A Useful Analytical Technique Singular-Value Decomposition Followed by Global Fitting

A Useful Analytical Technique Singular-Value Decomposition Followed by Global Fitting [23-25] 

The principles of SVD followed by global fitting are briefly described below taking the case of electronic absorption spectroscopy as an example The apparent optical density, D(X,t), of a system at wavelength X and at time t can be given as 

SVD serves to distinguish the signal from the noise components. With the help of the orthogonal relation of st shown in Eq. (3), Eq. (4) leads to 

This procedure is called global fitting. The origin of tp is also taken with a parameter, but for simplicity this parameter is omitted here. Since f is a nonlinear function of k, values, it can have many false minima. In order to avoid such false minima and to choose the true minimum, we need to obtain f that is closest to zero then Eq. (5) will be satisfied. Substituting Eq. (5) into the right-hand side of Eq. (4), and equating the coefficients of hftp) of both sides, we obtain 

The difference spectra thus obtained are called species-associated difference spectra (SADS).  

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