Inverse least-squares

In inverse calibration one models the properties of interest as a function of the predictors, e.g. analyte concentrations as a function of the spectrum. This reverses the causal relationship between spectrum and chemical composition and it is geared towards the future goal of estimating the concentrations from newly measured spectra. Thus, we write 

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 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. 

With this proviso a new unknown sample with the spectrum s translated into a concentration estimate  

It should be noted that B,ls is just a collection of separate multiple regression models. Each column b,Ls of B,ls is associated with a particular analyte and follows from a multiple regression of the corresponding column c of C on the predictor matrix of spectra S. 

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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)... 

Brown, C.W., "Classical and Inverse Least-Squares Methods in Quantitative Spectral Analysis", Spectrosc. 1986 (1) 23-37. 

Haaland, D.M. "Classical versus Inverse Least-Squares Methods in Quantitative Spectral Analyses", Spectrosc. 1987 (2) 56-57. 

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. 

Inverse least squares, 539-41 Inverse micelles, 25487 Inverse microemulsion polymerization, 20 461... 

Preface Introduction Basic Approach Creating Some Data Classical Least-Squares Inverse Least-Squares Factor Spaces... 

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

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 . ... 

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