Spatial Random Process Research Articles

A comparison of spatial prediction techniques for an exploratory analysis of human cortical motor representations.

We compare the use of two-dimensional Laplacian smoothing splines and median polish kriging for an exploratory analysis of spatial data sets. Splines were developed originally for modelling a process with determinstic mean structure, while median polish kriging was developed for predicting a spatial random process. We review the fundamentals of both methods, including smoothing parameter selection for splines and variogram estimation and solving the kriging equations for median polish kriging. We demonstrate application of the methods on maps of muscle representations on the human motor cortex (cortical maps). After applying both methods to 20 cortical maps (left and right hemispheres on each of 10 subjects), we compare the fitted surfaces on the basis of predictive ability, aesthetic appeal, and ease of computation.

Torsional response of building structures to spatial random ground excitations

This paper applies the random vibration method to analyse the coupled lateral and torsional responses of symmetric and asymmetric building structures subjected to spatially varying horizontal multiple ground excitations. The responses of a single-storey building model with different eccentricities and varying numbers of supports are analysed. Ground excitations at the multiple structural supports are assumed to be zero mean spatial random processes. Their intensities and spatial variations are modelled by a spectral density function and an empirical coherency function. The standard deviations of various response quantities such as base shear, torque and relative column displacements are calculated. Numerical results are presented in terms of dimensionless parameters for their general application. Discussions on the torsional responses induced by both spatial ground excitations and building eccentricities are made. The adequacy of the torsional provisions of current seismic codes is also examined.

Stochastic models of local structures versus global atomic configurations

Representative configurations of the atoms in a condensed system, produced by optimized fitting to given experimental pair distributions, for example by the reverse Monte Carlo method, contain largely redundant information. They can be replaced, successfully, by representative configurations of a set of atoms within a relatively small neighborhood (the local structure) obeying some spatial random process, such as structural diffusion, specified by one or more parameters. With increasing size of the local structure, accompanied by decreasing structural diffusion, the limit of a global representative configuration of the atoms is approached. This approach is illustrated for two simple liquids, where it is shown that a local structure of low complexity (n ≤ 4) already fits experimental data of the pair distribution to a high degree of accuracy.

Characterization of Preferential Flow Paths in Compacted Sand-Clay Mixtures

Laboratory soil-column experiments were conducted to study the distribution of preferential flow paths resulting from removal of colloidal-size clay particles. Flow paths were assumed to result when the fluid energy was able to overcome the cementitious bonding of particles along the pore walls. These experiments specifically studied the influence of clay (kaolinite) percentage in sand-clay mixtures and the effect of hydraulic gradients on pore evolution. Analysis of the effluent during the experiments indicated that clay particles were removed from the soil column, accompanied by an increase in porosity and hydraulic conductivity. Dye experiments were conducted on the same columns to stain the pathways where colloidal particle removal occurred. It was observed that pore distribution was fairly uniform in some cases, while other cases showed distinct preferential flow-path formation. Using concepts of image analysis, the dye pattern was converted to a binary file and analyzed as a spatial random process to characterize the spatial distribution of flow paths. The expected spatial density (λ) and the correlation length were chosen as the key parameters to quantify the random dye patterns and to describe the nature of flow-path evolution. A physically based model was used to identify a dimensionless parameter, G , which expresses the ratio of detachment and deposition forces at any space-time location. Soil detachability was identified as the other key parameter. This expresses the ease with which colloids can be removed from the pore walls. It was observed that λ exhibited an identifiable trend with G , while I was predominantly dependent on detachability of the mixture.

