Multivariate inverse models

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. 

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In multivariate calibration, selectivity is commonly used to measure the amount of signal that cannot be used for prediction because of the overlap between the signal of the analyte and the signal of the interferences [68,69]. For inverse models, such as PLS, selectivity is usually calculated for each calibration sample as... 

Faber, N.M., Efficient computation of net analyte signal vector in inverse multivariate calibration models, Anal. Chem., 70, 5108-5110, 1998. 

The different methods for multivariate calibration differ by the mathematical model that is based either on Beer s law, that is, the spectra are regressed on concentrations as with the /C-matrbc approach or on inverse models where the regression of concentrations on spectra is carried out. 

On the other hand, when latent variables instead of the original variables are used in inverse calibration then powerful methods of multivariate calibration arise which are frequently used in multispecies analysis and single species analysis in multispecies systems. These so-called soft modeling methods are based, like the P-matrix, on the inverse calibration model by which the analytical values are regressed on the spectral data ... 

Multivariate calibration tools are used to construct models for predicting some characteristic of future samples. Chapter 5 begins with a discussion of the reasons for choosing multivariate over univariate calibration methods. The most widely used multivariate calibration tools are then presented in two categories classical and inverse methods. 

The elements of fhe vector y are the reference values of the response variable, used for building the model. The uncertainty on the coefficient estimation varies inversely with the determinant of the information matrix (X X) which, in the case of a unique predictor, corresponds to its variance. In multivariate cases, the determinant value depends on the variance of fhe predictors and on their intercorrelation a high correlation gives a small determinant of the information matrix, which means a big uncertainty on the coefficients, that is, unreliable regression results. 

For most spectroscopic applications, the goal of multivariate calibration is to predict the concentration of a given analyte(s) in a future (prospective) sample using only its measured spectrum and a previously determined model. To do this, the inverse calibration method is used in which equation (12.2) is rewritten as... 

Constrained Regularization (CR) To understand constrained regularization, multivariate calibration can be viewed as an inverse problem. Given the inverse mixture model for a single analyte... 

From a practical perspective, this is the model that should be used to design a (multivariable) controller that manipulates the inputs us to fulfill the control objectives ys. It is important to note that the availability of a low-order ODE model of the process-level dynamics affords significant flexibility in designing the supervisory control system, since any of the available inversion- or optimization-based (e.g., Kravaris and Kantor 1990, Mayne et al. 2000, Zavala... 

Multivariate techniques are inverse calibration methods. In normal least-squares methods, often called classical least-squares methods, the system response is modeled as a function of analyte concentration. In inverse methods, the concentrations are treated as functions of the responses. The latter has some advantages in that concentrations can be accurately predicted even in the presence of chemical and physical sources of interference. In classical methods, all components in the system need to be considered in the mathematical model produced (regression equation). 

It is worth mentioning at this stage that the three-stage hierarchical model used in Bayesian analyses when undertaken within the framework provided by WinBUGS requires that normal distributions are parameterized as mean and precision. Precision is the inverse of variance. For example, when defining the prior for the population parameter vector 0, the multivariate normal distribution would be parameterized as the mean vector fi and the inverse of the variance-covariance matrix X such that,... 

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. 

Apart from discrete modelling of relaxation processes, ID and 2D Inverse Laplace Transformation (ILT) is gaining more and more interest. Moreover, soft and hard modelling data processing tools like PLS or multivariate curve resolution (MCR) are applied to low field NMR data. Special algorithms were developed for the needs in relaxation modelling, for example DOUBLESLICING l... 

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. 

In order to more precisely ascertain the degree of difficulty of the multivariable interactions, a 2 X 2 IMC controller [17] is considered next. The controller in this case is the inverse of the full 2x2 model matrix augmented with a diagonal filter block with first-order elements. The filter time constants are chosen to be the same as those used in the equivalent PI designs. The... 

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