Least squares regression basic algorithm

Multivariate analytical methods have also been applied to the analysis of drug substances. The methods have a significant component of matrix analysis, and the Beer-Lambert law is basically rewritten in matrix form, permitting matrix analysis of absorbance data. A number of other mathematical algorithms have also been developed for the quantitation of analytes in multicomponent mixtures. These have either been iterative methods or methods based on multiple least-squares regression. The multiple least-squares regression methods require a knowledge of all the components of the multicomponent mixture, whereas the iterative methods such as the Kalman or the simplex method are less restrictive in the sense that interferents whose spectra are not known need not be included in the database. 

In the basic form of the GMDH algorithm, all the possibiUties of two independent variables out of the total n input variables are taken in order to constract the regression polynomial in the form of equation (5) that best fits the dependent observations = i,2,...,A/) in a least squares sense (Nariman-Zadeh et al., 2002). Using the quadratic sub-expression in the form of equation (5) for each row of M data triples, the following matrix equation can be readily obtained as... 

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