Bayesian Inference and Prediction of Gaussian Random Fields Based on Censored Data

This work develops a Bayesian approach to perform inference and prediction in Gaussian random fields based on spatial censored data. These type of data occur often in the earth sciences due either to limitations of the measuring device or particular features of the sampling process used to collect the data. Inference and prediction on the underlying Gaussian random field is performed, through data augmentation, by using Markov chain Monte Carlo methods. Previous approaches to deal with spatial censored data are reviewed, and their limitations pointed out. The proposed Bayesian approach is applied to a spatial dataset of depths of a geologic horizon that contains both left- and right-censored data, and comparisons are made between inferences based on the censored data and inferences based on “complete data” obtained by two imputation methods. It is seen that the differences in inference between the two approaches can be substantial.     

On Fliess operators driven by L 2-Itô processes

Fliess operators, which are a type of functional series expansion, have been used to describe a broad class of nonlinear input–output maps driven by deterministic inputs. But in most applications, a system's inputs have noise components. This paper has three objectives. The first objective is to show that the notion of a Fliess operator can be generalized to admit a class of L ₂-Itô stochastic input processes. The next objective is to show that they converge absolutely over an arbitrarily large but finite time interval when a certain coefficient growth condition is met. However, a significant number of systems fail to meet this condition. Thus, the final objective is to consider an interval of convergence having a random length so that a Fliess operator might converge under less restrictive growth conditions.     

Identification of input random field samples causing extreme responses

Consider a physical system modeled by a differential equation that depends on a coefficient random field. The objective of this work is to identify samples of this random field which yield extreme response as a means to: study the law of the input conditioned on rare events and predict if a random field sample causes such an event. This differs from reliability engineering which focuses on computation of failure probabilities. We investigate two classification schemes that identify these samples of interest: physics-based indicators which are functionals of the input random field and surrogate models which approximate the response. As an alternative to these approaches, we propose a general framework consisting of two stages that combines the use of a physics-based surrogate model and a machine learning classifier. In the first stage, a multifidelity surrogate that requires infrequent evaluations of the full model is designed. This surrogate is then used to generate a sufficient number of samples of random fields that yield extreme events to train a machine learning classifier in the second stage. We study the analytical properties required of the surrogate model and demonstrate through numerical examples the synergy of the proposed approach.     

Modelling of spatial variability of soil undrained shear strength by conditional random fields for slope reliability analysis

Conditional random field model can make best use of limited site investigation data to properly characterize the spatial variation of soil properties. This paper aims to propose a simplified approach for generating conditional random fields of soil undrained shear strength. A numerical method is adopted to validate the effectiveness of the proposed approach. With the proposed approach, the analytical posterior statistics of spatially varying undrained shear strength conditioned on the known values at measurement locations can be obtained. The conditional random field model of undrained shear strength is constructed using the field vane shear test data at a site of the west side highway in New York and the probability of slope failure is estimated by subset simulation. A clay slope under undrained conditions is investigated as an example to illustrate the proposed approach. The effects of borehole location and borehole layout scheme on the slope reliability are addressed. The results indicate that the proposed approach not only can well incorporate the limited site investigation data into modelling of the actual spatial variation of soil parameters by conditional random fields, but also can capture the depth-dependent nature of soil properties. The realizations of conditional random fields generated by the proposed approach can be well constrained to the site investigation data.     

Influence of measurement data metrics on the identification of Interval Fields for the representation of spatial variability in finite element models

