Multivariate Random Fields Research Articles

Upscaling transport in heterogeneous media featuring local-scale dispersion: Flow channeling, macro-retardation and parameter prediction

Many theoretical treatments of transport in heterogeneous Darcy flows consider advection only. When local-scale dispersion is neglected, flux weighting persists over time; mean Lagrangian and Eulerian flow velocity distributions relate simply to each other and to the variance of the underlying hydraulic conductivity field. Local-scale dispersion complicates this relationship, potentially causing initially flux-weighted solute to experience lower-velocity regions as well as Taylor-type macrodispersion due to transverse solute movement between adjacent streamlines. To investigate the interplay of local-scale dispersion with conductivity log-variance, correlation length, and anisotropy, we perform a Monte Carlo study of flow and advective-dispersive transport in spatially-periodic 2D Darcy flows in large-scale, high-resolution multivariate Gaussian random conductivity fields. We observe flow channeling at all heterogeneity levels and quantify its extent. We find evidence for substantial effective retardation in the upscaled system, associated with increased flow channeling, and observe limited Taylor-type macrodispersion, which we physically explain. A quasi-constant Lagrangian velocity is achieved within a short distance of release, allowing usage of a simplified continuous-time random walk (CTRW) model we previously proposed in which the transition time distribution is understood as a temporal mapping of unit time in an equivalent system with no flow heterogeneity. The numerical data set is modeled with such a CTRW; we show how dimensionless parameters defining the CTRW transition time distribution are predicted by dimensionless heterogeneity statistics and provide empirical equations for this purpose.

On the properties of multivariate isotropic Random fields on the Ball

We consider multivariate isotropic random fields on the ball Bd. We first study their regularity properties in terms of Sobolev spaces. We further derive conditions guaranteeing the Hölder continuity of their covariance kernels and we prove the existence of sample Hölder continuous modifications for Gaussian random fields. Furthermore, we measure the error of truncated approximations of the corresponding series' representations. Moreover our developments are supported by numerical experiments. The majority of our results are new for multivariate random fields indexed over other domains, too. We express some of them for the case of the sphere.

Multivariate copula-based framework for stochastic analysis of landslide runout distance

The increasing frequency of landslide disasters worldwide highlights the importance of accurate prediction of post-failure runout distances for effective risk management. However, the prediction of runout distances of gravitational mass flow remains challenging due to the inherent complex heterogeneities of geomaterials and their inter-correlated spatially varying mechanical properties. To address this challenge, this study proposes a novel multivariate stochastic method based on the combination of the Generalised Interpolation Material Point (GIMP) analysis and the multivariate copula-based approach. The method considers geotechnical uncertainties by simulating copula-based cross-correlated random fields from sparse field data and incorporating them into the GIMP analysis through Monte Carlo simulations (MCS). Two slope cases with similar geometries but different sources of probability information are presented to illustrate the effectiveness of the proposed method. The results show that considering the influence of multivariate random fields can significantly affect the post-failure analysis of landslides. Both slope cases show that over 40 % of all MCS samples exceed the deterministic case, which indicates that the current deterministic analysis notably underestimates the risk associated with large runout distances of landslides. This study highlights the necessity of considering the interdependency of spatial heterogeneity in post-failure modelling of landslides.

Statistical Modeling of Right-Censored Spatial Data Using Gaussian Random Fields

Consider a fixed number of clustered areas identified by their geographical coordinates that are monitored for the occurrences of an event such as a pandemic, epidemic, or migration. Data collected on units at all areas include covariates and environmental factors. We apply a probit transformation to the time to event and embed an isotropic spatial correlation function into our models for better modeling as compared to existing methodologies that use frailty or copula. Composite likelihood technique is employed for the construction of a multivariate Gaussian random field that preserves the spatial correlation function. The data are analyzed using counting process and geostatistical formulation that led to a class of weighted pairwise semiparametric estimating functions. The estimators of model parameters are shown to be consistent and asymptotically normally distributed under infill-type asymptotic spatial statistics. Detailed small sample numerical studies that are in agreement with theoretical results are provided. The foregoing procedures are applied to the leukemia survival data in Northeast England. A comparison to existing methodologies provides improvement.

