Multiple linear regression inverse least squares model

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] ... 

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... 

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. 

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