Non-stationary Random Field Research Articles

A statistical modeling approach for air quality data based on physical dispersion processes and its application to ozone modeling

For many complex environmental processes such as air pollution, the underlying physical mechanism usually provides valuable insights into the statistical modeling. In this paper, we propose a statistical air quality model motivated by a commonly used physical dispersion model, called the scalar transport equation. The emission of a pollutant is modeled by covariates such as land use, traffic pattern and meteorological conditions, while the transport and decay of a pollutant are modeled through a convolution approach which takes into account the dynamic wind field. This approach naturally establishes a nonstationary random field with a space–time nonseparable and anisotropic covariance structure. Note that, due to the extremely complex interactions between the pollutant and environmental conditions, the space–time covariance structure of pollutant concentration data is often dynamic and can hardly be specified or envisioned directly. The relationship between the proposed spatial-temporal model and the physics model is also shown, and the approach is applied to model the hourly ozone concentration data in Singapore.

Asymptotic behaviour of binned kernel density estimators for locally non-stationary random fields

We investigate the asymptotic behaviour of binned kernel density estimators for dependent and locally non-stationary random fields converging to stationary random fields. We focus on the study of the bias and the asymptotic normality of the estimators. A simulation experiment conducted shows that both the kernel density estimator and the binned kernel density estimator have the same behavior and both estimate accurately the true density when the number of fields increases. We apply our results to the 2002 incidence rates of tuberculosis in the departments of France.

High resolution simulation of nonstationary Gaussian random fields

Simulation of random fields is a fundamental requirement for many spatial analyses. For small spatial networks, simulations can be produced using direct manipulations of the covariance matrix. Larger high resolution simulations are most easily available for stationary processes, where algorithms such as circulant embedding can be used to simulate a process at millions of locations. We discuss an approach to simulating high resolution nonstationary Gaussian processes that relies on generating a stationary random field followed by a nonlinear deformation to produce a nonstationary field. A spatially varying variance coefficient accounts for local scale effects. The nonstationary covariance function is estimated nonparametrically, and the deformation function is then estimated in a variational framework. We illustrate the proposed approach on synthetic datasets, a challenging temperature dataset over the state of Colorado and a regional climate model over North America.

A biorthogonal decomposition for the identification and simulation of non-stationary and non-Gaussian random fields

In this paper, a new method for the identification and simulation of non-Gaussian and non-stationary stochastic fields given a database is proposed. It is based on two successive biorthogonal decompositions aiming at representing spatio-temporal stochastic fields. The proposed double expansion allows to build the model even in the case of large-size problems by separating the time, space and random parts of the field. A Gaussian kernel estimator is used to simulate the high dimensional set of random variables appearing in the decomposition. The capability of the method to reproduce the non-stationary and non-Gaussian features of random phenomena is illustrated by applications to earthquakes (seismic ground motion) and sea states (wave heights).

Efficient slope reliability analysis at low-probability levels in spatially variable soils

Direct Monte Carlo simulation for the reliability analysis of slope stability with spatially variable soil properties suffers from a serious lack of efficiency when the probability of failure, pf, is low (e.g., pf<10-6). Based on the multiple response surfaces and Subset simulation, this paper proposes an efficient approach for the estimating of small probabilities of slope failure in spatially variable soils. An improved Cholesky decomposition technique is presented for the simulation of the globally non-stationary random fields of spatially variable soil properties in the multiple soil layers. Two slope examples are investigated to demonstrate the effectiveness of the proposed method. The efficiency of the proposed approach for parametric sensitivity analysis is also highlighted.

Almost Sure Convergence for the Maximum of Nonstationary Random Fields

We obtain an almost sure limit theorem for the maximum of nonstationary random fields under some dependence conditions.