Conditional Simulation of Spatially Correlated Earthquake Ground Motion

Methodology is presented for simulating properly correlated earthquake ground motions at an arbitrary set of closely spaced points, compatible with known or prescribed motions at other locations. The input consists of the spectral density function and frequency-dependent spatial correlation function in several nonoverlapping ground-motion segments. Linear-prediction estimators are used to generate a set of statistically independent, frequency-specific, spatial random processes, based on which ground motions are composed by means of a fast-Fourier-transform algorithm. The method’s advantage over existing linear-estimation techniques, known as Kriging, is that it correctly reproduces the specified space-time correlation structure of the ground motion. Examples are given to illustrate the features of the simulation methodology; in particular, response spectra and dynamic response ratios are compared for recorded and simulated motions at the site of the SMART 1 strong-motion accelerograph array.

Diffusive motion and structural correlations in condensed systems

A model of condensed disordered systems, based on theories of random spatial and temporal processes for local atomic structures, is used to derive analytic expressions for the contribution of the diffusive atomic motion to the van Hove time correlation function, G D( r, t), and its spectral representations. The spatial random process determines the static atomic correlations in space whereas the temporal diffusion process contributes to the central (Rayleigh) peak of the dynamic scattering function, S D( k, ω). The relaxation time of the intermediate scattering function, F D( k, t), exhibits the known phenomenon of slowing down in the neighborhood of the first peak of S( k), but in the limit of small k it remains finite, contrary to the infinite slowing down of heat dissipation predicted by linearized hydrodynamics.

Effect of Spatial Distribution of Material Properties on the Experimental Estimation

The purpose of the present study is to see if it is possible to estimate very short crack growth behavior using the data obtained from measurements on rather large scale fatigue crack growth. In this study, the material property was treated as a spatial random process along the path of the fatigue crack. The relation between the measured autocorrelation functions and averaging length was discussed for several correlation lengths. Applying the previously proposed method, the experimental data from stress-intensity-factor-controlled fatigue tests were analysed for different averaging lengths to determine the autocorrelation functions and probability distributions of parameters in Paris law.The main results obtained are : (1) The autocorrelation functions of Z (x) are almost same for different averaging lengths. (2) The variance decreases with the increasing averaging length δ. When the spatial correlation length is very small, the variance decreases significantly with the averaging length. (3) The autocorrelation functions of the material resistance Z (x) obtained from experiments are almost same for all δ, except for R (0). (4) The probability distribution functions of Z (x) show a slight dependence on the averaging length.As far as the present experiment is concerned, it can be concluded from these observations that a very small extension of a short crack can be predicted using the data obtained from measurements on rather large scale fatigue crack growth.

Diffusive motion and structural correlations in condensed disordered systems

A model of condensed disordered systems, based on theories of random spatial and temporal processes for local atomic structures, is used to derive analytic expressions for the contributions of the diffusive atomic motion to the van Hove time correlation function GD(r,t) and its spectral representations. The spatial random process determines the static atomic correlations in space whereas the temporal diffusion process contributes to the central (Rayleigh) peak of the dynamic scattering function SD(k,w). The relaxation time of the intermediate scattering function FD(k,t) exhibits the known phenomenon of slowing down in the neighbourhood of the first peak of S(k), but in the limit of small k it remains finite, contrary to the infinite slowing down of heat dissipation predicted by linearized hydrodynamics.

Statistical approach to X-ray CT imaging and its applications in image analysis. I. Statistical analysis of X-ray CT imaging

A statistical description of X-ray CT (computerized tomography) imaging, from the projection data to the reconstructed image, is presented. The Gaussianity of the pixel image generated by the convolution (image reconstruction) algorithm is justified. The conditions for two pixel images to be statistically independent (for a given probability) and the conditions for a group of pixel images to be a spatial stationary random process and ergodic in mean and autocorrelations are derived. These properties provide the basis for establishing the stochastic image model and conducting the statistical image analysis of X-ray CT images.

