Spatial Stochastic Process Research Articles

Finding spiral structures in images of galaxies

Much recent work in statistical image analysis has been concerned with ‘cleaning’ images by a bayesian statistical analysis incorporating a prior model, which reflects the spatial structure of the image. In almost all cases this has involved a description of the image at pixel level. In this paper we take the process further, and develop a spatial stochastic process of objects present in the image. The general theory is given and applied to images of spiral galaxies, with the aims of producing better schematic reconstructions and of automatically classifying galaxies.

Stochastic analysis of field measured unsaturated hydraulic conductivity

Unsaturated hydraulic conductivity K values as a function of soil‐water pressure head h were measured in the soil at 75 cm depth at 70 different sites separated from one another by a distance of l m along a horizontal transect. K field was viewed as a random function of spatial location x. Field data were analyzed (1) to examine the isotropy and stationarity of K, (2) to check the ergodicity of K in the mean and covariance functions, and (3) to characterize the distribution properties of K by estimating the higher‐order correlations, that is, third and fourth cumulants. The mean functions were estimated by averaging over h and x. The covariance function was studied to investigate its spatial origin dependency. Logs and square roots of K were used for estimating the third and fourth cumulants. Results showed that spatial covariance functions are anisotropic and both lag and origin dependent, that is, spatially nonhomogeneous. Because the stationarity (statistical homogeneity) of K is scale dependent, which was indicated by the identification of locally stationary covariance regions, the ergodic properties of K are also scale dependent at smaller spatial scales. Results related to the distribution characteristics of K indicated that although ln K is marginally Gaussian distributed, in the context of spatial stochastic processes the random field of ln K is not Gaussian because the third and fourth cumulants of the field are still significantly different from zero and have the same order of magnitude as the first and second cumulants. The square root transformation, however, resulted in a random field that is approximately Gaussian although marginal distributions of remained skewed. Analyses of ln K and indicated that better transformations which would result in both marginal and joint Gaussian behavior for the random field of K are needed.

Multivariate analysis of geomagnetic array data: 2. Random source models

In this paper I develop a spatial stochastic process model for randomly varying electromagnetic source fields observed over a one‐dimensional Earth. With this model the complex covariance between any two electromagnetic field components, measured at the same or different locations on the Earth's surface, can be computed. These calculations yield the expected values for the elements of the spectral density matrix (SDM; the matrix of averaged cross products of all components measured in the array) for synthetic arrays with arbitrary station locations. The model can thus be used to study the properties of the eigenvalues and eigenvectors of these synthetic array SDMs. For the case where both the array size and the skin depth of the electromagnetic fields in the conductive Earth are small compared to typical source length scales, the synthetic SDMs exhibit two larger eigenvalues, which correspond to nearly uniform source fields, followed by three smaller eigenvalues, which correspond to a set of canonical gradients. Qualitative consideration of more general models with three‐dimensional conductivity and arbitrary random sources shows that the eigenvalues of a geomagnetic array SDM represents a sort of discrete spatial power spectrum. The corresponding eigenvectors form a natural basis which any realizable field can be expanded in. These results provide strong justification for a generalized transfer function/response space model for the analysis of geomagnetic array data.

One‐dimensional steady state infiltration in heterogeneous soils

The effects of heterogeneity on one‐dimensional, steady state infiltration are studied using numerical simulations where the soil hydrologic properties are assumed to be spatial stochastic processes. Analytical solutions to one‐dimensional, steady state infiltration in heterogeneous soils are developed and applied to the stochastic random fields. The effects of spatial variability of parameters of an exponential unsaturated hydraulic conductivity model on the soil‐water pressure profiles are examined. The amount of variation in pressure heads is found to vary with infiltration rates and mean pressure heads, while the cross‐correlation between parameters is shown to have important influences on the value of the head variance. An inverse procedure is developed to determine the effective hydraulic conductivity parameters. The effective parameter is found to vary with mean pressures. Effective hydraulic conductivities and pressure head variances estimated from the numerical simulations were compared with those obtained from a spectral method by Yeh et al. (1985a, b, c). A unit mean gradient approach was used to estimate the effective unsaturated hydraulic conductivity, and the result shows that this approach is adequate for heterogeneous soils.

