Spatial Stochastic Process Research Articles

Intrinsic random functions on the sphere

Spatial stochastic processes that are modeled over the entire Earth’s surface require statistical approaches that directly consider the spherical domain. In practice, such processes rarely are second-order stationary — that is, they do not have constant mean values or the covariance function that depends only on their angular distance. Here, in order to model non-stationary processes on the sphere, we extend the notion of intrinsic random functions and show that low-frequency truncation plays an essential role. We show that these developments can be presented through the theory of reproducing kernel Hilbert space. In addition, the link between universal kriging and splines is carefully investigated, whereby we show that thin-plate splines are non-applicable for surface fitting on the sphere.

Improving satellite-based PM2.5 estimates in China using Gaussian processes modeling in a Bayesian hierarchical setting

Using satellite-based aerosol optical depth (AOD) measurements and statistical models to estimate ground-level PM2.5 is a promising way to fill the areas that are not covered by ground PM2.5 monitors. The statistical models used in previous studies are primarily Linear Mixed Effects (LME) and Geographically Weighted Regression (GWR) models. In this study, we developed a new regression model between PM2.5 and AOD using Gaussian processes in a Bayesian hierarchical setting. Gaussian processes model the stochastic nature of the spatial random effects, where the mean surface and the covariance function is specified. The spatial stochastic process is incorporated under the Bayesian hierarchical framework to explain the variation of PM2.5 concentrations together with other factors, such as AOD, spatial and non-spatial random effects. We evaluate the results of our model and compare them with those of other, conventional statistical models (GWR and LME) by within-sample model fitting and out-of-sample validation (cross validation, CV). The results show that our model possesses a CV result (R2 = 0.81) that reflects higher accuracy than that of GWR and LME (0.74 and 0.48, respectively). Our results indicate that Gaussian process models have the potential to improve the accuracy of satellite-based PM2.5 estimates.

R.B.シナジ(今野紀雄・林俊一訳), マルコフ連鎖から格子確率モデルへ, シュプリンガー・フェアラーク東京, 2001年

R.B.シナジ(今野紀雄・林俊一訳), マルコフ連鎖から格子確率モデルへ, シュプリンガー・フェアラーク東京, 2001年

Colonization and Collapse

Many species live in colonies that thrive for a while and then collapse. Upon collapse very few individuals survive. The survivors start new colonies at other sites that thrive until they collapse, and so on. We introduce a spatial stochastic process on graphs for modeling such population dynamic. We obtain conditions for the population to get extinct or to survive.

Model-Based Geostatistics for Prevalence Mapping in Low-Resource Settings

ABSTRACTIn low-resource settings, prevalence mapping relies on empirical prevalence data from a finite, often spatially sparse, set of surveys of communities within the region of interest, possibly supplemented by remotely sensed images that can act as proxies for environmental risk factors. A standard geostatistical model for data of this kind is a generalized linear mixed model with binomial error distribution, logistic link, and a combination of explanatory variables and a Gaussian spatial stochastic process in the linear predictor. In this article, we first review statistical methods and software associated with this standard model, then consider several methodological extensions whose development has been motivated by the requirements of specific applications. These include: methods for combining randomized survey data with data from nonrandomized, and therefore potentially biased, surveys; spatio-temporal extensions; and spatially structured zero-inflation. Throughout, we illustrate the methods with disease mapping applications that have arisen through our involvement with a range of African public health programs.

Scaling Limits in Models of Statistical Mechanics

The emphasis of the workshop was on the deep relations between, on the one hand, recent advances in probabilistic investigation of statistical mechanical models and spatial stochastic processes and, on the other hand, rigorous field-theoretic and analytic methods of mathematical physics. There were 52 participants, including 6 postdocs and graduate students, working in diverse intertwining areas of probability, statistical mechanics and field theory. Specific topics addressed during the 24 talks include: Universality and critical phenomena, disordered models, Gaussian free field (GFF), stochastic representation of classical and quantum-mechanical models and related random interchange and permutation processes, random planar graphs and unimodular planar maps, random walks on critical graphs and the Alexander-Orbach conjecture, reinforced random walks and non-linear -models, metastability, aging, equilibrium and dynamics for continuum particles with hard core interactions, non-equilibrium dynamics and Toom’s interfaces.

