Three-way (PARAFAC) factor analysis: examination and comparison of alternative computational methods as applied to ill-conditioned data

Four different approaches to solving the trilinear three-way factor analysis problem are compared, and their performance with `difficultʹ (i.e., ill-conditioned) data is tested. These approaches are represented by four different computer programs: one using a simple alternating least squares (ALS) algorithm with only minimal extrapolation (HL-PARAFAC), one in which the ALS is supplemented by a sophisticated extrapolation to speed convergence (TPALS), one using a non-linear curve fitting method (PMF3), and one using a non-iterative closed-form approximation (DTDMR). The options provided by these programs (e.g., with regard to missing values, weighted least squares, non-negativity and other constraints) are compared. Criteria for choosing synthesized test data and a method for synthesizing exponential test data are described. A numerical index is introduced to characterize the ill-conditioning of n-way arrays (n>2). Two well characterized synthetic data sets serve as `difficultʹ (ill-conditioned) test data. Intercomparisons among HL-PARAFAC, TPALS, DTDMR and PMF3 were implemented with these test data. Consequently, their limitations and strengths are determined. In addition, these trilinear analysis approaches are applied to a difficult set of ill-conditioned real data: a set of fluorescence spectroscopy measurements that characterize the steady-state fluorescence of an amino acid in aqueous solution. When converged, the results produced by the three least-squares techniques (but not DTDMR) agree. However, there are large differences in convergence speed when these difficult problems are solved: TPALS is faster than PARAFAC by a factor of ten, and PMF3 is faster than TPALS, again by a factor of ten. The program DTDMR is the fastest, but it only solves half of the problems.