Inverse least squares model

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 inverse least squares (ILS) method is sometimes referred to as the P-matrix method. The calibration model is transformed so that component concentrations are defined as a function of the recorded response values,... 

The improvement in computer technology associated with spectroscopy has led to the expansion of quantitative infrared spectroscopy. The application of statistical methods to the analysis of experimental data is known as chemometrics [5-9]. A detailed description of this subject is beyond the scope of this present text, although several multivariate data analytical methods which are used for the analysis of FTIR spectroscopic data will be outlined here, without detailing the mathematics associated with these methods. The most conunonly used analytical methods in infrared spectroscopy are classical least-squares (CLS), inverse least-squares (ILS), partial least-squares (PLS), and principal component regression (PCR). CLS (also known as K-matrix methods) and PLS (also known as P-matrix methods) are least-squares methods involving matrix operations. These methods can be limited when very complex mixtures are investigated and factor analysis methods, such as PLS and PCR, can be more useful. The factor analysis methods use functions to model the variance in a data set. 

Spectrophotometric monitoring with the aid of chemometrics has also been applied to more complex mixtures. To solve the mixtures of corticosteroid de-xamethasone sodium phosphate and vitamins Bg and Bi2, the method involves multivariate calibration with the aid of partial least-squares regression. The model is evaluated by cross-validation on a number of synthetic mixtures. The compensation method and orthogonal function and difference spectrophotometry are applied to the direct determination of omeprazole, lansoprazole, and pantoprazole in grastroresistant formulations. Inverse least squares and PCA techniques are proposed for the spectrophotometric analyses of metamizol, acetaminophen, and caffeine, without prior separation. Ternary and quaternary mixtures have also been solved using iterative algorithms. 

Notice that even if the concentrations of all the other constituents in the mixture are not known, the matrix of coefficients (P) can still be calculated correctly. This model, known as inverse least squares (ILS), multiple linear regression (MLR), or P matrix, seems to be the best approach for almost all quantitative analyses because no knowledge of the sample composition is needed beyond the concentrations of the constituents of interest. 

Inverse least squares in an example of a multivariate method. In this type of model, the dependent variable (concentration) is solved by calculating a solution from multiple independent variables (in this case, the responses at the selected wavelengths). It is not possible to work backwards from the concentration value to the independent spectral response values because an infinite number of possible solutions exist. However, the main advantage of a multivariate method is the ability to calibrate for a constituent of interest without having to account for any interferences in the spectra. 

Thus, the name for this type of model is principal components regression it combines principal components analysis and inverse least squares regression to solve the calibration equation for the model. All that remains is to come up with a single unified equation that represents the PCR model. Therefore, rearranging the previous matrix model equation to represent the scores as a function of the spectral absorbances and the eigenvectors produces... 

The alternative to the CLS calibration model is the inverse least squares (ILS) calibration model. Employing an ILS model alleviates the need for complete knowledge of the calibration set... 

Inverse least squares (ILS) is a least-squares method that assumes the inverse calibration model given in eqn (4.4). For this reason it is often also termed multiple linear regression (MLR). In the literature this calibration approach is... 

We have used the time-dependent model to develop a system (Cursim) which determines the transport parameters by curve-fitting of a single-ion transport experiment a exp as a function of time (24 hrs). From an initial starting point (D K ex) the parameters are varied in such a way that the model describes the experiment, i.e. the deviation between the experimental data and the model is minimized. The deviation is expressed in terms of the least squares value. Having obtained the best-fit parameters D and the optimum is examined by a search for other "best fits" around the best-fit parameter set (D ,K x)- This is represented by a 3-dimensional plot of the transport parameters and the inverse least squares value (1/precise estimation, a single and sharp peak in and K ex can be observed. 

Multiwavelength methods. Least squares curve fitting techniques may be used in the determination of multicomponent mixtures with overlapping spectral features. Two classical quantitation methods, the Classical Least Squares (CLS) mode and the Inverse Least Squares (ILS) model, are applied when wavelength selection is not a problem. CLS is based on Beer s law and uses large regions of the spec-tram for calibration but cannot cope with mixtures of interacting constituents. ILS (multivariate method) can accurately build models for complex mixtures when only some of the constituent concentrations are known. 

