field Observations ◽  
Mean Square ◽  
Local Behavior ◽  
Approximation Accuracy ◽  
Dimensional Cube ◽  
The Mean ◽  
Fractional Brownian Field

We consider a piecewise-multilinear interpolation of a continuous random field on a d-dimensional cube. The approximation performance is measured using the integrated mean square error. Piecewise-multilinear interpolator is defined by N-field observations on a locations grid (or design). We investigate the class of locally stationary random fields whose local behavior is like a fractional Brownian field, in the mean square sense, and find the asymptotic approximation accuracy for a sequence of designs for large N. Moreover, for certain classes of continuous and continuously differentiable fields, we provide the upper bound for the approximation accuracy in the uniform mean square norm.

scholarly journals Random fields of bounded variation and computation of their variation intensity

2016 ◽  
Vol 48 (4) ◽  
pp. 947-971
Author(s):  
Bruno Galerne
Keyword(s):  
Random Field ◽  
Bounded Variation ◽  
Random Fields ◽  
Gaussian Random Fields ◽  
Random Sets ◽  
Functions Of Bounded Variation ◽  
Boolean Models ◽  
Sample Paths ◽  
Directional Variation ◽  
The Mean

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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