1. Initial estimates for MCR-ALS method: EFA and SIMPLISMA
7th Iranian Workshop on Chemometrics
3-5 February 2008
                                       
Initial estimates for MCR-ALS method: EFA and SIMPLISMA
Bahram Hemmateenejad
Chemistry Department, Shiraz University, Shiraz, Iran

2. Fitting data to model (Hard model) Fitting model to data (soft model)
Chemical modeling
Fitting data to model (Hard model)
Fitting model to data (soft model)

3. Multicomponent Curve Resolution
Goal: Given an I x J matrix, D, of N species, determine N and the pure spectra of each specie.
Model: DIxJ = CIxN SNxJ
Common assumptions:
Non-negative spectra and concentrations
Unimodal concentrations
Kinetic profiles 20 40 60 10 30
0.5 1
sensors
samples S N x J C I = D

4. Basic Principles of MCR methods
PCA: D=TP
Beer-Lambert: D=CS
In MCR we want to reach from PCA to Beer-Lambert
D= TP = TRR-1P, R: rotation matrix
D = (TR)(R-1P)
C=TR, S=R-1P
The critical step is calculation of R

5. Multivariate Curve Resolution-Alternative Least Squares (MCR-ALS)
Developed by R. Tauler and A. de Juan
Fully soft modeling method
Chemical and physical constraints
Data augmentation
Combined hard model
Tauler R, Kowalski B, Fleming S, ANALYTICAL CHEMISTRY 65 (15): , 1993.
de Juan A, Tauler R, CRITICAL REVIEWS IN ANALYTICAL CHEMISTRY 36 (3-4):

6. MCR-ALS Theory Widely Applied to spectroscopic methods
UV/Vis. Absorbance spectra
UV-Vis. Luminescence spectra
Vibration Spectra
NMR spectra
Circular Dichroism …
Electrochemical data are also analyzed

7. MCR-ALS Theory In the case of spectroscopic data
Beer-Lambert Law for a mixture
D(mn) absorbance data of k absorbing species
D = CS
C(mk) concentration profile
S(kn) pure spectra

8. MCR-ALS Theory Initial estimate of C or S
Evolving Factor Analysis (EFA) C
Simple-to-use Interactive Self-Modeling Mixture Analysis (SIMPLISMA) S

9. MCR-ALS Theory Consider we have initial estimate of C (Cint)
Determination of the chemical rank
Least square solution for S: S=Cint+ D
Least square solution for C: C=DS+
Reproducing of Dc: Dc=CS
Calculating lack of fit error (LOF)
Go to step 2

10. Constraints in MCR-ALS
Non-negativity (non-zero concentrations and absorbencies)
Unimodality (unimodal concentration profiles). Its rarely applied to pure spectra
Closure (the law of mass conservation or mass balance equation for a closed system)
Selectivity in concentration profiles (if some selective zooms are available)
Selectivity in pure spectra (if the pure spectra of a chemical species, i.e. reactant or product, are known)

11. Constraints in MCR-ALS
Peak shape constraint
Hard model constraint (combined hard model MCR-ALS)

12. Rotational Ambiguity
Rank Deficiency

13. Evolving Factor Analysis (EFA)
Gives a raw estimate of concentration profiles
Repeated Factor analysis on evolving submatrices
Gampp H, Maeder M, Meyer CI, Zuberbuhler AD, CHIMIA 39 (10):
Maeder M, Zuberbuhler AD, ANALYTICA CHIMICA ACTA 181: , 1986
Gampp H, Maeder M, Meyer CJ, Zuberbuhler AD, TALANTA 33 (12): , 1986

14. Basic EFA Example Calculate Forward Singular Values1 1
___ 1st Singular Value
0.9
nd Singular Value
SVD
0.8
...… 3rd Singular Value S
0.7 i R i
0.6
0.5
0.4
0.3
0.2
0.1 I 5 10 15 20 25
I samples

15. Basic EFA Example Calculate Backward Singular Values1 1
0.9
___ 1st Singular Value
nd Singular Value
0.8
...… 3rd Singular Value
0.7 R
0.6
0.5 i
0.4
SVD
0.3 S
0.2 i
0.1 I 5 10 15 20 25 I
samples

