Sequential experimental design

If matrix A is ill-conditioned at the optimum (i.e., at k=k ), there is not much we can do. We are faced with a truly ill-conditioned problem and the estimated parameters will have highly questionable values with unacceptably large estimated variances. Probably, the most productive thing to do is to reexamine the structure and dependencies of the mathematical model and try to reformulate a better posed problem. Sequential experimental design techniques can also aid us in... 

SEQUENTIAL EXPERIMENTAL DESIGN FOR PRECISE PARAMETER ESTIMATION... 

The sequential experimental design can be made either for precise parameter estimation or for model discrimination purposes. 

As an example for precise parameter estimation of dynamic systems we consider the simple consecutive chemical reactions in a batch reactor used by Hosten and Emig (1975) and Kalogerakis and Luus (1984) for the evaluation of sequential experimental design procedures of dynamic systems. The reactions are... 

As a second example let us consider the fed-batch bioreactor used by Ka-logerakis and Luus (1984) to illustrate sequential experimental design methods for dynamic systems. The governing differential equations are (Lim et al., 1977) ... 

Buzzi-Ferraris, G. and P. Forzatti, "A New Sequential Experimental Design Procedure for Discriminating Among Rival Models", Chem. Eng. Sci., 38, 225-232(1983). 

Hosten, L.H. and G. Emig, "Sequential Experimental Design Procedures for Precise Parameter Estimation in Ordinary Differential Equations", Chem. Eng. Sci., 30, 1357 (1975)... 

Kalogerakis, N., "Sequential Experimental Design for Model Discrimination in Dynamic Systems", proc. 34th Can. Chem. Eng. Conference, Quebec City, Oct. 3-6 (1984). 

Kalogerakis, N., and R. Luus, "Sequential Experimental Design of Dynamic Systems Through the Use of Information Index", Can. J. Chem. Eng., 62, 730-737(1984). 

Procedures on how to make inferences on the parameters and the response variables are introduced in Chapter 11. The design of experiments has a direct impact on the quality of the estimated parameters and is presented in Chapter 12. The emphasis is on sequential experimental design for parameter estimation and for model discrimination. Recursive least squares estimation, used for on-line data analysis, is briefly covered in Chapter 13. 

Figure 17-1 Standard concept behind sequential experimental design (see text for definition of function/(a, /3)).

So how is this related to the nature of the graph used for the sequential experimental design We forgo the detailed math here, in deference to trying to impart an intuitive grasp of the topic, and we have already presented the equations involved [3],... 

Berridge, J.C. (1989), Chemometrics and Method Development in High-performance Liquid Chromatography. Part 2 Sequential Experimental Designs, Chemom. Intel. Lab. Sys., 5, 195-207. 

The experimental conditions to extract Sn from solid sediments were optimised with two sequential experimental designs... 

The text reviews the methodology of kinetic analysis for simple as well as complex reactions. Attention is focused on the differential and integral methods of kinetic modelling. The statistical testing of the model and the parameter estimates required by the stochastic character of experimental data is described in detail and illustrated by several practical examples. Sequential experimental design procedures for discrimination between rival models and for obtaining parameter estimates with the greatest attainable precision are developed and applied to real cases. 

Further reduction of the experimental effort is obtained by sequential experimental design, carefully planning the new experimental conditions [9, 10]. 

Figure 16. Flowchart of sequential experimental design in the process of model selection, followed by parameter improvement.

A typical flow diagram of the application of sequential experimental design is given in Fig. 16. It should be noted that both in the case of model discrimination and parameter improvement experimental settings at... 

Pinto, J. C., M. W. Lobao. and J. L. Monteiro, Sequential experimental design for parameter estimation analysis of relative deviations, Chem. Eng. Sci., 46, 3129-3138 (1991). 

Develop a priori factorial experimental design or sequential experimental design. 

When a reaction has many reactive species (which may be the case even for apparently simple processes such as pyrolysis of ethane or synthesis of methanol), a factorial or sequential experimental design should be developed and the data can be subjected to a response surface analysis (Box, Hunter, and Hunter, Statistics for Experimenters, 2d ed., Wiley Interscience, 2005 Davies, Design ana Analysis of Industrial Experiments, Oliver Boyd, 1954). This can result in a black box correlation or statistical model, such as a quadratic (limited to first- and second-order effects) for the variables Xi, x2, and X3 ... 

Sequential Experimental Design for Precise Parameter Estimation... 

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