PARAFAC - TUTORIAL AND APPLICATIONS

This paper explains the multi-way decomposition method PARAFAC and its use in chemometrics. PARAFAC is a generalization of PCA to higher ord er arrays, but some of the characteristics of the method are quite dif ferent from the ordinary two-way case. There is no rotation problem in PARAFAC, and e.g., pure spectra can be recovered from multi-way spect ral data. One cannot as in PCA estimate components successively as thi s will give a model with poorer fit, than if the simultaneous solution is estimated. Finally scaling and centering is not as straightforward in the multi-way case as in the two-way case. An important advantage of using multi-way methods instead of unfolding methods is that the es timated models are very simple in a mathematical sense, and therefore more robust and easier to interpret. All these aspects plus more are e xplained in this tutorial and an implementation in Matlab code is avai lable, that contains most of the features explained in the text. Three examples show how PARAFAC can be used for specific problems. The appl ications include subjects as: Analysis of variance by PARAFAC, a five- way application of PARAFAC, PARAFAC with half the elements missing, PA RAFAC constrained to positive solutions and PARAFAC for regression as in principal component regression.