Spatial Stochastic Process Research Articles

Geostatistical assessment of HadCM3 simulations via NCEP reanalyses over Europe

AbstractThis article attempts to quantify the spatial uncertainties associated with extreme temperature's response, by assessing extremes derived from climate model data. This is undertaken by a comparison of the spatial pattern of a long‐term time‐series aggregation (1960/61–1989/90) for extreme temperatures simulated by a particular GCM (HadCM3) to that of NCEP Reanalyses, which are considered as ‘truth’, over the MICE (Modelling the Impact of Climate Extremes—EU Project) spatial domain. Since evaluation of models is crucial to assessing future scenarios, the aim of this study is to investigate whether the extreme values predicted by the HadCM3 climate model can simulate those produced by NCEP Reanalyses, assuming that the extremes of both models are realisations of the same spatial stochastic process. To get more useful information about the uncertainties surrounding spatial climate projection, one also has to analyse the pattern of temperature extremes in terms of their anomalies. A common technical issue in the assessment of numerical spatial models is based on the geostatistical interpolation by kriging with the residual analysis. This methodology is very important and useful for guiding an evolutionary statistical model‐building process. This study leads to the conclusion that the HadCM3 Simulations do not realistically reproduce the NCEP Reanalyses, despite the fact that the climatology of extremes has demonstrated very similar spatial patterns. It is likely therefore that such instability may persist in the future. Copyright © 2004 Royal Meteorological Society

Statistical Method of Estimation of ESE in CBP

SYNOPTIC ABSTRACTA linear relationship between the expected squared error (ESE) in reconstruction of images by CBP method, and the covariance kernel of the spatial stochastic process of object function (images) has been established, where the covariance kernel is estimated by sample observations. And a method to estimate the error in reconstruction in practice using the linear relationship is given. When the object covariance kernel is not known an estimate of the covariance kernel is obtained from reconstructed images, and it is used to predict the error. The test cases have been simulated as random images from certain classes of spatial stochastic processes. It is empirically observed that the predicted error in both the cases of known and unknown (estimated) covariance kernel is within reasonable limits of the actual error.

On smoothness properties of spatial processes

For inferential analysis of spatial data, probability modelling in the form of a spatial stochastic process is often adopted. In the univariate case, a realization of the process is a surface over the region of interest. The specification of the process has implications for the smoothness of process realizations and the existence of directional derivatives. In the context of stationary processes, the work of Kent (Ann. Probab. 17 (1989) 1432) pursues the notion of a.s. continuity while the work of Stein (Interpolation of Spatial Data; Some Theory for Kriging, Springer, New York, 1999) follows the path of mean square continuity (and, more generally, mean square differentiability). Our contribution is to clarify and extend these ideas in various ways. Our presentation is self-contained and not at a deep mathematical level. It will be of primary value to the spatial modeller seeking greater insight into these smoothness issues.

Nonparametric variogram and covariogram estimation with Fourier–Bessel matrices

The nonparametric estimation of variograms and covariograms for isotropic stationary spatial stochastic processes is considered. It is shown that Fourier–Bessel matrices play an important role in this context because they provide an orthogonal discretization of the spectral representation of positive definite functions. Their properties are investigated and an example from a simulated two-dimensional spatial process is provided. It is shown that this approach provides a smooth and positive definite nonparametric estimator in the continuum, whereas previous methods typically suffer from spurious oscillations. A practical example from Astronomy is used for illustration.

Eigenstructures of Spatial Design Matrices

In estimating the variogram of a spatial stochastic process, we use a spatial design matrix. This matrix is the key to Matheron's variogram estimator. We show how the structure of the matrix for any dimension is based on the one-dimensional spatial design matrix, and we compute explicit eigenvalues and eigenvectors for all dimensions. This design matrix involves Kronecker products of second order finite difference matrices, with cosine eigenvectors and eigenvalues. Using the eigenvalues of the spatial design matrix, the statistics of Matheron's variogram estimator are determined. Finally, a small simulation study is performed.

Optimizing two- and three-stage designs for spatial inventories of natural resources by simulated annealing

This paper reviews design-based estimators for two- and three-stage sampling designs to estimate the mean of finite populations. This theory is then extended to spatial populations with continuous, infinite populations of sampling units at the latter stages. We then assume that the spatial pattern is the result of a spatial stochastic process, so the sampling variance of the estimators can be predicted from the variogram. A realistic cost function is then developed, based on several factors including laboratory analysis, time of fieldwork, and numbers of samples. Simulated annealing is used to find designs with minimum sampling variance for a fixed budget. The theory is illustrated with a real-world problem dealing with the volume of contaminated bed sediments in a network of watercourses. Primary sampling units are watercourses, secondary units are transects perpendicular to the axis of the watercourse, and tertiary units are points. Optimal designs had one point per transect, from one to three transects per watercourse, and the number of watercourses varied depending on the budget. However, if laboratory costs are reduced by grouping all samples within a watercourse into one composite sample, it appeared to be efficient to sample more transects within a watercourse.

