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	Random field -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-10-08)
		Several kinds of random fields exist, among them Markov random fields (MRF), Gibbs random fields (GRF) and (additional info and facts about Gaussian random field) Gaussian random fields.
		In other words, the probability a random variable assumes a value depends on the other random variables only through the ones that are its immediate neighbors.
		A probability of a random variable in a MRF is showed by the equation 1, Ω' is the same realization of Ω, except for random variable X
	www.absoluteastronomy.com /encyclopedia/r/ra/random_field.htm (223 words)

	[No title]
		It may be expressed by a simple formula in terms of the eigenvalues of the correlation matrix of the electric field and it reduces, for the two-dimensional case, to the usual well-known expression for the degree of polarization of beam-like fields.
		The effect of an incident field with a phase screw dislocation (a so-called optical vortex) on the shape of the enhanced backscattering cone was studied theoretically and demonstrated experimentally.
		Spectral degree of coherence of a random three-dimensional electromagnetic field
	random.creol.ucf.edu /home/publications/publications.html (7112 words)

	Web Site for Perfectly Random Sampling with Markov Chains:
		Another problem is that of generating a random spanning tree of a graph or spanning arborescence of a directed graph in accordance with the uniform distribution, or more generally in accordance with a distribution given by weights on the edges of the graph or digraph.
		However, the running time of their algorithm is an unbounded random variable whose order of magnitude is typically unknown a priori and which is not independent of the state sampled, so a naive user with limited patience who aborts a long run of the algorithm will introduce bias.
		For Markov random fields we show, using ideas of Marton and Shields, that the presence of a phase transition is an obstruction for the existence of the above coding: this yields a large class of Bernoulli shifts for which no such coding exists.
	dimacs.rutgers.edu /~dbwilson/exact.html (14686 words)

	Chapter Two, Random Fields (Site not responding. Last check: 2007-10-08)
		The random field function derives its spatial dependence by the use of a filter function summing together independent gaussian random deviates with a normal distribution.
		Random fields must be calibrated to the spatial continuity of the field being simulated.
		Random field models are fit by replicating the spatial statistics of application quality samples of each class using the class probability maps developed from lower quality map(s).
	www.wiu.edu /users/cre111/older/dissertation/20.htm (4356 words)

	Optimal Discretization of Random Fields (Site not responding. Last check: 2007-10-08)
		A new method for efficient discretization of random fields (i.e., their representation in terms of random variables) is introduced.
		The efficiency of the discretization is measured by the number of random variables required to represent the field with a specified level of accuracy.
		It represents the field as a linear function of nodal random variables and a set of shape functions, which are determined by minimizing an error variance.
	www.pubs.asce.org /WWWdisplay.cgi?9302205 (170 words)

	Random Fields (Site not responding. Last check: 2007-10-08)
		The idea behind random fields is that the value at each pixel is chosen by a two-dimensional stochastic process.
		Obviously, performing this calculation would be very expensive and of little use, since the random field model for texture segmentation assumes that different regions of the image have their pixel values drawn from different distributions.
		Fortunately, we have the Markov property, which is stated as follows: the probability that a pixel has a certain grey level given every other pixel in the image is equal to the probability that the pixel has that same grey level given only the pixels in a neighborhood surrounding that pixel.
	ai.stanford.edu /people/ruzon/tex_seg/node4.html (253 words)

	Unified univariate and multivariate random field theory (Site not responding. Last check: 2007-10-08)
		A smooth Gaussian random field with zero mean and unit variance is sampled on a discrete lattice, and we are interested in the exceedence probability (P-value) of the maximum in a finite region.
		If the random field is smooth relative to the mesh size, then the P-value can be well approximated by results for the continuously sampled smooth random field (Adler, 1981; Worsley, 1995a; Taylor & Adler, 2003).
		If the random field is not smooth, so that adjacent lattice values are nearly independent, then the usual Bonferroni bound is very accurate.
	www.math.mcgill.ca /keith/bubbles/dlm_abstract.htm (166 words)

