1. Multivariate Resolution in Chemistry
Lecture 1
Roma Tauler
IIQAB-CSIC, Spain

2. Lecture 1 Introduction to data structures and soft-modelling methods.
Factor Analysis of two-way data: Bilinear models.
Rotation and intensity ambiguities.
Pseudo-rank, local rank and rank deficiency.
Evolving Factor Analysis.

3. Chemical sensors and analytical data structures
one variable x1 e.g. pH
two variables x1,x2 e.g pH i T
three variables x1, x2 and x3
e.g. pH, T i P
n variables ?????
* ***** * ** *** * pH pH T * * * * pH T * * * * P

4. Data Structures: Zero order Zero-way datah
x; one sample gives one scalar (tensor 0th order)
Examples: - selective electrodes, pH
- absorption at one wavelength 
- height/area chromatographic peak
Assumptions: - total selectivity
- known lineal response
Tools: - univariate algebra and statistics
Advantages: - simple and easy to understand
Disadvantages: - only one compound information
- total selectivity
- one sensor for every analyte
- low information content
Time x x x x x hi x x x x ci

5. Data Structures: First order One-way data
x1, x2, ....., xn; one sample gives one vector (tensors of order 1)
Examples:
- matrix of sensors
- absorption at many  (spectra)
- chromatograms at a single 
- current intensities at many E
- readings with time (kinetics)
Assumptions: known lineal responses
- different and independent responses
Tools: - linear algebra
- multivariate statistics
- spectral analysis
- chemometrics (PCA,MLR, PCR, PLS...)
Advantages: Calibration in presence of interferences is possible
- Multicomponent analysis is possible
Disadvantages: Interferences should be present in calibration samples
Spectrum
Wavelength (nm)
Absorbance
min 10 20 30 40
Chromatogram
Time

6. Data Structures: Second order / Two-way data
xij; each sample gives a data
table/matrix; tensor of order 2
X =  xkykT
Examples: - LC-DAD; LC-FTIR; GC-MS; LC-MS;
FIA-DAD; CE-MS,.. (hyphenated techniques)
- esp. excitation/emission (fluorescence)
- MS/MS, NMR 2D, GCxGC-MS ...
- spectroscopic/voltammetric monitoring of chemical reactions/processes with pH, time, T, etc.
Assumptions: - linear responses
- sufficient rank (of the data matrices)
Tools: - linear algebra
- chemometrics
Advantages: - calibration for the analyte in the presence of interferences not modelled in calibration samples is possible
- full characterization of the analyte and interferents may be possible
- few calibration samples are needed (only one sample calibration)

7. Multi-way data analysis multivariate resolution
Data Structures: Third order Three-way data D Di 
time
xijk; each sample gives a data
cube; tensor of order 3
X =  xkykzk
Examples
- Several spectroscopic matrices
- Several hyphenated chromatographic
- Hyphenated multidimensional
chromatography (GC x GC / MS)
- excitation/emission/time
Assumptions: - bilinear/trilinear responses
- sufficient rank (of the data matrices)
Tools: - multilinear and tensor algebra
- chemometrics
Advantages: - unique solutions (no ambiguities)
- calibration for the analyte in the presence of interferences not modelled in calibration samples is possible
- full characterization of the analyte and interferents is possible
- few calibration samples are needed (only one sample calibration) D
time
Run nr. 
Multi-way data analysis
(PARAFAC, GRAM)
Extended
multivariate resolution

8. 0th order data: ISE, pH,..
1th order data: spectra
2nd order data: LC/DAD
GC/MS
fluorescence
3rf order data: time/
/excitation/
/emission

9. Examples
Chemical reaction systems monitored using spectroscopic measurements (even at femtosecond scale) to follow the evolution of a reaction with time, pH, temperature, etc., and the detection of the formation and disappearance of intermediate and transient species
Monitoring chemical
reactions. P e r i s t a l t i c p u m p
D (NR,NC) C o m p u t e r P i n S - 1 2 5 . 3 p H m e t r A u o b S p e c t r o p h o t o m e t e r pH . 5 m l
wavelength T = 3 7 o C T h e r m o s t a t i c b a t h