A Regression Model for Registering Multimodal Images

We propose a new multimodal image registration method when using different imaging modalities separately. The proposed method first aligns corresponding extracted geometric features (continuous curves), and then estimate the deformation vector field as spatial stochastic processes. The resulting deformation has the advantage of registering the given data in such a way that the corresponding curves-regions match as being sufficiently smooth over the whole image domain. Experimental results on both synthetic and real data show that the proposed method matches the state-of-the-art.

Stochastic characterization of the spectrum sharing game in ad-hoc networks

This work focuses on infrastructure-less ad hoc wireless networks where multiple transmitter/receiver pairs share the same radio resources (spectrum); transmitters have to choose how to split a total power budget across orthogonal spectrum bands with the goal to maximize their sum rate under cumulative interference from concurrent transmissions. We start off by introducing and characterizing the non-cooperative game among transmitter/receiver pairs when the network topology is deterministically given. The corresponding Nash equilibria are derived, highlighting their dependency on the topological parameters (distances between wireless nodes, propagation model, and background noise power). The analysis is then extended to the case of random network topologies drawn from a given spatial stochastic process. Tools of stochastic geometry are leveraged to derive a statistical characterization of the equilibria of the spectrum sharing game. Finally, a distributed algorithm is proposed to let the players of the spectrum sharing game converge to equilibria conditions. Numerical simulations show that the proposed algorithm drives the users to stable points that are close to the equilibria of the game requiring limited information exchange among nodes.

A Partial-differential Approximation for Spatial Stochastic Process Algebra

We study a spatial framework for process algebra with ordinary differential equation (ODE) semantics. We consider an explicit mobility model over a 2D lattice where processes may walk to neighbouring regions independently, and interact with each other when they are in same region. The ODE system size will grow linearly with the number of regions, hindering the analysis in practice. Assuming an unbiased random walk, we introduce an approximation in terms of a system of reaction-diffusion partial differential equations, of size independent of the lattice granularity. Numerical tests on a spatial version of the generalised Lotka-Volterra model show high accuracy and very competitive runtimes against ODE solutions for fine-grained lattices.

Universal Cokriging of Non stationary Spatial Stochastic Process with an Application to Wells Data in Wana District North of Iraq

This paper is concerned with the prediction ofnonstationaryspatial stochastic process by the multivariate approach which is known as universal cokriging. This approach is a generalization of kriging because universal cokriging depends on the underlying variable as well as other variables named as the secondary variables. These variables have a considerable effect on the process of prediction. In this work we obtained the final formulas of the prediction equations system through the derivation of universal cokriging equations in the form of matrices. The obtained universal cokriging equations are quiet similar to the known equations system, and summarise clearly the spatial weights and prediction error variance. The obtained equations system is applied to a real data which represent two variables which areelevation of water(primary variable) and depth of well(secondary variable)in Wana district in Ninavah governorate in north of Iraq.The results obtained are very encouraging, very near to the real values, and with minimum cokriging variance. We compute eight spatial predictions together with error variance of prediction. All algorithms are programmed by Matlab_7.12 package.

Universal Cokriging of Non stationary Spatial Stochastic Process with an Application to Wells Data in Wana District North of Iraq

Universal Cokriging of Non stationary Spatial Stochastic Process with an Application to Wells Data in Wana District North of Iraq

Numerical Assessment of Radiated Susceptibility of Twisted-Wire Pairs With Random Nonuniform Twisting

The influence of twist-pitch nonuniformity on the radiated susceptibility of a twisted-wire pair (TWP) running above ground and illuminated by a plane-wave field is investigated via a transmission-line model. The geometrical representation of the TWP involves a bifilar helix with a variable pitch function, which readily allows for the description of any unwanted/unknown deviations from the ideal twist geometry. Basic examples are used to show that such a deformation plays a fundamental role in the pickup of differential-mode (DM) noise, thus substantiating the need for the proposed prediction model. By describing the pitch random variations as a spatial stochastic process, a realistic representation of the twisting features is obtained (both for TWPs with imperfect uniform twists and for TWPs intentionally manufactured with random nonuniform twists). Repeated runs are used to characterize the sensitivity of the induced DM noise to random nonuniform twisting. Especially, it is shown that the twist-pitch nonuniformity, sometimes intentionally used to reduce pair-to-pair crosstalk in TWP cables, may however lead to a severe decrease of the TWP immunity to external fields.

