Sequential experimental design procedures for precise parameter estimation in ordinary differential equations

In this paper, it is shown how an experimental program for precise parameter estimation can be designed sequentially for the case that the mathematical model is given in the form of a set of ordinary differential equations. Two strategies are proposed. The first aims at minimizing the volume of the joint confidence region associated with the parameter estimates. The second attempts to alter as much as possible the shape towards a spherical region, by shortening the length of the longest principal axis of the confidence region to the maximum extent. The application of both criteria is illustrated by means of examples, representative for real problems in chemical reaction engineering. The techniques are easily applicable with our present day computing facilities. Qualitative indications are derived concerning the question when the use of an experimental design will result in an appreciable gain in significance for the parameter estimates.     

Nonlinear parameter estimation through particle swarm optimization

Parameter estimation procedures are very important in the chemical engineering field for development of mathematical models, since design, optimization and advanced control of chemical processes depend on model parameter values obtained from experimental data. Model nonlinearity makes the estimation of parameter and the statistical analysis of parameter estimates more difficult and more challenging. In this work, it is shown that many of these difficulties can be overcome with the use of heuristic optimization methods, such as the particle swarm optimization (PSO) method. Parameter estimation problems are solved here with PSO and it is shown that the PSO method is efficient for both minimization and construction of the confidence region of parameter estimates. Moreover, it is shown that the elliptical approximation of confidence regions of nonlinear model parameters can be very poor sometimes and that more accurate likelihood confidence regions can be constructed with PSO, allowing for more reliable statistical analysis of the significance of parameter estimates.     

A sequential experimental design procedure for precise parameter estimation based upon the shape of the joint confidence region

A sequential experimental design procedure which aims at obtaining precise parameter estimates of the mathematical model is proposed and illustrated. The design criterion modifies the shape of the joint confidence hyperellipsoid, rather than its volume. It consists of identifying those experimental conditions for the next experiment which maximize the smallest eigenvalue of X^TX, resulting in maximal contraction of the largest principal axis of the confidence hyperellipsoid. The conditions to be fulfilled for applicability of the criterion are that the mathematical model has to be sufficiently linear in the parameters in the vicinity of the maximum likelihood (least squares) parameter estimates and that the unobservable experimental error is distributed according to N(O, Iσ²).     

Estimating a minimum set of physically based dynamic parameters to enhance statistical inference in block-oriented modeling

In process identification (i.e., dynamic model development) information on the precision and reliability of a parameter estimate is conveyed by a confidence interval. The best confidence interval is the one with the shortest width for a given level of confidence. Confidence intervals widen as the standard error increases or as the number of estimated parameters increases. When the value of a parameter is needed for physical understanding of process characteristics, its precision and reliability, i.e., certainty, is crucial. Parameter certainty increases as the number of estimated parameters decreases because this causes confidence intervals to shorten and confidence levels to increase. Hence, this article focuses on maximizing parameter certainty of physically interpretable dynamic parameters under block-oriented modeling by obtaining accurate values for all the dynamic parameters from a minimum set of estimated parameters. This objective is accomplished by the development of a procedure that identifies equivalent sets of parameters and estimates one parameter for each set. For a seven (7) input, five (5) output, simulated CSTR, its 84 physically based dynamic parameters were accurately determined from 23 estimated parameters that resulted in an increase in confidence level from 50% to 99.9% for a fixed interval width.     

Statistical analysis of masked data in a hybrid system based on copula theory under progressive hybrid censoring

This article considers the reliability analysis of a hybrid system with dependent components, which are linked by a copula function. Based on Type I progressive hybrid censored and masked system lifetime data, we drive some probability results for the hybrid system and then the maximum likelihood estimates as well as the asymptotic confidence intervals and bootstrap confidence intervals of the unknown parameters are obtained. The effects of different dependence structures on the estimates of the parameter and the reliability function are investigated. Finally, Monte Carlo simulations are implemented to compare the performances of the estimates when the components are dependent with those when the components are independent.     

A JOINT EXPERIMENTAL DESIGN PROCEDURE FOR PARAMETER ESTIMATION

A joint experimental design procedure which aims at obtaining precise parameter estimates of a given mathematical model is proposed and illustrated by means of computer simulated experiments

This criterion is found both to modify the shape of the joint confidence hyperellipsoid and to reduce its volume significantly

Therefore, it behaves as a combination of two of the most representative techniques in use today, known as the minimum volume or the shape criteria. Numerical results are given which serve to display its salient features and to compare it with the above mentioned techniques.     