10. Examples
Quality control and optimisation of industrial batch reactions and processes, where on-line measurements are applied to monitor the process.
Process analysis C o m p u t e r
Spectrometer *
probe
wavelength
D (NR,NC)
time

11. Examples
Analytical characterisation of complex environmental, industrial and food mixtures using hyphenated (chromatography, continuous flow methods with spectroscopic detection)
Chromatographic Hyphenated techniques
LC-DAD, GC-MS, LC-MS, LC-MS/MS....
D (NR,NC)
time
wavelength

12. Examples
FIA-DAD-UV with pH gradient for the analysis of a mixture of drugs.
D (NR,NC) pH
wavelength

13. Examples
Analytical characterisation of complex sea-water samples by means of Excitation-Emission spectra for an unknown with tripheniltin (in the reaction with flavonol)
Excitation emission (fluorescence) EEM techniques
D (NR,NC)
excitation
emission

14. Examples
Protein folding and dynamic protein-nucleic acid interaction processes. In the post-genomic era, understanding these biochemical complex evolving processes is one of the main challenges of the current proteomics research.
Conformation changes
Primary
structure
Secondary
structure
Tertiary
structure
Quaternary
structure
Val
Leu
Ser
Ala
Asp
Trp
Gly
His
-helix
-sheet
turn
Random coil
Amino acids
Helix, sheet formation
Globule
formation
Assembled
subunits
D (NR,NC)
Temperature
wavelength

15. Examples
Image analysis of spatially distributed chemicals on 2D surfaces measured using coupled microscopy-spectroscopy techniques in geological samples, biological tissues or food samples. 
Spectroscopic Image analysis 
Total number of pixels (x  y) x y 

16. Data Structures in Chemistry two orders/ways/modes of measurement
Experimental Data
two orders/ways/modes of measurement
D(NR,NC)
row-order
(way,mode)
i.e. usually
change in
chemical
composition
(concentration
order)
column order (way,mode)
i.e usually change in system properties
like in spectroscopy, voltammetry,...
(spectral order)

17. D Chemical data tables (two-way data) J variables (wavelengths)
Instrumental
measurements
(spectra,
voltammograms,...)
Data table or
matrix
concentration changes
measurements
(time, tempera-ture, pH, ....
I spectra (times) D
Plot of spectra
(rows)
Plot of elution
profiles
(columns)

18. Chemical data modelling
Chemical data modelling methods may be divided in:
Hard- modelling methods (deterministic)
Soft-modelling methods (data driven)
Hybrid hard-soft modelling methods
Hard modelling
Soft modelling
Physical
Hard
Model
Analytical
Information
Data
Data
Data
driven
soft
model ?
Physical
Model
Analytical
Information

19. Hard-modelling
Hard-modelling approaches for chemical (stationary, dynamic, evolving…) systems are based on an accurate physical description of the system and on the solution of complex systems of (differential) equations fitting the experimental measurements describing the evolution and dynamics of these systems. They are deterministic models.
Hard-modelling methods usually use non-linear least squares regression (Marquardt algorithm) and optimisation methods to find out the best values for the parameters of the model.
Hard-modelling usually deal with univariate data. It has been often used in the past until the advent of modern instrumentation and computers giving large amounts of data outputs.
Hard-modelling is often successful for laboratory experiments, where all the variables are under control and the physicochemical nature of the dynamic model is known and can be fully described using a known mathematical model

20. Hard-modelling
However, and even at a laboratory level, there are examples where hard-modelling requirements and constraints are not totally fulfilled or no physicochemical model is known to describe the process (e.g. in chromatographic separations or in protein folding experiments).
Data sets obtained from the study of natural and industrial evolving processes are too complex and difficult to analyse using hard-modelling methods. In these cases, there is no known physical model available or it is too complex to be set in a general way.
Advanced hard-modelling in industrial applications has been attempted to model experimental difficulties, such as changes in temperature, pH, ionic strength and activity coefficients. This is a very difficult task!
Data Fitting in the Chemical Sciences
P. Gans, John Wiley and Sons, New York 1992

