Experimental Designs for Modeling

It is partly the fault of statistics that experimenters have misconstrued the value of the number and precision of data points relative to the value of the location of the points. The importance of the location of the data in the model specification stage can be seen from Fig. 1, which represents literature data (M3) on sulfur dioxide oxidation. The dashed and solid lines represent the predicted rates of two rival models, and the points are the results of two series of experimental runs. It can be seen that neither a greater number of experimental points nor data of greater precision will be of major assistance in discriminating between the two rival models, if data are restricted to the total pressure range from 2 to 10 atm. These data simply do not place the models in jeopardy, as would data below 2 atm and greater than 10 atm total pressure. This is presumably the problem in the water-gas shift reaction, which is classical in terms of the number of models proposed, each of which adequately represent given sets of data. 

The proper location of data is also important in parameter-estimation situations. For the nitric oxide reduction reaction (K11), for example, the relative sizes of the three-dimensional confidence regions calculated after each observation are shown in Fig. 27. The size of the confidence region after 12 points taken according to a one factor at a time variation of hydrogen and nitric oxide partial pressures is seen to be equivalent the size of the region 

The general scheme required here is shown in Fig. 29, a generalization of Fig. 28. With some ideas as to good areas of experimentation, the experimenter takes an initial set of data. These data are then analyzed to determine the best estimates of the parameters of the model or models under consideration. Since models that usually arise in these circumstances are nonlinear in the parameters, some version of nonlinear estimation will usually be employed in this analysis. Nonlinear estimation techniques, of course, almost always require the use of a computer. 

The criterion D is a measure of divergence among the models, obtained from information theory. The quantity nt is the prior probability associated with model / after the nth observation is obtained o2 is the common variance of the n observations y(l), y( 2), , y(n — 1), y(n) a2 is the variance for the predicted value of y(n + 1) by model i. When we have two models, D simplifies to 

An initial nine data points were taken at 35°C in an adiabatic flow reactor. The initial prior probabilities were taken to be equal (equal probability of 

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Buzzi-Ferraris, G., P. Forzatti, G. Emig and H. Hofmann, "Sequential Experimental Design for Model Discrimination in the Case of Multiple Responses", Chem. Eng. Sci., 39(1), 81-85 (1984). 

Figure 2.3. Two experimental designs for modeling a potentially nonlinear system, (a) Two point calibration does not detect curvature (b) three point calibration can account for quadratic behavior.

The rate constant /ct, determined by means of Eq. (6-47) or (6-48), may describe either general base or nucleophilic catalysis. To distinguish between these possibilities requires additional information. For example, in Section 3.3, we described a kinetic model for the N-methylimidazole-catalyzed acetylation of alcohols and experimental designs for the measurement of catalytic rate constants. These are summarized in Scheme XVIIl of Section 3.3, which we present here in slightly different form. 

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. 

Another approach for experimental design for inverse modeling is termed a natural design. This is where many samples are collected over a period of time until one has confidence that the variation has been adequately represented. In some cases, these natural designs are the only choice because neither R nor c can be controlled. Successful use of this approach requires some knowledge of the sj stem in order to make an intelligent assessment of how many samples are required and how long to sample. One rule of thumb is to have at least three times as many samples as the expected rank of the system. (Rank is a concept that is discussed in Chapters 4 and 5.)... 

Scores Plot (Sample Diagnostic) Figure 5-132 displays the Factor 2 versus Factor l scores for the MCB model (showing 98.41% of the spectral variance). The experimental design for this data set is not readily discernible because this plot shows only two dimensions. However, samples 1-4 define the extremes, which makes sense because these are the pure spectra. As expected, the center-point replicates lie very near each other and are in the middle of the plot. 

A. Dijkstra, and L. Kaufman, Evaluation and Optimization of Laboratory Methods and Analytical Procedures (Amsterdam Elsevier, 1978) G. E. P. Box, W. G. Hunter, and J. S. Hunter, Statistics for Experimenters An Introduction to Design Data Analysis and Model Building (New York Wiley, 1978) R. S. Strange, Introduction to Experimental Design for Chemists, J. Chem. Ed. 1990,67. 113. 

Most of these issues can be addressed through interpersonal skills, which are not the subject of this chapter. However, the staling process of chemometric models can be addressed by performing three important tasks during the project (1) publicize the limitations of your models as soon as you are aware of them, (2) do the hard work early on (experimental designs for calibration, collection and analysis of many samples) in order to avoid embarrassments through model extrapolations, and (3) keep an eye on your methods as they are operating, and update/adjust them promptly. 

The simplex-lattice type of experimental design for these models consists of points having coordinates that are combinations of the vth proportions of the variables ... 

Suppose we wish to construct an experimental design for a mixture consisting of four components and model it with a cubic polynomial model, as described by Equation 8.14. Our task is to construct a 4,3 simplex-lattice design with q = 4 and v = 3. The proportions of each of the components are calculated by Equation 8.16, giving the following values ... 

Symmetric experimental designs for mixture+process factor spaces are the cross products of symmetric designs for process variables and mixture variables. Figure 8.12 shows an experimental design in mixture+process factor space for a model where both types of variables are of the first order. In both of the examples shown in Figure 8.12, the process variables are described by a two-level full factorial... 

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