Multivariate receptor models and model uncertainty

Estimation of the number of major pollution sources, the source composition profiles, and the source contributions are the main interests in multivariate receptor modeling. Due to lack of identifiability of the receptor model, however, the estimation cannot be done without some additional assumptions.

A common approach to this problem is to estimate the number of sources, q, at the first stage, and then estimate source profiles and contributions at the second stage, given additional constraints (identifiability conditions) to prevent source rotation/transformation and the assumption that the q-source model is correct. These assumptions on the parameters (the number of sources and identifiability conditions) are the main source of model uncertainty in multivariate receptor modeling.

In this paper, we suggest a Bayesian approach to deal with model uncertainties in multivariate receptor models by using Markov chain Monte Carlo (MCMC) schemes. Specifically, we suggest a method which can simultaneously estimate parameters (compositions and contributions), parameter uncertainties, and model uncertainties (number of sources and identifiability conditions). Simulation results and an application to air pollution data are presented.     

Multivariate Spartan spatial random field models

This paper introduces a family of stationary multivariate spatial random fields with D scalar components that extend the scalar model of Gibbs random fields with local interactions (i.e., Spartan spatial random fields). We derive permissibility conditions for Spartan multivariate spatial random fields with a specific structure of local interactions. We also present explicit expressions for the respective matrix covariance functions obtained at the limit of infinite spectral cutoff in one, two and three spatial dimensions. Finally, we illustrate the proposed covariance models by means of simulated bivariate time series and two-dimensional random fields.     

Feasibility of producing prescribed distribution of temperature gradient with array of submerged jets

It is demonstrated that it is feasible to produce a region with a linear transverse temperature profile with the necessary temperature gradient along the jet height.     

Primitive elements with prescribed trace

Let \(q\) be a power of a prime number \(p\) . Let \(n\) be a positive integer. Let \(\mathbb {F}_{q^n}\) denote a finite field with \(q^n\) elements. In this paper, we consider the existence of the some specific elements in the finite field \(\mathbb {F}_{q^n}\) . We get that when \(n\ge 29\) , there are elements \(\xi \in \mathbb {F}_{q^n}\) such that \(\xi +\xi ^{-1}\) is a primitive element of \(\mathbb {F}_{q^n}\) , and \(\mathrm{Tr}(\xi ) = a, \mathrm{Tr}(\xi ^{-1}) = b\) for any pair of prescribed \(a, b \in \mathbb {F}_q^*\) .     

Multivariate GP-VAR models for robust structural identification under operational variability

While the concept of structural monitoring has been around for a number of decades, it remains under-exploited in practice. A main driver for this shortcoming lies in the difficulty to robustly and autonomously interpret the information that is extracted from dynamic data. This hindrance in properly deciphering the collected information may be attributed to the uncertainty that is inherent in i) the finite set of measured data, ii) the models employed for capturing the manifested dynamics, and more importantly, iii) the susceptibility of these systems to variations in Environmental and Operational Parameters (EOPs). In previous work of the authors, a Gaussian Process (GP) time-series approach has been introduced, which serves as a hierarchical input–output method to account for the influence of EOPs on structural response. This in turn enables a robust structural identification. In this scheme, the short-term dynamics are modeled by means of linear-in-the-parameters time-series models, while EOV dependence – acting on a long-term time scale – is achieved via GP regression of the model coefficients on measured EOPs. This work corresponds to a further advancement on this modeling approach, corresponding to its generalization to the vector response case. Particularly, the problem of global identification here is solved via an Expectation–Maximization algorithm tailored to the GP time-series model structure. Moreover, an EOP-dependent innovations covariance matrix is integrated in the model, which helps to capture variation in the vibration power. The resulting model does not only have the capability to represent the long-term response of a structure under variable EOPs, but also facilitates the enhanced tracking of modal quantities in contrast to traditional operational modal analysis techniques. The proposed approach is exemplified on the identification of the vibration response of a simulated wind turbine blade at different points along the blade axis in the flap-wise direction, under variability of both the acting wind speeds and ambient temperatures.     

