1. Multivariate Resolution in Chemistry Lecture 3 Roma Tauler IIQAB-CSIC, Spain e-mail: rtaqam@iiqab.csic.es

2. Lecture 3 Simultaneous resolution of multiple two-way data sets. Resolution of multivay data sets. Trilinear and multilinear models. Extension of MCR-ALS to multi-way data and to multi-set data. –Constraints. –Extension of resolution conditions. –Recovery of quantiative information. –Breaking rank deficiencies by matrix augmetation –Feasible bands Comparison of algorithms and methods. Examples of application. (1.5 hours)

3. Luminiscence excitacion /emission spectra/sample Process/Reaction spectroscopic monitoring time/pH/temperature wavelength sample/system/run Analytical Hyphenated Methods: LC/DAD; LC/FTIR; GC/MS; LC/MS time/wavelength/sample time/m/z ratios/sample Environmental monitoring samples/concentrations/time or conditions Spectroscopic imaging multiple spectroscopic images from different samples …… Examples of Three-way data in Chemistry

4. Three-way data in Chemistry Example: Multiple excitacion emission spectra (standards and unknown samples) Wavelengths wavelengths sample number emission excitation samples excitation emission samples. * * * *** ** * * * * *

5. Three-way data in Chemistry Example: Multiple HPLC-DAD-MS runs of a system (standards and unknown samples) Wavelengths Elution time Run number Spectrum Chromatogram runs chromatogram spectrum runs. * * * *** ** * * * * *

6. Three-way data in Chemistry Example: A chemical reaction or proces monitored spectrsocopically Process number time spectra reaction profiles process. Reaction profiles spectra process

7. Three-way data: Unfolding / Matricizing Matrix Augmentation NC NR NM NR NC x NM NR x NM NC NR x NC 3 Multiple data matrices in a cube (NR,NC,NM) Row-wise data matrix augmentation (NR,NCxNM) Column-wise data matrix augmentation (NC,NRxNM) Tube-wise data matrix augmentation (NM,NRxNC) NM

8. Multiple data sets (e.g. environmental data)

9. Extension of Bilinear Models (PCA or MCR) Matrix Augmentation The same experiment monitored with different techniques = D 1 D 2 D 3 D C1C1 = D 1 D 2 D 3 D Several experiments monitored with the same technique = D 1 D 2 D 3 D 4 D 5 D 6 = D 1 D 2 D C1C1 D 4 D 5 D 6 C D Several experiments monitored with several techniques Row-wise Column-wise Row and column-wise D = D 1 D 2 D 3 D = D 1 D 2 D 3 D = D 1 D 2 D 3 D 1 D 2 D 3 S1TS1T C C S2TS2T S3TS3T S1TS1T S2TS2T S3TS3T STST C2C2 STST STST C2C2 C3C3 C

10.  D1D2D3D1D2D3 DT1DT2DT3DT1DT2DT3 B A A B C Ex. Hyphenated Chromatography Column-wise data matrix augmentation = D 1 D 2 D 3 D C1C1 = D 1 D 2 D 3 D STST C2C2 C3C3 C D1. Mixture matrix formed by A, B (analytes) and C (interferent). D2. Standard of A. D3. Standard of B.

11. 050 0 0.02 0.04 0.06 0.08 0.1 050 -10 -5 0 5 10 15 20 050 0 0.02 0.04 0.06 0.08 0.1 050 -10 -5 0 5 10 15 20 0.05 0.15 050 0 0.1 0.2 050 -10 -5 0 5 10 15 02040 0 0.2 0.4 0.6 0.8 1 02040 0 0.2 0.4 0.6 0.8 1  Ex. CD-UV absorption monitoring of a protein folding process D 1,UV D 1,CD C1C1 S UV T S CD T C2C2 D 2,UV D 2,CD  UV CD 1212 Process

12. STST DkDk CkCk  (I x J)(I,n) STST (n,J) DkDk DkDk CkCk  (I x J)(I,n) STST (n,J) PCA: orthogonality; max. variance MCR: non-negativity, nat. constraints Stretched/unfolded representation ? D k = C k S T = C t k S T CkCk  D aug C aug Extension of Bilinear models for simultaneous analysis of multiple two way data sets Bilinear models to describe augmented matrices Matrix augmentation strategy

13. Lecture 3 Simultaneous resolution of multiple two-way data sets. Resolution of multivay data sets. Trilinear and multilinear models. Extension of MCR-ALS to multi-way data and to multi-set data. –Constraints. –Extension of resolution conditions. –Recovery of quantiative information. –Breaking rank deficiencies by matrix augmetation –Feasible bands Comparison of algorithms and methods. Examples of application.