用序贯法对反应速率模型中的参数作精确估算

对一个确定的模型,不仅要确定参数的估算值,还必须了解其可靠性。为了缩小参数的不定性,降低参数间的相关性,以扩大模型拟合的适定性和准确度,本文讨论了参数精确估算的序贯设计过程。 文内还详细讨论了精估参数的实验设计准则——最小联合置信容积准则和形状准则,以及序贯设计的实用价值。以氨合成为例,用序贯法精确估算了反应速度模型中的参数。这一模型是在序贯判别过程中选定的。把精估结果用于拟合大型生产过程的实测数据,得到了满意的结果。     

Precise parameter estimation in reaction modelling — sensitivity in choosing experimental conditions

A sequential experimental design developed by Box and coworkers was reviewed and the sensitivity of experimental conditions in increasing the preciseness of parameter estimates was examined. Continuous kinetic data were utilized for the examination. It is found that experimental conditions are in some cases rather sinsitive and slightly deviated experimental conditions from those specified by the criterion may induce some difficulty in reducing the size of confidence region of parameter estimates.     

Asymptotic Theory and Unified Confidence Region for an Autoregressive Model

Although some unified inferences for the coefficient in an AR(1) model have been proposed in the literature, it remains open as to how to construct a unified confidence region for the intercept and the coefficient jointly without a prior on whether the sequence is stationary or unit root or near unit root or moderate deviations from a unit root or explosive and whether the sequence has a zero or nonzero constant intercept. After deriving the joint limit of the least squares estimator for all of these cases, this article proposes a unified empirical likelihood confidence region by first splitting the data into two parts and then constructing some weighted score equations. The good finite sample performance of the proposed method is demonstrated via a simulation study. Real data applications are provided as well.     

Sequential statistical design strategy in a kinetic study of propylene oxidation

A sequential design strategy was used to direct an experimental kinetic study of propylene oxidation over bismuth molybdate. Two statistical criteria were used to design experiments. The first is a joint criterion for the dual problem of model discrimination and parameter estimation which was proposed by Hill et al. (1968), but which to the best of our knowledge has not been tested in an experimental program. Using this joint criterion design, it was demonstrated that both the objectives of model discrimination and precise parameter estimation were effectively accomplished. After model discrimination had been achieved, a design strategy for further improvement in parameter precision was found, as expected, to be effective for that purpose.     

Weak identification in the ESTAR model and a new model

Determining good parameter estimates in (exponential smooth transition autoregressive) models is known to be difficult. We show that the phenomena of getting strongly biased estimators is a consequence of the so‐called identification problem, the problem of properly distinguishing the transition function in relation to extreme parameter combinations. This happens in particular for either very small or very large values of the error term variance. Furthermore, we introduce a new alternative model – the TSTAR model – which has similar properties as the ESTAR model but reduces the effects of the identification problem. We also derive a linearity and a unit root test for this model.     

Characterization and material model definition of toughened adhesives for finite element analysis

Some toughened adhesives used for structural joints are characterised by non-linear behaviour prior to failure that may significantly influence the entire joint response. The determination of appropriate and accurate material models for use in analysis and design phases covering both nonlinearities and final material rupture constitutes one of the main challenges for the utilisation of adhesives and for offering designers the same confidence level as that offered by other joining techniques.The present research proposes the utilisation of both elasto-plastic and continuum damage models as a combination that can fully reproduce the mechanical response of toughened adhesives in finite element (FE) analysis. In this context, the Drucker-Prager exponential model has demonstrated to provide accurate fits with the nonlinearities of these materials, allowing the real plastic behaviour of the adhesives to be adjusted in the computational models with a high degree of correlation. On the other hand, a continuum damage model has been proposed to simulate the final material failure process introducing a displacement-based damage parameter into the constitutive equation of the damaged material. The definition of the parameters associated with the mentioned models has been carried out through the execution of an experimental programme combining traction and torsion tests, described in the present paper as part of the study developed. The research is finally completed with an experimental and FE analysis of a specific bonded joint that allows the operation of the material model to be checked in a real application.     

A SEQUENTIAL PROCEDURE FOR PRECISE ESTIMATION OF PARAMETERS IN KINETIC MODEL

For a determined mathematical model,it is not only necessary to obtain the estimates of parameters,but also their reliabilites.The present work deals with a sequential statistical design strategy for anexperimental program to obtain precise parameter estimetes for a nonlinear kinetic model.It aims at de-creasing the indeterminatness of the parameter estimates,in reducing the correlation between parameterestimates,and in inceasing the adequacy and accuracy of the model.The paper discusses the experimental design criterion for precise parameter estimates-the minimun vol-ume design criterion and the spherical shape design criterion,and the practical value of a sequential statis-tical design strategy.At last,an example was performed to precise estimate the parameters in the kinetic model of ammoniasynthesis reaction which was selected as being adequate in the previous work.Comparison of the results from the industrial unit and the corresponding values calculated with themodel using the estimates of parameters reveals the reasonable reliability of the model.     

