Uncertainty normalized

This normalized half width will be called the normalized uncertainty in the predicted response. It is bounded between 1 and . 

Thus, we will define the normalized information as the reciprocal of the normalized uncertainty. The normalized information is bounded between 1 and 0 ... 

The normalized uncertainty and normalized information are related to the variance function and information function, respectively, defined by Box and Draper (1987). 

One purpose of a good design is to minimize uncertainty and maximize information over the region of interest. We will use both normalized uncertainty and normalized information to discuss the effect of experimental design on the quality of information obtained from a two-factor FSOP model. 

The upper left panel shows a surface of normalized uncertainty (defined in Equation 13.7 above) as a function of factors x and x. The normalized uncertainty is relatively small in the center (approximately 1.1) and relatively large at the comers (approximately 4.0). Note that this surface generally reflects the underlying design the uncertainty surface is relatively low in those regions where experiments have been carried out and is relatively high in those regions where experiments have not been carried out. 

It is important to note that the normalized uncertainty surface shown in the upper left panel is not the response surface generated by the FSOP model itself. Instead, this upper left panel is a measure of how much the response surface might flap around in different regions of the factor space. Experiments serve to anchor the underlying model, to pin it to the data, and thereby reduce the amount of uncertainty in the model at those points. The large amount of uncertainty at the comers of this upper left panel is a reflection of the freedom the model has to move up and down in those regions where experiments have not been performed. 

One of the striking features of this central composite design is the flatness of the normalized uncertainty and normalized information surfaces near the center of the design. 

In Figure 13.2, the experimental design, the normalized uncertainty surface, and the normalized information surface each have four planes of mirror-image symmetry, all of which are perpendicular to the x, - Xj plane. One reflection plane contains the... 

Figure 13.3 shows a similar set of four panels for a slightly different central composite design. The lower left panel shows the placement of experiments in factor space (i.e., it shows the experimental design). The upper left panel shows the normalized uncertainty as a function of factors x, and x. The upper right panel shows the normalized information as a function of factors x, and Xj. The lower right panel plots normalized information as a function of factor x, for X2 = -5, -4, -3, -2, -1, and 0. The experimental design matrix is... 

Figure 13.4 shows four panels for still another central composite design. The lower left panel shows the experimental design itself. The upper left panel shows the normalized uncertainty associated with this design. The upper right panel shows the... 

This allocation of experiments has the effect of making the normalized uncertainty and normalized information contours more axially symmetric (the design isn t quite rotatable there are still only four mirror-image planes of reflection symmetry). However, because no experiments are now being carried out at the center point, the amount of uncertainty is greater there (and the amount of information is smaller there). The overall effect is to provide a normalized information surface that looks like a slightly square-shaped volcano. 

Because each of the pentagonal points is equidistant from the center of the design, the design is rotatable. This rotatability is seen in the axially symmetric surfaces for normalized uncertainty and normalized information. 

At a very basic level, the shapes of the normalized uncertainty and normalized information surfaces for a given model are a result of the location of points in factor space simply because carrying out an experiment provides information - that is, information is greatest in the vicinity of the design. But at a more sophisticated and often more important level, the shapes of the normalized uncertainty and normalized information surfaces are caused by the geometric vibrations of the response surfaces themselves - the more rigidly the model is pinned down by the experiments and the less it can squirm and thrash about, then the less will be the uncertainty and the greater will be the information content. 

The striking feature of this design is the set of six spikes in both the normalized uncertainty and normalized information surfaces. These spikes are an extreme expression of the basic idea that experiments provide information. Even if the experimental design is not a good match for the model even if the iX X) matrix is ill conditioned even if the model doesn t fit the data very well, there is still high-quality information at the points where experiments have been carried out. 

Find a report of a two-factor experimental design. Speculate about the shape of the normalized uncertainty and normalized information surfaces for the design. Sketch their shape. 

This is as close as the charts can be read, and probably as close as the normal uncertainties and approximations in such problems justify, so the required heating time is 135 s. 

A similar idea has been developed using chemical activation techniques hot H atoms, formed from the photolysis of HBr or H2S at certain wavelengths, are allowed to react with 1-butene to form vib-rationally hot n-butyl radicals with the wavelengths used in these experiments [80.G1 81.G], the n-butyl radicals, subject to the normal uncertainties in the assumed thermochemical data, were formed with excess internal energies of about 22, 28, 30, and 42 kcal mol" above the reaction threshold. ... 

The exact relationship between r] and 0 depends to a small extent on particle geometry and reaction order. However, the differences between Eqn. (9-15) and the exact solutions are not significant, given the normal uncertainties in ky and Dajss-... 

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