The Resolution Process Initial Estimates and Constraints

The resoluhon of data sets into their underlying bilinear models can be performed with algorithms based on very diverse backgrounds [71, 72]. Some of these algorithms rely essentially on the inner mathematical structure of the data set, and 

Multivariate curve resolution-alternating least squares (MCR-ALS) is an algorithm that fits the requirements for image resolution [71, 73-75]. MCR-ALS is an iterative method that performs the decomposition into the bilinear model D = CS by means of an alternating least squares optimization of the matrices C and according to the following steps  

The alternating least-squares procedure in steps 4 and 5 involves the operations C = DS(S S) and = (C C) C D, respectively. The end of the iterative process takes place when the reproduction of the original image from the product of the resolved concentration profiles and spectra has enough quality and there is no significant variation among the results of consecutive cycles. The quality in the data reproduction can be estimated through the lack of fit, expressed as  

The concept of ambiguity in curve resoluhon is linked to the fact that many CS products can reproduce the original data set with the same ophmal fit-that is, many sets of concentration profiles and spectra can be potenhaUy valid to describe the data. In mathemahcal notation, the bilinear model can be written as  

One way of reducing the uncertainty in the resoluhon results is by limihng the possible soluhons to those that fulfill the preset conshaints. Thus, the more efficient constraints are, the better defined are the resoluhon results. 

3) Constrained alternating least squares optimization of C and until convergence is achieved. 

The way to incorporate previous knowledge about the images in MCR-ALS is through the use of constraints. Constraints can be defined as chemical or mathematical properties that the concentration profiles or spectra should fulfill [114, 115,119). During the iterative process, the calculated concentration profiles and spectra are modified so that they obey the preselected conditions. The application of constraints is optional and flexible and takes into account the natural characteristics of the data set. Thus, concentration profiles and spectra can obey different constraints, and within the C or matrices, constraints can be applied profile-wise or even element-wise. Constraints play a double role in resolution methods. On the one hand, they ensure the chemical meaning of the recovered distribution maps and spectra and, on the other hand, they greatly decrease the ambiguity in the resolved profiles. 

The concept of ambiguity in curve resolution is linked to the fact that a range of CS products can reproduce the original data set with the same optimal fit. 

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