PARAFAC

PARAFAC

In statistics, parallel factor analysis (PARAFAC) also named canonical decomposition (candecomp) or candecomp/parafac decomposition (CP decomposition) is a multi-way method originating from psychometrics [Cite journal
author = J. D. Carroll & J. Chang
title = Analysis of individual differences in multidimensional scaling via an n-way generalization of 'Eckart-Young' decomposition
journal = Psychometrika
volume = 35
pages = 283–319
year = 1970] [Richard A. Harshman, (1970). [http://publish.uwo.ca/~harshman/wpppfac0.pdf "Foundations of the PARAFAC procedure: Models and conditions for an "explanatory" multi-modal factor analysis"] . UCLA Working Papers in Phonetics, 16, 84 pp. (University Microfilms, Ann Arbor, No. 10,085).] though going back to Hitchcock in 1927. [Cite journal
author = F. L. Hitchcock
title = The expression of a tensor or a polyadic as a sum of products
journal = Journal of Mathematical Physics
volume = 6
pages = 164–189
year = 1927] It is a well-used method in chemometrics and associated areas.

Multi-way data are characterized by several sets of categorical variables that are measured in a crossed fashion. Chemical examples could be fluorescence emission spectra measured at several excitation wavelengths for several samples, fluorescence lifetime measured at several excitation and emission wavelengths or any kind of spectrum measured chromatographically for several samples. Determining such variables will give rise to three-way data; i.e., the data can be arranged in a cube instead of a matrix as in standard multivariate data sets.

The PARAFAC model may be solved by alternating least squares.

PARAFAC may suffer from degeneracy.

Other decompositions

PARAFAC is one of several decomposition methods for multi-way data. The two main competitors are the Tucker3 method , and simply unfolding of the multi-way array to a matrix and then performing standard two-way methods as principal component analysis (PCA). The Tucker3 method should rightfully be called three-mode principal component analysis (or N-mode principal component analysis), but here the term Tucker3 or just Tucker will be used instead. PARAFAC, Tucker and two-way PCA are all multi- or bi-linear decomposition methods, which decompose the array into sets of scores and loadings, that hopefully describe the data in a more condensed form than the original data array. There are advantages and disadvantages with all the methods, and often several methods must be tried to find the most appropriate.

References

External links

* [http://www.models.kvl.dk/users/rasmus/presentations/parafac_tutorial/paraf.htm PARAFAC Tutorial]
* [http://www.fmrib.ox.ac.uk/analysis/techrep/tr04cb1/tr04cb1/node2.html 2 Parallel Factor Analysis (PARAFAC)]
* [http://factominer.free.fr/ FactoMineR] (free exploratory multivariate data analysis software linked to R)


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