A closed-form solution for the PARAFAC model

Appendix 6.A Closed-form Solution for the PARAFAC Model 

Rank annihilation factor analysis treats the special situation in which there are only two slices, where each slice contains the second-order measurements of one sample. One sample is usually a pure standard containing only the analyte of interest in a known concentration and the other is an unknown sample containing unknown interferents as well as the analyte of interest. The purpose in rank annihilation factor analysis is to estimate the concentration of the analyte of interest in the unknown sample. It is assumed that the signal from the analyte of interest corresponds to a rank-one component with the intensity proportional to the concentration. Thus the standard sample, say Xi, can be described 

Multi-way Analysis With Applications in the Chemical Sciences 

However, the second-order advantage is more naturally seen as a manifestation of the uniqueness of the PARAFAC model. In that context, it also provides means for separating not only concentration-related parameters but also, for example, separating a spectral part of a data set into the pure underlying spectra. 

The basic theory behind the generalized rank annihilation method is that the rank reduction can be re-expressed and automated. A scalar y (relative concentration of the analyte in the unknown sample) is sought such that the matrix pencil 

---