Differential Burn‐in and Reliability Screening Policy Using Yield Information Based on Spatial Stochastic Processes

Differential Burn‐in and Reliability Screening Policy Using Yield Information Based on Spatial Stochastic Processes

Modelling honey bee queen mating as a measure of feral colony density

Robust estimates of feral and wild honey bee (Apis mellifera) colony densities are essential for the understanding of the role of honey bees in an ecosystem, and for planning responses to an incursion of an exotic honey bee disease or pest. New genetic methods make it possible to identify the number of feral colonies with which a test queen mates. We present an agent-based model of the spatial process of honey bee mating that describes the relationship between feral colony densities and the number of unique colonies with which a queen mates. This model incorporates random, aggregated and overdispersed spatial distributions of feral colonies and drone congregation areas. We model different densities of feral colonies and drone congregation areas with various numbers of test queens and determine the range of counts of matings with drones from unique colonies. The model shows that the range of counts for a given set of parameters is consistent with a Poisson distribution, and that the most significant explanatory variable is density of the feral colonies. The results obtained reveal that 10 or more test queens may be needed in field studies to resolve order of magnitude differences in feral colony densities. We conclude that the new genetic methods provide a powerful tool for making indirect inferences about feral colony densities, so long as the field survey results are treated as outcomes of a stochastic spatial process.

Stochastic Dynamics on Hypergraphs and the Spatial Majority Rule Model

This article starts by introducing a new theoretical framework to model spatial systems which is obtained from the framework of interacting particle systems by replacing the traditional graphical structure that defines the network of interactions with a structure of hypergraph. This new perspective is more appropriate to define stochastic spatial processes in which large blocks of vertices may flip simultaneously, which is then applied to define a spatial version of the Galam’s majority rule model. In our spatial model, each vertex of the lattice has one of two possible competing opinions, say opinion 0 and opinion 1, as in the popular voter model. Hyperedges are updated at rate one, which results in all the vertices in the hyperedge changing simultaneously their opinion to the majority opinion of the hyperedge. In the case of a tie in hyperedges with even size, a bias is introduced in favor of type 1, which is motivated by the principle of social inertia. Our analytical results along with simulations and heuristic arguments suggest that, in any spatial dimensions and when the set of hyperedges consists of the collection of all n×⋯×n blocks of the lattice, opinion 1 wins when n is even while the system clusters when n is odd, which contrasts with results about the voter model in high dimensions for which opinions coexist. This is fully proved in one dimension while the rest of our analysis focuses on the cases when n=2 and n=3 in two dimensions.

Sampling rate of spatial stochastic processes with independent components in modeling random search paths

Continuous-time modeling of random searches is designed to be robust to the sampling rate while the spatial model is required to be of rotation-invariant type, which is often computationally prohibitive. Such computational difficulty may be circumvented by employing a model with independent components. We demonstrate that its disadvantages in statistical properties are blurred under lower frequency. We propose a quantitative criterion for choice of the sampling rate at which a spatial model with independent components resembles a rotation-invariant model. Our findings have the potential to assist the observer to employ simpler models in the continuous-time framework to avoid expensive computation required for statistical inference.

Affine Decompositions of Parametric Stochastic Processes for Application within Reduced Basis Methods

AbstractWe consider parameter dependent spatial stochastic processes in the context of partial differential equations (PDEs) and model order reduction. For a given parameter, a random sample of such a process specifies a sample coefficient function of a PDE, e.g. characteristics of porous media such as Li-ion batteries or random influences in biomechanical systems. To apply the Reduced Basis Method (RBM) to parametrized systems (with stochastic or deterministic parameter dependencies), it is necessary to obtain affine decompositions of the systems in parameter and space (cf. Patera and Rozza (2006); Wieland (2010)). For deterministic problems, it is common to use the Empirical Interpolation Method (EIM) (cf. Barrault et al. (2004)). For stochastic coefficients one can apply the Karhunen-Loève (KL) expansion (cf. Karhunen (1947)) where the terms with stochastic dependencies are assumed to satisfy certain distributions and are modeled using polynomial chaos (PC) expansions (cf. Ghanem and Spanos (1991)).In this paper, we extend the EIM to parametrized spatial stochastic processes. The goal is to develop efficiently computable affine decompositions of not only parameter dependent but also stochastic systems that separate spatial dependencies from parametric and probabilistic influences without any assumptions on the distribution of non-spatial terms. We will use the basic concept of the EIM together with ideas from Proper Orthogonal Decomposition (POD) as well as from the Discrete Empirical Interpolation Method (DEIM) (cf. Chaturantabut and Sorensen (2010)), and we use least-squares approxmations.

