Direct trilinear decomposition

RAFA and GRAM are limited to simultaneous analysis of one standard and one unknown. One way around this limitation is the direct trilinear decomposition (DTLD) [29], Aside from being a rapid method for solving Equation 12.1, DTLD also provides an excellent starting solution for least-squares and iterative methods. 

Direct Trilinear Decomposition as per Booksh, Lin, Wang, and Kowalski %... 

Direct trilinear decomposition (DTLD) is a direct non-iterative and therefore fast algorithm for solving the PARAFAC model providing a non-LS solution. If data are low-rank trilinear, the solution produced by DTLD will be close to identical to that resulting from the PARAFAC algorithm. 

In order to improve the calibration results, an analyte concentration in the mixture close to the analyte concentration in the standard would be favorable. This is not a practical solution, but an alternative is to use multiple standards. All five standards are used now. The calibration equations become slightly more complicated (i.e. a combination of cases (ii) and (iii) from above, see Appendix 10.C), but end up in a relatively straightforward three-way model. Generalized rank annihilation cannot be used anymore, since there are more than two slices in the three-way array. The authors used direct trilinear decomposition to solve this problem but an ordinary PARAFAC-ALS algorithm can also be used. The results are presented in Table 10.5 and show improvements at low concentrations. 

Gui M, Rutan SC, Agbodjan A, Kinetic detection of overlapped amino acids in thin-layer chromatography with a direct trilinear decomposition method, Analytical Chemistry, 1995, 67, 3293-3299. 

Li S, Gemperline PJ, Eliminating complex eigenvectors and eigenvalues in multiway analyses using the direct trilinear decomposition method, Journal of Chemometrics, 1993, 7, 77-88. 

Sanchez E, Kowalski BR, Tensorial resolution a direct trilinear decomposition, Journal of Chemometrics, 1990,4, 29 15. 

Zeng Y, Hopke PK, A new receptor model a direct trilinear decomposition followed by a matrix reconstruction, Journal of Chemometrics, 1992a, 6, 65-83. 

Sanchez E and Kowalski B R 1990 Tensorial resolution a direct trilinear decomposition J. Chemometr. 4 29... 

In the mid-1980s a new technique called rank annihilation factor analysis (RAFA and GRAM) was developed which performed this process non-iteratively [42, 43]. Now an exact solution can be estimated directly on the sample matrix if the pure component response matrix of the analyte of interest is known. The third technique uses a range of concentrations yielding several pure component data matrices. This approach, called trilinear decomposition (TLD), stabilizes the prediction of the unknown sample due to the range of known response matrices... 

MCR-ALS solutions can be additionally constrained to fulfill a trilinear model [82], When this trilinearity constraint is applied, the profiles in the three different modes (Useo, utemP, and VT) are directly recovered and can be compared with the profiles obtained using PARAFAC- or Tucker-based model decompositions. MCR-ALS results have already been compared with Tucker3-ALS and PARAFAC-ALS results in the resolution of different chemical systems [81],... 

If the underlying structure of a three-way data set is not trilinear, other three-way data analytical methods exist, but these do not usually result in pure component spectra. However, through application of appropriate constraints, pure component spectra may still be extracted, as argued by de Juan and Tauler.7 By using three-way decomposition methods, the data structure is maintained and used in the model. Interpretation is facilitated by relating the extracted patterns directly to the structure of the data. 

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