Karhunen–Loève expansion of Spartan spatial random fields

Random fields (RFs) are important tools for modeling space–time processes and data. The Karhunen–Loève (K–L) expansion provides optimal bases which reduce the dimensionality of random field representations. However, explicit expressions for K–L expansions only exist for a few, one-dimensional, two-parameter covariance functions. In this paper we derive the K–L expansion of the so-called Spartan spatial random fields (SSRFs). SSRF covariance functions involve three parameters including a rigidity coefficient η₁, a scale coefficient, and a characteristic length. SSRF covariances include both monotonically decaying and damped oscillatory functions; the latter are obtained for negative values of η₁. We obtain the eigenvalues and eigenfunctions of the SSRF K–L expansion by solving the associated homogeneous Fredholm equation of the second kind which leads to a fourth order linear ordinary differential equation. We investigate the properties of the solutions, we use the derived K–L base to simulate SSRF realizations, and we calculate approximation errors due to truncation of the K–L series.     

Cohesive crack propagation in a random elastic medium

The issue of generating non-Gaussian, multivariate and correlated random fields, while preserving the internal auto-correlation structure of each single-parameter field, is discussed with reference to the problem of cohesive crack propagation. Three different fields are introduced to model the spatial variability of the Young modulus, the tensile strength of the material, and the fracture energy, respectively. Within a finite-element context, the crack-propagation phenomenon is analyzed by coupling a Monte Carlo simulation scheme with an iterative solution algorithm based on a truly-mixed variational formulation which is derived from the Hellinger–Reissner principle. The selected approach presents the advantage of exploiting the finite-element technology without the need to introduce additional modes to model the displacement discontinuity along the crack boundaries. Furthermore, the accuracy of the stress estimate pursued by the truly-mixed approach is highly desirable, the direction of crack propagation being determined on the basis of the principal-stress criterion. The numerical example of a plain concrete beam with initial crack under a three-point bending test is considered. The statistics of the response is analyzed in terms of peak load and load–mid-deflection curves, in order to investigate the effects of the uncertainties on both the carrying capacity and the post-peak behaviour. A sensitivity analysis is preliminarily performed and its results emphasize the negative effects of not accounting for the auto-correlation structure of each random field. A probabilistic method is then applied to enforce the auto-correlation without significantly altering the target marginal distributions. The novelty of the proposed approach with respect to other methods found in the literature consists of not requiring the a priori knowledge of the global correlation structure of the multivariate random field.     

How to simulate normal data sets with the desired correlation structure

The Cholesky decomposition is a widely used method to draw samples from multivariate normal distribution with non-singular covariance matrices. In this work we introduce a simple method by using singular value decomposition (SVD) to simulate multivariate normal data even if the covariance matrix is singular, which is often the case in chemometric problems. The covariance matrix can be specified by the user or can be generated by specifying a subset of the eigenvalues. The latter can be an advantage for simulating data sets with a particular latent structure. This can be useful for testing the performance of chemometric methods with data sets matching the theoretical conditions for their applicability; checking their robustness when the hypothesized properties fail; or generating data from multi-stage or multi-phase processes.     

A biplot method for multivariate normal populations with unequal covariance matrices

Some previous ideas about non-linear biplots to achieve a joint representation of multivariate normal populations and any parametric function without assumptions about the covariance matrices are extended. Usual restrictions on the covariance matrices (such as homogeneity) are avoided. Variables are represented as curves corresponding to the directions of maximum means variation. To demonstrate the versatility of the method, the representation of variances and covariances as an example of further possible interesting parametric functions have been developed. This method is illustrated with two different data sets, and these results are compared with those obtained using two other distances for the normal multivariate case: the Mahalanobis distance (assuming a common covariance matrix for all populations) and Rao’s distance, assuming a common eigenvector structure for all the covariance matrices. This work is supported by DGICYT grant (Spain), BFM2000-0801 and also 1999SGR00059.     

A meshfree-Galerkin method in modelling and synthesizing spatially varying soil properties

Physical properties of soil vary from point to point in space and exhibit great uncertainty, suggesting random field as a natural approach in modelling and synthesizing these properties. The significance of considering spatial variability and uncertainty of soil properties is greatly manifested in the probabilistic seismic risk analysis of soil–structural system (nonlinear dynamic analysis under earthquake loading), where modelling and synthesis of the spatial variability and uncertainty of soil properties are necessary. This paper introduces a meshfree-Galerkin approach within the Karhunen–Loève (K–L) expansion scheme for representation of spatial soil properties modelled as the random fields. The meshfree shape functions are introduced and employed as a set of basis functions in the Galerkin scheme to obtain the eigen-solutions of integral equation of K–L expansion. An optimization scheme is proposed for the resulting eigenvectors in treating the compatibility between the target and analytical covariance models. Assessments of the meshfree-Galerkin method are conducted for the resulting eigen-solutions and the representation of covariance models for various homogeneous and nonhomogeneous random fields. The accuracy and validity of the proposed approach are demonstrated through the modelling and synthesis of the spatial field models inferred from the field measurements.     