Effect of specimen thickness on the statistical properties of fatigue crack growth resistance in BS4360 steel

In this paper the effect of specimen thickness on fatigue crack growth with the spatial distribution of material properties is presented. Basically, the material resistance to fatigue crack growth is treated as a spatial stochastic process, which varies randomly on the crack surface. The theoretical autocorrelation functions of fatigue crack growth resistance with specimen thickness are discussed for several correlation lengths. Constant ΔK fatigue crack growth tests were also performed on CT type specimens with three different thicknesses of BS 4360 steel. Applying the proposed stochastic model and statistical analysis procedure, the experimental data were analyzed for different specimen thicknesses for determining the autocorrelation functions and probability distributions of the fatigue crack growth resistance.

Book Reviews

Australian & New Zealand Journal of StatisticsVolume 42, Issue 1 p. 119-126 Book Reviews First published: 18 December 2002 https://doi.org/10.1111/1467-842X.00112AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat Abstract Books reviewed: R.R. Peck, L.D. Haugh and A. Goodman;Statistical Case Studies. A Collaboration between Academe and Industry Christian Gouriéroux;ARCH Models and Financial Applications E.L. Lehmann;Elements of Large-Sample Theory R.B. Schinazi;Classical and Spatial Stochastic Processes Volume42, Issue1March 2000Pages 119-126 RelatedInformation

Markov Property and Ergodicity of the Nonlinear Filter

In this paper we first prove, under quite general conditions, that the nonlinear filter and the pair (signal, filter) are Feller--Markov processes. The state space of the signal is allowed to be nonlocally compact and the observation function h can be unbounded. Our proofs, in contrast to those of Kunita [ J. Multivariate Anal., 1 (1971), pp. 365--393; Spatial Stochastic Processes, Birkhäuser, 1991, pp. 233--256] and Stettner [ Stochastic Differential Equations, Springer-Verlag, 1989, pp. 279--292], do not depend upon the uniqueness of the solutions to the filtering equations. We then obtain conditions for existence and uniqueness of invariant measures for the nonlinear filter and the pair process. These results extend those of Kunita and Stettner, which hold for locally compact state space and bounded h, to our general framework. Finally we show that the recent results of Ocone and Pardoux [ SIAM J. Control Optim., 34 (1996), pp. 226--243] on asymptotic stability of the nonlinear filter, which use the Kunita--Stettner setup, hold for the general situation considered in this paper.

Asymptotic transport parameters in a heterogeneous porous medium: Comparison of two ensemble-averaging procedures

We analyze the movement of a solute cloud in a saturated aquifer, resulting from a point-like instantaneous solute injection. Physical heterogeneities of the medium due to spatial variations of the hydraulic conductivity, as well as the chemical heterogeneities due to variations in the linear adsorption coefficient and the degradation rate, are modeled as spatial stochastic processes with exponential autocorrelation functions. Furthermore, cross-correlations between the chemical properties and the conductivity are taken into account. For large transport times, the movement of the solute cloud is characterized by its center-of-mass velocity, by the macroscopic dispersion constant, and the macroscopic degradation rate. These quantities are evaluated using perturbation theory and two different averaging procedures. The first procedure derives the large-scale properties from the central moments of the concentration distribution in a given aquifer realization, and averages over the ensemble afterwards. The second method which is mathematically less demanding obtains large scale transport coefficients from the central moments of the ensemble-averaged concentration distribution. Under the assumption that both prescriptions lead to the same macro-scale quantities, the second approach is usually preferred in literature. The present paper is an extension of the work of Metzger et al. (1996). We show that the two averaging procedures lead to different results in one-dimensional systems, whereas the difference vanishes for higher dimensions. Taking into account the influence of small scale dispersion, we give explicit results for the macroscopic parameters characterizing the solute plume. We analyze the various contributions to these parameters and show how the physical origin of these contributions can be traced back uniquely to fluctuations in the retardation factor, in the flow field, and in the degradation rate, and to the cross-correlations between these inhomogeneities, respectively.

A stochastic spatial process to model the persistence of sickle-cell disease

We consider a gene with two alleles. Allele A is normal, allele S is abnormal. Individuals with genotype SS have a severe disease called sickle-cell disease. Individuals with genotype AS are not sick, and it is thought that they are more resistant to malarial infection than individuals with genotype AA. This could explain why the allele S has persisted in regions where malaria is endemic. We use a stochastic spatial process to test this hypothesis. For our model, we show that if the genotype AS has an advantage over the genotype AA, then the allele S will persist in the population even if the genotype SS is not viable.