As was shown, the conventional method for data reconciliation is that of weighted least squares, in which the adjustments to the data are weighted by the inverse of the measurement noise covariance matrix so that the model constraints are satisfied. The main assumption of the conventional approach is that the errors follow a normal Gaussian distribution. When this assumption is satisfied, conventional approaches provide unbiased estimates of the plant states. The presence of gross errors violates the assumptions in the conventional approach and makes the results invalid. 

Figure 12.8 displays an organization chart of various quantitative methods, in an effort to better understand their similarities and differences. Note that the first discriminator between these methods is the direct versus inverse property. Inverse methods, such as MLR and partial least squares (PLS), have had a great deal of success in PAT over the past few decades. However, direct methods, such as classical least squares (CLS) and extensions thereof, have seen a recent resurgence [46-51]. The criterion used to distinguish between a direct and an inverse method is the general form of the model, as shown below ... 

Numerical simulations of the data were conducted with the algorithms discussed above, with the added twist of optimizing the model to fit the data collected in the laboratory by adjusting the collision efficiency and the fractal dimension (no independent estimate of fractal dimension was made). Thus, a numerical solution was produced, then compared with the experimental data via a least squares approach. The best fit was achieved by minimizing the least squared difference between model solution and experimental data, and estimating the collision efficiency and fractal dimension in the process. The best model fit achieved for the data in Fig. 10a is plotted in Fig. 10b, and that for Fig. 11a is shown in Fig. lib. The collision efficiencies estimated were 1 x 10-4 and 2 x 10-4, and the fractal dimensions were 1.5 and 1.4, respectively. As expected, collision efficiency and fractal dimension were inversely correlated. However, the values of the estimates are, in both cases, lower than might be expected. The lower values were attributed to the following ... 

Since this monograph is devoted only to the conception of mathematical models, the inverse problem of estimation is not fully detailed. Nevertheless, estimating parameters of the models is crucial for verification and applications. Any parameter in a deterministic model can be sensibly estimated from time-series data only by embedding the model in a statistical framework. It is usually performed by assuming that instead of exact measurements on concentration, we have these values blurred by observation errors that are independent and normally distributed. The parameters in the deterministic formulation are estimated by nonlinear least-squares or maximum likelihood methods. 

Butcher and Gauthier [18] used inverse modeling to estimate the residual DNAPL mass. The flux was estimated from observed down-gradient concentrations. For the inverse model, Butcher and Gauthier used a tractable analytical approximation to the problem and developed additional simplifications to yield a form that is easily solved for the parameters of interest. From the simplified set of analytical equations, the parameters desired were solved using a least-squares estimator. The authors concluded that the methods presented in... 

We can see from Eqs. (3.5) [see also Appendix in Ref. ] that the force constants F , F etc., are generally mass dependent quantities. To arrive at the iso-topically invariant potential function we must therefore express these quantities in terms involving mass independent valence force constants and to fit these to experimental spectra [cf. ]. For ammonia, this would represent a really formidable numerical problem. Taking into account the proposed limits of our model, the fact that we are mainly interested in the inversion—rotation structure of the spectra, we have overcome the above mentioned difficulties in the following way [see for details] (i) all the enharmonic force constants in Eq. (5.5) were neglected (ii) the p-dependent contributions to the harmonic force constants F [see Eq. (4.7)] were neglected (iii) the least squares fit of the double-minimum potential function parameters and the p-independent harmonic force constants were performed for light isotopes ( NHs, NHs) and heavy isotopes ( ND3, NTs) separately. 

We could of course attempt to adjust a potential function of ammonia using Eq. (5.4) in a least squares fit to the data extended to a set of energy levels with J = 0,k 0. However, it seems better to adjust a minimum number of potential function parameters using the vibration and inversion data alone and to check the validity of our model by comparing the calculated vibration—inversion—rotation transition frequencies with the observed data ... 

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