Empirical model building

Many phenomena in engineering are very complex and we do not have sufficient knowledge at the moment to develop a model from first principles instead we have to rely on empirical correlations. Today most process development is done using empirical or semi-empirical models. These models are usually accurate and very useful. The drawback is that they are only valid for specific equipment within an experimental domain where the parameters are determined. 

In the experimental designs we saw in the preceding chapters, each factor was studied at only two levels. Because of this economy, we had to be content with a hmited form of the function describing the effects of the factors on the response. Consider, for example, the variation in reaction yield with temperature discussed in Chapter 3. The average yields observed with catalyst A are 59% at 40 °C, and 90% at 60 °C, as given in Table 3.1. We can see in Fig. 5.1a that these two pairs of values are compatible with an infinite number of functions. In Chapter 3, we fitted a model with a linear part and an interaction term to the response values, but we had no guarantee that this was the correct model. To clarify this situation, more information is needed. 

If we make, say, three more measurements at intermediate temperatures and find that the plot of the five points is similar to Fig. 5.1b, we gain more confidence in the linear model. A graph like Fig. 5.1c, on the other hand, would be taken as evidence that the linear model is not the best one. 

These considerations also serve to remind us that two-level designs are appropriate only in the initial stages of an investigation. To improve the description of the response surface, we have to perform a larger number of experiments. 

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G. E. P. Box and N. R. Draper, Empirical Model Building and Response Surfaces, John Wiley Sons, New York, 1987. 

G.E.P. Box, Draper N.R., Empirical model-building and response surfaces, John... 

Box G.E.P. and Draper, N.R. Empirical Model-building and Response Surfaces, Wiley, Chichester, 1987. 

The principal reason that a test set is necessary for validation is that empirical model-building methods cannot readily distinguish between noise and information in data sets, so the methods are prone to adjusting the model parameters to reduce error beyond the point warranted by the information contained in the data. This problem is called overtraining and can be countered by a variety of techniques such as descriptor reduction and early stopping, and readers interested in those topics are referred to the more detailed reviews of numerical methods cited in each of the following sections. 

Empirical Model-Building and Response Surfaces WUey, New York 1987. 

Fernandez A, Cofiaudo J, Cunha EM, Ocio MJ, Martinez A (2002) Empirical model building based on Weibull distribution to describe the joint effect oh pH and temperature on the thermal resistance of Bacillus cereus in vegetale substrate. Int J Food Microbiol 77 147-153 Forney LJ, Zhou X, Brown CJ (2004) Molecular microbial ecology land of the one-eyed king. Curr Opin Microbiol 7 210-220... 

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