Model-Based Sampling Design for Multivariate Geostatistics

The quality of inferences made from geostatistical data is affected significantly by the spatial locations, or design, of the sites that are sampled. A large body of published work exists on sampling design for univariate geostatistics, but not for multivariate geostatistics. This article considers multivariate spatial sampling design based on criteria targeted at classical co-kriging (prediction with known covariance parameters), estimation of covariance (including cross-covariance) parameters, and empirical co-kriging (prediction with estimated covariance parameters). Through a combination of analytical results and examples, we investigate the characteristics of optimal designs with respect to each criterion, addressing in particular the design’s degree of collocation. We also consider the robustness of the optimal design to the strength of spatial correlation and cross-correlation; the effects of smoothness and/or separability of the sampled process on the optimal design; the relationship between optimal designs for the multivariate problems considered here and univariate problems considered previously; and the efficiency of optimal collocated designs. One key finding is that optimal collocated designs are highly efficient in many cases. Supplementary materials are available online.     

A spatial rank–based multivariate EWMA chart for monitoring process shape matrices

In this article, we propose a nonparametric EWMA control chart for monitoring the shape matrix of a multivariate process based on a spatial rank test and the exponentially weighted moving average scheme. The proposed control chart is essentially developed using an estimated spatial rank covariance matrix to test the shape matrix of the covariance matrix of multivariate distributions with heavy tails. Based on our simulation studies, the proposed control chart outperforms the only existing nonparametric control chart in many practical out‐of‐control scenarios for monitoring the shape matrix of the covariance matrix of many multivariate processes. Further, we point out the weaknesses of both the nonparametric EWMA control charts for monitoring the shape matrix of multivariate processes in real applications and propose one possible method to overcome these weaknesses. We also use an example from a white wine production process to demonstrate the applicability and implementation of the proposed control chart.     

Variation Charts for Multivariate Processes

Hotelling's T² is customarily used as the control chart for multivariate SPC analysis. This chart responds to changes in both the mean values and the covariance matrix of the responses. In this article, we propose the use of a chart that concentrates on changes in the covariance matrix. The use of this covariance chart in concert with the T² chart enables the user to better determine whether T² points out of control are due to changes in mean values or due to changes in the covariance matrix. Using this chart in conjunction with T² thus furnishes a suite of tools similar to the x-bar and standard deviation charts for univariate processes.     

A Fast Algorithm for the Minimum Covariance Determinant Estimator

The minimum covariance determinant (MCD) method of Rousseeuw is a highly robust estimator of multivariate location and scatter. Its objective is to find h observations (out of n) whose covariance matrix has the lowest determinant. Until now, applications of the MCD were hampered by the computation time of existing algorithms, which were limited to a few hundred objects in a few dimensions. We discuss two important applications of larger size, one about a production process at Philips with n = 677 objects and p = 9 variables, and a dataset from astronomy with n = 137,256 objects and p = 27 variables. To deal with such problems we have developed a new algorithm for the MCD, called FAST-MCD. The basic ideas are an inequality involving order statistics and determinants, and techniques which we call “selective iteration” and “nested extensions.” For small datasets, FAST-MCD typically finds the exact MCD, whereas for larger datasets it gives more accurate results than existing algorithms and is faster by orders of magnitude. Moreover, FASTMCD is able to detect an exact fit—that is, a hyperplane containing h or more observations. The new algorithm makes the MCD method available as a routine tool for analyzing multivariate data. We also propose the distance-distance plot (D-D plot), which displays MCD-based robust distances versus Mahalanobis distances, and illustrate it with some examples.     