Tailoring materials with prescribed elastic properties

This paper describes a method to design the periodic microstructure of a material to obtain prescribed constitutive properties. The microstructure is modelled as a truss or thin frame structure in 2 and 3 dimensions. The problem of finding the simplest possible microstructure with the prescribed elastic properties can be called an inverse homogenization problem, and is formulated as an optimization problem of finding a microstructure with the lowest possible weight which fulfils the specified behavioral requirements. A full ground structure known from topology optimization of trusses is used as starting guess for the optimization algorithm. This implies that the optimal microstructure of a base cell is found from a truss or frame structure with 120 possible members in the 2-dimensional case and 2016 possible members in the 3-dimensional case. The material parameters are found by a numerical homogenization method, using Finite-Elements to model the representative base cell, and the optimization problem is solved by an optimality criteria method.

Numerical examples in two and three dimensions show that it is possible to design materials with many different properties using base cells modelled as truss or frame works. Hereunder is shown that it is possible to tailor extreme materials, such as isotropic materials with Poisson's ratio close to − 1, 0 and 0.5, by the proposed method. Some of the proposed materials have been tested as macro models which demonstrate the expected behaviour.     

Correct voltage distribution for axisymmetric sinusoidal modelingof induction heating with prescribed current, voltage, or power

We consider the problem of determining the voltage in coil rings, which arise as an axisymmetric approximation of a single connected induction coil during modeling of induction heating. Assuming axisymmetric electromagnetic fields with sinusoidal time dependence, we compute the voltages from the condition that the total current must be equal in each ring. Depending on whether total current, total voltage, or total power is prescribed, different linear systems of complex equations represent the ring voltages. In two sets of numerical simulations, varying the number of coil rings, we compare results using the correct voltage distribution to the corresponding results using the simple homogeneous voltage distribution     

Variances, covariances and artifacts in image deconvolution

In this paper we investigate the role of estimator variance and covariance in signal and image deconvolution, where artificial structures and large fluctuations are frequently observed. The deconvolution procedure adopted in this paper is based on the Least Squares Method and the regularization, on a minimum variance property. The whole covariance matrices were taken into account in every step of the deconvolution and regularization procedures. We show that the covariances created by the deconvolution procedure are at the origin of the observed artifacts in the recovered signal.     

Multivariate probability distribution of ordered peaks of vector Gaussian random processes

The problem of determining the joint probability distribution of ordered peaks of jointly stationary Gaussian random processes is considered. The solution is obtained by modeling the number of times a specified threshold is crossed by the component processes as a multivariate Poisson process. Based on this, the joint probability distribution of the time required for the nth crossing of a specified level with positive slope is derived. This formulation is further extended to derive the joint distribution of ordered peaks in a given time interval. An illustrative example on a bivariate Gaussian random process is presented and the analytical predictions are shown to compare reasonably well with corresponding results from Monte Carlo simulations. Also presented is an analysis of response of a randomly driven multi-degree of freedom system with emphasis on the sensitivity of ordered peak characteristics with respect to changes in system parameters. It is demonstrated that higher order statistics are generally more sensitive to changes in system characteristics—a property that has potential for application in structural model updating and damage detection.     

Multivariate analysis of retention and distribution of aroma compounds in barrier dispersion coatings

The influence of drying conditions, latex type, talc content and amount of added aroma compounds on the retention and distribution of aroma compounds in barrier dispersion coatings was explored using static headspace gas chromatography combined with multivariate data analysis. The model aroma system consisted of an alcohol, an ester, an aldehyde, a ketone and two terpenes. Four latex grades based on styrene‐acrylate and styrene‐butadiene were studied. It was found that non‐polar aroma compounds were retained best in styrene‐butadiene latex, while polar ones were retained more in the styrene‐acrylate latex. Talc increased the retention of all the aroma compounds. The retention of polar compounds was higher with shorter drying times, which is probably due to greater losses during the later steps of the film formation process when internal water was evaporated. Polarity and talc also affected the partition of the aroma compounds between air and film. Copyright © 2007 John Wiley & Sons, Ltd.     