ANALYSIS OF SPATIAL VARIABILITY IN SODIC SOILS

Structural analyses of sodium adsorption ratio (SAR), pH of saturation paste (pHs), CaCO3, and CO3 + HCO3 ions (in saturation extract) were stochastically modeled at different depths. One hundred four (13 x 8) samples were drawn at 12− x 12-m grid nodes in each of four depths of a Fine Loamy Typic/Natric Haplustalf. Each sample was a composite of four samples drawn within 1-m2 area (support) around grid nodes. The coefficient of variation (CV) in different layers varied from 78 to 91 for SAR, 60 to 130 for CaCO3 and 29 to 43% for CO3 + HCO3 ions. Geostatistical analysis of soil heterogeneity revealed that the spatially dependent stochastic component was generally predominant over the nugget effect. Structured variation was modeled into isotropic, positive, and definite spherical functions. In the upper layers, the spatially dependent component (C1) explained most of the variability in SAR and pHs. However, beyond the 60− to 90-cm depth, the unstructured or nugget component also became significant. Distribution of CaCO3 in the 0− to 30-cm depth, however, was almost a random phenomenon. However, below 30-cm depths its distribution was also a spatial stochastic process. For CO3 + HCO3 ions, spatial dependence and nugget variances were almost in equal proportions. Range, i.e., the scale of autorelationship was almost the same in different depths and for most of the properties studied. There was a linear relationship (two-parameter fit) between variance and mean oyer all the depths. Hence, scaled variograms were computed accordingly, and an intrinsic function of variability for the super region of four depths was arrived at. Cross-variability among depths was established by treating them as simultaneous intrinsic random fields.

Stochastic Finite Element Analysis of Thermal Deformation and Thermal Stresses of CFRP Laminated Plates

The effect of probabilistic fiber orientations on the response of carbon-fiber-reinforced plastic laminated plates due to thermal load is analyzed by the stochastic finite element method based on the first-order perturbation technique. The fiber orientations are taken as a two-dimensional spatial stochastic process, and the stochasticity is represented by a trigonometrical autocorrelation function. Numerical examples are carried out in regard to the rectangular CFRP laminated plates subjected to a uniform temperature rise or a constant temperature gradient through the thickness. The variance of the thermal deformation and stresses is discussed for the input statistical property of the orientations.

Application of Stochastic FEM in the Analysis of Magnetic Fields

The stochastic finite element method designed to deal with indeterminate elements in actual magnetic materials was studied. Permeability is considered as a variable in a homogeneous two-dimensional spatial stochastic process, and the analysis results are compared with those of the usual determinate finite-element method. Both methods produced identical results over a wide range. But the advantage of the probabilistic approach is that the variance can be evaluated directly without performing recursion.

Depth‐averaging effects on hydraulic head for media with stochastic hydraulic conductivity

Hydraulic conductivity of a porous medium frequently is considered to be a single realization of a three‐dimensional spatial stochastic process. The most common observations of flow in porous media are hydraulic‐head measurements obtained from wells which are screened over extensive sections of the medium. These measurements represent, approximately, a one‐dimensional spatial average of the actual three‐dimensional head distribution, the actual head distribution being a stochastic process resulting from flow through a random hydraulic‐conductivity field. This paper examines, via ensemble averages, the effect of such spatial averages of groundwater flow on the spatial autocovariance function for a simple, yet viable, stochastic model of a bounded medium. The model is taken to be three‐dimensional flow in a medium that is bounded above and below and in which the hydraulic conductivity is a second‐order stationary stochastic process. Spatial averaging of the hydraulic heads is considered to take place over the entire thickness of the medium. Ensemble variances and autocorrelations for depth‐averaged heads are computed for the resulting two‐dimensional flow system and compared with those from a fully three‐dimensional flow system.

Stochastic finite element analysis of thermal deformation and thermal stresses of CFRP laminated plates. In problems of probabilistic fiber orientations.

The effect of probabilistic fiber orientations on the response of carbon fiber reinforced plastic laminated plates due to thermal load is analysed by the stochasic finite element method based on the first order perturbation technique. The fiber orientations are taken as two-dimensional spatial stochastic process, and the stochasticity is represented by a trigonometrical autocorrelation function. Numerical analyses are carried out in regard to rectangular CFRP laminated plates subjected to a uniform temperature rise or constant temperature gradient through the thickness. The variance of thermal deformation and stresses is discussed for the input statistical property of the orientations.

A study of the dimensional invariance of the reliability index in Advanced First-Order Second-Moment Method.