Prediction by Regression and kriging for Spatial Data with Application

This study deals with the prediction of the non-stationary spatial stochastic process. The prediction is done by two techniques which are regression technique (generalized least square estimation) and universal kriging technique. As it is familiar, that the non-stationary stochastic process has a trend (mean) as a linear or non-linear model. By this process we can find covariance function from knowing the variogram function and the latter is attributed to a spherical variogram model, also in order to estimate parameters of spherical model by minimum norm quadratic unbiased estimator which requires that the covariance function must be linear in the parameters, then changing the spherical model into an approximated linear model by Taylor series in the linear approximation The prediction in these two techniques is applied to real data which represent height levels of ground water of 47 wells with their regional coordinates in Sinjar district in Ninevah Governorate in Iraq. The results were so encouraging where we show the approximation between the predictive values and the real values as well as computing the variance of prediction in these two techniques. It is shown that the prediction variance of universal kringing is less than that of regression.

Accounting for spatial autocorrelation in null models of tree species association

A commonly used null model for species association among forest trees is a well‐mixed community (WMC). A WMC represents a non‐spatial, or spatially implicit, model, in which species form nearest‐neighbor pairs at a rate equal to the product of their community proportions. WMC models assume that the outcome of random dispersal and demographic processes is complete spatial randomness (CSR) in the species’ spatial distributions. Yet, stochastic dispersal processes often lead to spatial autocorrelation (SAC) in tree species densities, giving rise to clustering, segregation, and other nonrandom patterns. Although methods exist to account for SAC in spatially‐explicit models, its impact on non‐spatial models often remains unaccounted for. To investigate the potential for SAC to bias tests based upon non‐spatial models, we developed a spatially‐heterogeneous (SH) modelling approach that incorporates measured levels of SAC. Using the mapped locations of individuals in a tropical tree community, we tested the hypothesis that the identity of nearest‐neighbors represents a random draw from neighborhood species pools. Correlograms of Moran's I confirmed that, for 50 of 51 dominant species, stem density was significantly autocorrelated over distances ranging from 50 to 200 m. The observed patterns of SAC were consistent with dispersal limitation, with most species occurring in distinct patches. For nearly all of the 106 species in the community, the frequency of pairwise association was statistically indistinguishable from that projected by the null models. However, model comparisons revealed that non‐spatial models more strongly underestimated observed species‐pair frequencies, particularly for conspecific pairs. Overall, the CSR models projected more significant facilitative interactions than did SH models, yielding a more liberal test of niche differences. Our results underscore the importance of accounting for stochastic spatial processes in tests of association, regardless of whether spatial or non‐spatial models are employed.

Geostatistical methods to identify and map spatial variations of soil salinity

The problem of estimating and predicting spatial distribution of a spatial stochastic process, observed at irregular locations in space, is considered in this paper. Environmental variables usually show spatial dependencies among observations, with lead one to use geostatistical methods to model the spatial distributions of those observations. This is particularly important in the study of soil properties and their spatial variability. In this study geostatistical techniques were used to describe the spatial dependence and to quantify the scale and intensity of spatial variations of soil properties, which provide the essential spatial information for local estimation. In this contribution, we propose a spatial Gaussian linear mixed model that involves (a) a non-parametric term for accounting deterministic trend due to exogenous variables and (b) a parametric component for defining the purely spatial random variation due possibly to latent spatial processes. We focus here on the analysis of the relationship between soil electrical conductivity and Na content to identify spatial variations of soil salinity. This analysis can be useful for agricultural and environmental land management.