21. Hard modelling D C ST Output: C, S and model parameters.10 20 30 40 50 60 70 80 90
100
0.5 1
1.5 2
2.5
Wavelength
Absorbance 3 4 5 6 7 8 9
0.1
0.2
0.3
0.4
0.6
0.7
0.8
0.9
Time
Concentration
Non-linear model
fitting
min(D(I-CC+)
C = f(k1, k2) D C 10 20 30 40 50 60 70 80 90
100
0.5 1
1.5 2
2.5 3
x 10 4
Wavelengths
Absortivities
LS (D, C)
(ST) ST
Output: C, S and model parameters.
The model should describe all the variation in the experimental measurements.

22. Soft-modelling
Soft-modelling instead, attempts the description of these systems without the need of an a priori physical or (bio)chemical model postulation. The goal of the latter methods is the explanation of the variations observed in the systems using the minimal and softer assumptions about data. They are data driven models.
Soft models usually give an improved analytical description of the analysed process.
Soft modelling needs more data than hard-modelling. Soft modelling methods deal with multivariate data. Its use has augmented in the recent years because of the advent of modern analytical instrumentation and computers providing large amounts of data outputs.
The disadvantage of soft models is their poorer extrapolating capabilities (compared with hard-modelling).

23. Soft-modelling
A soft model is hardly able to predict the behaviour of the system under very different conditions from which it was derived.
Complex multivariate soft-modelling data analysis methods have been introduced for the study of chemical processes/systems like Factor Analysis derived methods.
Factor Analysis is a multivariate technique for reducing matrices of data to their lowest dimensionality by the use of orthogonal factor space and transformations that yield predictions and/or recognizable factors.
Factor Analysis in Chemistry 3rd Edition, E.R.Malinowski, Wiley, New York 2002

24. Constrained ALS optimisation
Soft modelling D C ST 10 20 30 40 50 60 70 80 90
100
0.5 1
1.5 2
2.5 3
x 10 4
Wavelengths
Absortivities 10 20 30 40 50 60 70 80 90
100
0.5 1
1.5 2
2.5
Wavelength
Absorbance 1 2 3 4 5 6 7 8 9 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Time
Concentration ,
Constrained ALS optimisation
LS (D,C)  S*
LS (D,S*)  C*
min (D –C*S*)
Output: C and S.
All absorbing contributions in and out of the process are modelled.

25. Lecture 1 Introduction to data structures and soft-modelling methods.
Factor Analysis of two-way data: Bilinear models.
Rotation and intensity ambiguities.
Pseudo-rank, local rank and rank deficiency.
Evolving Factor Analysis.

26. Soft-modelling Factor Analysis (Bilinear Model) experimental data
is modelled as a
linear sum of weighted
(scores) factors (loadings)
In matrix form
data scores loadings

27. Soft-modelling BILINEARITY= +
... A B C E
Assumption: Bilinearity (the contributions of the components in the two orders of measurement are additive) 7

28. Soft-modelling GOALS OF BILINEAR MODEL =
0.35
0.35
0.3
0.3
0.25
0.25 =
0.2
0.2
0.15
0.15
0.1
0.1
0.05
0.05 20 40 60 50
100
Recovery of the responses of every component (chemical species) in the different modes of measurement 8

29. Soft-modelling: Factor Analysis
Experimental
Data Matrix
Principal
Components
Cluster
Analysis
Factor
Identification
Target
testing
Real Factor
Models
Predictions

30. Soft-modelling: Factor Analysis (traditional approach)
matrix
multiplication
Covariance
matrix
Data
matrix
decomposition
combination
abstract
reproduction
Abstract
Factors
Real
Factors
target
transformation
abstract rotation
New Abstract
Factors

31. Soft-modelling methods (I)
Factor Analysis methods based on the use of latent variables or eigenvalue/singular value data matrix decompositions. Examples
PCA, SVD, rotation FA methods
Evolving Factor Analysis methods
Rank Annihilation methods
Window Factor Analysis methods
Heuristic Evolving Latent Projections methods
Subwindow Factor Analysis methods
…..