The dimensional invariance related to the reliability index is examined. The reliability indices β, based on the Advanced First-Order Second Moment (AFOSM) method by use of the stochastic finite element method, are evaluated in the cases of bars and square plates under tension, whose Young's modulus is regarded as spatial homogeneous stochastic process. The reliability index for the bars is obtained by means of the analytical formulation for continuum. The numerical results show that the reliability indices for several autocorrelation fucitons converge well with the increase of the finite element division, and the index for the bar agrees well with the analytical one. This suggests strongly that the dimensional invariance holds for the reliability index evaluated by the finite element discretization and AFOSM method.

Stochastic finite element analysis of thermal deformation of fiber reinforced plastic laminated plate.

The thermal deformation of a fiber reinforced plastic laminated plate due to its material constant variability is formulated by the stochastic finite element method. Four elastic constants and two thermal expansion coefficients are taken as representing the spatial stochastic process. The rates of change of the finite element solution are obtained with regard to the said constants on the basis of the first order perturbation technique, and are employed to evaluate the variance of the plate deflection due to temperature distribution by the first order approximation in conjunction with the second moments of the probabilistic material constants. The numerical examples are carried out in regard to a boron/epoxy plate consisting of up to six layers, and the effects correlation and orthotropy of the probabilistic variables on the deflection variance are evaluated.

Time series analysis of fatigue crack growth rate data

Previous treatments of fatigue crack growth rate (FCGR) statistics have sought to determine the best testing and data analysis techniques with which to process fatigue crack growth data. The best techniques were denned as the ones that produced the least scatter in the data. Recent studies suggest that the scatter in FCGR data has physical significance, which must be understood in order to predict the growth of small cracks. This paper presents a stochastic model which treats the material's resistance to fatigue crack growth as a spatial stochastic process evolving along the path of the crack. The parameters of the model are found by a time series analysis which accounts for the statistical correlation that has been observed between adjacent FCGR measurements. The data to be analyzed is assumed to consist of an ensemble of replicate crack growth tests sampled at equal increments of crack growth. The model is shown to be useful for explaining the moderate scatter observed in mid-range ΔK tests. It is also likely to be useful for understanding the large scatter observed in small crack tests.

Workshop on stochastic spatial processes mathematical theories and biological applications : 10–14 September 1984 Heidelberg, F.R.G.

Workshop on stochastic spatial processes mathematical theories and biological applications : 10–14 September 1984 Heidelberg, F.R.G.

Modelling and Evaluation of Generally Corroded Surface by Filtered Poisson Process

Irregular configuration of the corroded surface is taken as spatial stochastic process and two-dimensional filtered Poisson process is newly developed as a numerical model of corrosion. In the present model any corroded surface is interpreted as being made by the superposition of pits. This enables us to characterize any general corrosion quantitatively, which has been hardly achieved. The availability of the proposed method is demonstrated through the analyses of two corroded specimens and discussions are give in the light of the above interpretation. Also the sensitivity study is exemplified making use of the flexibility of the model, which yields the basis of prediction of the development of corrosion.

A System of World Mammal Faunal Regions. II. The Distance Decay Effect Upon Inter-Regional Affinities

Regional/historical biogeography is often considered descriptive and/or highly idiographic by nature. This point of view is challenged here through an analysis in which a newly developed mammal faunal regions classification system is linked to a simple model of evolutionary process. Variations in subregional characteristics are used to examine the view that evolution represents a stochastic spatial process. This is investigated by starting with the proposition that the present spatial arrangement of mammalian faunal elements (families) should be explainable solely on the basis of chance interaction rates among subregions and a deterministic distance decay effect on the diffusion of evolutionary innovations across the system over time. A derivative of the entropy maximization model developed by Alan G. Wilson is employed to separate out the effects of chance interaction among fauna subregions from the second order faunal characteristics used as a surrogate for the distance decay effect. The present distribution of mammal families is shown to be well represented through this model, which accounts for almost all the variance in the original inter-subregional faunal similarities matrix. The normative character of the model and the classification underlying it is further pursued through an examination of within-system characteristics of subregional interrelation. It is shown that, despite variation among subregions with respect to their absolute diversities and other attributes, the characteristics of any individual unit can be used equally well to specify the general properties of the rest of the system, a fact that substantiates the claim that the component units are logically equivalent. It is concluded that a carefully specified system of world faunal regions can be used as a tool through which to infer system level relationships among these interacting units.