Due to the exponential growth in computing power, numerical modelling techniques method have gained an increasing amount of interest for engineering and design applications. Nowadays, the deterministic finite element (FE) method, an efficient tool to accurately solve the Partial Differential Equations (PDE) that govern most real-world problems, has become an indispensable tool for an engineer in various design stages. A more recent trend herein is to use the ever increasing computing power incorporate uncertainty and variability, which is omnipresent is all real-live applications, into these FE models. Several advanced techniques for incorporating either variability between nominally identical parts or spatial variability within one part into the FE models, have been introduced in this context. For the representation of spatial variability on the parameters of an FE model in a possibilistic context, the theory of Interval Fields (IF) was proven to show promising results. Following this approach, variability in the input FE model is introduced as the superposition of base vectors, depicting the spatial ‘patterns’, which are scaled by interval factors, which represent the actual variability. Application of this concept, however, requires identification of the governing parameters of these interval fields, i. e. the base vectors and interval scalars. Recent work of the authors therefore was focussed on finding a solution to the inverse problem, where the spatial uncertainty on the output side of the model is known from measurement data, but the spatial variability on the input parameters is unknown. Based on an a priori knowledge on the constituting base vectors of the interval field, the simulated output of the IFFEM computation is compared to measured data, and the input parameters are iteratively adjusted in order to minimize the discrepancy between the variability in simulation and measurement data. This discrepancy is defined based on geometric properties of the convex sets of both measurement and simulation data. However, the robustness of this methodology with respect to the size of the measurement data set that is used for the identification, as yet remains unclear. This paper therefore is focussed on the investigation of this robustness, by performing the identification on different measurement sets, depicting the same variability in the dynamic response of a simple FE model, which contain a decreasing amount of measurement replica. It was found that accurate identification remains feasible, even under a limited amount of measurement replica, which is highly relevant in the context of a non-probabilistic representation of variability in the FE model parameters. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Hedging of Spatial Temperature Risk with Market-Traded Futures

Abstract

The main objective of this work is to construct optimal temperature futures from available market-traded contracts to hedge spatial risk. Temperature dynamics are modelled by a stochastic differential equation with spatial dependence. Optimal positions in market-traded futures minimizing the variance are calculated. Examples with numerical simulations based on a fast algorithm for the generation of random fields are presented.     

Bayesian Analysis of Geostatistical Models With an Auxiliary Lattice

The Gaussian geostatistical model has been widely used for modeling spatial data. However, this model suffers from a severe difficulty in computation: it requires users to invert a large covariance matrix. This is infeasible when the number of observations is large. In this article, we propose an auxiliary lattice-based approach for tackling this difficulty. By introducing an auxiliary lattice to the space of observations and defining a Gaussian Markov random field on the auxiliary lattice, our model completely avoids the requirement of matrix inversion. It is remarkable that the computational complexity of our method is only O(n), where n is the number of observations. Hence, our method can be applied to very large datasets with reasonable computational (CPU) times. The numerical results indicate that our model can approximate Gaussian random fields very well in terms of predictions, even for those with long correlation lengths. For real data examples, our model can generally outperform conventional Gaussian random field models in both prediction errors and CPU times. Supplemental materials for the article are available online.     

A robust data envelopment analysis model with different scenarios

Input and output data, under uncertainty, must be taken into account as an essential part of data envelopment analysis (DEA) models in practice. Many researchers have dealt with this kind of problem using fuzzy approaches, DEA models with interval data or probabilistic models. This paper presents an approach to scenario-based robust optimization for conventional DEA models. To consider the uncertainty in DEA models, different scenarios are formulated with a specified probability for input and output data instead of using point estimates. The robust DEA model proposed is aimed at ranking decision-making units (DMUs) based on their sensitivity analysis within the given set of scenarios, considering both feasibility and optimality factors in the objective function. The model is based on the technique proposed by Mulvey et al. (1995) for solving stochastic optimization problems. The effect of DMUs on the product possibility set is calculated using the Monte Carlo method in order to extract weights for feasibility and optimality factors in the goal programming model. The approach proposed is illustrated and verified by a case study of an engineering company.     

基于Monte-Carlo随机有限元方法的随机边界条件下自然对流换热不确定性研究

为分析边界条件不确定性对方腔内自然对流换热的影响,发展了一种求解随机边界条件下自然对流换热不确定性传播的Monte-Carlo随机有限元方法.通过对输入参数场随机边界条件进行Karhunen-Loeve展开及基于Latin(拉丁)抽样法生成边界条件随机样本,数值计算了不同边界条件随机样本下方腔内自然对流换热流场与温度场,并用采样统计方法计算了随机输出场的平均值与标准偏差.根据计算框架编写了求解随机边界条件下方腔内自然对流换热不确定性的MATLAB随机有限元程序,分析了随机边界条件相关长度与方差对自然对流不确定性的影响.结果表明:平均温度场及流场与确定性温度场及流场分布基本相同;随机边界条件下Nu数概率分布基本呈现正态分布,平均Nu数随着相关长度和方差增加而增大;方差对自然对流换热的影响强于相关长度的影响.     