Atomic pair correlations in Lennard-Jones fluids: Analysis in terms of structural diffusion models

We present an analysis of pair correlation data of a Lennard-Jones fluid (argon), covering a wide range of thermodynamic states, in terms of structural diffusion (SD) models. These are based on the notion of a locally observed ordered structure and of spatial disorder as an outcome of a random spatial process. A general recipe is developed for analysing structural data by such models. First, we analyse fluid argon in terms of the simple SD model (SDM1). Nine lattices are chosen for the description of the local order in the fluid. It is found that close packed lattices are better suited for describing the high density fluid structure while open lattices are better suited for the description of the low density fluid. The SDM parameters obtained by constrained optimization describe the dependence of the pair correlation on density and temperature, and constitute a structural map of the fluid in the (ρ, T) plane. Two possible ways are presented for improving the fitting of g(r) data: (1) distortion of a chosen symmetric lattice (FCC in our case) by varying the basic cell angles and optimizing within the SDM1. It is found that the higher symmetry lattices, BCC and FCC, give the best fitting to the experimental g(r); (2) use of an extended version of the model (SDM2) that enables us to obtain a refined fitting through a built-in contraction function, f(r). A simple choice for f(r) in the model expression of g(r), applied to the four most dense lattices, gives an improved fit for high densities. The best results are obtained with the hcp lattice.

On the flow of a thin liquid film over a rotating disk

The depletion of thin liquid films due to the combined effect of centrifugation, surface roughness, and air shear has recently been studied. While surface roughness of a rotating solid disk can be represented by deterministic curves, it has been argued that spatial random processes provide a more realistic description. Chiefly because of surface roughness, there is an asymptotic limit of retention of a thin film flowing on the rotating disk. The aim of this article is twofold. First, the effect of disjoining pressure on the retention of a thin film is investigated. It is found that incorporating disjoining pressure term has small but still appreciable effect on the asymptotic limits of lubricant retention. For a partially wetting lubricant, the two components of the disjoining pressure function tend to have opposite influence on the lubricant retention; one enhances the retention, while the other diminishes it. Second, the robustness of stochastic description of surface roughness is examined. For a given mean and variance, it is noted that different probability distributions of the surface fluctuations lead, in the absence of air shear, to significantly different asymptotic limits of thin-film retention. If air shear is incorporated, this sensitivity is substantially attenuated.

Effect of Spatial Distribution of Material Properties on the Experimental Estimation

The purpose of the present study is to see if it is possible to estimate very short crack growth behavior using the data obtained from measurements on rather large scale fatigue crack growth. In this study, the material property was treated as a spatial random process along the path of the fatigue crack. The relation between the measured autocorrelation functions and averaging length was discussed for several correlation lengths. Applying the previously proposed method, the experimental data from stress-intensity-factor-controlled fatigue tests were analysed for different averaging lengths to determine the autocorrelation functions and probability distributions of parameters in Paris law.The main results obtained are : (1) The autocorrelation functions of Z (x) are almost same for different averaging lengths. (2) The variance decreases with the increasing averaging length δ. When the spatial correlation length is very small, the variance decreases significantly with the averaging length. (3) The autocorrelation functions of the material resistance Z (x) obtained from experiments are almost same for all δ, except for R (0). (4) The probability distribution functions of Z (x) show a slight dependence on the averaging length.As far as the present experiment is concerned, it can be concluded from these observations that a very small extension of a short crack can be predicted using the data obtained from measurements on rather large scale fatigue crack growth.

Stochastic optimal control of non-stationary response of a single-degree-of-freedom vehicle model

An active suspension system to control the non-stationary response of a single-degree-of-freedom (sdf) vehicle model with variable velocity traverse over a rough road is investigated. The suspension is optimized with respect to ride comfort and road holding, using stochastic optimal control theory. The ground excitation is modelled as a spatial homogeneous random process, being the output of a linear shaping filter to white noise. The effect of the rolling contact of the tyre is considered by an additional filter in cascade. The non-stationary response with active suspension is compared with that of a passive system.

