Quantitative model building

In PAT, the most common purpose of quantitative model building is to convert a nonselective analyzer into a selective one, to enable its effective use for a specific application. There are several different quantitative model building tools that have been shown to be effective for PAT applications, and these will be discussed in this section. 

Where C is a matrix of component concentrations or sample properties, S is a matrix of basis vectors (pure component spectra, or spectral profiles reflecting a pure sample property), and E and Ec are model residuals. The direct model expresses the analyzer responses (X) as a function of concentrations, whereas the inverse model expresses concentrations as a function of the analyzer responses. Because the former is more in line with the Beer-Lambert Law (absorbance = concentration x absorptivity), it is given the label direct . 

Finally, the specific methods discussed below were chosen based on their utility for PAT applications and their availability in current commercial software packages. For those who are interested in a more complete list of quantitative model building methods, further details can be found in the following references [1,46-56]. 

MLR is an inverse method that uses the multiple linear regression model that was discussed earlier [1,46]  

Once the m variables have been identified, the MLR regression coefficients are determined from a matching set of analyzer (X) and reference (y) calibration data nsing least squares  

This section discusses common methods for building quantitative chemometric models in PAC. In this field, the user most often desires to build a model that converts values generated by an analytical instrument into values of properties or concentrations of interest for use in process control, quality control, industrial hygiene, safety, or other value-adding purposes. There are several chemometric techniques that can be used to build quantitative models, each of which has distinct advantages and disadvantages. 

When MLR is used to provide a predictive model that uses multiple analyzer signals (e.g. wavelengths) as inputs and one property of interest (e.g. constituent concentration) as output, it can be referred to as an inverse linear regression method. The word inverse arises from the spectroscopic application of MLR because the inverse MLR model represents an inverted form of Beer s Law, where concentration is expressed as a function of absorbance rather than absorbance as a function of concentration 1,38 

Once the regression coefficients of the inverse MLR model are determined (according to Equation 8.24), the property (Y-value) of an unknown sample can be estimated from the values of its selected X-variables (Xse] p)  

The fit of an MLR model can be assessed using both the RMSEE and the correlation coefficient, discussed earlier (Equations 8.10 and 8.11). The correlation coefficient has the advantage that it takes into account the range of the Y-data, and the RMSEE has the advantage that it is in the same units as the property of interest. 

The only significant implication of the inverse model is that all of the model error is assumed to be present in the Y-data, rather than in the X-data. Although this is not strictly true in any real applications (where the X-variables have noise ), this limitation has not prevented the effective use of inverse MLR in many process analytical applications. 

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Figure 12.9 The NIR transmission spectra of 70 different styrene-butadiene copolymers over the range 1570-1850nm, for use in demonstrating the quantitative model building methods.

Although the above-mentioned methods cover most of the quantitative model building tools that are available in current commercial software, there are several other useful methods that were not mentioned. Many of these are simply extensions of the methods already discussed. A more complete list of such tools is also available. l>44>47 9... 

The abbreviation QSAR stands for quantitative structure-activity relationships. QSPR means quantitative structure-property relationships. As the properties of an organic compound usually cannot be predicted directly from its molecular structure, an indirect approach Is used to overcome this problem. In the first step numerical descriptors encoding information about the molecular structure are calculated for a set of compounds. Secondly, statistical methods and artificial neural network models are used to predict the property or activity of interest, based on these descriptors or a suitable subset. A typical QSAR/QSPR study comprises the following steps structure entry or start from an existing structure database), descriptor calculation, descriptor selection, model building, model validation. 

This extension in the laboratory can be seen as the fantastic hypothesis testing application of molecular modeling. It is rare to find a chemical problem where there are not at least a few theories of the molecular mechanism involved. How many times has each of us heard steric affect or hydrogen bonding invoked as the explanation of a variety of experimental observations made at the bench level How useful would it be to be able to actually build accurate, quantitative models to investigate such ideas ... 

In this chapter, we consider how to construct quantitative models of the dynamics of microbial communities, building on our discussion of microbial kinetics in Chapter 18. In our modeling, we take care to account for how the ambient geochemistry controls microbial growth, and the effect of the growth on geochemical conditions. 

Building Product Models. The next step in product optimization deals with model building. A model summarizes the relations between formula variables in a succinct, quantitative way. 

Many 2D or 3D QSAR models rely on multiple structural and/or property attributes to build a quantitative model, and this makes them difficult to interpret qualitatively as design guidelines. 

Transaction costs in employment are apparently balanced against the difficulty of finding a sufficiently good match between employer and employed. Here we build a simple quantitative model using straightforward engineering reasoning. 

As discussed earlier, the two figures of merit for a linear regression model, the RMSEE and the correlation coefficient (Equations 8.11 and 8.10), can also be used to evaluate the fit of any quantitative model. The RMSEE, which is in the units of the property of interest, can be used to provide a rough estimate of the anticipated prediction error of the model. However, such estimates are often rather optimistic because the exact same data are used to build and test the model. Furthermore, they cannot be used effectively to determine the optimal complexity of a model because increased model complexity will always result in an improved model fit. As a result, it is very dangerous to rely on this method for model validation. 

Once one builds a quantitative model and assesses its performance using either the validation methods discussed above or actual on-line implementation, the unavoidable question is Can I do better In many cases, the answer is quite possibly. There are several different actions that one could take to attempt to improve model performance. The following is a list of such actions, which is an expansion of a previously published guide by Martens and Naes.1... 

When one builds a quantitative model using PCR or PLS, one is often not aware that the model parameters that are generated present an opportunity to learn some useful information. Information extracted from these model parameters cannot only be used to better understand the process and measurement system, but also lead to improved confidence in the validity of the quantitative method itself. 

Building Quantitative Models for the Hofmeister Series and Cohn-Edsall and Setschenow Equations... 

While the majority of published models are based on a limited number of drug molecules, especially the study of Zhao et al. (2001) provides the most extensive compilation from available literature data and a statistical model derived from those using Abraham descriptors. We used this carefully selected dataset to build a quantitative model for human intestinal absorption employing VolSurf descriptors (see Cruciani et al. 2000). 

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