14. D = C STST T PARAFAC (trilinear model) The same number of components In the three modes: Ni = Nj = Nk = N No interactions between components Different slices D k are decomposed In bilinear profiles having the same shape!

15. PARAFAC trilinear model N N D = C STST T NR NC N NM D = ++ E +... + comp 1 comp 2 comp 3...... error/noise

16. Three-way data Trilinear Data There is a unique response profile for each component in all three measurement orders/modes. The two response profiles of the common components in every simultaneously analyzed data matrix are equal (have the same shape)

17. d ijk is the concentration of chemical contaminant j in sample I at time (condition) k n=1,...,N are a reduced number of independent environmental sources c in is the amount of source n in sample i; f nj is the amount of contaminant j in source n D k is the data matrix of the measured concentrations of j=1,...,J contaminants in i=1,...,I samples at time k=1,…,K C is the factor matrix describing the row (sample) profiles. Scores. Map of the samples S T is the factor matrix describing the column (spectra) profiles. Loadings. Map of variables T is the factor matrix describing the third mode (conditions, situations,) T={T k } Chemometric models to describe chemical measurements Trilinear models for three-way data: k=1,...,K i=1,...,I j=1,...,J DkDk

18. Three Way data models CS T Np Nq Nr I J K C-mode D S-mode T-mode ( I, J, K) variables samples conditions In general N p, N q and N r may be different,

19. D C-mode S-mode T-mode CSTST T Np NqNr Three-way data models variables samples conditions

20. D C STST G T (Np,Nq,Nr) = Three-way data general model Tucker3 model Data cube decomposition Decomposition gives different number of components in the three modes/orders G (Np,Nq,Nr) is a cube of reduced dimensions, giving the interaction between the factors in the different modes/orders

21. D C STST G T = Different number of components in the different modes Np  Nq  Nr Interaction between components in different modes is possible In PARAFAC N p = N q = N r = N and core array G is a superdiagonal identity cube Tucker3 models

22. D = C STST G T (N x N x N) Three-way trilinear restricted model PARAFAC model Data cube decomposition It is the Identity cube G = I It may be omitted!!! Decomposition gives the same number of components in all three modes/orders!!!

23. Three-way data: Tucker models

24. Lecture 3 Simultaneous resolution of multiple two-way data sets. Resolution of multivay data sets. Trilinear and multilinear models. Extension of MCR-ALS to multi-way data and to multi-set data. –Constraints. –Extension of resolution conditions. –Recovery of quantiative information. –Breaking rank deficiencies by matrix augmetation –Feasible bands Comparison of algorithms and methods. Examples of application. (1.5 hours)

25. D1D1 D2D2 D3D3 STST C1C1 C2C2 C3C3 T = DC Multivariate Curve resolution Alternating Least Squares MCR-ALS quantitative information row-, concentration profiles column-, spectra profiles column-wise augmented data matrix NR 1 NR 2 NR 3 NC NM = 3 Different row sizes

26. Bilinear Model MCR-ALS of column-wise augmented data matrices

27. Unconstrained Alternating Least Squares solution Optional constraints are applied at each ALS iteration!!! + matrix pseudoinverse calculation

28. MCR-ALS constraints for three-way data (simultaneous analysis of a set of correlated bilinear data matrices) Same constraints as those applied to individual data matrices (non- negativity, unimodality, closure, local rank,...). Correspondence between common species in the different data matrices Extension of resolution theorems to augmented data matrices (local rank conditions) Non-trilinear Data –Column profiles (spectra) of the common components are forced to be equal in all the simultaneously analyzed data matrices Trilinear data (trilinearity constraint) –Column and row profiles of the common components are forced to be equal in all the simultaneously analyzed data matrices (trilinearity)

29. Constraints applied to individual data matrices Like in MCR-ALS for two-way data, but separately for each data matrix and species non-negative profiles (concentration, spectra, elution,...) unimodal profiles closure, mass-balance,... shape (gaussian, assimetric,...) selectivity, local rank.............. MCR-ALS constraints for three-way data

30. Correspondence between common species in the different data matrices Species B + C Species A + B Species A+B+C+D = Species A B C D 0 0 0 D1 D2 D3 D1 D2 D3 S T 0 0 0 0 0 0 0 0 0 [C1;C2;C3][D1;D2;D3] x x x x x x x x x x x x x x x x x x x x x x x x Zero values give selectivity and local rank resolution conditions!!!! Appropriate design of experiments will help for total resolution and remove of rotational ambiguities!!