Localized/Shrinkage Kriging Predictors

The objective of the study is to improve the robustness and flexibility of spatial kriging predictors with respect to deviations from spatial stationarity assumptions. A predictor based on a non-stationary Gaussian random field is defined. The model parameters are inferred in an empirical Bayesian setting, using observations in a local neighborhood and a prior model assessed from the global set of observations. The localized predictor exhibits a shrinkage effect and is termed a localized/shrinkage kriging predictor. The predictor is compared to traditional localized kriging predictors in a case study on observations of annual accumulated precipitation. A cross-validation criterion is used in the comparison. The shrinkage predictor appears as clearly preferable to the traditional kriging predictors. A simulation study on prediction in non-stationary Gaussian random fields is conducted. The results from this study confirm that the shrinkage predictor is preferable to the traditional one. Moreover, the cross-validation criterion is found to be suitable for selection of the local neighborhood in the predictor. Lastly, the computational demands of localized predictors are very modest, hence these localized/shrinkage predictors are suitable for large scale spatial prediction problems.

A Central Limit Theorem for Non-stationary Strongly Mixing Random Fields

In this paper we extend a central limit theorem of Peligrad for uniformly strong mixing random fields satisfying the Lindeberg condition in the absence of stationarity property. More precisely, we study the asymptotic normality of the partial sums of uniformly \(\alpha \)-mixing non-stationary random fields satisfying the Lindeberg condition, in the presence of an extra dependence assumption involving maximal correlations.

Does non-stationary spatial data always require non-stationary random fields?

A stationary spatial model is an idealization and we expect that the true dependence structures of physical phenomena are spatially varying, but how should we handle this non-stationarity in practice? We study the challenges involved in applying a flexible non-stationary model to a dataset of annual precipitation in the conterminous US, where exploratory data analysis shows strong evidence of a non-stationary covariance structure.The aim of this paper is to investigate the modelling pipeline once non-stationarity has been detected in spatial data. We show that there is a real danger of over-fitting the model and that careful modelling is necessary in order to properly account for varying second-order structure. In fact, the example shows that sometimes non-stationary Gaussian random fields are not necessary to model non-stationary spatial data.

Rainfall infiltration in a landslide soil deposit: Importance of inverse particle segregation

Coarse particles move towards the front and surface of the deposit, while fine particles stay in the rear and bottom. Accordingly the coefficient of permeability of the soil statistically tends to decrease from the surface to the bottom and from the toe to the apex of the deposit. This feature can be described by trend functions that describe the changes in particle size and permeability inside the soil deposit. In this paper, attempts are made to include the physical background in the characterization of geotechnical uncertainties. The objectives of this paper are to simulate non-stationary fields of particle size and coefficient of permeability in a landslide soil deposit, and to analyse rainfall infiltration in the non-stationary random fields. The soil deposit is assumed to consist of inter-layers parallel to the slope surface. The soil deposit is also assumed to be axisymmetric; therefore, any vertical section of the deposit can be selected to generate two dimensional non-stationary random fields. Each of the non-stationary random fields is formulated by adding a trend surface to a residual stationary random field with orthogonal major and minor scales of fluctuation. The produced random fields follow an exponential correlation structure. Subsequently, steady-state rainfall infiltration in the anisotropic non-stationary heterogeneous random fields is analysed. The effects of variations of saturated permeability and scale of fluctuation on the pore-water pressure regime are also investigated.

Reliability analysis of strip footing considering spatially variable undrained shear strength that linearly increases with depth

This paper aims to investigate the reliability of strip footing in the presence of spatially variable undrained shear strength that linearly increases with depth. A non-stationary random field is used to model the spatially varying undrained shear strength. A strip footing example is presented to investigate the effect of spatially variable undrained shear strength on the performance of strip footing. The results indicate that the mean bearing capacity for spatially variable undrained shear strength is smaller than that obtained from a deterministic analysis. Both the mean and standard deviation of bearing capacity increase with increasing autocorrelation length. Ignoring the trend of undrained shear strength linearly increasing with depth will significantly overestimate the probability of failure of the strip footing. A factor of safety significantly below 3.0 may be used for designing strip footings if the trend of undrained shear strength linearly increasing with depth is considered properly.