	Conditional Random Fields (Site not responding. Last check: 2007-10-08)
		Conditional random fields (CRFs) are a probabilistic framework for labeling and segmenting structured data, such as sequences, trees and lattices.
		Linear-chain conditional random fields (CRFs) have been shown to perform well for information extraction and other language modelling tasks due to their ability to capture arbitrary, overlapping features of the input in a Markov model.
		Kernel conditional random fields are introduced as a framework for discriminative modeling of graph-structured data.
	www.inference.phy.cam.ac.uk /hmw26/crf (5057 words)

	DC MetaData pour: Contribution à l'étude asymptotique des estimateurs du maximum de la Pseudo-vraisemblance ...
		The first chapter is devoted to the presentation of Gibbs random fields, and we give some methods of estimations of the parameters of these fields, as well as, some methods of image reconstruction from the observed records.
		In the second chapter, we present a review, on the one hand, on the central limit theorems of the random fields on a network and, on the other hand, on the asymptotic properties of the estimations of parameters of Markov random fields.
		The consistency and the asymptotic normality of the maximum pseudo-likelihood estimations for Gibbs Markov random fields are studied in the third chapter.
	www-mathdoc.ujf-grenoble.fr /archive-publis/theses/falaha70.Thu_Dec_13_16_33_37.html (467 words)

	A consistent model selection procedure for Markov random fields based on penalized pseudolikelihood, Chuanshu Ji, Lynne ...
		The procedure is shown to be consistent for choosing the true model, even for Gibbs random fields with phase transitions.
		COMETS, F. On consistency of a class of estimators for exponential families of Markov ´ random fields on the lattice.
		Markov Random Fields: Theory and Applications R. Chellappa and A. Jain, eds.
	projecteuclid.org /Dienst/UI/1.0/Display/euclid.aoap/1034968138 (551 words)

	Markov Random Fields
		It should perhaps be stressed that, whereas in image processing there is a predetermined `correct answer' to the reconstruction problem (the correct answer being the original image or `true scene'), this is not necessarily the case in the rainfall disaggregation problem.
		The underlying concept behind statistical image reconstruction techniques is that of the Markov Random Field, in which the probability distribution for rainfall at a site (`site' being taken here to mean a pixel at a particular spatial scale) is specified conditionally on the pattern of rainfall in the neighbourhood of that site.
		For a fuller discussion of the theory of Markov Random Fields, see for example Besag (1974, 1986), [Isham1981].
	www.ucl.ac.uk /Stats/research/Resrprts/rr176/node25.html (595 words)

	Applications of Random Fields in Human Brain Mapping (ResearchIndex)
		Abstract: The goal of this short article is to summarize how random field theory has been used to test for activations in brain mapping applications.
		Tables of most widely used random fields, examples of their applications, as well as references to distributions of some of their relevant statistics are provided.
		8 Upcrossings of random fields (context) - Hasofer - 1978
	citeseer.ist.psu.edu /278265.html (486 words)

	ipedia.com: Stochastic process Article (Site not responding. Last check: 2007-10-08)
		In practical applications, the domain over which the function is defined is a time interval (a stochastic process of this kind is called a time series in applications) or a region of space (a stochastic process being called a random field).
		Note, however, that the definition of stochastic process as an indexed collection of random variables is much more general than the case where the indices are points of the domain of the random function.
		The measurable space and the probability measure arise from the random variables and expectations by means of well-known representation theorems of analysis.
	www.ipedia.com /stochastic_process.html (1600 words)

	Excursion Sets of Random Fields With Applications to Human Brain Mapping (ResearchIndex)
		105 The Geometry of Random Fields (context) - Adler - 1981
		16 Estimating the number of peaks in a random field using the H..
		5 Local structure of Gaussian random fields in the vicinity of..
	citeseer.ist.psu.edu /cao97excursion.html (509 words)