31. X aug D  YTYT contaminants compartments sites F S W F S W contaminants sites 1 2 3 4 5 6 PCA MCR-ALS Bilinear modelling of three-way data (Matrix Augmentation, matricizing, stretching, unfolding ) SVD 123xixi 456x ii zizi z ii Scores refolding strategy!!! (applied to augmented Scores) X Y Z sites contaminants compartment s (F,S,W) xixi x ii zizi z ii Loadings recalculation in two modes from augmented scores Chemometrics and Intelligent Laboratory Systems, 2007, 88, 69-83

32. D contaminants compartments sites F S W F S W contaminants sites  X aug YTYT 1 2 3 MCR-ALS TRILINEARITY CONSTRAINT (ALS iteration step) Selection of species profile 1 2 3 Folding every augmented scored wnated to follow the trilinear model is refolded MA-MCR-ALS Trilinearity constraint SVD Substitution of species profile Rebuilding augmented scores 1’ 2’ 3’ Loadings recalculation in two modes from augmented scores X YTYT contaminants Z sites compartments (F,S,W) This constraint is applied at each step of the ALS optimization and independently for each component individually

33. STST C = D D1D1 D2D2 D3D3 Trilinearity can be implemented independently for each component (chemical species) in MCR-ALS ! 1st score loadings PCA, SVD Folding species profile 1st score gives the common shape Loadings give the relative amounts! Trilinearity Constraint Unfolding species profile Unique Solutions ! Substitution of species profile C Selection of species profile

34. Effect of application of the trilinearity constraint Profiles with different shape Profiles with equal shape Trilinearity constraint 050100150200250 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 050100150200250 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Run 2 Run1 Run 3 Run 4 Run 2 Run1 Run 3 Run 4 one profile in C augmented data matrix

35. D = X aug Y contaminants compartments sites F S W F S W metals sites 1 2 3 4 5 6 MCR-ALS Folding 123456 component interaction constraint (ALS iteration step) interacting augmented scores are folded together 1’ 2’ 3’ 4’ 5’ 6’ = Loadings recalculation in two modes from augmented scores MA-MCR-ALS component interaction constraint SVD = This constraint is applied at each step of the ALS optimization and independently and individually for each component i X Y Z compartments (F,S,W) This is analogous to a restricted Tucker3 model

36. Lesson 3 Simultaneous resolution of multiple two-way data sets. Resolution of multivay data sets. Trilinear and multilinear models. Extension of MCR-ALS to multi-way data and to multi-set data. –Constraints. –Extension of resolution conditions. –Recovery of quantiative information. –Breaking rank deficiencies by matrix augmentation –Feasible bands Comparison of algorithms and methods. Examples of application.

37. Extension of resolution theorems to augmented data matrices Resolution local rank conditions are more easily achieved for augmented ata matrices When resolution conditions are achieved for some component/species present in one of the single matrices, the resolution is also achieved for the same component/species in the rest of matrices (due to the correspondence between component/species!) MCR-ALS constraints for three-way data

39. Lecture 3 Simultaneous resolution of multiple two-way data sets. Resolution of multivay data sets. Trilinear and multilinear models. Extension of MCR-ALS to multi-way data and to multi-set data. –Constraints. –Extension of resolution conditions. –Recovery of quantitative information. –Breaking rank deficiencies by matrix augmetation –Feasible bands Comparison of algorithms and methods. Examples of application.

40. In the simultaneous analysis of multiple data matrices intensity/scale ambiguities can be solved a) in relative terms (directly) b) in absolute terms using external knowledge Solving intensity ambiguities in MCR-ALS dcscs ijin nj n innj n   k 1 k k is arbitrary. How to find the right one?

