Inverse multiple linear regression

The previous section alludes to the most common problems in quantitative Raman spectroscopic calibrations Most models require that all components in a system to be known and modeled in the calibration data to accurately predict any one component. Inverse calibration techniques such as inverse multiple linear regression (inverse MLR), principal component regression (PCR) and partial least squares (PLS also known as principal latent structures) avoid this problem by forcing the calibration steps to utilize only the spectral features which are either changing (PCR) or directly correlated to the property of interest (PLS). More so, not all components in a sample need to be known to perform an inverse calibration. The basic form of an inverse calibration centers around an equation of the form... 

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)... 

In Chapters 2 and 3, we discussed the rules related to solving systems of linear equations using elementary algebraic manipulation, including simple matrix operations. The past chapters have described the inverse and transpose of a matrix in at least an introductory fashion. In this installment we would like to introduce the concepts of matrix algebra and their relationship to multiple linear regression (MLR). Let us start with the basic spectroscopic calibration relationship ... 

MLR is an inverse method that uses the multiple linear regression model that was discussed earlier [1,46] ... 

Like MLR, PCR [63] is an inverse calibration method. However, in PCR, the compressed variables (or PCs) from PCA are used as variables in the multiple linear regression model, rather than selected original X variables. In PCR, PCA is first done on the calibration x data, thus generating PCA scores (T) and loadings (P) (see Section 12.2.5), then a multiple linear regression is carried out according to the following model ... 

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... 

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] ... 

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. 

Equation [9] highlights the most frequent problem in multiple linear regression, namely, the matrix whose inverse must be calculated, (C C), may suffer from collinearity or singularity problems. The matrix may not possess... 

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. 

As stated earlier, Matlab s philosophy is to read everything as a matrix. Consequently, the basic operators for multiplication, right division, left division, power (, /,, A) automatically perform corresponding matrix operations (A will be introduced shortly in the context of square matrices, / and will be discussed later, in the context of linear regression and the calculation of a pseudo inverse, see The Pseudo-Inverse, p.117). 

Multiple Enear regression with variable selection makes the matrix inversion possible by selecting a subset of the original variables. Both PCR and PLS reduce the number of variables by calculating linear combinations of the original variables (factors) and using a small enough number of these factors to allow for the matrix inversion. 

When MLR is used to provide a predictive model that uses multiple analyzer signals (e.g. wavelengths) as inputs and one property of interest (e.g. constituent concentration) as output, it can be referred to as an inverse linear regression method. The word inverse arises from the spectroscopic application of MLR because the inverse MLR model represents an inverted form of Beer s Law, where concentration is expressed as a function of absorbance rather than absorbance as a function of concentration 1,38... 

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