Generating non-stationary random fields of auto-correlated, normally distributed CPT profile by matrix decomposition method

This paper emphasises the true realisation of Cone Penetration Test (CPT) profiles considering non-stationary nature of the data. Formulation of stationary random field theory has been modified and adapted to non-stationary state in order to take into account the mean and variance variability for soil properties. Multi-variance correlation matrix along with the Cholesky decomposition technique was employed to produce realisations of non-homogenous and non-stationary random fields of CPT profiles. A piecewise and segmental data realisation according to the lithology and site class specifications acquired directly from CPT data is adopted in this study so as to render an accurate data simulation. For validation of proposed method 8 CPT test profiles collected from Urmia Lake site have been introduced and simulated by the stationary and non-stationary algorithms. The mean correlation coefficient between the actual CPT data profiles and related realisations along with some other important statistical parameters and their coefficients of variation strongly demonstrate that non-stationary random field generation technique gives quite better accuracy, by comparison to the conventional stationary random field generation scheme.

Effects of the spatial variability of permeability on rainfall-induced landslides

The saturated hydraulic conductivity (Ks) of soil is characterized by strong spatial variability and often decreases with depth. The main objective of this paper is to investigate how the trend function of the mean of saturated hydraulic conductivity (μKs) and its spatial variability affect the distribution of critical failure surfaces and the probability of slope failure during a period of rainfall. A random field method is used to simulate the spatial variability structure in the numerical analysis of a slope, and a numerical procedure for a probabilistic slope stability analysis based on Monte Carlo simulations is presented, in which Ks is modeled as a non-stationary lognormal random field. For a hypothetical weathered soil slope subjected to rainfall, a deterministic analysis and a sequence of probabilistic analyses are conducted using an infinite unsaturated slope model. The results show that these shallow failures are attributed to the reduction of matric suction and the development of positive pore pressure. The probability of slope failure increases with an increase in the variation of the trend component (Δk), but the rate of increase levels off at the later stage of rainfall. Ignoring the trend component of μKs would lead to less conservative estimates of the failure probability. The coefficient of variability has a more significant influence on the distribution of the critical failure surfaces and the probability of failure than does the correlation length of Ks.

Spatial reliability analysis of explosive blast load damage to reinforced concrete columns

Columns are the key load-bearing elements in frame structures and exterior columns are probably the most vulnerable structural components to terrorist attack. In this paper, a spatial reliability analysis is conducted to predict the damage for reinforced concrete (RC) columns subject to explosive blast loading. The spatial variability of material and dimensional properties of RC columns are modelled by stationary and non-stationary random fields. The variability of blast loading is also taken into consideration. Monte Carlo simulation and numerical methods are used to derive Blast Reliability Curves for RC columns under explosive loading for a number of terrorism threat scenarios, based on a high–fidelity physics-based computer programme LS-DYNA to estimate design and residual axial load-carrying capacity of RC columns. It was found that spatial variability has a significant effect on structural reliabilities and the spatial model will lead to more accurate predictions of damage and safety risks.

Learning Sparse FRAME Models for Natural Image Patterns

It is well known that natural images admit sparse representations by redundant dictionaries of basis functions such as Gabor-like wavelets. However, it is still an open question as to what the next layer of representational units above the layer of wavelets should be. We address this fundamental question by proposing a sparse FRAME (Filters, Random field, And Maximum Entropy) model for representing natural image patterns. Our sparse FRAME model is an inhomogeneous generalization of the original FRAME model. It is a non-stationary Markov random field model that reproduces the observed statistical properties of filter responses at a subset of selected locations, scales and orientations. Each sparse FRAME model is intended to represent an object pattern and can be considered a deformable template. The sparse FRAME model can be written as a shared sparse coding model, which motivates us to propose a two-stage algorithm for learning the model. The first stage selects the subset of wavelets from the dictionary by a shared matching pursuit algorithm. The second stage then estimates the parameters of the model given the selected wavelets. Our experiments show that the sparse FRAME models are capable of representing a wide variety of object patterns in natural images and that the learned models are useful for object classification.