	On the morphological analysis of binary random fields (Site not responding. Last check: 2007-10-08)
		Modeling image data by means of random fields and developing statistical techniques for the processing and analysis of these random fields is an important problem in image processing and analysis.
		Since morphological transformations of continuous space binary random fields are not measurable in general, we employ intermediate steps which require generation of an equivalent random closed set.
		The relationship between binary random fields and random closed sets is thoroughly investigated.
	csdl2.computer.org /persagen/DLAbsToc.jsp?resourcePath=/dl/proceedings/&toc=comp/proceedings/icip/1995/7310/01/7310toc.xml&DOI=10.1109/ICIP.1995.529759 (268 words)

	Representation of Gibbs fields with Synchronous Random Fields. (ResearchIndex)
		Abstract: We address the issue of the representation of Gibbs random fields over some configuration set by means of synchronous random fields, which lend themselves more efficiently to sampling on parallel devices.
		After describing the class of synchronous fields which is considered, we introduce a parametrization of synchronous fields by means of a potential.
		30 The description of a random field by means of conditional pr..
	citeseer.ist.psu.edu /154384.html (649 words)

	Random fields (Site not responding. Last check: 2007-10-08)
		There exist several types of random fields, such as Markov random field (MRF) and Gibbs randomfield (GRF).
		In other words, the probability a random variableassumes a value does not depend on all of the random variables.
		A probability of a random variable in a MRF is showed by theequation 1, Ω' is the same realization of Ω, except for random variable X
	www.therfcc.org /random-fields-128743.html (164 words)

	Citebase - Nonparametric regression estimation for random fields in a fixed-design
		Citebase - Nonparametric regression estimation for random fields in a fixed-design
		Nonparametric regression estimation for random fields in a fixed-design
		The description of a random fields by mean of conditional probabilities and condition of its regularity.
	www.citebase.org /cgi-bin/citations?id=oai:arXiv.org:math/0502091 (541 words)

	VanMarcke: Random Fields (Site not responding. Last check: 2007-10-08)
		"Random Fields is a book which I found both technically interesting and a pleasure to read...
		The presentation is clear and the book should be useful to almost anyone who uses random processes to solve problems in engineering or science...
		Both Chapter 2, which provides general bacground on random fields, and Chapter 3, which summarizes second-order theory, are well-written.
	www.princeton.edu /~evm/randomfields.html (237 words)

	R: Gaussian Random Fields (Site not responding. Last check: 2007-10-08)
		These functions simulate stationary spatial and spatio-temporal Gaussian random fields using turning bands/layers, circulant embedding, direct methods, and the random coin method.
		An array of the dimension of the random field is returned.
		An array of dimension d+1, where d is the dimension of the random field, is returned.
	www.maths.lth.se /help/R/.R/library/RandomFields/html/GaussRF.html (669 words)

	Random fields of multivariate test statistics (Site not responding. Last check: 2007-10-08)
		Our data are random fields of multivariate normal observations, and we fit a multivariate linear model with common design matrix at each point.
		The problem is to find the P-value of the maximum of such a random field of test statistics.
		The results are applied to a problem in shape analysis, where we look for brain damage due to non-missile trauma, and to detecting functional magnetic resonance imaging (fMRI) activation allowing for unknown latency of the hemodynamic response.
	www.math.mcgill.ca /keith/roy/roy_abstract.htm (145 words)

	Graphical Models
		Probabilistic graphical models are graphs in which nodes represent random variables, and the (lack of) arcs represent conditional independence assumptions.
		The simplest kind is importance sampling, where we draw random samples x from P(X), the (unconditional) distribution on the hidden variables, and then weight the samples by their likelihood, P(yx), where y is the evidence.
		In principle, it is straightforward to use graphical models to do Bayesian learning: the parameters, being random variables, become nodes as well, and the goal is the standard inference problem of computing posterior distributions on the (parameter) nodes.
	www.cs.ubc.ca /~murphyk/Bayes/bayes.html (6598 words)

