On the avoidance of crossing of singular values in the evolving factor analysis

Abstract

Evolving factor analysis (EFA) investigates the evolution of the singular values of matrices formed by a series of measured spectra, typically, resulting from the spectral observation of an ongoing chemical process. In the original EFA, the logarithms of the singular values are plotted for submatrices that include an increasing number of spectra. A typical observation in these plots is that pairs of trajectories of the singular values are on a collision course, but finally, the curves seem to repel each other and then run in different directions. For parameter-dependent square matrices, such a behaviour is known for the eigenvalues under the keyword of an avoidance of crossing. Here, we adjust the explanation of this avoidance of crossing to the curves of singular values of EFA. Further, a condition is studied that breaks this avoidance of crossing. We demonstrate that the understanding of this noncrossing allows us to design model data sets with a predictable crossing behaviour. © 2020 John Wiley & Sons, Ltd.