Vector Random Fields Research Articles

Stationary and Isotropic Vector Random Fields on Spheres

This paper presents the characterization of the covariance matrix function of a Gaussian or second-order elliptically contoured vector random field on the sphere which is stationary, isotropic, and mean square continuous. This characterization involves an infinite sum of the products of positive definite matrices and Gegenbauer’s polynomials, and may not be available for other non-Gaussian vector random fields on spheres such as a χ2 or log-Gaussian vector random field. We also offer two simple but efficient constructing approaches, and derive some parametric covariance matrix structures on spheres.

Reducing the number of orthogonal factors in linear coregionalization modeling

The simulation of vector random fields whose spatial correlation structure is represented by a linear coregionalization model can be performed by decomposing the vector components into spatially orthogonal factors and by simulating each factor separately. However, when the number of basic nested structures is large, so is the number of factors, making simulation computationally demanding.This paper proposes a methodology to construct linear coregionalization models with as many nested structures as desired, together with as few orthogonal factors as possible. The construction rests on the decomposition of the model coregionalization matrices into pairwise commuting matrices, followed by a factorization by principal component analysis. The proposed approach is illustrated through a case study in mineral resources evaluation and compared to the traditional fitting procedure, obtaining an equally good fit of the direct and cross variograms but with significantly less factors.

Shock waves in random viscoelastic media

Determining the effects of material spatial randomness on the evolution of shocks is the objective of this study. Considering a linear viscoelastic-type material, a very general class of random media is modeled by a three-component random vector field: the instantaneous relaxation function (E), its derivative (E � ), and the mass density (ρ). The reason for considering the randomness of the said material coefficients is the fact that a wavefront's length scale (thickness) is most likely smaller than the Representative Volume Element, thus contradicting a "separation of scales" condition tacitly assumed in deterministic continuum mechanics, even in the context of wavefront analyses. In effect, the wavefront is an object much more appropriately analyzed as a Statistical Volume Element and therefore to be treated via a stochastic dynamical system rather than a deterministic one. Various cases of the random vector field (E, E � ,ρ )x are examined: white versus correlated noises as well as the independence versus coupling of these random fields to the shock amplitude.

Special Issue on Spatial Multivariate Methods

This special issue is dedicated to spatial multivariate methods and addresses the emergence of the latest practical and theoretical aspects on this topic. The papers herein present a wide spectrum of recent advances in methods and applications in diverse areas. It grew out of an invited paper session of the International Statistical Institute meeting that was held in Durban, South Africa, in 2009. This special issue, comprised of seven papers, a teaching aid and short note, aims to broaden interest and offer insights into current trends and future directions of spatial multivariate methods. The issue begins with Bailey and Krzanowski’s overview of the existing approaches for the analysis of geostatistical multivariate data. These approaches are divided into classes of factor models or spatial random field models; and their differences are discussed. Their paper introduces and discusses the minimum/maximum autocorrelation factors (MAF), which is a common theme throughout this special issue. The next paper by Christensen and Sain considers an alternate view of multimodel ensembles for use with the North American Regional Climate Change Assessment Program. Numerical models of atmosphere–ocean circulation are widely used to understand past climate and to project future climate change. The authors propose a spatially-correlated latent variable model in order to facilitate the exploration of similarities between regional climate models and what factors best predict observed locations of model convergence. In their paper, Mueller and Ferreira contrast the MAF method with the Gauss iterations (U-WEDGE) method and present a comparison with a full cosimulation of the attributes. Du and Ma propose an efficient approach in constructing variogram matrix functions in order to measure the dependence of a vector random field with secondorder increments. This paper further demonstrates that various dependence structures

Variogram Matrix Functions for Vector Random Fields with Second-Order Increments

The variogram matrix function is an important measure for the dependence of a vector random field with second-order increments, and is a useful tool for linear predication or cokriging. This paper proposes an efficient approach to construct variogram matrix functions, based on three ingredients: a univariate variogram, a conditionally negative definite matrix, and a Bernstein function, and derives three classes of variogram matrix functions for vector elliptically contoured random fields. Moreover, various dependence structures among components can be derived through appropriate mixture procedures demonstrated in this paper. We also obtain covariance matrix functions for second-order vector random fields through the Schoenberg–Levy kernels.