32. Soft-modelling methods (II)
Multivariate Resolution methods do a data matrix decomposition into their ‘pure’ components without using explicitly latent variables analysis techniques. Examples:
SIMPLISMA
Orthogonal Projection Approach (OPA),
Positive Matrix Factorization methods (and Multilinear Engine extensions)
Multivariate Curve Resolution-Alternating Least Squares (MCR-ALS)
Gentle
.....

33. Soft-modelling methods (III)
Three-way and Multiway methods
which decompose three-way or multiway data structures. Examples:
Multiway and multiset extensions of PCA
Genralized rank Annihilation, GRAM; Direct Trilear Decomposition (DTD, TLD)
Multiway and multiset extensions of MCR-ALS methods       
PARAFAC-ALS
Tucker3-ALS

34. Soft-modelling
Factor Analysis in Chemistry, 3rd Ed., E.R.Malinowski, John Wiley & Sons, New York, 2002
Principal Component Analysis, I.T. Jollife, 2nd Ed., Springer, Berlin, 2002
Multiway Analysis, Applications in the Chemical Sciences, A.Smilde, R.Bro and P.Geladi, John Wiley & Sons, New York, 2004
Multivariate Image Analysis, P.Geladi, John Wiley and Sons, 1996
Soft modeling of Analytical Data. A.de Juan, E.Casassas and R.Tauler, Encyclopedia of Analytical Chemistry: Instrumentation and Applications, Edited by R.A.Meyers, John Wiley & Sons, 2000, Vol 11,

35. Soft-modelling Data structures Type of Models
One way data (vectors)  Linear and non-linear models
di = b0 + b ci;
di = fnon-linear(ci)
Two way data (matrices)  Bilinear and non-bilinear models
Non-bilinear data can still be linear in one of the two modes
Three-way data (cubes)  Trilinear and non-trilinear models
Non-trilinear data can still be bilinear in two modes di
I samples
J variables
dij
I samples D
k=1,...,K
conditions
i=1,...,I
j=1,...,J

36. D Soft-modelling Bilinear models for two way data: IJ
dij I D
dij is the data measurement (response) of variable j in sample i
n=1,...,N are the number of components (species, sources...)
cin is the concentration of component n in sample i;
snj is the response of component n
at variable j

37. D Soft-modelling Bilinear models for two way data U VT or ST E C  I +J J J U or C
VT or ST N D E  I + I I
N << I or J N
PCA
D = UVT + E
U orthogonal, VT orthonormal
VT in the direction of maximum
variance
Unique solutions
but without physical meaning
Useful for interpretation
but not for resolution!
MCR
D = CST + E
Other constraints (non-negativity,
unimodality, local rank,… )
U=C and VT =ST non-negative,...
C or ST normalization
Non-unique solutions
but with physical meaning
Useful for resolution
(and obviously for interpretation)!

38. PCA Model (Principal Component Analysis)
X = U VT + E
U ‘scores’ matrix (orthogonal)
VT loadings matrix (orthonormal)
SVD Model (Singular Value Decomposition)
D = U* S VT + E
U* ‘scores’ matrix (orthonormal)
S diagonal matrix of the singular values s
s = 1/2
 eigenvalues of the covariances matrix DDT
VT ‘loadings’ matrix (orthonormal)

39. PCA Model: D = U VT unexplained variance VT D E = loadings
(projections) + U
scores
D = u1v1T + u2v2T + ……+ unvnT + E
n number of components (<< number of variables in D) D =
u1v1T +
u2 v2T
+….+
unvnT + E
rank 1
rank 1
rank 1