Multivariate Random Fields Evolving Temporally Over Hyperbolic Spaces

AbstractGaussian random fields are completely characterised by their mean value and covariance function. Random fields on hyperbolic spaces have been studied to a limited extent only, namely for the case of scalar-valued fields that are not evolving over time. This paper challenges the problem of the second-order characteristics of multivariate (vector-valued) random fields that evolve temporally over hyperbolic spaces. Specifically, we characterise the continuous space–time covariance functions that are isotropic (radially symmetric) over space (the hyperbolic space) and stationary over time (the real line). Our finding is the analogue of recent findings that have been shown for the case where the space is either the n-dimensional sphere or more generally a two-point homogeneous space. Our main result can be read as a spectral representation theorem, and we also detail the main result for the subcase of covariance functions having a spectrum that is absolutely continuous with respect to the Lebesgue measure (technical details are reported below).

Spatial variability characterization and modelling of 2.5D woven SiO2f/SiO2 composites

2.5D woven SiO2f/SiO2 composites possess geometric variabilities across scales, such as the spatial fluctuations of fibers and yarns or the random distributions of defects and pores formed in the manufacturing process, which induces a complex stress and strain distribution in material testing. In this study, a multivariate cross-correlated non-Gaussian random field based on Vine Copula is proposed to characterize the spatial variability of mechanical properties. A database of material mechanical properties is acquired by stochastic representative volume element (SRVE) simulation results which contains realistic meso-geometry characteristics and spatially random voids. Based on this database, the cross- and auto-correlations of the mechanical properties are quantified by Vine Copula and non-Gaussian random field model. In a comparison of different numerical models and experimental results, the proposed method is verified, which shows great advantages in characterizing the spatial variability and big potential abilities in damage or reliability analysis.

Matrix-valued isotropic covariance functions with local extrema

Multivariate random fields are commonly used in spatial statistics and natural science to model coregionalized variables. In this context, the matrix-valued covariance function plays a central role in capturing their spatial continuity and interdependence. This study aims to contribute to the literature on covariance modeling by proposing new parametric families of isotropic matrix-valued functions exhibiting non-monotonic behaviors, namely hole effects and cross-dimples. The benefit of the proposed models is shown on a bivariate data set consisting of concentrations of airborne particulate matter.

Exploring the stability of unsaturated soil slope under rainfall infiltration conditions: A study based on multivariate interrelated random fields using R-vine copula

Estimating the instability of soil slopes due to rainfall in highly heterogeneous materials poses a considerable challenge. In this study, a general framework for coupled hydro-mechanical modeling of rainfall-induced instability in unsaturated slopes with multivariate random fields is developed. The R-vine copula is introduced to simulate the intricate non-Gaussian dependencies between soil parameters. UMAP is utilized for visualizing these dependencies. The study compares the fitting performances of R-vine and Gaussian copulas using LOWESS Regression. Subsequently, deterministic model computations are conducted in Abaqus, along with batch random field model analysis based on the R-vine copula. The instability probability of soil slopes under rainfall infiltration conditions is evaluated through direct Monte Carlo simulations, and statistically investigate the relationships between groundwater level, sliding volume, plastic zone volume, and safety factor. The findings indicate that: 1) The R-vine copula model proficiently captures the non-Gaussian dependencies in soil data, demonstrating superior fitting performance over the Gaussian copula; 2) deterministic simulations might overestimate the safety factor at certain instances; 3) as rainfall progresses, a growing negative correlation is observed between groundwater levels and slope instability; 4) continuous rainfall leads to a re-equilibration of the sliding mass, heightening the influence of the sliding volume on stability.

Modeling tensile failure of concrete considering multivariate correlated random fields of material parameters