42. Recovery of quantitative information Relative Quantitation Unknown reference concn. C r C 1 /C r = A 1 / A r C 2 /C r = A 2 / A r Absolute Quantitation Known reference concn. C r C 1 = (A 1 / A r ) C r C 2 = (A 2 / A r ) C r

43. D1D1 D2D2 D3D3 = NR NS=4 NC C1C1 C2C2 C3C3 STST E1E1 E2E2 E3E3 + Quantitative MCR-ALS for three-way data c11c11 c21c21 c 31 unfolding profile 1 c11c11 c21c21 c 31 Relative Quantitation ratio of conc. profile areas: A 12 /A 11, A 13 /A 11.... ratio of conc. profile maximum intensities m 21 /m 11, m 31 /m 11,... other..... A 11 A 21 A 31 m 11 m 21 m 31

45. Quantitative information in iterative three-way methods (PARAFAC-ALS and Tucker-ALS) DkDk CTkTk STST = (m x n)(m x c) (c x c)(c x n) tktk Quantitative information is available from matrix T k (third mode)!!

46. Lecture 3 Simultaneous resolution of multiple two-way data sets. Resolution of multivay data sets. Trilinear and multilinear models. Extension of MCR-ALS to multi-way data and to multi-set data. –Constraints. –Extension of resolution conditions. –Recovery of quantiative information. –Breaking rank deficiencies by matrix augmentation –Feasible bands Comparison of algorithms and methods. Examples of application. (1.5 hours)

47. Rank augmentation by matrix augmentation Matrix augmentation allows the study of rank deficient systems Rank deficient systems are systems where the number of linearly independent components is lower than the number of the true contributions. In reaction based systems: D = C S T rank(D) = min(rank (C,S T )) rank(D)= min (R+1, Q) R num. of reactions, Q num. of species Rank augmentation can be obtained by matrix augmentation! A  B2 species, 1 reaction, rank is 2 A + B  > C 3 species, 1 reaction, rank is only 2 (rank deficiency) A  > B + C 3 species, 1 reaction, rank is only 2 (rank deficiency) A + B  > C + D 4 species, 1 reaction, rank is only 2 (rank deficiency) A  B C  D4 species, 2 reactions, rank is 3 (rank deficiency)............................................................................................

48. [ACU;A] pH = 9.4 pH = 10.5 pH = 13.3 R1R1 R2R2 R3R3 Kinetic determinations Journal of Chemometrics, 1998, 12, 183-203 Acid-base spectrometric titrations: mixtures of nucleic bases : HA; U, HU; H, HH; T, HT Chemometrics and Laboratory Systems, 1997, 38, 183- 197 Rank deficiency is broken By means of matrix augmentation Quantitative determinations with errors < 3% ACU A  MCR- ALS

49. Lecture 3 Simultaneous resolution of multiple two-way data sets. Resolution of multivay data sets. Trilinear and multilinear models. Extension of MCR-ALS to multi-way data and to multi-set data. –Constraints. –Extension of resolution conditions. –Recovery of quantiative information. –Breaking rank deficiencies by matrix augmentation –Feasible bands Comparison of algorithms and methods. Examples of application. (1.5 hours)

50. Calculation of band boundaries of feasible solutions for three-way data  The same general optimization problem as for two-way data can be easily implemented and extended to column-wise augmented data matrices (three-way data).  Constraints are implemented in the same way as for two-way data (natural, local rank, selectivity...)  Additional constraints for trilinear data: Trilinearity constraint!!!

51. Extensión to ‘multiway’ data: 4 chromatographic runs of 4 coeluting components Trilinear data 050100150200 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 Run 2 Run1 Run 3 Run 4 050100150 200 0 1 2 3 0.5 02040 0 0.1 0.2 0.3 0.4 a) Matrix augmentation, non-negativity and spectra normalization constraints c) Matrix augmentation, non-negativity, spectra normalization and trilinearity constraints b) Matrix augmentation, non-negativity, spectra normalization and selectivity constraints

52. Resolution local rank/selectivity conditions are achieved in many situations for well designed experiments (unique solutions!) Rank deficiency problems can be more easily solved Constraints (local rank/selectivity and natural constraints) can be applied independently to each component and to each individual data matrix. Total resolution is achieved for three-way trilinear and for most of non-trilinear data systems The multilinear structure can be introduced in a flexible way as an additional constraint in the ALS algorithm (even for Tucker models with interaction among components) J,of Chemometrics 1995, 9, 31-58; J.of Chemometrics and Intell. Lab. Systems, 1995, 30, 133 Advantages of MCR-ALS of Three-way Data