40. X = structure + noise PCA Model X = U VT + E
It is an approximation to the experimental data matrix X
Loadings, Projections: VT relationships between original variables
and the principal components (eigenvectors of the covariances matrix).
Vectors in VT (loadings) are orthonormals (orthogonal and normalized).
Scores, Targets: U relationships between the samples (coordinates of
samples or objects in the space defined by the principal components
Vectors in U (scores) are orthogonal
Noise E Experimental error, non-explained variances

41. Summary of Principal Component Analysis PCA
Formulation of the problem to solve
Plot of the original data
3. Data pretreatment.
(data centering, autoscaling, logarithmic transformation…)
4. Built PCA model. Determination of the number of
components. Graphical inspection of explained/residual
plots)
5. Study of the PCA model PCA. Multivariate data exploration
- ‘loadings’ plot ==> map of the variables
- ‘scores’ plot ==> map of the samples
Interpretation of the PCA mode. Identification of the
main sources of data variance
7. Analysis of the residuals matrix E = D -U VT

42. D PCA U VT scores loadings Data set Scores plot Loadings plot Biplot
Site 1
Site 2
Site 3
Site 22
Sampling sites
[org]1
[org]2
[org]3
[org]96
Pollutant concentration
Data set D
PCA -2 -3 -1 1 2 3
PC2 (27%)
Scores plot
-0.8
-0.6
-0.4
-0.2
0.2
Loadings plot 4
PC1 (41%) 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22
-0.5
0.5 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 B A
Biplot U
scores VT
loadings

43. Multivariate Curve Resolution (MCR)D
Mixed information tR 
Pure component information s1  sn ST c c 1 n C
Wavelengths
Retention times
Pure concentration profiles
Chemical model
Process evolution
Compound contribution
relative quantitation
Pure signals
Compound identity
source identification and Interpretation

44. Lecture 1 Introduction to data structures and soft-modelling methods.
Factor Analysis of two-way data: Bilinear models.
Rotation and intensity ambiguities.
Pseudo-rank, local rank and rank deficiency.
Evolving Factor Analysis.

45. Factor Analysis Ambiguities in the analysis of a data matrix (two-way data)
Rotation and scale/intensity ambiguities
Rotation Ambiguities
Factor Analysis (PCA) Data Matrix Decomposition
D = U VT + E
‘True’ Data Matrix Decomposition
D = C ST + E 10

46. How to find the rotation matrix T?
Factor Analysis Ambiguities in the analysis of a data matrix (two-way data)
Rotation and scale/intensity ambiguities
Rotation Ambiguities
D = U T T-1 VT + E = C ST + E
C = U T; ST = T-1 VT
How to find the rotation matrix T?

47. Matrix decomposition is not unique!
Rotation and scale/intensity ambiguities
D = C ST + E = D* + E
Cnew = C T
( NR,N) (NR,N) (N,N)
STnew = T-1 ST
(N,NC) (N,N) (N,NC)
D* = C ST = CnewSTnew
Matrix decomposition is not unique!
T(N,N) is any non-singular matrix
Rotational freedom for any T 6

48. = Rotation and scale/intensity ambiguities
Rotation ambiguities and rotation matrix T(N,N) =
Cnew,2
Cold,1, Cold,2
t1,2
t2,2
STnew,1
STold,1
STold,2
STnew,2
t-12,2
t-12,1 T
t-11,1
t-11,2
T-1
Cnew,1
t1,1
t2,1

49. Intensity (scale) ambiguities:
Rotation and scale/intensity ambiguities
Intensity (scale) ambiguities:
For any scalar k d c s ij in nj n   k 1
Intensity/scale ambiguities make difficuly to obtain quantitative information
When they are solved then it is also possible to have quantitative information 11

50. cold sold = = ( cold x k)(1/k x sold) = cnew snew
Rotation and scale/intensity ambiguities
Intensity (scale) ambiguities:
cold x k = cnew
cold sold =
= ( cold x k)(1/k x sold) =
cnew snew x x
1/k x sold = snew

51. Rotation and scale/intensity ambiguities
Questions to answer:
Is it possible to have unique solutions?
What are the conditions to have unique solutions?
If total unique solutions are not possible:
Is it still possible at least to find out some of the possible solutions?
Is it possible to have an estimation of the band or range of possible/feasible solutions?
How this range of feasible solutions can be reduced?