Rotating porous convection with prescribed heat flux

The problem of thermal convection in a rotating horizontal layer of porous medium is considered. The porous medium is described by the equations of Darcy. A novel aspect of this work is to consider boundary conditions for the temperature of Newton–Robin type with heat flux prescribed as a limiting case. The effect of rotation is found to be crucial. For the Taylor number small enough the critical wave number is zero but we find a threshold such that for Taylor numbers beyond this non-zero critical wave numbers are found. The threshold is verified via a weakly nonlinear analysis. Finally, a sharp global nonlinear stability analysis is given.     

Generating data with prescribed power spectral density

Data generation is straightforward if the parameters of a time series model define the prescribed spectral density or covariance function. Otherwise, a time series model has to be determined. An arbitrary prescribed spectral density will be approximated by a finite number of equidistant samples in the frequency domain. This approximation becomes accurate by taking more and more samples. Those samples can be inversely Fourier transformed into a covariance function of finite length. The covariance in turn is used to compute a long autoregressive (AR) model with the Yule-Walker relations. Data can be generated with this long AR model. The long AR model can also be used to estimate time series models of different types to search for a parsimonious model that attains the required accuracy with less parameters. It is possible to derive objective rules to choose a preferred type with a minimal order for the generating time series model. That order will generally depend on the number of observations to be generated. The quality criterion for the generating time series model is that the spectrum estimated from the generated number of observations cannot be distinguished from the prescribed spectrum.     

Operational planning models for distribution networks

The design and optimisation of a logistic network deals with a wide set of decisions, e.g. the determination of the best location and capacity of the different logistic facilities (production plants, distribution centres, transit points, wholesalers, etc.), the allocation of the product demand coming from customers in presence (or absence) of fractionable flows of material, the determination of the best transportation mode (truck, rail, etc.) as well as loading and routing of vehicles. These decisions involve multiple stages of a distribution network: customers-regional distribution centres (RDC), RDCs-central distribution centres (CDC) and CDCs-production plants and sources, in presence of multiple products and the variable time (i.e. time-dependent product demand and flows of material). This paper presents a top-down methodology that joins the strategic planning, the tactical planning and the operational planning of distribution networks with a special focus on the development of effective heuristic methods to face the vehicle routing problem. Original models and heuristic algorithms for the operational planning are illustrated. The impact of the strategic and tactical decisions on the performance of the operational planning is evaluated by the application of the proposed hierarchical approach to two realistic case studies. Obtained results are illustrated in a what-if experimental analysis conducted on multiple problem settings and realistic scenarios.     

Optical filters with prescribed optical thickness and refined refractive indices

We propose to refine the refractive index of the layers composing optical filters while keeping their optical thicknesses constant. Using this technique, one can optimize filters made of quarter-wave layers using conventional optimization techniques, while preserving the possibility to use turning-point monitoring during their fabrication. Application of this method to the design of a dual narrowband filter and a tilted edge filter demonstrates its effectiveness.     

Multivariate sensitivity analysis to measure global contribution of input factors in dynamic models

Many dynamic models are used for risk assessment and decision support in ecology and crop science. Such models generate time-dependent model predictions, with time either discretised or continuous. Their global sensitivity analysis is usually applied separately on each time output, but Campbell et al. (2006 [1]) advocated global sensitivity analyses on the expansion of the dynamics in a well-chosen functional basis. This paper focuses on the particular case when principal components analysis is combined with analysis of variance. In addition to the indices associated with the principal components, generalised sensitivity indices are proposed to synthesize the influence of each parameter on the whole time series output. Index definitions are given when the uncertainty on the input factors is either discrete or continuous and when the dynamic model is either discrete or functional. A general estimation algorithm is proposed, based on classical methods of global sensitivity analysis.The method is applied to a dynamic wheat crop model with 13 uncertain parameters. Three methods of global sensitivity analysis are compared: the Sobol'-Saltelli method, the extended FAST method, and the fractional factorial design of resolution 6.     