53. Lecture 3 Simultaneous resolution of multiple two-way data sets. Resolution of multivay data sets. Trilinear and multilinear models. Extension of MCR-ALS to multi-way data and to multi-set data. –Constraints. –Extension of resolution conditions. –Recovery of quantitative information. –Breaking rank deficiencies by matrix augmentation –Feasible bands Comparison of algorithms and methods. Examples of application. (1.5 hours)

54. D = C STST T DkDk CTkTk STST = (m x n)(m x c) (c x c)(c x n) tktk PARAFAC

55. D C STST G T (r x c x t) = DkDk C = (m x n)(m x r) STST (c x n) NkNk (r x c) TUCKER

56. = STST DC T DkDk CkCk = (m x n)(m x c) STST (c x n) MCR-ALS C k * = C k T k

57. Resolution of three-way data Trilinear data: factor analysis rotational ambiguities are totally solved –Examples of methods: GRAM, TLD, PARAFAC- ALS, Tucker-ALS, MCR-ALS,... Non-trilinear data: Factor analysis rotational ambiguities can still be present but they are solved in many situations under some constraints –Examples of methods: Tucker-ALS, MCR-ALS

58. Non-iterative (Eigenvector Decomposition) GRAM (Generalized Rank Annihilation) TLD (Trilinear Data Decomposition) Iterative (Alternating Least Squares, ALS) PARAFAC-ALS Tucker-ALS MCR-ALS Resolution methods for trilinear data

59. Non-iterative three-way methods (GRAM and TLD) A.Lorber, Anal. Chim. Acta, 164 (1984) 293 E.Sanchez and B.R.Kowalski, Anal. Chem., 58 (1986) 496-9 E.Sanchez, B.R.Kowalski, J.of Chemometrics, 4 (1990)29-45 Solving the generalized eigenvalue- eigenvector equation M is the unknown mixture to estimate data matrix N is the standard data matrix C concn profiles S T spectra  concn ratio of the analyte in N (  ) compared to M (  ), it is obtained by generalized eigenvalue-eigenvector equation generalized eigenproblem

60. PARAFAC-ALS R.Bro, Chemolab, (1997) 149-171 Alternating Least Squares Algorithm: 1.Determination of the number of chemical compounds (N) in the original three-way array. 2.Calculation of initial estimates for C and S T. 3.Estimation of T, given D T, C and S T. 4.Estimation of C, given D R, S T and T. 5.Estimation of S T, given D C, C and T. 6.Go to 3 until convergence is achieved. This data decomposition gives the same number of components in the different modes/orders!! Find the minimum of

61. PARAFAC-ALS R.Bro, Chemolab, (1997) 149-171 Step 4 of the algorithm (example) 4. Estimation of C, given D R, S T and T. D DRDR Row-wise augmented data matrix D R * * * *** ** * * * * * STST T ALS C C = D R Z + Z = T  S T  Kronecker product

62. Tucker-ALS P.M.Kroonenberg and J.DeLeeuw, Psychometrika, 45 (1980) 9 1.Determination of the number of components in each order. 1.Calculation of initial estimates for C, S and T. 2.Estimation of G, given C, S and T. 3.Estimation of C, given G, S and T. 4.Estimation of S T, given G, C and T. 5.Estimation of T, given G, C and S T 6.Go to 3 until convergence is achieved. This data decomposition allows different umber of components in the different orders!! Find the minimum of

63. General comparison of three-way methods for resolution of three-way chemical data GRAM is fast and works well for (only) 2 data matrices of trilinear data DTLD is fast and works for trilinear data (algorithm may fail; complex solutions; not Least Squares) PARAFAC gives least-squares solutions but it is too restrictive for multivariate resolution of chemical data (it is very good for trilinear data) Tucker3 imposses a too complex data structure model for multivariate resolution of usually found chemical data

64. General comparison of three-way methods for resolution of three-way chemical data MCR-ALS model is similar to a Tucker2 or a Tucker1 model (depending on the case): a)it is very flexible and easy to use and interpret b)only needs one order/mode/direction in common c)different number of rows are allowed in differnt matrices d)constraints can be applied for each individual species and matrix e)it adapts easily to chemical data with a simple bilinear model and constraints; e) it may assume simple interaction between components (like in Tucker models).

