Inverse Least Squares ILS

ILS is a least-squares method that assumes the inverse calibration model given in eqn (3.4). For this reason it is often also termed multiple linear regression (MLR). In this model, the concentration of the analyte of interest, k, in sample i is regressed as a linear combination of the instrumental measurements at J selected sensors [5,16-19]  

Calibration consists of estimating the J+ 1 coefficients of the model by solving a system of I (I J+ 1) equations. For I calibration samples, the model becomes 

The predicted concentration of analyte k in a sample from its vector of 

The number of spectral variables and the collinearity problem are the two main drawbacks of the ILS model when applied to spectroscopic data. 

Note that both the collinearity problem and the requirement of having more samples than sensors can be solved by using regression techniques which can handle collinear data, such as factor-based methods such as PCR and PLS. These use linear combinations of all the variables and reduce the number of regressor variables. PCR or PLS are usually preferred instead of ILS, although they are mathematically more complex. 

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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 ... 

Multiple Linear Regression (MLR), Classical Least-Squares (CLS, K-matrix), Inverse Least-Squares (ILS, P-matrix)... 

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. 

Dine reported that three chemometric techniques, classical least squares (CLS), inverse least square (ILS) and principal component... 

In the inverse least squares (ILS) approach [6], the matrix equation expressing Beer s law is rearranged into equation (19) ... 

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. 

Both classical least squares (CLS) and inverse least squares (ILS) approaches should be considered. The main advantages and disadvantages of both approaches are listed in Table 1, with detailed descriptions of the various techniques within both approaches described below. 

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. 

Unfortunately, this method is not perfect. In fact, there are a number of drawbacks. As with the inverse least squares (ILS) quantitative analysis method, this approach to discriminant analysis relies on selecting a subset of wavelengths to represent the entire spectrum. Again, if any impurities or aberrations appear in the spectra of the unknowns that do not appear at the selected wavelengths, the discriminant analysis will determine that the sample matches the group, when in fact it does not ... 

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... 

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. 

This method is commonly known as inverse least-squares (ILS) regression, but it is also referred to as the P-matrix method or multiple linear regression (MLR) when the number of analytical wavenumbers is small. ILS is a multivariate technique that has some advantages over CLS, but ILS also has some shortcomings. 

ATR-FTIR is a useful analytical tool for multicomponent analysis that employs a mathematical data-treatment process. Also, Carolei and Gutz (2005) have used this technique combined with chemometrics, to determine three surfactants and water simultaneously in shampoo and in liquid soap without either sample dilution or pretreatment. The surfactants analysed were an amphoteric one (cocoamidopropyl betaine), two nonionic ones (coco diethanolamide in shampoo and alkylpolyglucoside in liquid soap), (minor components) and an anionic one (sodium lauryl ether sulfate). Overlapping bands and water absorption were resolved by two multivariate quantification methods classical least squares (CLS) and inverse least squares (ILS) (Massart et al., 1997, 1998). The wave numbers chosen for the calculation process were preferably those of maximum absorption of the minor components. This method can be applied during the production process but not in final product analysis because of interference caused by the fragrance added in the last step (Figure 7.1.2). 

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. 

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