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.     

Resolution Adaptive Fixed Rank Kriging

The spatial random effects model is flexible in modeling spatial covariance functions and is computationally efficient for spatial prediction via fixed rank kriging (FRK). However, the model depends on a class of basis functions, which if not selected properly, may result in unstable or undesirable results. Additionally, the maximum likelihood (ML) estimates of the model parameters are commonly computed using an expectation-maximization (EM) algorithm, which further limits its applicability when a large number of basis functions are required. In this research, we propose a class of basis functions extracted from thin-plate splines. The functions are ordered in terms of their degrees of smoothness with higher-order functions corresponding to larger-scale features and lower-order ones corresponding to smaller-scale details, leading to a parsimonious representation of a (nonstationary) spatial covariance function with the number of basis functions playing the role of spatial resolution. The proposed class of basis functions avoids the difficult knot-allocation or scale-selection problem. In addition, we show that ML estimates of the random effects covariance matrix can be expressed in simple closed forms, and hence the resulting FRK can accommodate a much larger number of basis functions without numerical difficulties. Finally, we propose to select the number of basis functions using Akaike’s information criterion, which also possesses a simple closed-form expression. The whole procedure, involving no additional tuning parameter, is efficient to compute, easy to program, automatic to implement, and applicable to massive amounts of spatial data even when they are sparsely and irregularly located. Proofs of the theorems and an R package autoFRK are provided in supplementary materials available online.     

The posterior predictive p-value for the problem of goodness of fit

The aim of this paper is to explore some features and possible uses of the posterior predictivep-value for the problem of goodness of fit. First, the behaviour of the posterior predictivep-value is compared with the behaviour of the classicalp-value through some interesting examples. Then, we consider a decision problem for simultaneously deciding to accept/reject a modelM and to accept/reject a null hypothesis (if we have accepted the modelM); the posterior predictivep-value is used for estimating the posterior probability of the model. Research partially supported by DGESIC (Spain) under grant number PB97-0021.     

Reference Stress Approximations for J and COD of Circumferential Through-Wall Cracked Pipes

Reference stress approximations for the J integral and crack tip opening displacement (COD) for circumferential through-wall cracked pipes under tension and under bending are reported. The proposed J estimation equation is fully compatible with the existing reference stress based J estimation, currently embedded in the R6 assessment procedure, but involves a slightly different definition of the reference stress, using an optimised reference load instead of the limit load. This modification enhances the accuracy of the J estimation for circumferentially cracked pipes. Confidence in the proposed equation is gained from the significantly reduced hardening dependence of the plastic influence functions in the GE/EPRI method. The proposed COD estimation equation includes two further modifications. One is the use of a power-law fit to the plastic portion of the stress strain data, instead of the use of the actual stress strain data. In this context, a robust estimation equation for the strain hardening index is given. The other modification is to the plasticity correction term in contained yielding. A lower bound COD estimation equation is also given, similar to the R6 option 1 Jestimation curve, which is suitable when only limited tensile properties are available. The resulting estimation equations are simple to use. Comparisons with experimental pipe test data show that the proposed COD estimation equations provide overall good agreement, which gives confidence in applying them to Leak-before-Break (LBB) analyses.     

Optimal allocation of bioassays in the case of parametrized covariance functions: an application to Lung’s retention of radioactive particles

In this paper, we investigate the effect of employing a parametrized covariance function in a regression experiment on corresponding optimum designs. We demonstrate these effects in the framework of a real example for measuring the lung’s retention of radioactive particles. Also, two different covariance functions are considered, and it is shown that this choice can play a crucial role.      

A mechanical measurement of textural relaxation in superfluid3He-A at very low temperatures

We have studied superfluid³He-A textures by a mechanical method in the very low temperature limit, T/T_c 0.14. The damping of a vibrating wire viscometer is affected by the structure of the 1-vector texture near the wire. The texture is disturbed by a violent motion of the wire. The relaxation of the texture, back to equilibrium, is then observed through changes in the damping of the wire's motion.We gratefully acknowledge discussions with G.R. Pickett and J.R. Hook. J.D.Close helped with early preparation of the experiment. This work was supported by NSF grant DMR 9120277.     

