Calibration inverse least-squares

Inverse least-squares (ILS), sometimes known as P-matrix calibration, is so called because, originally, it involved the application of multiple linear regression (MLR) to the inverse expression of the Beer-Lam be rt Law of spectroscopy ... 

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 priority phenols (Table 4) in tap and river waters were determined by SPE on line with SEC wi DA-UVD. Tetrabutylammonium bromide was used in the extraction process to increase breakthrough volumes. The mobile phase was CO2 at 40 °C, modified by a gradient of MeOH. LOD was 0.4 to 2 tig L , for 20 mL samples, with good repeatability and reproducibility between days (n = 3) for real samples spiked with 10 [xgL . Seven pollutant phenols, 107a-f and pentachlorophenol, were determined by lEC with a basic SAX resin (styrene-divinylbenzene copolymer with quaternary ammonium groups) and single channel UVD. Resolution of overlapping peaks was carried out by inverse least-squares multivariate calibration. LOD was 0.6 to 6.6 ng, with better than 90% recovery from spiked pure water and 83% from river water. No extensive clean-up was necessary . ... 

There are several mathematical limitations inherent in the inverse least squares method. The number of frequencies employed cannot exceed the number of calibration standards in the training set. The selection of frequencies is further limited by the problem of collinearity that is, the solution of the matrix equation tends to become unstable as more frequencies that correspond to absorptions of a particular component x are included because the absorbances measured at these frequencies will change in a collinear manner with changes in the concentration of x. Thus, the possibilities for averaging out errors through the use of over-determination are greatly reduced by comparison with the classical least squares method, in which there are no limitations on the number of frequencies employed. 

PLS is a powerful technique that shares the advantages of both the CLS and ILS methods hut does not suffer from the limitations of either these methods. A PLS calibration can, in principle, he based on the whole spectrum, although in practice the analysis is restricted to regions of the spectrum that exhibit variations with changes in the concentrations of the components of interest. As such, the use of PLS can provide significant improvements in precision relative to methods that use only a limited number of frequencies [9]. In addition, like the inverse least squares method, PLS treats concentration rather than spectral intensity as the independent variable. Thus, PLS is able to compensate for unidentified sources of spectral interference, although all such interferences that may be present in the samples to be analysed must also be present in the calibration standards. The utility of PLS will be demonstrated by several examples of food analysis applications presented in Section 4.7. 

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,... 

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. 

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... 

Once the PCA has been calculated from the spectral data, the concentration data can be regressed against the scores matrix using the inverse least squares method to generate the matrix of constituent calibration coefficients. A usual practice in performing PCR regression is to add an extra unit vector column to the scores matrix to allow for inclusion of an offset coefficient in the regression. 

The inverse least-squares method (ILS) assumes that concentration is a function of absorbance. For m calibration standards and n digitized absorbances,... 

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... 

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. 

Inverse least squares Internal standardisation (calibration)... 

In a variation of this type of analysis, the concentrations are expressed as functions of the various absorbances rather that vice-versa as before. This is called the inverse least squares (ILS) (or the Pmatrix method). An advantage of this method is that a quantitative analysis can be performed on some components using calibrated standards, even if some other components with unknown concentrations are present in the standards in amounts bracketing those in the samples. A disadvantage is that it is not a full-spectrum method. In the analysis, there must be at least as many standards for calibration as there are analytical wave numbers used. 

This means a quantitative analysis can be performed even if the concentration of only one component in the calibration mixtures is known. The disadvantage of the inverse least-squares method is that the analysis is restricted to a small number of frequencies because the matrix that must be inverted has dimensions equal to the number of frequencies, and this number cannot exceed the number of calibration mixtures used in the analysis. 

The P-matrix is chosen to fit best, in a least-squares sense, the concentrations in the calibration data. This is called inverse regression, since usually we fit a random variable prone to error (y) by something we know and control exactly x). The least-squares estimate P is given by... 

The inverse calibration regresses the analytical values (concentrations), x, on the measured values, y. Although with it a prerequisite of the GAussian least squares minimization is violated because the y-values are not error-free, it has been proved that predictions with inverse calibration are more precise than those with the classical calibration (Centner et al. [1998]). This holds true particularly for multivariate inverse calibration. 

The inverse calibration method of Projection to Latent Structures (PLS, also known as partial least squares ), is very similar to PCR, and has been a highly utilized tool in PAT [1]. Like the PCR method, PLS uses the... 

If the system is not simple, an inverse calibration method can be employed where it is iKst necessary to obtain the spectra of the pure analytes. The three inverse methods discussed later in this chapter include multiple linear regression (MLR), jirincipal components regression (PCR), and partial least squares (PLS). Wlien using. MLR on data sees found in chemlstiy, variable. sciectson is... 

Partial least squares (PLS) and principal component regression (PCR) are the most widely used multivariate calibration methods in chemometrics. Both of these methods make use of the inverse calibration approach, where it i.s... 

Calibration of Caustic/Salt systems using NIR Spearoscopy description of 227-228 inverse classical least squares (ICLS). 227-243... 

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