Spherically Invariant Vector Random Fields in Space and Time

This paper is concerned with spherically invariant or elliptically contoured vector random fields in space and/or time, which are formulated as scale mixtures of vector Gaussian random fields. While a spherically invariant vector random field may or may not have second-order moments, a spherically invariant second-order vector random field is determined by its mean and covariance matrix functions, just like the Gaussian one. This paper explores basic properties of spherically invariant second-order vector random fields, and proposes an efficient approach to develop covariance matrix functions for such vector random fields.

Enhanced coregionalization analysis for simulating vector Gaussian random fields

This paper deals with the simulation of a stationary vector Gaussian random field whose spatial correlation structure is given by a linear model of coregionalization. Traditionally, simulation is performed by decomposing the vector random field into a set of independent vector random fields with coregionalization models that contain a single nested structure, and a factorization of these fields into principal components. A variation of this approach is proposed, by considering the minimum/maximum autocorrelation factors associated with groups of two nested structures. This variation reduces the total number of independent factors by one-half, thus considerably decreases memory requirements and CPU time for simulation, without any loss of accuracy for reproducing the linear model of coregionalization, regardless of how many nested structures are contained in this model. The proposed approach is implemented in a set of computer programs and illustrated through a synthetic example and a case study in mineral resources evaluation.

Hanbury Brown–Twiss effect with electromagnetic waves

The classic Hanbury Brown-Twiss experiment is analyzed in the space-frequency domain by taking into account the vectorial nature of the radiation. We show that as in scalar theory, the degree of electromagnetic coherence fully characterizes the fluctuations of the photoelectron currents when a random vector field with Gaussian statistics is incident onto the detectors. Interpretation of this result in terms of the modulations of optical intensity and polarization state in two-beam interference is discussed. We demonstrate that the degree of cross-polarization may generally diverge. We also evaluate the effects of the state of polarization on the correlations of intensity fluctuations in various circumstances.

Cokriging random fields with means related by known linear combinations

Traditional approaches to predict a second-order stationary vector random field include simple and ordinary cokriging, depending on whether or not the mean values of the vector components are assumed to be known. This paper explores a variant of cokriging, in which the mean values of the vector components are related by linear combinations with known coefficients. Equations for the cokriging predictor and for the variance–covariance matrix of prediction errors are presented. A set of computer programs is provided and illustrated with applications to mineral resources evaluation, in which the proposed cokriging variant compares favorably with traditional approaches.

Covariance Matrices for Second-Order Vector Random Fields in Space and Time

This paper deals with vector (or multivariate) random fields in space and/or time with second-order moments, for which a framework is needed for specifying not only the properties of each component but also the possible cross relationships among the components. We derive basic properties of the covariance matrix function of the vector random field and propose three approaches to construct covariance matrix functions for Gaussian or non-Gaussian random fields. The first approach is to take derivatives of a univariate covariance function, the second one is to work on the univariate random field whose index domain is in a higher dimension and the third one is based on the scale mixture of separable spatio-temporal covariance matrix functions. To illustrate these methods, many parametric or semiparametric examples are formulated.

Vector Random Fields with Second-Order Moments or Second-Order Increments

This article is concerned with vector (multivariate, or multidimensional) random fields with second-order moments or second-order increments. Two crucial questions for such a random field are what kind of the square matrix function can be employed as its covariance matrix or variogram matrix, and, in particular, what type of the functions can be employed as its cross covariances or cross variograms. We attempt to explore the relationships between the direct covariance and the cross covariance in a covariance matrix and the relationships between the direct variogram and the cross variogram in a variogram matrix. Necessary and sufficient conditions are obtained for a given square matrix function to be the covariance matrix or variogram matrix of a vector Gaussian or elliptically contoured random field, and some parametric or nonparametric examples are given for stationary and nonstationary cases in a temporal, spatial, or spatio-temporal domain.

A random transport-diffusion equation

In this paper we study the integral curve in a random vector field perturbed by white noise. It is related to a stochastic transport-diffusion equation. Under some conditions on the covariance function of the vector field, the solution of this stochastic partial differential equation is proved to have moments. The exact p-th moment is represented through integrals with respect to Brownian motions. The basic tool is Girsanov formula.

A Class of Variogram Matrices for Vector Random Fields in Space and/or Time

This paper studies vector (multivariate, multiple, or multidimensional) random fields in space and/or time with second-order increments, for which the variogram matrix is an important tool to measure the dependence within each component and between each pair of distinct components. We introduce an efficient approach to construct Gaussian or non-Gaussian vector random fields from the univariate random field with higher dimensional index domain, and particularly to generate a class of variogram matrices.

