A fast polygon inflation algorithm to compute the area of feasible solutions for three‐component systems. II: Theoretical foundation, inverse polygon inflation, and FAC‐PACK implementation. (27th March 2014)

• Record Type: Journal Article
• Authors: Sawall, Mathias; Neymeyr, Klaus; Åberg, Magnus
• Is Part Of: Journal of chemometrics. Volume 28:Number 8(2014:Aug.)
• Page Start: 633
• Page End: 644
• Publication Date: 2014-03-27
• Subjects: Chemistry -- Mathematics -- Periodicals; Chemistry -- Statistical methods -- Periodicals; 542.85
• DOI: 10.1002/cem.2612
• Languages: English
• ISSNs: 0886-9383

Abstract

<abstract abstract-type="main" id="cem2612-abs-0001"> <title> <x xml:space="preserve">Abstract</x> </title> <p id="cem2612-para-0001">The area of feasible solutions (AFS) of a multivariate curve resolution method is the continuum of feasible solutions under the given constraints. In the current paper, the AFS is computed only on the condition of nonnegative solutions. This work is a continuation of a paper (J. Chemometrics 28:106–116, 2013) on the polygon inflation algorithm for AFS computations. In this second part, various properties of the AFS are analyzed. First, its boundedness is proved, which is a necessary condition for its numerical computation. Second, it is shown that the origin is never contained in the area of feasible solutions. This fact is the basis for the inverse polygon inflation algorithm, which allows to compute specific types of an AFS containing a hole.</p> <p id="cem2612-para-0002">The numerical computation of the AFS is a complicated and computationally expensive process. The construction of proper objective functions for the AFS optimization problem appears to be decisive. The paper contains a comparative analysis of two objective functions and describes the ideas of the new <bold><italic>FAC‐PACK</italic></bold> toolbox for <sc>MATLAB</sc>. This freely available toolbox contains a numerical implementation of the polygon inflation and of the inverse polygon inflation algorithm. Copyright © 2014 John Wiley &amp; Sons, Ltd.</p> </abstract>