65. Deviations from trilinearity Mild Medium Strong Array size PARAFAC Small PARAFAC2 MediumTUCKER LargeMCR, PCA, SVD,.. Guidelines for selection of resolution method Journal of Chemometrics, 2001, 15, 749-771 Software 1. N-way toolbox by C. Andersson and R. Bro. http://www.models.kvl.dk/source/nwaytoolbox 2. MCR-ALS by R. Tauler and A. de Juan. http://www.ub.es/gesq/mcr/mcr.htm

66. Lecture 3 Simultaneous resolution of multiple two-way data sets. Resolution of multivay data sets. Trilinear and multilinear models. Extension of MCR-ALS to multi-way data and to multi-set data. –Constraints. –Extension of resolution conditions. –Recovery of quantiative information. –Breaking rank deficiencies by matrix augmentation –Feasible bands Comparison of algorithms and methods. Examples of application.

67. Run1 Run 2 Run 3 Run 4

68. Check of trilinear data structure: SVD analysis of concentration profiles svd of trilinear data 1.5018e-004 1.0421e-004 3.8935e-005 1.7183e-005 1.7569e-020 9.7494e-021 8.5585e-021 5.9053e-021 5.1355e-021 4.5152e-021

69. Example 1 Four chromatographic runs following a trilinear model lof % R 2 a) Theoretical 1.634 0.99973 (added noise) b) MA-MCR-ALS-tril 1.6240.99974 c) PARAFAC 1.6130.99974 (small overfitting) O PARAFAC + MA-MCR-ALS tril - theoretical O PARAFAC + MA-MCR-ALS tril - theoretical

70. Three-way trilinear data: spectra recovery speciesTLD (cos)ALS (cos)TLD (sin)ALS (sin) 10,99950,99990,0330,0107 2110,00690,0068 30,99980,99990,02210,0136 40,999910,01240,0086

71. Trilinear data: quantitative recovery

72. Calculation of feasible bands in the simultaneous resolution of several chromatographic runs (runs 1, 2, 3 and 4) Matrix augmentation, non-negativity and spectra normalization constraints

73. Calculation of feasible bands in the simoultaneous resolution of several chromatographic runs (runs 1, 2, 3 and 4) Matrix augmentation, non-negativity, spectra normalization and selectivity constraints Totally unique solutions are not achieved in this case!

74. Feasible bands for the 4 th spectrum obtained under selectivity constraints after the simultanous analysis of the 4 runs (this is the profile with more rotational ambiguity)

75. Trilinearity gives unique solutions! Calculation of feasible bands in the simoultaneous resolution of several chromatographic runs (runs 1, 2, 3 and 4) Matrix augmentation, non-negativity, spectra normalization and trilinearity constraints

76. 4 Non-trilinear data 050100150200 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Run1 Run 2 Run 3 Run 4 050100150200 0 0.5 1 1.5 2 x 10 -5 050100 0 1 2 3 4 x 10

77. Non-trilinear data 050100150200 0 0.5 1 1.5 2 x 10 -5 The chromatographic profiles of the common components in every simultaneously analyzed data matrix are different (in shape and position)

78. Test of three-way non-trilinear data structure svd non-trilinear 1.3933e-004 7.5324e-005 3.8957e-005 1.9943e-005 9.3868e-006 7.8565e-006 6.0801e-006 2.2149e-006 1.1052e-006 7.4765e-007

79. Detection of trilinear structure by SVD of augmented matrices SVD tri row SVD tri col SVD ntril row SVD ntri col 2.0524e+01 2.0593e+001 1.8918e+01 1.9148e+001 3.8184e+00 3.4987e+000 3.1731e+00 2.5268e+000 1.2735e+00 8.7933e-001 2.2716e+00 9.0939e-001 5.0908e-001 7.7666e-001 1.0068e+00 7.5818e-001 7.8332e-002 6.8924e-002 4.0698e-001 6.9556e-002 7.7272e-002 6.7916e-002 3.0997e-001 6.8167e-002 7.5234e-002 6.5720e-002 1.9856e-001 6.6348e-002 7.4882e-002 6.5390e-002 1.0443e-001 6.5728e-002 7.3814e-002 6.4768e-002 8.0703e-002 6.5172e-002 7.1760e-002 6.4072e-002 7.6440e-002 6.4753e-002