An EWMA chart for monitoring the covariance matrix of a multivariate process based on dissimilarity index

In this article, we propose an exponentially weighted moving average (EWMA) control chart for monitoring the covariance matrix of a multivariate process based on the dissimilarity index of 2 matrices. The proposed control chart essentially monitors the covariance matrix by comparing the individual eigenvalues of the estimated EWMA covariance matrix with those of the estimated covariance matrix from the in‐control (IC) phase I data. It is different from the conventional EWMA charts for monitoring the covariance matrix, which are either based on comparing the sum or product or both of the eigenvalues of the estimated EWMA covariance matrix with those of the IC covariance matrix. We compare the performance of the proposed chart with that of the best existing chart under the multivariate normal process. Furthermore, to prevent the control limit of the proposed EWMA chart developed using the limited IC phase I data from having extensively excessive false alarms, we use a bootstrap resampling method to adjust the control limit to guarantee that the proposed chart has the actual IC ARL(average run length) not less than the nominal level with a certain probability. Finally, we use an example to demonstrate the applicability and implementation of the proposed EWMA chart.     

Fast Computing for Distance Covariance

Distance covariance and distance correlation have been widely adopted in measuring dependence of a pair of random variables or random vectors. If the computation of distance covariance and distance correlation is implemented directly accordingly to its definition then its computational complexity is O(n²), which is a disadvantage compared to other faster methods. In this article we show that the computation of distance covariance and distance correlation of real-valued random variables can be implemented by an O(nlog?n) algorithm and this is comparable to other computationally efficient algorithms. The new formula we derive for an unbiased estimator for squared distance covariance turns out to be a U-statistic. This fact implies some nice asymptotic properties that were derived before via more complex methods. We apply the fast computing algorithm to some synthetic data. Our work will make distance correlation applicable to a much wider class of problems. A supplementary file to this article, available online, includes a Matlab and C-based software that realizes the proposed algorithm.     

Conditional narrowing modulo a set of equations

Narrowing is a universal unification procedure for equational theories given by a canonical term rewrite system. In this paper we introduce conditional narrowing modulo a set of conditional equations and give a full proof of its correctness and completeness for equational conditional rewrite systemsR, E without extravariables whereE is regular andR, E is Church-Rosser moduloE and decreasing moduloE. This result can be seen as the theoretical foundation of a special form of constraint logic and functional programming.     

On the availability of a k-out-of-N system given limited spares and repair capacity under a condition based maintenance strategy

This paper considers a k-out-of-N system with identical, repairable components. Maintenance is initiated when the number of failed components exceeds some critical level. After a possible set-up time, all failed components are replaced by spares. A multi-server repair shop repairs the failed components. The system availability depends on the spare part stock level, the maintenance policy and the repair capacity. We present a mathematical model supporting the trade-off between these three parameters. We present both an exact and an approximate approach to analyse our model. In some numerical experiments, we provide insight on the impact of repair capacity, number of spares and preventive maintenance policy on the availability.     

Study of elastomers behavior at rupture based on the first Seth strain measures invariant

In this work, we are interested in the study of the mechanical behavior of the elastomers materials. We analyze mainly their ultimate properties. This study is based on the first Seth strain measures invariant [1] and the literature experimental data [2] and [3], allowing to take into account of the multiaxial loading effects. With this intention, the step that we adopted is translated by the division of this work into two parts:In the first time, we developed the relation of Seth–Hill based to the strain measures. In the second time, we exposed the principal experimental results used in this work, including versus loading modes (uniaxial tension, equal biaxial tension and biaxial tension), realized on two types of elastomers: a natural rubber (NR) and a styrene butadiene rubber (SBR).Indeed, the installation the first invariant of the Seth strain measures and the experimental results data made it possible to obtain an exploitable data base to determine the non-linear evolution of elastomers behavior at rupture in order to identify its parameter’s materials after.     

A solution to Kronecker's problem

Consider [x ₁,...,x _n], the multivariate polynomial ring over integers involvingn variables. For a fixedn, we show that the ideal membership problem as well as the associated representation problem for [x ₁,...,x _n] are primitive recursive. The precise complexity bounds are easily expressible by functions in the Wainer hierarchy.Thus, we solve a fundamental algorithmic question in the theory of multivariate polynomials over the integers. As a direct consequence, we also obtain a solution to certain foundational problem intrinsic to Kronecker's programme for constructive mathematics and provide an effective version of Hilbert's basis theorem. Our original interest in this area was aroused by Edwards' historical account of theKronecker's problem in the context of Kronecker's version of constructive mathematics.Supported by an Italian grant Italian MURST 40% Calcolo Algebraico e Simbolico 1993 and an NSF grant: #CCR-9002819.     

Nonlinear surface impedance in “low” and “high”T c superconductors

The degradation of the surface impedanceZ = R + iX with rf power is a principal limit for measurements and applications. Such nonlinearities occur for normal conducting cavities by heating or electron loading. But superconductors show a much richer spectrum of causes for nonlinearities. The nonlinearities in the surface resistance δR ∞ γ H ⁿ and surface reactance δR ∞ α H ⁿ can be classified by the ratior = δX/δ R. This ratio differs in value, temperature, or frequency dependence for the different mechanisms, allowing a unique identification of those mechanisms. This identification is a prerequisite for an improved rf cavity design and improvement of superconducting materials.     