Fractional Brownian Fields for Response Surface Metamodeling

Kriging, a widely used metamodeling method for computer-simulation data, models the response surface as a realization of a random field. Stationary covariance functions such as the Gaussian, power exponential, or Matérn class are the most common choice for the underlying random field model. Nonstationary versions of these same covariance functions with scale parameters that vary spatially are also sometimes used. Fractional Brownian fields (FBFs) are a different form of nonstationary random field model having stationary increments, an example of more general intrinsic stationary processes. Although FBFs have been considered for intrinsic kriging in the spatial statistics literature, they have received little attention for computer-simulation response surface metamodeling. For use in the latter context, we argue that they have some attractive (as well as some unattractive) properties that mitigate certain problems inherent to many stationary covariance models, such as reversion to the mean, numerical issues due to near-singularity of covariance matrices, and difficulties in handling abrupt response surface features.

Almost sure asymptotics for extremes of non-stationary Gaussian random fields

In this paper, the authors prove an almost sure limit theorem for the maxima of non-stationary Gaussian random fields under some mild conditions related to the covariance functions of the Gaussian fields. As the by-products, the authors also obtain several weak convergence results which extended the existing results.

Exploring a New Class of Non-stationary Spatial Gaussian Random Fields with Varying Local Anisotropy

Gaussian random elds (GRFs) play an important part in spatial mod- elling, but can be computationally infeasible for general covariance structures. An ecient approach is to specify GRFs via stochastic partial dierential equations (SPDEs) and derive Gaussian Markov random eld (GMRF) approximations of the solutions. We consider the construction of a class of non-stationary GRFs with varying local anisotropy, where the local anisotropy is introduced by allowing the coecients in the SPDE to vary with position. This is done by using a form of diusion equation driven by Gaussian white noise with a spatially varying diu- sion matrix. This allows for the introduction of parameters that control the GRF by parametrizing the diusion matrix. These parameters and the GRF may be considered to be part of a hierarchical model and the parameters estimated in a Bayesian framework. The results show that the use of an SPDE with non-constant coecients is a promising way of creating non-stationary spatial GMRFs that allow for physical interpretability of the parameters, although there are several remaining challenges that would need to be solved before these models can be put to general practical use.

Ubiquity of, and geostatistics for, nonstationary increment random fields

[1] Nonstationary random fields such as fractional Brownian motion and fractional Levy motion have been studied extensively in the hydrology literature. On the other hand, random fields that have nonstationary increments have seen little study. A mathematical argument is presented that demonstrates processes with stationary increments are the exception and processes with nonstationary increments are far more abundant. The abundance of nonstationary increment processes has important implications, e.g., in kriging where a translation-invariant variogram implicitly assumes stationarity of the increments. An approach to kriging for processes with nonstationary increments is presented and accompanied by some numerical results.

Statistical analysis of high density diffuse optical tomography

High density diffuse optical tomography (HD-DOT) is a noninvasive neuroimaging modality with moderate spatial resolution and localization accuracy. Due to portability and wear-ability advantages, HD-DOT has the potential to be used in populations that are not amenable to functional magnetic resonance imaging (fMRI), such as hospitalized patients and young children. However, whereas the use of event-related stimuli designs, general linear model (GLM) analysis, and imaging statistics are standardized and routine with fMRI, such tools are not yet common practice in HD-DOT. In this paper we adapt and optimize fundamental elements of fMRI analysis for application to HD-DOT. We show the use of event-related protocols and GLM de-convolution analysis in un-mixing multi-stimuli event-related HD-DOT data. Statistical parametric mapping (SPM) in the framework of a general linear model is developed considering the temporal and spatial characteristics of HD-DOT data. The statistical analysis utilizes a random field noise model that incorporates estimates of the local temporal and spatial correlations of the GLM residuals. The multiple-comparison problem is addressed using a cluster analysis based on non-stationary Gaussian random field theory. These analysis tools provide access to a wide range of experimental designs necessary for the study of the complex brain functions. In addition, they provide a foundation for understanding and interpreting HD-DOT results with quantitative estimates for the statistical significance of detected activation foci.