Sequential experimentation

Stolzberg, R. J. Screening and Sequential Experimentation Simulations and Elame Atomic Absorption Spectrometry Experiments, /. Chem. Educ. 1997, 74, 216-220. 

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],... 

If the null hypothesis was wrong, then the data would vary around a line offset from the line representing /x0, and get closer and closer to it, instead. Eventually, at some value of n, this line would cross the converging lines representing the critical limits around /x0, indicating the result. This is the basic picture, shown in Figure 20-2. For a sequential experimental plan, the sequence is terminated at the first significant experiment, as shown. 

In designing sequential experimental measurements or groups of them, other functions that might be performed with appropriate calculations are ... 

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. 

Figure 4.3 The seventh sequential experimental reaction spectrum (a) and the seventh...

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... 

The decision to use orthogonal plans or to allow nonorthogonal plans, and the decision to allow or disallow the reuse of runs from stage to stage, are related operational issues. They individually and jointly affect the performance of a screening plan and the complexity of calculations required to assess analytically that performance. Depending on characteristics of the application, such as the degree of measurement error and the need to account for block effects in sequential experimentation, either or both may be important considerations. 

The mixture experiment counterpart to conventional screening/fractional factorial experimentation also is possible. So-called axial designs have been developed for the purpose of providing screening-type mixture data for use in rough evaluation of the relative effects of a large number of mixture components on a response variable. The same kind of sequential experimental strategy illustrated in the process improvement example is applicable in mixture contexts as well as contexts free of a constraint such as (5-15). 

Investigation of the Great Curvature Domain of the Response Surface Sequential Experimental Planning... 

Hosten, L. H.. A sequential experimental procedure for precise parameter estimation based on the shape of the joint confidence region, Chem. Eng. Sci.. 29, 2247-2252 (1974). 

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