80. Concentration (elution) profiles: non-trilinear data It is very difficult to resolve each chromatographic run individually! Local rank resolution conditions are now present in run 4 Run 1 Run 2 Run 3 Run 4

81. Elution feasible bands: matrix augmentation, non-negative, spectra normalization and selectivity constraints blue = no selectivity (feasible bands no-unimodal) red = selectivity (unique solutions)

82. Spectra feasible bands: matrix augmentation, non-negative, spectra normalization and selectivity constraints blue = no selectivity (feasible bands) red = selectivity (unique solutions) one of the bounds of feasible bands (no selectivity) is equal to the real solution

83. Example 2 Four chromatographic runs not following a trilinear model lof %R 2 a) Theoretical 0.97540.99990 (added noise) b) MA-MCR-ALS-tril 17.0960.97077 (the data system is far from trilinear, and impossing trilinearity gives a much worse fit and wrong shapes of the recovered profiles) + MA-MCR-ALS tril - theoretical + MA-MCR-ALS tril - theoretical

84. Example 2 Four chromatographic runs not following a trilinear model lof %R 2 a) Theoretical 0.97540.99990 (added noise) b) PARAFAC lof (%) 14.34 0.97941 (the data system is far from trilinear, and impossing trilinearity gives a much worse fit and wrong shapes of the recovered profiles) O PARAFAC - theoretical O PARAFAC - theoretical

85. Example 2 Four chromatographic runs not following a trilinear model lof %R 2 a) Theoretical 0.97540.99995 (added noise) b) MA-MCR-ALS-non-tril 0.99590.99990 (good MA and local rank conditions for total resolution without ambiguities) + MA-MCR-ALS non tril - theoretical + MA-MCR-ALS non tril - theoretical

86. SpeciesTLDALS (cos)ALS (sin) 1complex0,99840,0567 2complex0,99970,0246 3complex10,008 4complex10,008 Three-way non-trilinear data: spectra recovery

87. Non-trilinear data: quantitative recovery

88. Example of Quantiative determinations Determination of triphenyltin in sea-water by excitation-emission matrix fluorescence and multivariate curve resolution A method for the determination of triphenyltin (TPhT) in sea- water was proposed: 1) Solid phase exctraction (SPE) of sea-water samples; 2)Reaction with a fluorogenic reagent (flavonol in a micellar medium); 3)Excitation-emission fluorescence measurements (giving an EEM data matrix); 4)MCR-ALS analysis of EEM data matrices 5)Quantitation of TPhT J.Saurina, C.Leal, R.Compañó, M.Granados, R.Tauler and M.D.Prat. Analytica Chimica Acta, 2000, 409, 237-245

89. Example of Quantiative determinations Determination of triphenyl in sea-water by excitation-emission matrix fluorescence and multivariate curve resolution. Difficulties were: - low concentrations of TPht ng/l - strong background (fulvic acids) emission - strong reagent emission - lack of selective emission/excitation wavelengths - to have sea-water TPhT standards available

90. Excitation-Emission spectra for an unknown sea- water sample

91. U R B S ex,m ex,1 em,1 em,m em,1 em,n em,m em,1 em,m em,1 = em,m em,1 em,n em,m em,1 em,m em,1 XTXT EUEU ERER EBEB ESES em,m em,1 em,n em,m em,1 em,m em,1 ex,m ex,1 ex,m ex,1 + YUYU YSYS YRYR YBYB U unknown sea water; S TPhT pure standard; R reagent (flavonol); B sea-water background (fulvic acids) EEM D aug emission Y aug excitation X T noise E aug =+ MCR-ALS resolution of EEM data

92. Model: [U;S;R;B] =D aug = Y aug X T + E aug Resolution: (emission)Y aug = D aug (X T ) + Constraints: - non-negativity (excitation)X T = (Y aug ) + D aug - trilinearity Quantitation: c U = [Area(y 1,U ) / Area(y 1,S )] c S

94. Plot of the emission profiles areas for TPhT species in standards, synthetic and sea-water samples respect the analyte concentration MCR-ALS resolution/quantitation of EEM data

95. Comparison between 'true' and MCR-ALS calculated TPhT concentrations in sea-water samples overall prediction errors were always below 13%! Quantitation: c U = [Area(y U ) / Area(y S )] c S