	AN INNOVATION APPROACH TO RANDOM FIELDS (Site not responding. Last check: 2007-10-08)
		A random field is a mathematical model of evolutional fluctuating complex systems parametrized by a multi-dimensional manifold like a curve or a surface.
		As the parameter varies, the random field carries much information and hence it has complex stochastic structure.
		The authors of this book use an approach that is characteristic: namely, they first construct innovation, which is the most elemental stochastic process with a basic and simple way of dependence, and then express the given field as a function of the innovation.
	www.worldscibooks.com /mathematics/5046.html (188 words)

	Effective hydraulic conductivity in multiscale random fields with truncated power variograms
		In recent years, it became common and widely accepted to view hydraulic conductivities as random fields, and the corresponding flow and transport equations as stochastic.
		In this chapter we address this question by developing an expression for the equivalent hydraulic conductivity of a box-shaped porous block, embedded within such a multiscale hydraulic conductivity field, Using this expression we rigorously show that hydraulic conductivity varies with the observation scale in a manner conjectured earlier by S. Neuman.
		In particular, as the observation scale increases, hydraulic conductivity increases in three dimensions, remains the same in two dimensions, and decreases for one-dimensional flow.
	math.lanl.gov /Research/Publications/difederico-2000-effective.shtml (241 words)

	Inducing Features Of Random Fields - Pietra, Pietra, Lafferty (ResearchIndex)
		Abstract: We present a technique for constructing random fields from a set of training samples.
		The learning paradigm builds increasingly complex fields by allowing potential functions, or features, that are supported by increasingly large subgraphs.
		Della Pietra, V. Della Pietra, and J. Lafferty, "Inducing features of random fields," In IEEE Transactions on Pattern Analysis and Machine Intelligence, vol.
	citeseer.ist.psu.edu /dellapietra95inducing.html (656 words)

	Markov Random Fields (Site not responding. Last check: 2007-10-08)
		F is said to be a Markov random field on
		of any random field is uniquely determined by its local conditional probabilities [Besag 1974].
		Fortunately, a theoretical result about the equivalence between Markov random fields and Gibbs distribution [Hammersley and Clifford 1971 ; Besag 1974] provides a mathematically tractable means of specifying the joint probability of an MRF.
	www.vision.ee.ethz.ch /~rpaget/Markov/Chapter_1/node11.html (544 words)

	Murad S. Taqqu -- Articles
		``Random processes with long-range dependence and high variability.'' Journal of Geophysical Research, 92, D8 (1987) 9683-9696.
		``Stochastic monotonicity and Slepian-type inequalities for infinitely divisible and stable random vectors'' (with Gennady Samorodnitsky).
		``Lévy measures of infinitely divisible random vectors and Slepian inequality'' (with Gennady Samorodnitsky).
	math.bu.edu /people/murad/articles.html (3443 words)

	Properties of random electromagnetic fields (Site not responding. Last check: 2007-10-08)
		We show that these spectral modifications depend on the coherence properties of an equivalent planar source, which, in turn, relates to the statistical characteristics of the interface.
		Mujat and A. Dogariu, "Polarimetric and Spectral Changes in Random Electromagnetic Fields", Opt.
		Abstract: A theory is formulated for analyzing interference experiments with two correlated wave fields, each of an arbitrary state of coherence and is illustrated by the analysis of a typical detection scheme involving two random fields.
	random.creol.ucf.edu /home/research/field_properties.html (467 words)

	Random field (Site not responding. Last check: 2007-10-08)
		Several kinds of random fields exist, among them Markov random fields (MRF), Gibbs random fields (GRF) and Gaussian random fields.
		Teenager's random thoughts on computers, cars, politics, religion, and other topics.
		Short paper describing planar anisotropy and other structural characteristics of nonwovens, based on the random field theory and spatial autocorrelation devices.
	www.omniknow.com /common/wiki.php?in=en&term=Random_fields (489 words)

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