Non-Negative Least Squares

Matlab supplies the function Isqnonneg that performs a non-negative least-squares fit of the kind y=Ca+r, where y and a are column vectors. The function computes the best vector a with only positive entries. This equation corresponds to data acquired at only one wavelength. In our application, the columns of A have to be computed individually in a loop over all wavelengths, in each instance using the appropriate column of Y. C is the complete matrix of concentrations. It is, of course, the same for all wavelengths. 

Figure 4. Reanalysis of the data of Figure 1 (with noise) via the method of non-negative least squares, making no assumptions about the nuaflser of knowns.

Paatero P, A weighted non-negative least squares algorithm for three-way PARAFAC factor analysis, Chemometrics and Intelligent Laboratory Systems, 1997, 38, 223-242. 

A method for obtaining G(T, 9) from ACF, gn)(tcorr,0), suitable fortreating broad and multi-modal distributions, was developed by S. Provencher [47,48]. This method, based on regularized non-negative least-squares technique, also known as the CONTIN algorithm, has won common acceptance and is utilized in commercial PCS instruments. While the detailed mathematical description of this method is rather cumbersome and is beyond the scope of this book, we will still briefly explain the essence of it. 

The amplitudes of the histogram of the distribution function are calculated by a non-negative least square method. This procedure is known as the exponential sampling method and is applicable to both monomodal and bimodal distributions. However, in view of the limitations of the Laplace inversion, it is difficult to resolve bimodal distributions with a ratio between the two particle species below 2. 

In Eq. 5.15, A, q, g and a are the mxn kernel matrix, the distribution vector, the ACF vector, and the error vector, respectively. The regularizor G is a linear constraint operator that defines the additional constraint to the process. The best possible fit is sacrificed in order to find a reasonable and stable solution. G can be set in various values the identity matrix, the first derivative operator, or the second derivative operator. The coefficient a, called the regularization parameter, allows the process to define the strength of the constraint in the solution. A value of zero leads the algorithm back to a normal non-negative least-squares result. A small value does not introduce any effective constraint and so the result may be unstable and may have little relation to the actual solution. Values that are too large are insensitive to the measured data, resulting... 

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