96. FIGURES OF MERIT IN SECOND ORDER MULTIVARIATE CURVE RESOLUTION From MCR-ALS resolution of the pure response profiles of the analyte in different known and unknown mixures (data matrices), a Calibration Curve is built. Figures of merit such as Limit of Detection, Sensitivity, Precision and Accuracy are calculated from the calibration curve like in univariate calibration! J. Saurina *, C. Leal, R. Compañó, M. Granados, M. D. Prat and R.Tauler

97. 0 0.5 1 1.5 2 2.5 3 3.5 051015 TPhT concentration (µg / L) Relative Area Approach (a) [U;S2;R] r i = 0.260 c i + 0.014 (r = 0.998) Approach (b) [U1;U2;U3;U4;U5;U6;U7;U8;U9;U10;U11;U12;S2;R;B] r i = 0.244 c i + 0.201 (r = 0.987) Building the Calibration Curve and Sensitivity r i = a i / a std = f(c std )

98. Precision bands ± s R t ( 1/m + 1/n + (r i - ) 2 /  (c i - ) 2 ) 1/2 LOD = + t s R / b ( 1/m + 1/n + + ((r i - ) / b) 2 /  (c i - ) 2 ) 1/2 Limit of detection (a) and (b) LOD = 0.7  g l -1 (a) and (b) s R = 0.0404 Precision:

99. Accuracy of the method in the prediction of TPhT in real samples 0 10 20 30 40 010203040 Actual Concentration Calculated Concentration (ng/L) Sea Water A Sea Water B Sea Water C Sea Water D Error % = 5.5 % for strategy (A) Error % = 12.7 % for strategy (B) overall prediction error

100. Solving matrix effects in the analysis of triphenyltin in sea-water samples by three-way multivariate curve resolution Three strategies were compared for the recovery of the analyte response in the sea-water samples: (i) using pure standards (ii) using sea-water standards; and (iii) using the standard addition method The combination of standard addition with multivariate curve resolution method improved the accuracy of predictions in the presence of matrix effects. J.Saurina and R.Tauler, The Analyst, 2000, in press

101. Standard addition strategy: For each unknown sample, MCR-ALS is applied to the following aug- mented matrices (i.e A4, the same for the other A1, A2, A3, A5 and A6) augmented matricesidentification [A4;S2;R;B]=>A4 unknown sample [A4SA1;S2;R;B]=>A4SA1 = A4 + 0.20 µg l -1 TPhT [A4SA2;S2;R;B]=>A4SA2 = A4 + 0.75 µg l -1 TPhT [A4SA3;S2;R;B]=>A4SA3 = A4 + 1.05 µg l -1 TPhT [A4SA4;S2;R;B] =>A4SA4 = A4 + 1.87 µg l -1 TPhT [A4SA5;S2;R;B]=>A4SA5 = A4 + 3.30 µg l -1 TPhT [A4SA6;S2;R;B]=>A4SA6 = A4 + 4.52 µg l -1 TPhT [A4SA7;S2;R;B] =>A4SA7 = A4 + 7.42 µg l -1 TPhT S2 EMM response matrix of an standard of TPhT R EMM response matrix of the reagent B EMM response matrix of the background

102. Standard addition calibration graph in a sea-water analyte determination (sea-water sample A4)

103. -100 -50 0 50 100 A1A2A3A4A5A6 Sample Reference Prediction Error (%) standards Pure standards Sea-water Standard addition Prediction errors in the determination of TPhT in sea-water samples A1-A6 using MCR-ALS and three calibration approaches:

104. Recent advances and current research on MCR-ALS method Hybrid soft- hard- (grey) bilinear models (kinetic and equilibrium chemical reactions, profile responses shape...) Extension to multiway data analysis (PARAFAC, Tucker3 models....) Multivariate Image Analysis.(MIA) Weighted Alternating Least Squares (WALS) Calculation of feasible band boundaries (rotation ambiguity) Error propagation in MCR-ALS solutions …… Applications:  Bioanalytical: polynucleotides, proteins, u-array...  Environmental: contamination sources resolution and apportionemnt  Analytical: Hyphenated methods(LC-DAD, LC-MS, GC-MS, FIA-DAD,…), multidimensional spectroscopies (2D-NMR, EEM,…  ON-line spectroscopic monitoring of (bio)chemical processes and reactions......  …. New user interface: http://www.ub.es/gesq/mcr/mcr.htm J. Jaumot,et al., Chemometrics and Intelligent Laboratory Systems, 2005, 76(1) 101-110