Spatial connectivity in mosquito-borne disease models: a systematic review of methods and assumptions

Spatial connectivity plays an important role in mosquito-borne disease transmission. Connectivity can arise for many reasons, including shared environments, vector ecology and human movement. This systematic review synthesizes the spatial methods used to model mosquito-borne diseases, their spatial connectivity assumptions and the data used to inform spatial model components. We identified 248 papers eligible for inclusion. Most used statistical models (84.2%), although mechanistic are increasingly used. We identified 17 spatial models which used one of four methods (spatial covariates, local regression, random effects/fields and movement matrices). Over 80% of studies assumed that connectivity was distance-based despite this approach ignoring distant connections and potentially oversimplifying the process of transmission. Studies were more likely to assume connectivity was driven by human movement if the disease was transmitted by an Aedes mosquito. Connectivity arising from human movement was more commonly assumed in studies using a mechanistic model, likely influenced by a lack of statistical models able to account for these connections. Although models have been increasing in complexity, it is important to select the most appropriate, parsimonious model available based on the research question, disease transmission process, the spatial scale and availability of data, and the way spatial connectivity is assumed to occur.     

Statistical Analysis for Decision Making

In this article we consider a generalization of the univariate g-and-h distribution to the multivariate situation with the aim of providing a flexible family of multivariate distributions that incorporate skewness and kurtosis. The approach is to modify the underlying random variables and their quantiles, directly giving rise to a family of distributions in which the quantiles rather than the densities are the foci of attention. Using the ideas of multivariate quantiles, we show how to fit multivariate data to our multivariate g-and-h distribution. This provides a more flexible family than the skew-normal and skew-elliptical distributions when quantiles are of principal interest. Unlike those families, the distribution of quadratic forms from the multivariate g-and-h distribution depends on the underlying skewness. We illustrate our methods on Australian athletes data, as well as on some wind speed data from the northwest Pacific.     

Multivariate performance reliability prediction in real-time

This paper presents a technique for predicting system performance reliability in real-time considering multiple failure modes. The technique includes on-line multivariate monitoring and forecasting of selected performance measures and conditional performance reliability estimates. The performance measures across time are treated as a multivariate time series. A state–space approach is used to model the multivariate time series. Recursive forecasting is performed by adopting Kalman filtering. The predicted mean vectors and covariance matrix of performance measures are used for the assessment of system survival/reliability with respect to the conditional performance reliability. The technique and modeling protocol discussed in this paper provide a means to forecast and evaluate the performance of an individual system in a dynamic environment in real-time. The paper also presents an example to demonstrate the technique.     

MCRC software: A tool for chemometric analysis of two-way chromatographic data

This paper describes the development and implementation of MCRC software, chemometric software for Multivariate Curve Resolution of two-way Chromatographic data. MCRC software is developed for chemometric analysis of chromatographic data; however, it may also be used for other types of multivariate data. It consists of five groups of techniques for preprocessing, chemical rank determination, local rank analysis, multivariate resolution and peak integration. This software has the ability of the analysis of complex multi-component chromatographic signals of gas chromatography-mass spectrometry (GC-MS) and high performance liquid chromatography-diode array detection (HPLC-DAD). The software allows a user to apply the implemented methods in an easy way and it gives a straightforward possibility to visualize the obtained results. The main features of the presented software are providing a number of preprocessing techniques, implementation of different chemical rank determination methods, usage of iterative and non-iterative resolution techniques and a user-friendly environment with a variety of graphical outputs. The implementation of the MCRC software is demonstrated by the analysis of an overlapped peak cluster of simulated GC-MS data.     

Nonstationary multivariate process modeling through spatially varying coregionalization

Models for the analysis of multivariate spatial data are receiving increased attention these days. In many applications it will be preferable to work with multivariate spatial processes to specify such models. A critical specification in providing these models is the cross covariance function. Constructive approaches for developing valid cross-covariance functions offer the most practical strategy for doing this. These approaches include separability, kernel convolution or moving average methods, and convolution of covariance functions. We review these approaches but take as our main focus the computationally manageable class referred to as the linear model of coregionalization (LMC). We introduce a fully Bayesian development of the LMC. We offer clarification of the connection between joint and conditional approaches to fitting such models including prior specifications. However, to substantially enhance the usefulness of such modelling we propose the notion of a spatially varying LMC (SVLMC) providing a very rich class of multivariate nonstationary processes with simple interpretation. We illustrate the use of our proposed SVLMC with application to more than 600 commercial property transactions in three quite different real estate markets, Chicago, Dallas and San Diego. Bivariate nonstationary process inodels are developed for income from and selling price of the property. The work of the first and second authors was supported in part by NIH grant R01ES07750-06.     

