Sequential experimental design based on multiobjective optimization procedures

Model-based sequential experimental designs are frequently applied for discrimination of rival models and/or estimation of precise model parameters. Although the development and use of a single design criterion to perform the simultaneous model discrimination and precise parameter estimation seem appealing, published material indicates that previous attempts to develop such a single design criterion have not been successful. Despite that, this problem has rarely been analyzed with the help of multiobjective optimization procedures. In this work, a multiobjective optimization method based on the particle swarm optimization procedure is used to build the Pareto fronts in experimental design problems where distinct design criteria used for discrimination of rival models and/or estimation of precise model parameters are considered simultaneously. It is shown through the rigorous analysis of the Pareto sets that both design objectives are frequently conflicting, which means that optimum discrimination of rival models and estimation of precise model parameters cannot be performed simultaneously in many cases. However, it is also shown that the use of the posterior covariance matrix of estimated model parameters for model discrimination makes the design of experiments for the simultaneous optimum model discrimination and estimation of model parameters possible in many experimental design problems.     

Experimental design for the joint model discrimination and precise parameter estimation through information measures

Experimental design procedures for model discrimination and for estimation of precise model parameters are usually treated as independent techniques. In order to conciliate the objectives of both experimental design procedures, the present paper proposes the use of experimental design criteria that are based on measures of the information gain when new experiments are carried out. The proposed criterion depends on the volumes of the confidence regions of the model parameters and presents a number of advantageous aspects, such as the conciliation of the usual experimental design objectives and the fact that the obtained criterion values can be easily interpreted in terms of the information eliminated after carrying out additional experiments. Besides, the proposed design criterion can easily accommodate multiobjective experimental design approaches, as shown in the examples.     

Use of expected likelihood in sequential model discrimination in multiresponse systems

Use of Bayes' theorem for sequential model discrimination requires the knowledge of the likelihood of each model. However, the likelihood, being a function of parameters, which are considered to be random variables, is in itself a random variable. The standard procedure in such a case is to use the expected likelihood in computing the posterior model probabilities. In this work an expression for the expected likelihood for a multiresponse system is developed. Computational results are presented using multiresponse data for the first time to demonstrate the utility of the expected likelihood in efficient model discrimination in a catalyst fouling system. Two discriminatory design criteria, viz., the Box-Hill criterion and the Roth's criterion, are compared.     

Iterative improvement of parameter estimation for model migration by means of sequential experiments

Determining an optimal design for estimation of parameters of a class of complex models expected to be built at a minimum cost is a growing trend in science and engineering. We adopt a scale-bias adjustment migration strategy for integrating base and new models based on similar nature underlying processes. Further, we propose a Bayesian sequential algorithm for obtaining the statistically most informative data about the migrated model for use in parameter estimation. The benefits of the proposed strategy over traditional approaches presented in recent reported work are demonstrated using Monte Carlo simulations.     

Sequential design including initial experimental runs for a kinetic study of propylene oxidation

A sequential design strategy for selecting experimental runs to obtain model discrimination and precise parameter estimation is tested via a simulation study of propylene oxidation kinetics. The strategy is used to design all runs including the preliminary ones which were arbitrarily chosen by earlier researchers. To design initial runs, crude initial parameter guesses may be used in the rival models until least squares estimates can be calculated. Even under conditions of very bad initial guesses and high error variances, this procedure selects whichever model is the correct one and estimates with precision its parameters, in fewer runs than previously reported.     

A new sequential experimental design procedure for discriminating among rival models

A new procedure to be used in the sequential design of experiments for discriminating among rival models is presented. The procedure consists of a new design criterion and a model adequacy test based on statistical principles. The method is illustrated with examples and compared to previous methods.     

Sequential model selection method for nonparametric autoregression

Abstract

In this article, the nonparametric autoregression estimation problem for quadratic risks is considered. To this end, we develop a new adaptive sequential model selection method based on the efficient sequential kernel estimators proposed by Arkoun and Pergamenshchikov (2016). Moreover, we develop a new analytical tool for general regression models to obtain the non-asymptotic sharp oracle inequalities for both usual quadratic and robust quadratic risks. Then, we show that the constructed sequential model selection procedure is optimal in the sense of oracle inequalities.     