52. Lecture 1 Introduction to data structures and soft-modelling methods.
Factor Analysis of two-way data: Bilinear models.
Rotation and intensity ambiguities.
Pseudo rank, local rank and rank deficiency.
Evolving Factor Analysis.

53. Definitions
Mathematical rank of a data matrix is the minimum number of linearly independent rows or columns describing the variance of the whole data set. Minimum number of basis vectors spanning the row and column vector spaces. It may be obtained by SVD or PCA.
Pseudo-rank or Chemical rank is the mathematical rank in absence of experimental error/noise. Usually it is equal to the number of chemical/physical components contributing to the observed data variance apart from experimental noise/error. Obtained from the number of larger components from PCA, SVD or other FA methods
Local Rank is the chemical rank of data submatrices. Obtained from EFA, EFF, SIMPLISMA, OPA, or other FA submatrix analysis methods
Rank deficiency when chemical rank is lower than the known number of contributions. Rank deficiency may be broken/solved by data matrix augmentation and perturbation strategies.
Rank overlap rank deficiency caused by equal vector profiles of different chemical/physical components in one or more modes.

54. Pseudo Rank: Number of contributions (factors, components)
Principal Component Analysis
Gives an abstract (orthogonal) bilinear model to describe optimally the variation in our data set. D = U un u1 VT vn v1
Useful chemical information
Size of the model (chemical rank)
Number of chemical contributions

55. Pseudo Rank: Number of contributions (factors, components)
Principal Component Analysis (SVD algorithm)
D = TPT = USVT
Diagonal matrix (singular values)
Magnitude of singular value
Importance of contribution

56. Pseudo Rank: Number of contributions (factors, components)
Principal Component Analysis (SVD algorithm)
D = TPT = USVT
Diagonal matrix (singular values) 1 2 3 4 5 6 7 8 9 10 18 19 20 21 22 23 24 25
Number of components
log(eigenvalues)
4 contributions
Plot log(eigenvalues)
Plot singular values
Eigenvalue = (sing. value)2
large size 
small size 

57. Pseudo Rank: Number of contributions (factors, components)
Overestimations of rank (overfitting).
Large overestimation: the measurements may not follow a bilinear model.
Small overfestimation: presence of structured noise or high noise levels.
Underestimations of rank (rank deficiency).
Linear dependencies
Contributions with very similar signals or concentration profiles.
Compounds with non-measurable signals.
Minor compounds.

58. Rank deficiency No  Rank- deficient systems
Are all the signals distinguishable and independent?
Are all the concentration profiles distinguishable and independent?
No  Rank- deficient systems
Detectable rank < nr. of process contributions
Examples:
1) 2nd order reaction A = B + C, [B] = [C], 3 chemical species/contributions, but Rank =2
2) Enantiomer conversion monitored by UV and the spectrum D = spectrum L, two chemical species/components but Rank =1 (Rank overlap)

59. Rank deficiency
Closed reaction systems. Some concentration profiles are described as linear combinations of others.
System HA / A-, HB / B-
CA = [HA] + [A-]
CB = [HB] + [B-]
CB = kCA
[HA], [HB]
[A-] = CA - [HA]
[B-] = CB - [HB] = kCA - [HB] CA CB
[HA], [HB], [A-], [B-]  f ([HA], [HB], CA)
Rank 3

60. Breaking rank-deficiency by matrix augmentation
Matrix Augmentation in the rank-deficient direction
Data set HA
HA / HB  pH
CB  kCA
[B-] = CB - [HB]  kCA - [HB]
[HA], [HB], [A-], [B-]  f ([HA], [HB], CA)
Rank 4
Breaking rank-deficiency by matrix augmentation

61. Lecture 1 Introduction to data structures and soft-modelling methods.
Factor Analysis of two-way data: Bilinear models.
Rotation and intensity ambiguities.
Pseudo-rank and rank deficiency.
Local Rank and Evolving Factor Analysis.