Spatial estimation of snow water equivalent at different dates within the Adamello Park of Italy

Spatial estimation of snow water equivalent SWE at six different dates from February 1st to June 1st is tackled using Kriging from a sparse network of 14 snow stakes with density within the Adamello Natural Park of Italy. Therein, SWE is measured at these six dates for the period 1967-2009. Second order statistics of SWE are evaluated and linked to physiographic features. The covariance of the SWE field within the Park is studied, necessary for Kriging, and its regularization provided based upon geomorphic attributes. Seasonal dependence of the covariance of the SWE field is observed, and taken into account for optimal estimation. Then, a Kriging procedure based upon the so obtained covariance fields is developed and cross-validated. The accuracy of Kriging estimates is then compared against that of other commonly adopted methods for spatial interpolation. Kriged SWE maps are then produced at the six dates for two sample years, to demonstrate use of the method. Snow Cover Area SCA from the MODIS® satellite is used to constrain Kriging procedure upon snowed areas. The procedure provides well estimated, least variance SWE values and it is relatively simple and fast because it uses only information of physiography of the area. The so obtained maps can be used for spatial estimation of SWE within the investigated region for water availability conjectures, for constraining hydrological models simulating runoff at thaw, for ecological conjectures upon snow cover related species within the Park, and to evaluate snowpack dynamics for avalanche risk assessment.     

Nonparaxial fields with maximum joint spatial-directional localization. II. Vectorial case

The monochromatic nonparaxial vector fields that achieve a minimum spatial spread for a given directional spread are found. The derivation of these fields is analogous to the one presented in part I of this series for the case of scalar fields. This derivation is based on a variational treatment and multipolar expansion. The resulting lower bounds for the spreads of vector fields turn out to be considerably more restrictive than for scalar fields.     

The Role of Convection During the Self-Focusing of Nonlinear Second Sound Pulses Near the Lambda Point

Second sound in superfluid helium is made up of scalar temperature and vector convective velocity fields. The local sound velocity is altered by these fields, leading to finite amplitude effects. The vector convective field and the direction of propagation are colinear for infinite plane waves and the amplitude dependent velocity is described by the Khalatnikov nonlinear coefficient. For finite sources of second sound, the convective velocity and the local sound velocity may not be colinear, leading to unusual nonlinear effects. We present a vector representation of the finite amplitude velocity of sound emitted from arbitrary sources of second sound, and apply it to a model of two point sources. This simple model is used to show how the linear superposition of signals delayed by the scalar temperature and vector convective fields of second sound may describe a finite amplitude phenomenon: The distortion, due to self-focusing, of pulsed nonplanar second sound near the lambda point.     

A Monitoring and Diagnostic Approach for Stochastic Textured Surfaces

We develop a supervised-learning-based approach for monitoring and diagnosing texture-related defects in manufactured products characterized by stochastic textured surfaces that satisfy the locality and stationarity properties of Markov random fields. Examples of stochastic textured surface data include images of woven textiles; image or surface metrology data for machined, cast, or formed metal parts; microscopy images of material microstructure samples; etc. To characterize the complex spatial statistical dependencies of in-control samples of the stochastic textured surface, we use rather generic supervised learning methods, which provide an implicit characterization of the joint distribution of the surface texture. We propose two spatial moving statistics, which are computed from residual errors of the fitted supervised learning model, for monitoring and diagnosing local aberrations in the general spatial statistical behavior of newly manufactured stochastic textured surface samples in a statistical process control context. We illustrate the approach using images of textile fabric samples and simulated two-dimensional stochastic processes, for which the algorithm successfully detects local defects of various natures. Supplemental discussions, results, data and computer codes are available online.     

A multivariate spatial crash frequency model for identifying sites with promise based on crash types

Many studies have proposed the use of a systemic approach to identify sites with promise (SWiPs). Proponents of the systemic approach to road safety management suggest that it is more effective in reducing crash frequency than the traditional hot spot approach. The systemic approach aims to identify SWiPs by crash type(s) and, therefore, effectively connects crashes to their corresponding countermeasures. Nevertheless, a major challenge to implementing this approach is the low precision of crash frequency models, which results from the systemic approach considering subsets (crash types) of total crashes leading to higher variability in modeling outcomes. This study responds to the need for more precise statistical output and proposes a multivariate spatial model for simultaneously modeling crash frequencies for different crash types. The multivariate spatial model not only induces a multivariate correlation structure between crash types at the same site, but also spatial correlation among adjacent sites to enhance model precision. This study utilized crash, traffic, and roadway inventory data on rural two-lane highways in Pennsylvania to construct and test the multivariate spatial model. Four models with and without the multivariate and spatial correlations were tested and compared. The results show that the model that considers both multivariate and spatial correlation has the best fit. Moreover, it was found that the multivariate correlation plays a stronger role than the spatial correlation when modeling crash frequencies in terms of different crash types.