A Comparative Study of Some Stochastic Methods and Autoprojective Models for Spatial Processes

Spatial stochastic processes often show a distinct persistence, which becomes noticeable in a nonrandom spatial grouping of similar or regularly dissimilar values. Various methods have been developed for statistically analyzing such persistence effects which are dependent upon distance and direction. Important differences and similarities in the concept of spatial autocorrelation and the theory of regionalized variables will be pointed out and explained by means of examples. The use of these geostatistical methods in modelling spatial processes leads to the SAR (spatial auto-regressive) models and to the Kriging concept, respectively. Although SAR models are used in particular to represent spatial persistences as characteristic endogenous dependency structures, the Kriging method is more pragmatically orientated to the interpolation and extrapolation of spatial data. The suitability of such stochastic models in purely spatial forecasting and also in the estimation of areal means will be described by use of examples, and compared with conventional methods.

Competition for Light in a Plant Monoculture Modelled as a Spatial Stochastic Process

SUMMARY In a glasshouse experiment closely spaced plants were regularly sub-irrigated with nutrient solution and it was assumed that competition between individuals was for light. Interaction between neighbours is modelled as a spatial process in relation to differences in plant height. The influence one plant has on another is calculated as the ratio of the angle s between their apices, and an angle 6 which represents the intensity of competition in the population as a whole. When s are considered to die. Parameter values were estimated for a sequence of fortnightly harvests of glasshouse grown Tagetes patula. There was a decrease in /i and 6, (j> increased and plants came to compete with their second nearest neighbours as well as their nearest as the community developed. The greatest or was at an intermediate stage, suggesting that more frequent assessments of the competition process are required at some stages than at others. Model parameters fitted to one data set were used in a Monte Carlo testing procedure with a second, independent data set. The importance of this technique is stressed because plants in a single community are not independent realization of the competition process. Simulations with the model reproduced both bimodality in the frequency distributions of plant size and an even spatial distribution of large or surviving plants, features which have been observed in a range of plant monocultures. To achieve these features it was essential that plant interaction be modelled as a one-sided process.

Stochastic analysis of spatial variability in subsurface flows: 1. Comparison of one‐ and three‐dimensional flows

The complex variation of hydraulic conductivity in natural aquifer materials is represented in a continuum sense as a spatial stochastic process which is characterized by a covariance function. Assuming statistical homogeneity, the theory of spectral analysis is used to solve perturbed forms of the stochastic differential equation describing flow through porous media with randomly varying hydraulic conductivity. From analyses of unidirectional mean flows which are perturbed by one‐and three‐dimensional variations of the logarithm of the hydraulic conductivity, local relationships between the head variance and the log conductivity variance are obtained. The results show that the head variance produced by three‐dimensional statistical isotropic conductivity perturbations is only 5% of that in the corresponding one‐dimensional case. The head variance is also strongly dependent on the correlation distance of the log conductivity covariance function. These results emphasize the importance of including spatial correlation structure and multidimensional effects in stochastic simulation of groundwater flow.

Markovian contact processes

Markovian contact processes (mcp's) include some of the commoner models for the spatial spread of population processes, such as the ‘simple’ and ‘general’ epidemics, percolation processes, and birth, death and migration processes. They are here set in a framework which relates them to each other, and to a particular basic model, the contact birth process (cbp); indeed they are defined as modifications of the cbp. The immediate advantage of this is that bounds for the velocity of the cbp, which can be obtained from the linear equation describing its expected numbers (provided the contact distribution has exponentially bounded tail) apply also to general mcp's. Existence of moments (and for some processes their time-derivatives) of St (θ), the distance to the furthest individual in an arbitrary direction θ, can also be deduced whenever the corresponding moment of the contact distribution exists. An important subclass consists of population-monotone mcp's, for which the distributions of St can be shown to be subconvolutive, so that the work of Hammersley (1974) can be applied to obtain convergence theorems for St /t; and these can be extended to convergence of the convex hull of the set of inhabited points. These results are particularly valuable because they apply to some non-linear processes, e.g. the simple epidemic. Some special results on the cbp (Section 6) emphasize the differences between linear and non-linear spatial stochastic processes. Although the paper is written throughout in terms of continuous-time processes, the results on subconvolutive distributions are actually more easily applied in the discrete-time case. (Conversely, the approach used in the accompanying paper by Biggins (pp. 62–84), which is written in terms of discrete-time processes, can be extended to deal also with continuous-time processes.)

Velocity of stochastic processes in two dimensions

A basic class of spatial stochastic processes is that in which the phenomenon spreads between static individuals with probability of ‘infection’ falling off with distance according to a contact distribution (Mollison (1972)). The following questions then arise:(1) How does the velocity of spread depend on the contact distribution? In particular, for which contact distributions is it finite?(2) Does the phenomenon spread at a steady velocity, or does it spread in jumps?