Nonlinear filtering for observations on a random vector field along a random path. Application to atmospheric turbulent velocities

To filter perturbed local measurements on a random medium, a dynamic model jointly with an observation transfer equation are needed. Some media given by PDE could have a local probabilistic representation by a Lagrangian stochastic process with mean-field interactions. In this case, we define the acquisition process of locally homogeneous medium along a random path by a Lagrangian Markov process conditioned to be in a domain following the path and conditioned to the observations. The nonlinear filtering for the mobile signal is therefore those of an acquisition process contaminated by random errors. This will provide a Feynman-Kac distribution flow for the conditional laws and an N particle approximation with a O( 1 √ N ) asymptotic convergence. An application to nonlinear filtering for 3D atmospheric turbulent fluids will be described.

Fractional Brownian Vector Fields

This work puts forward an extended definition of vector fractional Brownian motion (fBm) using a distribution theoretic formulation in the spirit of Gel′fand and Vilenkin's stochastic analysis. We introduce random vector fields that share the statistical invariances of standard vector fBm (self-similarity and rotation invariance) but which, in contrast, have dependent vector components in the general case. These random vector fields result from the transformation of white noise by a special operator whose invariance properties the random field inherits. The said operator combines an inverse fractional Laplacian with a Helmholtz-like decomposition and weighted recombination. Classical fBm's can be obtained by balancing the weights of the Helmholtz components. The introduced random fields exhibit several important properties that are discussed in this paper. In addition, the proposed scheme yields a natural extension of the definition to Hurst exponents greater than one.

An Asbestos Detection Method from Microscope Images Using Support Vector Random Field of Local Color Features

In this paper, an asbestos detection method from microscope images is proposed. The asbestos particles have different colors in two specific angles of the polarizing plate. Therefore, human examiners use the color information to detect asbestos. To detect the asbestos by computer, we develop the detector based on Support Vector Machine (SVM) of local color features. However, when it is applied to each pixel independently, there are many false positives and negatives because it does not use the relation with neighboring pixels. To take into consideration of the relation with neighboring pixels, Conditional Random Field (CRF) with SVM outputs is used. We confirm that the accuracy of asbestos detection is improved by using the relation with neighboring pixels.

Hydrodynamic Turbulence and Intermittent Random Fields

In this article, we construct two families of multifractal random vector fields with non-symmetrical increments. We discuss the use of such families to model the velocity field of turbulent flows.

On superdiffusive behavior of a passive tracer in a random flow

In this note we consider a passive tracer model describing particle dispersion in a turbulent flow. The trajectory of the particle is given by the solution of an ordinary differential equation x ̇ ( t ) = F ( x ( t ) ) , x ( 0 ) = x 0 , where F ( x ) is a divergence-free, random vector field that is spatially homogeneous and isotropic. We show that trajectories of the tracer display superdiffusive behavior when the random velocity F ( x ) decorrelates, at large distances, but does it not rapidly but rather at some moderate rate. The main tools used in the proofs are variational principles and Tauberian-type theorems.

Block Simulation of Multiple Correlated Variables

Numerical representations of multivariate natural phenomena, including characteristics of mineral deposits, petroleum reservoirs and geo-environmental attributes, need to consider and reproduce the spatial relationships between correlated attributes of interest. There are, however, only a few methods that can practically jointly simulate large size multivariate fields. This paper presents a method for the conditional simulation of a non-Gaussian vector random field directly on block support. The method is derived from the group sequential simulation paradigm and the direct block simulation algorithm which leads to the efficient joint simulation of large multivariate datasets jointly and directly on the block support. This method is a multistage process. First, a vector random function is orthogonalized with minimum/maximum autocorrelation factors (MAF). Blocks are then simulated by performing LU simulation on their discretized points, which are later back-rotated and averaged to yield the block value. The internal points are then discarded and only the block value is stored in memory to be used for further conditioning through a joint LU, resulting in the reduction of memory requirements. The method is termed direct block simulation with MAF or DBMAFSIM. A proof of the concept using an exhaustive data set demonstrates the intricacies and the performance of the proposed approach.

Dynamic interaction models of economic equilibrium

The paper develops a stochastic dynamic model of economic equilibrium with locally interacting agents. The main focus of the study is on the modeling of market interactions – those arising in connection with commodity exchange and regulated by price mechanisms. The mathematical framework is a control theory for random vector fields on directed graphs. The graphs involved serve to describe the spatio-temporal structure of commodity flows in the system. The main results are concerned with the existence, uniqueness and stability of stochastic dynamic equilibria.