Methyl methacrylate-chloroprene copolymerization: an evaluation of copolymerization models

The composition and triad fraction data of Ebdon (Polymer 1974, 15, 782) for the copolymerization of methyl methacrylate with chloroprene have been re-evaluated using non-linear least squares curve-fitting procedures and statistical testing of conclusions. Using the compositions and triad fractions separately, best estimates of reactivity ratios together with the joint confidence intervals for these parameters for terminal and penultimate models have been calculated. It has been shown that the terminal model cannot describe adequately the experimental compositions or triad fractions. While the allowance of a penultimate effect for methyl methacrylate chain end radicals provides a significant improvement over the fit of the terminal model to the copolymer compositions, there is only a small probability that both the triad fraction and the composition data can be described adequately by one set of penultimate reactivity ratios.     

A rate—controlling step in langmuir—hinshelwood kinetic models

Langmuir‐Hinshelwood models with and without rate controlling step assumptions have been compared. Hydrogenations of toluene, cinnamaldehyde and mixture of methylesters of fatty acids were used as model systems. Integral data were obtained in semibatch reactors. The proposed models have been simplified by neglecting statistically insignificant parameters. The predicted mixture composition and surface coverages have been compared and the reliability of parameter estimates tested.     

Parameter estimation and prediction uncertainties for multi-response kinetic models with uncertain inputs

Error-in-variables model (EVM) methods are used for parameter estimation when independent variables are uncertain. During EVM parameter estimation, output measurement variances are required as weighting factors in the objective function. These variances can be estimated based on data from replicate experiments. However, conducting replicates is complicated when independent variables are uncertain. Instead, pseudo-replicate runs may be performed where the target values of inputs for repeated runs are the same, but the true input values may be different. Here, we propose a method to estimate output-measurement variances for use in multivariate EVM estimation problems, based on pseudo-replicate data. We also propose a bootstrap technique for quantifying uncertainties in resulting parameter estimates and model predictions. The methods are illustrated using a case study involving n-hexane hydroisomerization in a well-mixed reactor. Case-study results reveal that assumptions about input uncertainties can have important influences on parameter estimates, model predictions and their confidence intervals.     

Distribution theory of group sequential t,χ2and F-tests for general linear models

We derive the joint distribution of the sequence of estimates of the parameter vector θ in a normal general linear model when data accumulate over a series of analyses. This seclllerlce of estimates has a remarkably simple covariance structure, even when observations are correlated, allowing standard group sequential tests to be applied in very general settings. If observations variaices and covariances depend on an unknown scale factor σ², the joint distribution of the sequence of estimates of θ σ² has a simple form. again even in the case of correlated observations. From these results, we establish a general treatment of group sequential t,χ²and F-tests.     

High Resolution Optical Interference Investigation of Swelling Due to Water Uptake by Model Adhesive Joints

Fully cured resin adhesives swell when they absorb water. By manufacturing a model butt joint between a thin, and therefore flexible, microscope cover slip as one adherend and a massive, and therefore rigid, slab of glass as the other adherend, development of the swelling can be observed and measured by generating Moire images from photographs of the pattern of optical interference fringes formed in the gap between an optical flat and the free surface of the cover slip.

The swelling is strongly inhomogeneous and this inhomogeneity gives rise to a distribution of stress normal to the joint which is compressive near both the rim and centre, and tensile within an annular region located between the rim and centre. It has been demonstrated that swelling stresses large enough to cause fracture of the cover slip can be developed. Regularly spaced radial perturbations in the boundary between unswollen and swollen adhesive have been observed in a film which contains a carrier cloth.     

钢制薄壁外压圆筒的可靠性稳定系数

应用基于概率统计理论的可靠性方法，对钢制薄壁圆筒形外压容器的临界失稳压力进行了分析，认为临界失稳压力实测值与理论预测值之比是符合正态分布的随机变量；从设计、制造、检验、操作和安全监察等角度，讨论了有关因素对临界失稳压力的影响，运用强度-载荷干涉模型，定量分析了设计公式置信度、常规稳定系数、可靠性稳定系数的关系。     

Estimation of parameters in monte carlo modelling

This paper presents a general method for estimating model parameters from experimental data when the model relating the parameters and input variables to the output responses is a Monte Carlo simulation. From a statistical point of view a Bayesian approach is used in which the distribution of the parameters is handled in discretized form as elements of an array in computer storage. The stochastic nature of the Monte Carlo model allows only an estimate of the distribution to be calculated from which the true distribution must then be estimated. For this purpose an exponentiated polynomial function has been found to be useful. The method provides point estimates as well as joint probability regions. Marginal distributions and distributions of functions of the parameters can also be handled. The motivation for exploring this alternative parameter estimation technique comes from the recognition that for some systems, particularly when the underlying process is stochastic in nature, Monte Carlo simulation often is the most suitable way of modelling. As such, the Monte Carlo approach increases the range of problems which can be handled by mathematical modelling. The technique is applied to the modelling of binary copolymerization. Two models, the Mayo-Lewis and the Penultimate Group Effects models, are considered and a method for discriminating between these models in the light of sequence distribution data is proposed.