Alternating Least-Squares and Constraints

By far the most important aspect of the ALS algorithm is the ease of implementing restrictions. In the following we demonstrate this using a number of examples. 

The program Ma in ALS, m forms the backbone of the ALS algorithm. It reads in the data set Data Chrom2a (p.251) which simulates an overlapping chromatogram of three components. It is the data set we used previously in Chapter 5.3.3 to demonstrate the concepts of iterative and explicit computation of the concentration profiles, based on the window information from EFA. 

EFA b (isnan (EFA b) == 1) =0 % replace NaN s by zeros % combined singular value curves 

There are several types of constraints that can be used and generally the more constraints applied, the better the convergence and the better defined the results. 

The most important and almost universally applicable constraint is the nonnegativity of all elements of C and A. Obviously, neither concentrations nor molar absorptivities can be negative. In many ALS algorithms, this constraint is enforced by simply setting all negative entries in C and A to zero  

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