An Analysis of Crack Growth Rate Due to Variation of Fatigue Crack Growth Resistance

Reliability analysis of structures based on fracture mechanics requires knowledge on statistical characteristics of the parameter C and m in the fatigue crack growth law, . The purpose of the present study is to investigate if it is possible to predict fatigue crack growth rate by only the fluctuation of the parameter C. In this study, Paris-Erdogan law is adopted, where the author treat the parameter C as random and m as constant. The fluctuation of crack growth rate is assumed only due to the parameter C. The growth resistance coefficient of material to fatigue crack growth (Z=1/C) was treated as a spatial stochastic process, which varies randomly on the crack path. The theoretical crack growth rates at various stress intensity factor range are discussed. Constant fatigue crack growth tests were performed on the structural steel, SM45C. The experimental data were analyzed to determine the autocorrelation function and Weibull distributions of the fatigue crack growth resistance. And also, the effect of the parameter m of Paris' law due to variation of fatigue crack growth resistance was discussed.

Model-Based Geostatistics

SUMMARY Conventional geostatistical methodology solves the problem of predicting the realized value of a linear functional of a Gaussian spatial stochastic process S(x) based on observations Yi = S(xi) + Zi at sampling locations xi, where the Zi are mutually independent, zero-mean Gaussian random variables. We describe two spatial applications for which Gaussian distributional assumptions are clearly inappropriate. The first concerns the assessment of residual contamination from nuclear weapons testing on a South Pacific island, in which the sampling method generates spatially indexed Poisson counts conditional on an unobserved spatially varying intensity of radioactivity; we conclude that a conventional geostatistical analysis oversmooths the data and underestimates the spatial extremes of the intensity. The second application provides a description of spatial variation in the risk of campylobacter infections relative to other enteric infections in part of north Lancashire and south Cumbria. For this application, we treat the data as binomial counts at unit postcode locations, conditionally on an unobserved relative risk surface which we estimate. The theoretical framework for our extension of geostatistical methods is that, conditionally on the unobserved process S(x), observations at sample locations xi form a generalized linear model with the corresponding values of S(xi) appearing as an offset term in the linear predictor. We use a Bayesian inferential framework, implemented via the Markov chain Monte Carlo method, to solve the prediction problem for non-linear functionals of S(x), making a proper allowance for the uncertainty in the estimation of any model parameters.

Geostatistical modelling of environmental data

Many studies in environmental research include the analysis of spatially distributed measurements. The spatial representativity of the data can be quantified by modelling its autocorrelation structure with geostatistical methods based on the theory of spatial stochastic processes. The estimation of variogram functions — a measure to describe spatial variability — is illustrated by using precipitation and deposition data. The main charactristics of the variogram are discussed and interpreted. Kriging, a weighted moving average interpolation technique with weights depending on the specified variogram model, takes into account the spatial autocorrelation of the data. In addition kriging provides information about the accuracy of interpolation. That's why kriging and variogram analysis are useful tools for data interpolation and for network design in environmental studies. Umweltstudien erfordern oft die Analyse räumlich verteilter Messungen. Mit Methoden der Geostatistik wird die räumliche Abhängigkeitsstruktur der Daten modelliert und somit Aufschlußüber deren Flächenrepräsentanz gewonnen. Ausgehend von der Theorie räumlich stochastischer Prozesse schätzt man das Variogramm — ein Maß zur Beschreibung der räumlichen Variabilität —, dessen wichtigste Eigenschaften am Beispiel von Niederschlags- und Depositionsdaten dargestellt und interpretiert werden. Das Kriging-Verfahren ist eine Technik der gewichteten moving-average Interpolation, bei der durch die vom Variogramm-Modell beeinflußte Gewichtung die Autokorrelationsstruktur der Daten berücksichtigt wird. Darüberhinaus ermöglicht es Aussagen zur Genauigkeit der Interpolation. Somit stehen mit der Variogramm-Analyse und dem darauf aufbauenden Kriging-Verfahren wertvolle Hilfsmittel zur Interpolation und zur Meßnetzplanung in Umweltstudien zur Verfügung.

