Modeling multi-way data with linearly dependent loadings

Journal of Chemometrics

Volumn 23, Issue 7-8, 2009, Pages 324-340

  Bro, Rasmus ^a   Harshman, Richard A ^b   Sidiropoulos, Nicholas D ^c   Lundy, Margaret E ^b  

^a UNIVERSITY OF COPENHAGEN   (Denmark)

^b UNIVERSITY OF WESTERN ONTARIO   (Canada)

^c University Campus Kounoupidiana   (Greece)

Author keywords

Constrained ALS estimation; Equality constraints; Linear dependence; Multi mode factor analysis; PARAFAC; PARATUCK2; Uniqueness

Indexed keywords

MATRIX ALGEBRA;

CONSTRAINED ALS ESTIMATION; EQUALITY CONSTRAINTS; FACTORS ANALYSIS; LINEAR DEPENDENCE; MULTI-MODE FACTOR ANALYSE; MULTIMODES; NONUNIQUENESS; PARAFAC; PARATUCK2; UNIQUENESS;

FLUORESCENCE SPECTROSCOPY;

EID: 70349274902     PISSN: 08869383     EISSN: 1099128X     Source Type: Journal    
DOI: 10.1002/cem.1206     Document Type: Article

References (44)

1. Kiers HAL, Smilde AK. Constrained three-mode factor analysis as a tool for parameter estimation with second-order instrumental data. J. Chemom. 1998; 12: 125-147.
2. Harshman RA. Foundations of the PARAFAC procedure: models and conditions for an'explanatory' multi-modal factor analysis. UCLA Working Papers in Phonetics 1970; 16: 1-84.
3. Harshman RA, Lundy ME. PARAFAC: parallel factor analysis. Comput. Stat. Data Anal. 1994; 18: 39-72.
4. Carroll JD, Chang J. Analysis of individual differences in multidimensional scaling via an N-way generalization of'Eckart-Young'decomposition. Psychometrika 1970; 35: 283-319.
5. Brereton RG, Gurden SP, Groves JA. Use of eigenvalues for determining the number of components in window factor analysis of spectroscopic and chromatographic data. Chemom. Intell. Lab. Syst. 1995; 27: 73-87.
6. Izquierdo-Ridorsa A, Saurina J, Hernandez-Cassou S, Tauler R. Second-order multivariate curve resolution applied to rank-deficient data obtained from acid-base spectrophotometric titrations of mixtures of nucleic bases. Chemom. Intell. Lab. Syst. 1997; 38: 183-196.
7. Ross RT, Leurgans SE. Component resolution using multilinear models. Methods Enzymol. 1995; 246: 679-700.
8. Sidiropoulos ND, Dimic GZ. Blind multiuser detection in W-CDMA systems with large delay spread. IEEE Signal Process. Lett. 2001; 8: 87-89.
9. Sidiropoulos ND, Liu XQ. Identifiability results for blind beamforming in incoherent multipath with small delay spread. IEEE Trans. Signal Process. 2001; 49: 228-236.
10. van der Veen AJ. Algebraic methods for deterministic blind beamforming. Proc. IEEE 1998; 86: 1987-2008.
11. Amrhein M, Srinivasan B, Bonvin D, Schumacher MM. On the rank deficiency and rank augmentation of the spectral measurement matrix. Chemom. Intell. Lab. Syst. 1996; 33: 17-33.
12. Garrido M, Lazaro I, Larrechi MS, Rius FX. Multivariate resolution of rank-deficient near-infrared spectroscopy data from the reaction of curing epoxy resins using the rank augmentation strategy and multivariate curve resolution alternating least squares approach. Anal. Chim. Acta 2004; 515: 65-73.
13. Amrhein M. Reaction and flow variants/invariants for the analysis of chemical reaction data. PhD Thesis, École Polytechnique Fédé rale de Lausanne, 1998.
14. Smilde AK, Tauler R, Saurina J, Bro R. Calibration methods for complex second-order data. Anal. Chim. Acta 1999; 398: 237-251.
15. Amrhein M, Srinivasan B, Bonvin D, Schumacher MM. Calibration of spectral reaction data. Chemom. Intell. Lab. Syst. 1999; 46: 249-264.
16. Smilde AK, Tauler R, Henshaw JM, Burgess LW, Kowalski BR. Multicomponent determination of chlorinated hydrocarbons using a reaction-based chemical sensor. 3. Medium-rank second-order calibration with restricted Tucker models. Anal. Chem. 1994; 66: 3345-3351.
17. Levin J. Three-mode factor analysis. Psychol. Bull. 1965; 64: 442-452
