Inverse models/modeling advantages

The multivariate quantitative spectroscopic analysis of samples with complex matrices can be performed using inverse calibration methods, such as ILS, PCR and PLS. The term "inverse" means that the concentration of the analyte of interest is modelled as a function of the instrumental measurements, using an empirical relationship with no theoretical foundation (as the Lambert Bouguer-Beer s law was for the methods explained in the paragraphs above). Therefore, we can formulate our calibration like eqn (3.3) and, in contrast to the CLS model, it can be calculated without knowing the concentrations of all the constituents in the calibration set. The calibration step requires only the instrumental response and the reference value of the property of interest e.g. concentration) in the calibration samples. An important advantage of this approach is that unknown interferents may be present in the calibration samples. For this reason, inverse models are more suited than CLS for complex samples. 

Condition 2 either requires a large number of extra experiments or a reduction to 25 wavelengths. There have been a number of algorithms that have been developed to reduce the wavelengths to the most significant ones, so enabling inverse models to be used, but there is no real advantage over classical models unless very specific information is available about error distributions. 

In analysing the non-uniqueness problem, we should distinguish betw een the two classes of inverse problems we introduced above the inverse model (or inverse scattering) problem and the inverse source problem. The advantage of the inverse... 

The approach based on the inverse model in the selection of components has several advantages. This enables, in one simulation run, relevant information to be obtained in the selection process. In fact the backward transportation takes into account the dynamic feature of the specifications. It clearly shows possible over-sizing margins or, if manufacturer s data limits are crossed over, the amplitude and duration over the limits since the curves obtained are time parametrized. The duration over the limits can even be used in the case of sizing based on intermittent operation. Also, the simulation results are obtained in a completely independent way from what is still unknown in the system to design and from the control inputs of the system. Finally, the approach does not necessitate to take any a priori option on the component technology. Thus technology comparison can also be undertaken in an easy way. This selection phase can be summarized in Fig. 6.16. 

The model of eq. (36.3) has the considerable advantage that X, the quantity of interest, now is treated as depending on Y. Given the model, it can be estimated directly from Y, which is precisely what is required in future application. For this reason one has also employed model (36.3) to the controlled calibration situation. This case of inverse calibration via Inverse Least Squares (ILS) estimation will be treated in Section 36.2.3 and has been treated in Section 8.2.6 for the case of simple straight line regression. 

The advantage of the inverse calibration approach is that we do not have to know all the information on possible constituents, analytes of interest and inter-ferents alike. Nor do we need pure spectra, or enough calibration standards to determine those. The columns of C (and P) only refer to the analytes of interest. Thus, the method can work in principle when unknown chemical interferents are present. It is of utmost importance then that such interferents are present in the Ccdibration samples. A good prediction model can only be derived from calibration data that are representative for the samples to be measured in the future. 

The main advantage of PCR over inverse MLR is that it accounts for covariance between different x variables, thus avoiding any potential problems in the model computation mathematics, and removing the burden on the user to choose variables that are sufficiently independent of one another. From a practical viewpoint, it is much more flexible than CLS, because it can be used to develop regression models for any property, whether it is a concentration property or otherwise. Furthermore, one needs only the values of the property of interest for the calibration samples (and not the concentrations of aU components in the samples). 

Gi cn th.iu ail tiircc assiiiiiptions hold (linearity, linear addithin, and all pure spectra known), CIS has an advantage over the inverse methods (see Section 5.3) in that the calibration models are often easier to determine. For a simple system with three components, calibration may be as simple as obtaining the spectra of the three pure components. 

Before looking at the results we mention that, as an alternative to the Fourier transforms just described, one may take advantage of the fact that both the classical line shape, Gc (correlation function, Cci(t), may be represented very closely by an expression as in Eq. 5.110 [70]. The parameters ti T4, e and S of these functions are adjusted to match the classical line shape. These six parameter model functions have Fourier transforms that may be expressed in closed form so that the inverse and forward transforms are obtained directly in closed form. We note that the use of transfer functions is merely a convenience, certainly not a necessity as the above discussion has shown. 

Aside from univariate linear regression models, inverse MLR models are probably the simplest types of models to construct for a process analytical application. Simplicity is of very high value in PAC, where ease of automation and long-term reliability are critical. Another advantage of MLR models is that they are rather easy to communicate to the customers of the process analytical technology, since each individual X-variable used in the equation refers to a single wavelength (in the case of NIR) that can often be related to a specific chemical or physical property of the process sample. 

At this stage we should confess that we are stressing statistical models very much because our example has so few objects compared to the number of features. Especially for the nonelementary discrimination functions df discussed above there are recommendations for ensuring a ratio between the number of objects n and the number of original features m of n/m > 3 or even nk/m > 3. Therefore attempts are made to develop classifiers which work well in the case of n/m < 3. One example is the EUCLIDean distance classifier of MARCO et al. [1987], the additional advantages of which are no covariance matrix is necessary, inversions are skipped, and correlated training data cause no problems. 

In summary, the SQMF technique proposes several important advantages over the traditional empirical approaches to the vibrational dynamics. The relative magnitudes and signs of all the elements in the force-constant matrix are calculated by means of realistic quantum-mechanical calculations. The Puley s scaling scheme is based on a small number of adjustable parameters and therefore the inverse vibrational problem is well defined, contrary to the VFF model, where additional conditions on the adjustable force constants have to be imposed. The scale factors are transferable in a much wider classes of molecules than the force constants themselves. This makes SQMF a powerful predicting tool for the vibrational assignment of novel materials. 

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