Park M. Reilly

A Bayesian Study of the Error-in-Variables Model

This article presents the results of a study of the functional case of the problem of parameter estimation when there is error in all the variables. There is consequently no distinction between independent and dependent variables. Posterior probability density functions are developed for the parameters with both linear and nonlinear, and possibly multiple, relations among the true values of the variables. There is no distinction between models that are linear or nonlinear in the parameters. The results are equivalent to generalizations of the work of some previous authors, but lead to new and efficient algorithms for finding point estimates and their precisions. For most of the results the error covariance matrix is assumed known, though a case is treated where it is known except for a scalar multiplier. The results are also shown to be valid if the covariance matrix is singular. Geometric interpretations are described.

A General Model for Prediction of Molecular Weight Distributions of Degraded Polymers. Development and Comparison with Ultrasonic Degradation Experiments

Abstract A general model has been developed for the process of polymer chain breakage. The molecular weight distributions (MWDs) of the degraded polymer, after a specified number of chain ruptures, can be calculated from the model, given the MWD of the initial material. The calculations can easily be handled numerically on a digital computer. The model is applicable to any process in which polymer chains break, e.g., ultrasonation and high shear mechanical action. The course of degradation is described in terms of the probability that a molecule of a given length will break and the probability that a molecule of a particular length will result from this rupture. Any form of both these probability distributions and the initial MWD can be used in the model.

An approach to interval estimation in partial least squares regression

Abstract Although partial least squares regression (PLS) is widely used in chemometrics for quantitative spectral analysis, little is known about the distribution of the prediction error from calibration models based on PLS. As a result, we must rely on computationally intensive procedures like bootstrapping to produce confidence intervals for predictions, or, in many cases, we must do with no interval estimates at all, only point estimates. In this paper we present an approach, based on the linearization of the PLS estimator, that allows us to construct approximate confidence intervals for predictions from PLS.

Non-linear parameter estimation for a dynamic model in photocatalytic reaction engineering

Abstract Kinetics in photocatalytic reactions typically follow some form of the Langmuir–Hinshelwood rate law, and the kinetic parameters have usually been estimated using the method of initial rates. This method has serious drawbacks, primarily related to the subjectivity in estimating initial rates from experimental data. These drawbacks are discussed and illustrated using literature data. A superior approach using all the experimental data is described and compared to the method of initial rates. This technique is based on the Box–Draper method of non-linear estimation, coupled with a variable metric algorithm employing an alternative method for calculation of the gradient. It is shown that this approach results in better and more objective parameter estimates in these kinetic models.

The asymptotic variance of the univariate PLS estimator

In this short note, we derive an expression for the asymptotic covariance matrix of the univariate partial least squares (PLS) estimator. In contrast to M.C. Denham [J. Chemometrics 11 (1997) 39], who provided a locally linear approximation based on a recursive definition of the estimator, we derive a more compact expression for the asymptotic covariance matrix by combining a standard convergence result with matrix differential calculus, in particular the approach of J.R. Magnus and H. Neudecker [Matrix Differential Calculus with Applications in Statistics and Econometrics, revised ed., Wiley, Chichester, UK, 1991]. We also describe some theoretical and practical aspects of calculating the asymptotic covariance matrix, and illustrate its use on spectroscopic data.

Uncertainty in incident rates for trucks carrying dangerous goods.

This paper addresses the uncertainty associated with release and fire incident rates for trucks in transit carrying dangerous goods. The research extends the treatment of uncertainty beyond sensitivity analysis, low-best-high estimates and confidence intervals, and represents the uncertainty through probability density functions. The analysis uses Monte Carlo simulations to propagate the uncertainty in the input variables through to the resulting release and fire incident rates. The paper illustrates how we can combine information on accident and non-accident releases and fires to generate probability density functions for the total expected releases and fires per billion vehicle kilometres for trucks carrying dangerous goods.

A systematic approach to the study of multicomponent polymerization kinetics: The butyl acrylate/methyl methacrylate/vinyl acetate example. IV. Optimal Bayesian design of emulsion terpolymerization experiments in a pilot plant reactor

A systematic study of the terpolymerization of butyl acrylate/methyl methacrylate/vinyl acetate (BA/MMA/VAc) is being conducted. In this stage of the study, emulsion terpolymerizations were performed in a 5 L stainless steel pilot plant reactor. The experimental trials were of the two-level factorial type and were designed optimally using a Bayesian method. The design procedure allowed us to improve our knowledge about the process using our prior knowledge and our subjective judgement. The polymers produced were characterized for conversion, composition, molecular weight, and particle size. The Bayesian design of experiments is shown to have several advantages over conventional factorial designs.

The geometry of 2-block partial least squares regression

In this paper, we outline a geometric interpretation of both univariate and multivariate partial least squares regression (PLS) that illustrates very clearly the mathematical description of the PLS algorithms. In addition, we show how the concept of continuum regression arises quite naturally out of a geometric interpretation of ordinary least squares, principal component regression, and PLS. We also derive a simple expression that relates the first PLS dimension to the correlational- and eigen-structure of the data and suggest a property of PLS subspaces as a whole, one that is defined with respect to the corresponding subspace in principal component regression.

An application of the Error-in-Variables Model—parameter estimation from Van Ness-type vapour-liquid equilibrium experiments

Abstract The Error-in-Variables Model (EVM) provides a means for estimating parameter values in mathematical models where there is error in every measured variable. This is a distinct improvement over the Method of Least Squares in most situations because the latter requires that there be error in measuring only one of the variables. As an example of the use of EVM, a method is presented for estimating Wilson equation parameters in binary and ternary vapour-liquid equilibrium where the data are obtained using the Van Ness experimental technique. The application presents unusual difficulties in that the compositions of the phases are not measured directly and the total quantity of one component is only measured as the sum of a series of increments. Data for the ternary system n-heptane/n-propanol/1-chlorobutane and the binary system n-heptane/1-chlorobutane are analysed by the proposed method, and the results are compared with the empirical results of Sayegh et al. (1979). The results of the EVM analysis are considerably more accurate. Analysis of the residuals obtained by the EVM procedure suggest serious deficiencies in the thermodynamic model.

A Stochastic Simulation Model of A Chemical Plant

AbstractChemical plants are usually very complex and simulation is increasingly being used to study the complex interactions that occur. This paper describes the simulation of a batchchemical plant with series and parallel sets of discrete events. Event duration is determined stochastically from prior probability distributions.This paper discusses the beneficial results obtained from such simulation in an actualcase, although details of the case are not provided for proprietary reasons. Some areas for extending this type of work are mentioned.