18. Bro R. Multi-way analysis in the food industry. models, algorithms, and applications. PhD Thesis, University of Amsterdam (NL), 1998. http://www.models.life.kvl.dk/research/theses/
19. Harshman RA, Lundy ME. Uniqueness proof for a family of models sharing features of Tucker's three-mode factor analysis and PARAFAC/CANDECOMP. Psychometrika 1996; 61: 133-154.
20. Harshman RA. Generalizations, extensions and structured specialpurpose modifications of PARAFAC with possible applications in chemistry. Talk presented to TRIC (Three-way analysis methods in Chemistry) in Epe, Holland, 1993.
21. Rao CR, Mitra S. Generalized Inverse of Matrices and its Applications. Wiley: New York, 1971.
22. Kruskal JB. Three-way arrays: rank and uniqueness of trilinear decompositions with applications to arithmetic complexity and statistics. Linear Algebra Appl. 1977; 18: 95-138.
23. Kruskal JB. More factors than subjects, tests and treatments: an indeterminacy theorem for canonical decomposition and individual differences scaling. Psychometrika 1976; 41: 281-293.
24. Harshman RA. Determination and proof of minimum uniqueness conditions for PARAFAC1. UCLA Working Papers in Phonetics 1972; 22: 111-117.
25. ten Berge JMF. Partial uniqueness in CANDECOMP/PARAFAC. J. Chemom. 2004; 18: 12-16.
26. Faber NM, Buydens LMC, Kateman G. Generalized rank annihilation method. I: derivation of eigenvalue problems. J. Chemom. 1994; 8: 147-154.
27. Lorber A. Quantifying chemical composition from two-dimensional data arrays. Anal. Chim. Acta 1984; 164: 293-297.
28. Sanchez E, Kowalski BR. Generalized rank annihilation factor analysis. Anal. Chem. 1986; 58: 496-499.
29. Sidiropoulos ND, Giannakis GB, Bro R. Blind PARAFAC receivers for DS-CDMA systems. IEEE Trans. Signal Process. 2000; 48: 810-823.
30. Swindlehurst A, Kailath T. Azimuth/elevation direction finding using regular array geometries. IEEE Trans. Aerosp. Electron. Syst. 1993; 29: 145-156.
31. Leurgans SE, Ross RT. Multilinear models: applications in spectroscopy. Stat. Sci. 1992; 7: 289-319.
32. Paatero P. The multilinear engine-a table-driven, least squares program for solving multilinear problems, including the n-way parallel factor analysis model. J. Comput. Graph. Stat. 1999; 8: 854-888.
33. Jiji RD, Booksh KS. Mitigation of Rayleigh and Raman spectral interferences in multiway calibration of excitation-emission matrix fluorescence spectra. Anal. Chem. 2000; 72: 718-725.
34. Rinnan A, Andersen CM. Handling of first-order Rayleigh scatter in PARAFAC modelling of fluorescence excitation-emission data. Chemom. Intell. Lab. Syst. 2005; 76: 91-99.
35. Bro R. Exploratory study of sugar production using fluorescence spectroscopy and multi-way analysis. Chemom. Intell. Lab. Syst. 1999; 46: 133-147.
36. Nørgaard L, Ridder C. Rank annihilation factor analysis applied to flow injection analysis with photodiode-array detection. Chemom. Intell. Lab. Syst. 1994; 23: 107-114.
37. Andersson CA, Bro R. The N-way Toolbox for MATLAB. Chemom. Intell. Lab. Syst. 2000; 52: 1-4.
38. Bro R, Andersson CA. Improving the speed of multiway algorithms part II: compression. Chemom. Intell. Lab. Syst. 1998; 42: 105-113.
39. Kiers HAL. A three-step algorithm for CANDECOMP/PARAFAC analysis of large data sets with multicollinearity. J. Chemom. 1998; 12: 155-171.
40. Bro R, de Jong S. A fast non-negativity-constrained least squares algorithm. J. Chemom. 1997; 11: 393-401.
41. Bro R, Sidiropoulos ND. Least squares algorithms under unimodality and non-negativity constraints. J. Chemom. 1998; 12: 223-247.
42. Henderson HV, Searle SR. The vec-permuation matric, the vec operator and the Kronecker products: a review. Linear and Multilinear Algebra 1981; 9: 271-288.
43. Bro R. PARAFAC. Tutorial and applications. Chemom. Intell. Lab. Syst. 1997; 38: 149-171.
44. Harshman, RA, Hong, S.'Stretch' vs.'slice' methods for representing three-way structure via matrix notation. J. Chemom. 2002; 16, 198-205.