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.     

The analysis of ordered categorical data: An overview and a survey of recent developments

This article review methodologies used for analyzing ordered categorical (ordinal) response variables. We begin by surveying models for data with a single ordinal response variable. We also survey recently proposed strategies for modeling ordinal response variables when the data have some type of clustering or when repeated measurement occurs at various occasions for each subject, such as in longitudinal studies. Primary models in that case includemarginal models andcluster-specific (conditional) models for which effects apply conditionally at the cluster level. Related discussion refers to multi-level and transitional models. The main emphasis is on maximum likelihood inference, although we indicate certain models (e.g., marginal models, multi-level models) for which this can be computationally difficult. The Bayesian approach has also received considerable attention for categorical data in the past decade, and we survey recent Bayesian approaches to modeling ordinal response variables. Alternative, non-model-based, approaches are also available for certain types of inference. This work was partially supported by a grant for A. Agresti from NSF and by a research study leave grant from Victoria University for I. Liu.     

Sensitivity analysis using anchored ANOVA expansion and high‐order moments computation

An anchored analysis of variance (ANOVA) method is proposed in this paper to decompose the statistical moments. Compared to the standard ANOVA with mutually orthogonal component functions, the anchored ANOVA, with an arbitrary choice of the anchor point, loses the orthogonality if employing the same measure. However, an advantage of the anchored ANOVA consists in the considerably reduced number of deterministic solver's computations, which renders the uncertainty quantification of real engineering problems much easier. Different from existing methods, the covariance decomposition of the output variance is used in this work to take account of the interactions between non‐orthogonal components, yielding an exact variance expansion and thus, with a suitable numerical integration method, provides a strategy that converges. This convergence is verified by studying academic tests. In particular, the sensitivity problem of existing methods to the choice of anchor point is analyzed via the Ishigami case, and we point out that covariance decomposition survives from this issue. Also, with a truncated anchored ANOVA expansion, numerical results prove that the proposed approach is less sensitive to the anchor point. The covariance‐based sensitivity indices (SI) are also used, compared to the variance‐based SI. Furthermore, we emphasize that the covariance decomposition can be generalized in a straightforward way to decompose higher‐order moments. For academic problems, results show the method converges to exact solution regarding both the skewness and kurtosis. Finally, the proposed method is applied on a realistic case, that is, estimating the chemical reactions uncertainties in a hypersonic flow around a space vehicle during an atmospheric reentry. Copyright © 2015 John Wiley & Sons, Ltd.     

Effect of Incommensurate Charge-Density Wave Scattering on the Electronic Structure of High-Tc Cuprates

We argue that the collective mode as observed in angle resolved photoemission spectroscopy (ARPES) on a large class of cuprates can be associated with dynamic incommensurate CDW fluctuations present in these materials. This scenario is substantiated by a comparison of calculated spectra with experimental ARPES data where we obtain a mode frequency which decreases towards optimal doping thus strongly supporting the existence of a quantum critical point around this concentration. Moreover we extract the temperature dependence of the associated bosonic spectrum from ARPES data where it turns out that there is a continuous evolution from mode-type behavior below T _c to a marginal Fermi liquid structure well above T _c.     

A two channel model as a possible microscopic configuration of the “barrier” in high-Tc grain boundary junctions

High-T _c superconductors are intensively studied for applications such as biomagnetism, but the great difficulties in making integrated dc SQUIDs have slowed down applications in the biomedical field. Moreover, magnetic noise and energy resolution are not always low enough to permit measurements of human body magnetic signals. Noise in bicrystal and biepitaxial grain boundary junctions has been extensively analyzed, and both structures showed similar 1/f noise behaviors. In order to account for the experimental results, different models describing grain boundary junctions have been made, each able to explain some aspects of the phenomenology. In this work we suggest that the barrier is constituted by a large number of microscopic weak links in parallel, and we analyze the effects of such a model on noise properties and the temperature dependences of the critical current, finding a good agreement with most experiments carried out on grain boundary junctions.     

Minimal cut sets of s–t networks with k-out-of-n nodes

In this paper, we extend traditional directed s–t network by letting nodes have k-out-of-n property: To generate output flows, a node must receive at least k flows from its n input links, where k is an integer assigned for the node and its value can be any number from 1 to n. To evaluate the system reliability, minimal cut sets for the extended network are defined for nodes. Under this definition, an extended network and its sink node have the same minimal cut sets. A new algorithm is designed to generate minimal cut sets for all nodes, starting with the source node and ending with the sink node. With different initializations, the algorithm can be applied for extended s–t networks with or without node failures.