A joint model-based experimental design approach for the identification of kinetic models in continuous flow laboratory reactors

Continuous flow laboratory reactors are typically used for the development of kinetic models for catalytic reactions. Sequential model-based design of experiments (MBDoE) procedures have been proposed in literature where experiments are optimally designed for discriminating amongst candidate models or for improving the estimation of kinetic parameters. However, the effectiveness of these procedures is strongly affected by the initial model uncertainty, leading to suboptimal design solutions and higher number of experiments to be executed. A joint model-based design of experiments (j-MBDoE) technique, based on multi-objective optimization, is proposed in this paper for the simultaneous solution of the dual problem of discriminating among competitive kinetic models and improving the estimation of the model parameters. The effectiveness of the proposed design methodology is tested and discussed through a simulated case study for the identification of kinetic models of methanol oxidation over silver catalyst.     

Sequential experimental design for model discrimination: Taking into account the posterior covariance matrix of differences between model predictions

Techniques for experimental design of experiments for model discrimination constitute important tools for scientists and engineers, as analyzed phenomena can very often be described fairly well by different mathematical models. As interpretation and use of available experimental data depend on the model structure, techniques for design of experiments for selection of the best model are of fundamental importance. Besides, experiments must often be designed for estimation of model parameters and reduction of variances of model predictions (or parameter estimates). These two classes of experimental design techniques generally lead to different experimental designs, although model discrimination and reduction of variances of parameter estimates are closely related to each other. In this work the posterior covariance matrix of difference between model predictions is taken into account during the design for model discrimination for the first time. The obtained results show that the model discrimination power becomes much higher when the posterior covariance matrix of difference between model predictions are considered during the experimental design, increasing the capability of model discrimination and simultaneously leading to improved parameter estimates.     

Sequential procedures for comparing several medical treatments

The use of sequential methods in clinical trials allows inferior treatments to be eliminated early. From an ethical point of view, the advantages are substantial. However, early stopping induces estimation bias and a deterioration in precision because of reduced sample sizes. This paper considers the problem of determining which of k ≥ 2 treatments with Bernoulli responses has the highest probability of success. Two sequential procedures are investigated and compared with a fixed—sample procedure. Various properties are derived and illustrated for the cases k =2,3 and 5. It is shown that the sequential procedures can achieve a pattern of error probabilities equivalent to the fixed—sample procedure for a much lower level of expected successes lost. Approximations for the bias and standard deviation of estimators of treatment differences are obtained by using results about the distribution of stopping times for a normal process.     

用序贯法判别“A”系催化剂上氨合成反应的速率模型

本文详细讨论了用序贯法对“A”系催化剂上氨合成反应的速率模型最佳判别的过程;对不同速率模型中的参数进行初估,按设定的准则作序贯判别,最后选出优惠的模型方程。 文中还讨论了最小二乘法的选择;判别过程的具体技术问题及模型的判别准则。     

钒催化剂上二氧化硫氧化过程的本征动力学模型

在二段反应器内,模拟工业条件下,测定了细颗粒S101型钒催化剂上二氧化硫氧化过程的数据,并首次将序贯法应用于二氧化硫氧化过程动力学的研究.将基本实验测定的数据,用非线型最小二乘法回归,算出模型的参数估算值,利用离散度最大原理选择补充实验条件,从十一个竞争模型中筛选出适定的最佳模型.应用最小联合置信容积准则精估了模型参数.推荐S101型钒催化剂上二氧化硫氧化过程的本征动力学速率方程是     

Integration of Design and Control: A Robust Control Approach Using MPC

This paper presents a new method to integrate process control with process design. The process design is based on steady‐state costs, .i.e., capital and operating costs. Control is incorporated into the design in terms of a variability cost. This term is calculated based on the non‐linear process model, represented here as a nominal linear model supplemented with model parameter uncertainty. Robust control tools are then used within the approach to assess closed‐loop robust stability and to calculate closed‐loop variability. The integrated method results in a non‐linear constrained optimization problem with an objective function that consists of the sum of the steady costs and the variability cost. Optimization using the traditional sequential approach and the new integrated method was applied to design a multi‐component distillation column using a Model Predictive Control (MPC) algorithm. The optimization results show that the integrated method can lead to significant cost savings when compared to the traditional sequential approach. In addition, an RGA analysis was performed to study the effects of process interactions on the optimization results.     

The use of sequential discontinuous procedures for the model discrimination in biological wastewater treatment

Culture procedures are discussed in view of mode discrimination and parameter estimation. The favorable possibilities of sequential discontinuous culture procedures are reviewed. Some necessary conditions for discerning growth- and uptake-inhibition are given. Mathematical handling of the steady-state data requires numerical integration, since treatment in terms of steady-state data for continuous cultures is found to be inconvenient. The procedure is illustrated for mode discrimination of algal growth, under single nutrient limitation.     