Spatial Expectile Predictions for Elliptical Random Fields

In this work, we consider an elliptical random field. We propose some spatial expectile predictions at one site given observations of the field at some other locations. To this aim, we first give exact expressions for conditional expectiles, and discuss problems that occur for computing these values. A first affine expectile regression predictor is detailed, an explicit iterative algorithm is obtained, and its distribution is given. Direct simple expressions are derived for some particular elliptical random fields. The performance of this expectile regression is shown to be very poor for extremal expectile levels, so that a second predictor is proposed. We prove that this new extremal prediction is asymptotically equivalent to the true conditional expectile. We also provide some numerical illustrations, and conclude that Expectile Regression may perform poorly when one leaves the Gaussian random field setting.     

Structural optimization of an acoustic horn

The objective of the present article is to find an optimal design of an acoustic horn in the case that the magnitude of the reflection wave integrated over the inflow boundary is to be minimized meanwhile the Helmholtz equation models the wave propagation. In contrast to the current approaches such as gradient-based optimization algorithms, we employ here a non-iterative method based on measure theory which dose not require any information of gradients and the differentiability of objective function in the optimization problem is not as a rule. Implementation of our fast convergence approach shows that the resulting horns, not only for single frequency optimization but also for a band of frequencies, are very efficient.     

Regularly varying random fields

We study the extremes of multivariate regularly varying random fields. The crucial tools in our study are the tail field and the spectral field, notions that extend the tail and spectral processes of Basrak and Segers (2009). The spatial context requires multiple notions of extremal index, and the tail and spectral fields are applied to clarify these notions and other aspects of extremal clusters. An important application of the techniques we develop is to the Brown–Resnick random fields.     

Computing Scan Statistic p Values Using Importance Sampling,With Applications to Genetics and Medical Image Analysis

We present an importance sampling method for deciding, based on an observed random field, if a scan statistic provides significant evidence of increased activity in some localized region of time or space. Our method allows consideration of scan statistics based simultaneously on multiple scan geometries. Our approach yields an unbiased p value estimate whose variance is typically smaller than that of the naive hit-or-miss Monte Carlo technique when the p value is small. Furthermore, our p value estimate is often accurate for critical values that are not far enough in the tails of the null distribution to allow for accurate approximations via extreme value theory. The importance sampling approach unifies the analysis of various random field models, from (spatial) point processes to Gaussian random fields. For a scan statistic M, the method produces a p value of the form P[M ≥ τ] = Bρ, where B is the Bonferroni upper bound and the correction factor ρ measures the conservativeness of this upper bound. We present the application of our importance sampling estimator to multinomial sequences (molecular genetics), spatial point processes (digital mammography), and Gaussian random fields (PET scan brain imagery).     

A numerical model for magnetic chromatography

In this paper, we present details of a mathematical model for magnetic chromatography (MC) systems where strong distorted magnetic fields are used to separate particles from a colloidal mixture. The model simulates the effect of magnetic field gradients on particle motion, and includes calculation of the fluid flow, magnetic field, and particle concentration field. It is based on the finite-volume method (FVM) and uses an expanding-grid technique to handle domains with large aspect ratios. The model has been validated against the results from an analytical model. The numerical model has been used to simulate the performance of a real MC system under various operating conditions.     