16. Basic EFA
Use ‘forward’ and ‘backward’ singular values to estimate initial concentration profiles
Area under both nth forward and (K-n+1)th backward singular values is estimate for initial concentration of nth component. 1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1 5 10 15 20 25 I
samples

17. Basic EFA First estimated spectra Area under 1st forward5 10 15 20 25
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9 1 I
samples
First estimated spectra
Area under 1st forward
and 3rd backward singular
value plot. (Blue)
Compare to true component
(Black)

18. Basic EFA First estimated spectra Area under 2nd forward5 10 15 20 25
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9 1 I
samples
First estimated spectra
Area under 2nd forward
and 2nd backward singular
value plot. (Red)
Compare to true component
(Black)

19. Basic EFA First estimated spectra Area under 3rd forward5 10 15 20 25
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9 1 I
samples
First estimated spectra
Area under 3rd forward
and 1st backward singular
value plot. (Green)
Compare to true component
(Black)

20. Example data
Spectrophotometric monitoring of the kinetic of a consecutive first order reaction of the form of
A B C k1 k2

22. Pseudo first-order reaction with respect to A A + R B C
[R]1 k1= k2=0.02
[R]2 k1= k2=0.08
[R]3 k1= k2=0.32 k1 k2

29. K1=0.2
K2=0.02
K1=0.3
K2=0.08
K1=0.45
K2=0.32

33. K1=0.30
K2=0.08
K1=0.20
K2=0.02
K1=0.45
K2=0.32

34. Noisy data

35. EFA Analysis The m.file is downloadable from the MCR-ALS home page:

51. Simple-to-use Interactive Self-Modeling Mixture Analysis (SIMPLISMA)
W. Windigm J. Guilment, Anal. Chem. 1991, 63,
F.C. Sanchez, D.L. Massart, Anal. Chim. Acta 1994, 298,

52. SIMPLISMA is based on the selection of what are called pure variables or pure objects.
Data matrix
Variable (i.e. wavelength)
Object (i.e. time or sample)
A pure variable is a wavelength at which only one of the compounds in the system is absorbing.
A pure object is an analysis time at which only one compound is eluting.

53. Absorbance spectra
Chromatographic profile
Pure object
Pure variable

54. 1 2

55. 35 20

56. Standard deviation vector
Mean vector
Standard deviation vector 
t = 0 µ0 µm . 0 m .
t = m

57. Standard deviation vector
t = m  . µ0 µm
Mean vector .
Standard deviation vector 0 m

58. chromatogram
Pure spectra

59. Pure spectra
Mean
Standard deviation

60. chromatogram
Mean
Standard deviation

61. λ1 λ2 µi vi xi

62. SIMPLISMA steps
1) The ratio between the standard deviation, σi, and the mean, μi, of each spectrum is determined

63. To avoid attributing a high purity value to spectra with low mean absorbances, i.e., to noise spectra, an offset is included in the denominator
0<offset<3

64. 2) Normalisation of the data matrix: Each spectrum xi is normalised by dividing each element of a row xij by the length of the row ||xi||:
When an offset is added, the same offset is also included in the normalisation of the spectra.

65. 3) Determination of the weight of each spectrum, wi
3) Determination of the weight of each spectrum, wi. The weight is defined as the determinant of the dispersion matrix of Yi, which contains the normalised spectra that have already been selected and each individual normalised spectrum zi of the complete data matrix.
Yi = [Zi H]
Initially, when no spectrum has been selected, each Yi contains only one column, zi (H=1), and the weight of each spectrum is equal to the square of the length of the normalized spectrum

66. When the first spectrum has been selected, p1, each matrix Yi consists of two columns: p1 and each individual spectrum zi, and the weight is equal to
Yi = [Zi p1]
When two spectra have been selected, pl and p2, each Yi consists of those two selected spectra and each individual zi, and so on.
Yi = [Zi [p1 p2]]

67. i= H=I
i= H=p1
i= H=[p1 p2]
i= H=[p1 p2 p3] …

72. Offset=0

73. Offset=1

74. * *

75. * *

76. * *

77. Example data HPLC-DAD data of a binary mixture

78. chromatogram

79. Pure spectra