Bayesian deconvolution in optical astronomy

view Abstract Citations (10) References (19) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS Bayesian Deconvolution in Optical Astronomy Molina, R. ; del Olmo, Ascension ; Perea, Jaime ; Ripley, B. D. Abstract In this work we use Bayesian methods and spatial stochastic processes in the deconvolution of images of galaxies. Under very simple but realistic prior assumptions about the true underlying image of a galaxy the Bayesian framework is put to work. The method is tested in CCD images of extragalactic objects of different morphological types and an analysis of the deconvolutions obtained is performed emphasizing the comparison with other observational results. Publication: The Astronomical Journal Pub Date: February 1992 DOI: 10.1086/116092 Bibcode: 1992AJ....103..666M Keywords: Bayes Theorem; Image Processing; Optical Correction Procedure; Seeing (Astronomy); Convolution Integrals; Electromagnetic Noise; Stochastic Processes; Astronomy; ATMOSPHERIC EFFECTS; TECHNIQUES: IMAGE PROCESSING full text sources ADS | data products SIMBAD (5) NED (4)

Empirical bayes regionalization methods for spatial stochastic processes : Butcher, J B; Medina, M A; Marin, C M Water Resour ResV27, N1, Jan 1991, P7–15

Empirical bayes regionalization methods for spatial stochastic processes : Butcher, J B; Medina, M A; Marin, C M Water Resour ResV27, N1, Jan 1991, P7–15

Empirical Bayes regionalization methods for spatial stochastic processes

Many geophysical properties can be described as spatial stochastic processes, including spatially correlated hydraulic conductivity fields. Use of regional data can potentially improve estimation of such processes. We consider the case in which observations at each of several sites are described by a general linear model, while the parameters of these models arise from a common regional distribution. Parametric empirical Bayes methods enable the determination of the parameters of the regional distribution via maximum likelihood. However, such methods have not been utilized for spatial stochastic processes. We develop the application of a simple iterative technique for maximum likelihood estimation of the regional parameters, and demonstrate its use with a common parameterization of the spatial covariance structure. Synthetic data tests show the potential for substantial reduction in estimation risk through use of such techniques.

Spectral analysis of fatigue crack propagation resistance and its application to evaluation of the distribution of crack propagation fatigue life.

Fatigue crack propagation can be represented by a stochastic process such as a Markov process, which accounts for the distribution of crack propagation fatigue life. However, since we normally are able to obtain only limited information about the actual stochastic properties of the fatigue crack propagation process, it is necessary to represent the process by an appropriate model. This paper presents a new stochastic model which treats the material's resistance against fatigue crack growth as a spatial stochastic process evolving along the path of the crack. To obtain the stochastic properties of the fatigue crack propagation process, constant ΔK tests were conducted for several ΔK cases, and the parameters of the model were calculated by a spectral analysis using the maximum entropy method (MEM), which accounts for the statistical correlation that has been observed between adjacent fatigue crack growth rates. Using a proposed random crack propagation model, Monte-Carlo simulations were also performed and satisfactory agreement with the results of experiments were obtained.

Monte Carlo Simulation of Linear Two-Phase Flow in Heterogeneous Media

In this paper an algorithmic formulation is given for the synthesis of linear two-phase systems with inherent variability. It is assumed that uncertainties in the flow system arise due to the heterogeneities of the porous medium. Methods of stochastic mechanics are adopted to offer a novel approach, whereby the absolute permeability is taken as a spatial stochastic process with a log-normal distribution at each point of the flow domain. A correlation structure is built into the model so as to achieve a realistic representation. In addition, a constraint on the invariance of the harmonic means of the random permeabilities is attained by way of topological scaling. A computer-synthesized waterflooding experiment is conducted to explore the effect of heterogeneity on output variables such as the oil pressure head and cumulative recovery. It is observed that output uncertainties can become significant even when the heterogeneities are small. By employing a least-squares procedure, the sensitivity of relative permeabilities to induced stochastic variations is also qualitatively investigated.

Stochastic inversion of thermal data in a sedimentary basin: resolving spatial variability

The concept of spatial stochastic processes and the techniques of geostatistics and stochastic inversion are presented as a means for evaluating the spatial variability of thermal properties and temperature in sedimentary basins. Data from the Uinta Basin of NE Utah are used to demonstrate the analysis. Bottom hole temperatures from oil and gas wells provide the best opportunity for a spatial analysis because of the large number and wide distribution of data. Errors in bottom hole temperatures are not too large to preclude their use and stochastic inversion provides an optimum method for removing random noise while retaining spatial resolution. Corrected bottom hole temperatures from 235 wells in the Uinta Basin are used to estimate: (a) average geothermal gradient as a function of position in the basin and (b) temperature gradients in individual formations, also as a function of position in the basin. The latter analysis yields a high-resolution estimate of temperature gradient and temperature throughout the basin. The temperature field shows variations of up to 50°C at a depth of 4 km and is estimated to be accurate to better than 5°C. This variation in temperature is consistent with conductive and advective mechanisms of heat transfer in a heterogeneous medium.