Random Elds Research Articles

COMPARISON OF EXPONENTIAL COVARIANCE FUNCTIONS FOR BIVARIATE GEOSTATISTICAL DATA

In the analysis of multivariate spatial random elds, it is essential to dene a covariance structure that adequately models the relationship between the variables under study. We propose a covariance structure with exponential correlation function for bivariate random elds, the SEC model. We compare the SEC model fits with the bivariate separable exponential model and the bivariate exponential model with constraints, which are particular cases of the full bivariate Matern model, presented in the literature. A simulation study assess characteristics of the proposed model. The model is tted to a weather data set from Brazil, bearing in mind the importance of analyzing climate data to predict adverse environmental conditions. Predictive measures are used to compare the models under study. The satisfactory results compared to the models considered and the simpler structure makes the SEC model an alternative for the analysis of bivariate spatial elds.

Exploring a New Class of Non-stationary Spatial Gaussian Random Fields with Varying Local Anisotropy

Gaussian random elds (GRFs) play an important part in spatial mod- elling, but can be computationally infeasible for general covariance structures. An ecient approach is to specify GRFs via stochastic partial dierential equations (SPDEs) and derive Gaussian Markov random eld (GMRF) approximations of the solutions. We consider the construction of a class of non-stationary GRFs with varying local anisotropy, where the local anisotropy is introduced by allowing the coecients in the SPDE to vary with position. This is done by using a form of diusion equation driven by Gaussian white noise with a spatially varying diu- sion matrix. This allows for the introduction of parameters that control the GRF by parametrizing the diusion matrix. These parameters and the GRF may be considered to be part of a hierarchical model and the parameters estimated in a Bayesian framework. The results show that the use of an SPDE with non-constant coecients is a promising way of creating non-stationary spatial GMRFs that allow for physical interpretability of the parameters, although there are several remaining challenges that would need to be solved before these models can be put to general practical use.

Description of Extreme Gibbs Measures for the Ising Model with Three Interactions

In this paper, we consider an Ising model with three competing interactions (nearest neighbor, next-nearest neighbor, and ternary prolonged neighbor) on the Cayley tree of order two, investigated by Ganikhodjaev et al. We study translation-invariant Gibbs measures of the Ising model with these competing interactions. Also, we investigate the set of the extreme Gibbs measures called Markov random �elds with memory 2 of the model.

Some asymptotic results of Gaussian random fields with varying mean functions and the associated processes

In this paper, we derive tail approximations of integrals of exponential functions of Gaussian random elds with varying mean functions and approximations of the associated point processes. This study is motivated naturally by multiple applications such as hypothesis testing for spatial models, study of the distribution of Bayesian marginal likelihood and Bayes factor, and nancial applications.

Conditional spectral models of extreme ocean waves

In this paper, we make an attempt to apply numerical conditional spectral models of random elds to study features of formation and development of the extreme ocean waves.

ON THE MARCINKIEWICZ ZYGMUND LAWS OF LARGE NUMBERS FOR NEGATIVELY DEPENDENT RANDOM FIELDS

Abstract. In this paper we provide extensions of the MarcinkiewiczZygmund laws of large numbers for i.i.d random variables with mul-tidimensional indices to the case of negatively dependent random elds. 1. Introduction Let Z d+ , where dis a positive integer, denote the positive integer d-dimensional lattice points. The notation m n, where m = (m 1 ;m 2 ; ;m d ) and n = (n 1 ;n 2 ; ;n d ) in Z d+ , means that m i n i for all 1 id. The following multiindex version of the Marcinkiewicz Zygmundstrong laws of large numbers was given in Gut(1978).Theorem A Let 0 <r<2, and suppose that X, fX i ;i 2Z d+ gis a eld of i.i.d random variables with partial sums S n =P in X i ;i 2Z d+ :If EjXj r (log + jXj) d 1 <1and EX= 0 when 1 r<2, then(1:1)S n jnj 1r !0 a:s:asn !1:Conversely, if (1.1) holds, then EjXj r (log + jXj) d 1 <1and EX= 0when 1 r<2. Here jnj=Q di=1 n i and n !1means min 1id n i !1, that is, all coordinates tend to in nity. Also, throughout the paper,log + x= maxf1;logxg.Next, we turn to our attention to the negative dependence. Tworandom variables Xand Y are negative quadrant dependent(NQD) ifP(Xx;Y y) P(Xx)P(Y y) for all x;y2R. A nite family

A review of some methods to estimate the tail of the distribution of the maximum of a Gaussian field

We review two methods aiming to compute the distribution of the maximum of a Gaussian process: an updated version of the Rice series and the so-called \direct method. The Rice series applies to one-parameter processes while the direct method can be used also for multi-parameter random elds.

Stochastic perturbation approach to the wavelet‐based analysis

AbstractThe wavelet‐based decomposition of random variables and fields is proposed here in the context of application of the stochastic second order perturbation technique. A general methodology is employed for the first two probabilistic moments of a linear algebraic equations system solution, which are obtained instead of a single solution projection in the deterministic case. The perturbation approach application allows determination of the closed formulas for a wavelet decomposition of random fields. Next, these formulas are tested by symbolic projection of some elementary random field. Copyright © 2004 John Wiley &amp; Sons, Ltd.

Algorithms from statistical physics for generative models of images

A general framework for dening generative models of images is Markov random elds (MRF’s), with shift-invariant (homogeneous) MRF’s being an important special case for modeling textures and generic images. Given a dataset of natural images and a set of lters from which lter histogram statistics are obtained, a shiftinvariant MRF can be dened (as in Zhu [12]) as a distribution of images whose mean lter histogram values match the empirical values obtained from the data set. Certain parameters in the MRF model, called potentials, must be determined in order for the model to match the empirical statistics. Standard methods for calculating the potentials are computationally very demanding, such as Generalized Iterative Scaling (GIS), an iterative procedure that converges to the correct potential values. We dene a fast approximation, called BKGIS, which uses the Bethe-Kikuchi approximation from statistical physics to speed up the GIS procedure. Results are demonstrated on a model using two lters, and we show synthetic images that have been sampled from the model. Finally, we show a connection between GIS and our previous work on the g-factor.

Influence of wave noise on frequencies and amplitudes of the solar $\vec{p}$-modes

The influence of space- and time-dependent random mass density eld, associated with granules, on frequencies and amplitudes of the solar p-modes is examined in the limit of weak random elds and small amplitude oscillations. The p-modes are approximated by the sound waves which propagate in the gravity-free medium. Using a perturbative method, we derive a dispersion relation which is solved for the case of wave noise for which the spectrum E(k;!) E(k)(! crk), where is the Dirac's delta-function and cr is the random phase speed. We nd that at cr = !=k a resonance occurs at which the cyclic frequency ! tends to innity. For values of cr which are close to the resonance point, the frequency shift may be both negative or positive and the imaginary part of the frequency attains the negative (positive) sign for cr != k).

Estimation problems for random fields from noisy data

The problem of the least mean-square linear estimation of the functional ∫ Sn a(t;x) (t;x)m n(dx) dt of the unknown values of a random eld (t;x); t ∈ R 1 , x ∈ Sn that is time homogeneous and isotropic on a sphereSn from observations of the eld (t;x)+ (t;x) fort < 0, x ∈ Sn, where (t;x) is a random eld that is time homogeneous and isotropic on a sphere Sn uncorrelated with (t;x), is considered. Formulas are obtained for computing the value of the mean-square error and the spectral characteristic of the optimal linear estimate of the functional A. The least favorable spectral densities and the minimax (robust) spectral characteristics of the optimal estimates of the functionals are determined for some classes of random elds.