On the asymptotic distribution of residual autocovariances in VARX models with applications

In this paper, we derive the asymptotic distribution of residual autocovariance matrices in the class of vector autoregressive models with explanatory variables (VARX). The asymptotic distribution of residual autocorrelation matrices is also obtained. New test statistics are proposed as an application of our main result. Test statistics at individual lags and portmanteau statistics are introduced and their asymptotic distributions are established. The proposed test, statistics are illustrated in a small simulation study. This work was supported by grants from the NSERC (Canada), FQRNT (Québec) and the Institut de Finance Mathématique de Montréal IFM²     

Multivariate models for the concentration of oxygen-18 in precipitation based on meteorological and geographical features

Concentrations of the stable isotope oxygen-18 in precipitation samples have been modeled by geographical and meteorological features of the sampling stations. Precipitation samples are from 30 locations in Austria; oxygen-18 concentrations have been determined by isotope ratio mass spectrometry; each location has been characterized by three geographical features (longitude, latitude and elevation), and five meteorological features (relative humidity, fresh snow, wind speed, precipitation, and air temperature). All data are monthly means for the summer period April to September computed as long term averages for a time period of typically 20 years. The basic feature set has been augmented by 49 nonlinear transformations of the original features giving a data set with 57 features. The number of objects was 180 corresponding to the 30 sampling stations times six months. Different methods of feature selection (correlation coefficient, backward elimination, all subsets regression, and genetic algorithm) have been applied, and for the 20 best feature subsets the prediction errors of PLS models have been estimated from test sets. The final best model for a prediction of oxygen-18 concentrations contains ten features selected by a genetic algorithm. This model has been applied to the computation of a geographical oxygen-18 distribution map from Austria based on computed oxygen-18 values for 200 locations, and experimental values of 30 locations.     

多元约束Welsch-Kuh距离与多元约束Cook距离

对多元约束线性回归模型，本文讨论了约束最小二乘估计的影响问题，定义了度量影响的多元约束Ｃｏｏｋ距离和多元约束Ｗ－Ｋ统计量，给出了其分布，建立了约束Ｗ－Ｋ统计量与约束广义相关系数之间的联系。     

Exact solutions of non-Newtonian fluid flows with prescribed vorticity

Summary The equations of motion of a non-Newtonian second-grade fluid flow are highly nonlinear partial differential equations. For this reason, there exists only a limited number of exact solutions. Due to the complexity of the equations, inverse methods described by Nemenyi [1] have become attractive in the study of non-Newtonian fluids. In these methods, certain physical or geometrical properties of the flow field are assumed a priori.Lin and Tobak [2] studied steady plane viscous incompressible flows for a chosen vorticity function by decomposing the nonlinear fourth-order partial differential equation in the streamfunction. This excellent approach yielded two second-order linear partial differential equations in the streamfunction. Hui [3] used this approach to study unsteady plane viscous incompressible flows.During the past decade, there has been substantial interest in flows of viscoelastic liquids due to the occurrence of these flows in industrial processes. In this paper, we study the steady and unsteady incompressible viscous non-Newtonian second-grade fluid flows in which the vorticity is proportional to the streamfunction perturbed by a uniform stream. The solutions obtained are exact solutions and represent various non-parallel flows of second-grade fluids.The plan of this paper is as follows: In Sect. 2, the equations of motion of an unsteady plane incompressible second-grade fluid are given, and the vorticity function is assumed to be ² =A(–Ux–BUy ²). In Sect. 3, solutions for the steady flow are obtained. In Sect. 4, solutions for unsteady flow are obtained.