Rapid detection of spatial pattern by Fourier analysis

Fast Fourier transform analysis can be used to accurately and rapidly detect pattern in large two-dimensional arrangements of points, such as the locations of cells in culture or plants in a unit square. We present here a sample study of pattern in spatial random point processes. This is evidently the first time that Fourier transform-based cross-correlation techniques have been applied to the analysis of point processes of biological origin. Radial profiles of the power spectra and autocorrelation estimates revealed a nearly constant interpore distance of 0.49 ± 0.04 mm in the locations of eccrine gland pores on the surface of human skin. Additionally, gland-free areas may exist near hair follicles.

On sampling nonstationary spatial autocorrelated data

Sampling dependent random variables constituting a nonstationary spatial two-dimensional random process due to the presence of drift is a difficult problem consisting of determining the optimum sampling design and sampling density or number of samples, in order for one to attain a desirable precision as it is expressed by the Mean Square Error (MSE). The complexity of the problem prevents the development of a closed form solution expressing the number of samples needed to attain a given precision. Thus a user friendly FORTRAN program solving this problem for the situation of spherical semivariogram, the equilateral, square and hexagon designs, and no drift, linear, and quadratic, is given.

Interacting Particle Systems and Random Media: An Overview

Summary We have learned much about spatial random processes during the last 15 years. Physical science is a rich source of interesting questions, some of which have motivated probabilists to develop related mathematical theory. We describe something of two such theories here, concerning interacting particle systems and random media. In the first category, we concentrate on the stochastic Ising model and the exclusion process, and in the second on the percolation process and the question of flows through random media. We attempt to describe the general flavours of these topics, with brief accounts of recent developments and important open problems.

Quasicrystalline structural diffusion in amorphous systems

The representation of aperiodic, globally ordered, structures by means of hyperlattices is utilized in a general theory of random spatial processes to obtain an extended structural diffusion model of disordered systems. Two characteristic parameters of the model, determining the spatial rate of decay of coherence between local structures, lead to a distinction between two extreme types of disordered systems: (1) normal liquids and glasses and (2) disordered quasicrystals whose bond orientations and lengths are preserved over macroscopic distances. A continuous range of disordered phases exists between the two extremes. With a proper choice of the local structure, the model offers a concise description of pair distributions and structure factors of condensed disordered phases.

Comment on the use of “reciprocal space expansion” in the analysis of structure factors of liquids

A formulation of the local lattice model of liquid structure, due to Prins, was applied recently to the interpretation of experimental scattering data for liquids. However the theoretical expression obtained for S( k) has a wrong asymptotic behavior in the small- k range. A different formulation based on a general theory of spatial random processes produces an expression for S( k) which has the correct asymptotic behavior. The possible extensions and modifications of the model within this theory are pointed out.

Use of indicator variograms for an enhanced spatial analysis

Flat variograms often are interpreted as representing a lack of spatial autocorrelation. Recent research in earthquake engineering shows that nearby field noise can substantially mask a prominent spatial autocorrelation and result in what appears to be a purely random spatial process. A careful selection of threshold in assigning an indicator function can yield an indicator variogram which reveals underlying spatial autocorrelation. Although this application involves use of seismic data, the results are relevant to geostatistical applications in general.

Spatial random processes and local plane wave modes in vibrational states of an amorphous solid

A quasicrystalline model of the vibrational motion of an amorphous solid is considered. According to it the locally observed motion of atoms in a vibrational mode is assumed to have a plane wave character specified by a certain wavevector K=Klambda +k, where Klambda is characteristic of the vibrational state and k is a random spatial modulation of Klambda varying within the first Brillouin zone of a suitably chosen reciprocal lattice. The random function k(r) is assumed to be a spatial random process of a Markoffian type, leading to a spatial Fokker-Planck-type equation for the conditional displacement amplitude correlation function. The correlation function obtained exhibits a spatial decay of the form exp(-c Gamma r3) where Gamma is a reciprocal coherence volume characteristic of the solid and c is a bounded slowly varying factor. Assuming the k process to be independent of structural correlations leads to a dispersion relation omega lambda = omega (Klambda ) which differs from the 'quasicrystalline approximation' result by taking into account the spatial decay of plane wave modes.