Three-way (PARAFAC) factor analysis: examination and comparison of alternative computational methods as applied to ill-conditioned data
✍ Scribed by Philip K. Hopke; Pentti Paatero; Hong Jia; Robert T. Ross; Richard A. Harshman
- Book ID
- 104309612
- Publisher
- Elsevier Science
- Year
- 1998
- Tongue
- English
- Weight
- 502 KB
- Volume
- 43
- Category
- Article
- ISSN
- 0169-7439
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✦ Synopsis
Four different approaches to solving the trilinear three-way factor analysis problem are compared, and their performance Ž . with 'difficult' i.e., ill-conditioned data is tested. These approaches are represented by four different computer programs: Ž . Ž . one using a simple alternating least squares ALS algorithm with only minimal extrapolation HL-PARAFAC , one in which Ž . the ALS is supplemented by a sophisticated extrapolation to speed convergence TPALS , one using a non-linear curve fit-Ž . Ž . ting method PMF3 , and one using a non-iterative closed-form approximation DTDMR . The options provided by these Ž . 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 Ž . index is introduced to characterize the ill-conditioning of n-way arrays n ) 2 . Two well characterized synthetic data sets Ž . serve as 'difficult' 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 pro-Ž . duced 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.