Spectral representation and asymptotic properties of certain deterministic fields with innovation components

In this paper we derive the spectral and ergodic properties of a special class of homogeneous random fields, which includes an important family of evanescent random fields. Based on a derivation of the resolution of the identity for the operators generating the homogeneous field, and using the properties of measurable transformations, the spectral representation of both the field and its covariance sequence are derived. A necessary and sufficient condition for the existence of such representation is introduced. Using an analysis approach that employs the solution to the linear Diophantine equations, further characterization and modeling of the spectral properties of evanescent fields are provided by considering their spectral pseudo-density function, defined in this paper. The geometric properties of the spectral pseudo-density of the evanescent field are investigated. Finally, necessary and sufficient conditions for mean and strong ergodicity of the first and second order moments of these fields are derived. The analysis, initially carried out for complex valued random fields, is later extended to include the case of real valued fields.This work was supported in part by the EU 5th Framework IHP Program, MOUMIR Project, under Grant RTN-1999-0177. Mathematics Subject Classification (2000):62M40, 62J05     

Fractional-order regularization and wavelet approximation to the inverse estimation problem for random fields

The least-squares linear inverse estimation problem for random fields is studied in a fractional generalized framework. First, the second-order regularity properties of the random fields involved in this problem are analysed in terms of the fractional Sobolev norms. Second, the incorporation of prior information in the form of a fractional stochastic model, with covariance operator bicontinuous with respect to a certain fractional Sobolev norm, leads to a regularization of this problem. Third, a multiresolution approximation to the class of linear inverse problems considered is obtained from a wavelet-based orthogonal expansion of the input and output random models. The least-squares linear estimate of the input random field is then computed using these orthogonal wavelet decompositions. The results are applied to solving two important cases of linear inverse problems defined in terms of fractional integral operators.     

Spatiotemporal Models for Gaussian Areal Data

We introduce a class of spatiotemporal models for Gaussian areal data. These models assume a latent random field process that evolves through time with random field convolutions; the convolving fields follow proper Gaussian Markov random field (PGMRF) processes. At each time, the latent random field process is linearly related to observations through an observational equation with errors that also follow a PGMRF. The use of PGMRF errors brings modeling and computational advantages. With respect to modeling, it allows more flexible model structures such as different but interacting temporal trends for each region, as well as distinct temporal gradients for each region. Computationally, building upon the fact that PGMRF errors have proper density functions, we have developed an efficient Bayesian estimation procedure based on Markov chain Monte Carlo with an embedded forward information filter backward sampler (FIFBS) algorithm. We show that, when compared with the traditional one-at-a-time Gibbs sampler, our novel FIFBS-based algorithm explores the posterior distribution much more efficiently. Finally, we have developed a simulation-based conditional Bayes factor suitable for the comparison of nonnested spatiotemporal models. An analysis of the number of homicides in Rio de Janeiro State illustrates the power of the proposed spatiotemporal framework.

Supplemental materials for this article are available online in the journal’s webpage.     

Structural reliability analysis with imprecise random and interval fields

This paper investigates the issue of reliability assessment for engineering structures involving mixture of stochastic and non-stochastic uncertain parameters through the Finite Element Method (FEM). Non-deterministic system inputs modelled by both imprecise random and interval fields have been incorporated, so the applicability of the structural reliability analysis scheme can be further promoted to satisfy the intricate demand of modern engineering application. The concept of robust structural reliability profile for systems involving hybrid uncertainties is discussed, and then a new computational scheme, namely the unified interval stochastic reliability sampling (UISRS) approach, is proposed for assessing the safety of engineering structures. The proposed method provides a robust semi-sampling scheme for assessing the safety of engineering structures involving multiple imprecise random fields with various distribution types and interval fields simultaneously. Various aspects of structural reliability analysis with multiple imprecise random and interval fields are explored, and some theoretically instructive remarks are also reported herein.     

Image classification based on Markov random field models with Jeffreys divergence

This paper considers image classification based on a Markov random field (MRF), where the random field proposed here adopts Jeffreys divergence between category-specific probability densities. The classification method based on the proposed MRF is shown to be an extension of Switzer's soothing method, which is applied in remote sensing and geospatial communities. Furthermore, the exact error rates due to the proposed and Switzer's methods are obtained under the simple setup, and several properties are derived. Our method is applied to a benchmark data set of image classification, and exhibits a good performance in comparison with conventional methods.