关于模型筛选的适应性准则问题——兼与“用序贯法判别在‘A’系催化剂上氨合成反应的速率模型”的作者商榷

<正>一、引言 序贯法是一新的实验设计方法,文献“用序贯法判别在‘A’系催化剂上氨合成反应的速率模型”将序贯法应用到实际中去,并取得较为理想的效果。本文拟对模型筛选的适应性准则这一问题作进一步的探讨。     

State-preserving nonlinear model reduction procedure

Model reduction has proven to be a powerful tool to deal with challenges arising from large-scale models. This is also reflected in the large number of different reduction techniques that have been developed. Most of these methods focus on minimizing the approximation error; however, they usually result in a loss of physical interpretability of the reduced model. A new reduction technique, which preserves a non-prescribed subset of the original state variables in the reduced model, is presented in this work. The technique is derived from the Petrov–Galerkin projection by adding constraints on the projection matrix. This results in a combinatorial problem of which states need to be selected. A sequential algorithm has been developed based on the modified Gram–Schmidt orthogonalization procedure. The presented technique is applied to two examples where the reduction error is found to be comparable to the traditional POD method. At the same time, the technique has the advantage that the physical interpretation of the remaining states is retained.     

A backoff strategy for model‐based experiment design under parametric uncertainty

Model‐based experiment design techniques are an effective tool for the rapid development and assessment of dynamic deterministic models, yielding the most informative process data to be used for the estimation of the process model parameters. A particular advantage of the model‐based approach is that it permits the definition of a set of constraints on the experiment design variables and on the predicted responses. However, uncertainty in the model parameters can lead the constrained design procedure to predict experiments that turn out to be, in practice, suboptimal, thus decreasing the effectiveness of the experiment design session. Additionally, in the presence of parametric mismatch, the feasibility constraints may well turn out to be violated when that optimally designed experiment is performed, leading in the best case to less informative data sets or, in the worst case, to an infeasible or unsafe experiment. In this article, a general methodology is proposed to formulate and solve the experiment design problem by explicitly taking into account the presence of parametric uncertainty, so as to ensure both feasibility and optimality of the planned experiment. A prediction of the system responses for the given parameter distribution is used to evaluate and update suitable backoffs from the nominal constraints, which are used in the design session to keep the system within a feasible region with specified probability. This approach is particularly useful when designing optimal experiments starting from limited preliminary knowledge of the parameter set, with great improvement in terms of design efficiency and flexibility of the overall iterative model development scheme. The effectiveness of the proposed methodology is demonstrated and discussed by simulation through two illustrative case studies concerning the parameter identification of physiological models related to diabetes and cancer care. © 2009 American Institute of Chemical Engineers AIChE J, 2010     

MULTIVARIABLE PROCESS IDENTIFICATION AND CONTROL OF CONTINUOUS FLUIDIZED BED DRYERS

Dynamic models that rigorously describe fluidized bed dryers based on the fundamental principles of the process are usually so complex to be employed in control system design. To obtain simple reduced-order models for such systems, a sequence of step changes in the manipulated and load variables is introduced into the rigorous model. The obtained input-output dynamic response data are used for off-line model identification. Different types of linear models are generated, which are shown to be adequately representing the fluidized bed drying dynamics. The derived models are useful to develop model-based control algorithms such as Internal Model Control (IMC) and Model Predictive Control (MPC). Performance and robustness properties of these controllers are analyzed. Simulation results demonstrate a good performance in terms of tracking and load rejection capabilities.     

Minimum risk sequential point estimation of the stress-strength reliability parameter for exponential distribution

Abstract

In this article, using purely and two-stage sequential procedures, the problem of minimum risk point estimation of the reliability parameter (R) under the stress–strength model, in case the loss function is squared error plus sampling cost, is considered when the random stress (X) and the random strength (Y) are independent and both have exponential distributions with different scale parameters. The exact distribution of the total sample size and explicit formulas for the expected value and mean squared error of the maximum likelihood estimator of the reliability parameter under the stress–strength model are provided under the two-stage sequential procedure. Using the law of large numbers and Monte Carlo integration, the exact distribution of the stopping rule under the purely sequential procedure is approximated. Moreover, it is shown that both proposed sequential procedures are finite and for special cases the exact distribution of stopping times has a degenerate distribution at the initial sample size. The performances of the proposed methodologies are investigated with the help of simulations. Finally, using a real data set, the procedures are clearly illustrated.