Lawton matrix

If some stopping criterion has been reached, then the algorithm proceeds to Step 10 where the Lawton matrix condition is verified. Provided Conditions 1 and 2 of Section 4 hold, then the Lawton matrix condition will not be satisfied for exceptional degenerate cases, thus the Lawton matrix is verified after the adaptive wavelet has been found. Finally, the multivariate statistical procedure can be performed using the coefficients X " (to). The optimizer used in the adaptive wavelet algorithm is the default unconstrained MAT-LAB optimizer [12]. 

In previous methods no pre-knowledge of the factors was used to estimate the pure factors. However, in many situations such pre-knowledge is available. For instance, all factors are non-negative and all rows of the data matrix are nonnegative linear combinations of the pure factors. These properties can be exploited to estimate the pure factors. One of the earliest approaches is curve resolution, developed by Lawton and Sylvestre [7], which was applied on two-component systems. Later on, several adaptations have been proposed to solve more complex systems [8-10]. 

In their fundamental paper on curve resolution of two-component systems, Lawton and Sylvestre [7] studied a data matrix of spectra recorded during the elution of two constituents. One can decide either to estimate the pure spectra (and derive from them the concentration profiles) or the pure elution profiles (and derive from them the spectra) by factor analysis. Curve resolution, as developed by Lawton and Sylvestre, is based on the evaluation of the scores in the PC-space. Because the scores of the spectra in the PC-space defined by the wavelengths have a clearer structure (e.g. a line or a curve) than the scores of the elution profiles in the PC-space defined by the elution times, curve resolution usually estimates pure spectra. Thereafter, the pure elution profiles are estimated from the estimated pure spectra. Because no information on the specific order of the spectra is used, curve resolution is also applicable when the sequence of the spectra is not in a specific order. 

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