This paper presents a novel approach for modeling the tensile failure of quasi-brittle materials by incorporating a multivariate random field to represent material parameters in the phase field model. The aim is to characterize and propagate uncertainties in the behaviors of materials, taking into account the spatial variability and probabilistic dependence. Copula theory is employed to investigate the probability distributions and dependence configuration of the parameters. The proposed framework combines the generation approach for the multivariate random field with the phase field model, resulting in a complete numerical analysis methodology. The governing equations of the deterministic phase field model for quasi-brittle solids are outlined, followed by a description of the multivariate random field for quasi-brittle media and its uncertainty characterization procedure using copula theory. A numerical approach for generating samples of the multivariate random field is introduced. The numerical analysis procedure for the tensile fracture of concrete is presented, and the effects of material uncertainties on failure patterns and macroscopic responses are discussed. The results demonstrate that the developed methodology effectively captures the random damage and fracture processes in concrete specimens. The interaction between the correlation length of the random field and the characteristic length scale of the phase field significantly influences the probabilistic characteristics of the random responses. The research findings emphasize the importance of determining the correlation length values accurately. This study contributes to the field of structural reliability analysis by providing a comprehensive framework for modeling the stochastic behavior of quasi-brittle materials. The integration of a multivariate random field and copula theory allows for the realistic representation of material uncertainties and their probabilistic dependence. The proposed methodology, combined with the probability density evolution method (PDEM), offers a powerful tool for analyzing and predicting the probabilistic evolution of material behaviors under tensile loading conditions.

Probabilistic Analysis of Passive Trapdoor in c-ϕ Soil Considering Multivariate Cross-Correlated Random Fields

Probabilistic Analysis of Passive Trapdoor in c-ϕ Soil Considering Multivariate Cross-Correlated Random Fields

An extended two-scale random field model for stochastic response analysis and its application to RC Short-leg shear wall structure

For the stochastic response analysis of concrete structures, it is necessary to model the heterogeneous concrete with significant spatial variability by random fields. In practical applications, multiple material parameters are typically required to characterize the nonlinear constitutive relationship of concrete. As a result, the material parameters of concrete will be represented by a multivariate random field in the nonlinear stochastic response analysis. Nevertheless, a vast number of random variables are frequently involved in representing the multivariate random field, and it is difficult to quantify the probabilistic dependence among the material parameters. To address the preceding issues, this paper proposes an extended two-scale random field model (ETSRF) to quantify the spatial variability of the constitutive relationship of concrete. The model can accurately depict the fluctuation of material properties at the micro-meso scale and the spatial correlation at the macro scale. Besides, it is proved that there is a linear relationship between the mesoscopic random field and ETSRF. Furthermore, a new framework of stochastic response analysis is proposed, where the probability density evolution method (PDEM) is applied to obtain the probabilistic information of the structural responses, and ETSRF is adopted to describe the spatial variability of the material properties. Finally, the proposed model is validated by two illustrative examples, one of which simulates the random fields of compressive and tensile strength of a plain concrete slab, and the other of which deals with the stochastic response analysis of a RC short-leg shear wall structure. The results indicate that the proposed model can depict the spatial variability of the nonlinear constitutive relationship of concrete effectively.

Non-parametric generation of multivariate cross-correlated random fields directly from sparse measurements using Bayesian compressive sensing and Markov chain Monte Carlo simulation

Non-parametric generation of multivariate cross-correlated random fields directly from sparse measurements using Bayesian compressive sensing and Markov chain Monte Carlo simulation

Multivariate exponential power Lévy processes and random fields

Two multivariate exponential power Lévy processes are introduced in this paper via compound Poisson representations, which are so termed because their characteristic exponents are the linear combination of certain positive or negative power functions. They can be represented as the time changed multivariate Brownian motions using the time changes that are two exponential power subordinators, respectively. Two multivariate exponential power random fields are also proposed as elliptically contoured random fields, whose finite-dimensional characteristic functions are made up of the exponential power functions and enjoy the infinitely divisible property.

Modified Matrix Decomposition Method for Generating Multivariate Three-Dimensional Cross-Correlated Random Fields

Modified Matrix Decomposition Method for Generating Multivariate Three-Dimensional Cross-Correlated Random Fields

Multivariate geometric anisotropic Cox processes

AbstractThis paper introduces a new modeling and inference framework for multivariate and anisotropic point processes. Building on recent innovations in multivariate spatial statistics, we propose a new family of multivariate anisotropic random fields, and from them a family of anisotropic point processes. We give conditions that make the proposed models valid. We also propose a Palm likelihood‐based inference method for this type of point process, circumventing issues of likelihood tractability. Finally we illustrate the utility of the proposed modeling framework by analyzing spatial ecological observations of plants and trees in the Barro Colorado Island data.

Covariance models for multivariate random fields resulting from pseudo cross-variograms

So far, the pseudo cross-variogram is primarily used as a tool for the structural analysis of multivariate random fields. Mainly applying recent theoretical results on the pseudo cross-variogram, we use it as a cornerstone in the construction of valid covariance models for multivariate random fields. In particular, we extend known univariate constructions to the multivariate case, and generalize existing multivariate models. Furthermore, we provide a general construction principle for conditionally negative definite matrix-valued kernels, which we use to reinterpret previous modeling proposals.

Characterization theorems for pseudo cross-variograms

AbstractPseudo cross-variograms appear naturally in the context of multivariate Brown–Resnick processes, and are a useful tool for analysis and prediction of multivariate random fields. We give a necessary and sufficient criterion for a matrix-valued function to be a pseudo cross-variogram, and further provide a Schoenberg-type result connecting pseudo cross-variograms and multivariate correlation functions. By means of these characterizations, we provide extensions of the popular univariate space–time covariance model of Gneiting to the multivariate case.

Test and Visualization of Covariance Properties for Multivariate Spatio-Temporal Random Fields

The prevalence of multivariate space-time data collected from monitoring networks and satellites, or generated from numerical models, has brought much attention to multivariate spatio-temporal statistical models, where the covariance function plays a key role in modeling, inference, and prediction. For multivariate space-time data, understanding the spatio-temporal variability, within and across variables, is essential in employing a realistic covariance model. Meanwhile, the complexity of generic covariances often makes model fitting very challenging, and simplified covariance structures, including symmetry and separability, can reduce the model complexity and facilitate the inference procedure. However, a careful examination of these properties is needed in real applications. In the work presented here, we formally define these properties for multivariate spatio-temporal random fields and use functional data analysis techniques to visualize them, hence, providing intuitive interpretations. We then propose a rigorous rank-based testing procedure to conclude whether the simplified properties of covariance are suitable for the underlying multivariate space-time data. The good performance of our method is illustrated through synthetic data, for which we know the true structure. We also investigate the covariance of bivariate wind speed, a key variable in renewable energy, over a coastal and an inland area in Saudi Arabia. The supplementary material is available online, including the R code for our developed methods.

A Bayesian model for multivariate discrete data using spatial and expert information with application to inferring building attributes

When modeling sparsely observed multivariate data, strong prior information elicited from experts can be used to bolster predictive accuracy and counteract sampling bias. Similarly, modeling autocorrelation in space can help make use of co-occurrence patterns present in many types of spatial data. To make use of both expert prior information and spatial structure, we propose a novel graphical model for a spatial Bayesian network developed specifically to address challenges in inferring the attributes of buildings from geographically sparse observational data. This model is implemented as the sum of a spatial multivariate Gaussian random field and a tabular conditional probability function in real-valued space prior to projection onto the probability simplex. This modeling form is especially suitable for the usage of prior information in the form of sets of atomic rules obtained from experts. To perform inference with missing data, we implement a Markov chain Monte Carlo scheme composed of alternating steps of Gibbs sampling of missing entries and Hamiltonian Monte Carlo for model parameters. A case study in building attribution is presented to highlight the advantages and limitations of this approach.

Simulation of Cross-Correlated Random Fields for Transversely Anisotropic Soil Slope by Copulas

Multi-source uncertainties yielded by randomness, spatial variability and cross-correlation of soil parameters severely affect the realization of random fields. However, current studies rarely account for these simultaneously, leading to inevitable bias in random field simulation and subsequent structure analysis. In this paper, copula-based cross-correlated random fields for transversely anisotropic soil slope are proposed. Firstly, based on the traditional probabilistic method and random field theory, the effect of the cross-correlation of soil parameters on the random field is comprehensively analyzed. Then copulas, which mainly characterize the dependent structures of random variables, are further expanded to connect multivariate random fields. Four special algorithms associated with Gaussian, Frank, Plackett and No. 16 copulas are subsequently developed. At last, the performance and effectiveness of copula-based cross-correlated random fields are illustrated by means of assumed and engineering slope cases. The results show that the proposed method is amenable to characterizing spatial variability comprising multiple cross-correlated soil parameters of transversely anisotropic slope. Soil profiles can be represented with a relatively high accuracy. Moreover, the performance of copula-based CCRF is simultaneously governed by margins, cross-correlated coefficients and copulas. The proper selection of these crucial factors can considerably reduce multi-source uncertainties. Overall, the proposed method could provide a useful guideline for accurately modeling cross-correlation random fields of soil slope.