62. Local exploratory analysis
Study of the variation of the number of contributions in the process or system. Study of the rank variation during the process.
Evolving Factor Analysis (EFA)
Fixed Size Moving Window - Evolving Factor Analysis (FSMW-EFA)

63. Evolving Factor Analysis
Stepwise chemometric monitoring of a process.
Forward Evolving FA (from beginning to end)
Backward Evolving FA (from end to beginning)
Working procedure
Display of subsequent PCA analyses along gradually increasing data set windows.

64. Evolving Factor Analysis
HPLC-DAD example
Wavelengths
Retention times
Spectrum
Chromatogram D

65. Evolving Factor Analysis
Forward Evolving Factor Analysis
PCA 5 10 15 20 25 30 35 40 45 50
7.5 8
8.5 9
9.5
10.5 11
Retention times
log(eigenvalues)

66. Evolving Factor Analysis
Forward Evolving Factor Analysis
Location of the emergence of compounds
Total number of compounds (PCA in last window) 5 10 15 20 25 30 35 40 45 50
7.5 8
8.5 9
9.5
10.5 11
Retention times
log(eigenvalues)
Selective zone
Noise level

67. Evolving Factor Analysis
Backward Evolving Factor Analysis
PCA 5 10 15 20 25 30 35 40 45 50
7.5 8
8.5 9
9.5
10.5 11
Retention times
log(eigenvalues)

68. Evolving Factor Analysis
Backward Evolving Factor Analysis
Location of the disappearance of compounds
Total number of compounds (PCA last total window) 5 10 15 20 25 30 35 40 45 50
7.5 8
8.5 9
9.5
10.5 11
Retention times
log(eigenvalues)
Selective zone
Noise level

69. Evolving Factor Analysis
Combined EFA plot (forward and backward EFA) 5 10 15 20 25 30 35 40 45 50
7.5 8
8.5 9
9.5
10.5 11
Retention times
log(eigenvalues)
Total number of components (PCA of extreme windows)
Detection of selective zones (extremes)
Location of emergence and decay of compounds

70. Evolving Factor Analysis
Consecutive emergence-decay profiles. No embedded compounds.
Sequential processes
Approximate concentration profiles
Noise level 5 10 15 20 25 30 35 40 45 50
7.5 8
8.5 9
9.5
10.5 11
Retention times
log(eigenvalues)
Concentration window
Zero-component windows

71. Evolving Factor Analysis
Approximate concentration profiles
EFA derived concentration profiles
Real concentration profiles

72. Fixed Size Moving Window-Evolving FA (FSMW-EFA)
Local rank map along the process direction or the signal direction.
Working procedure
Subsequent PCA in fixed size windows moving stepwisely along the data set.
Window size  min(number of components + 1)

73. FSMW-EFA PCA log(eigenvalues) Retention times 5 10 15 20 25 30 35 405 10 15 20 25 30 35 40 45 50
3.5 4
4.5
5.5
Retention times
log(eigenvalues)

74. Detection of selective zones along the whole process
FSMW-EFA 5 10 15 20 25 30 35 40 45 50
3.5 4
4.5
5.5
Retention times
log(eigenvalues)
Detection of selective zones along the whole process 1 2
Variation of local rank along the process direction
(complexity, degree of overlap among compounds)
Noise level

75. Local rank detection EFA EFA FSMW-EFA FSMW-EFA window size 520 40 60
0.2
0.4
0.6
0.8 1
1.2
1.4
1.6
1.8 2
x 10 -5 80
100
0.5
1.5
2.5 3
3.5 4
0.05
0.1
0.15
0.25
0.3 -3 -2 -1 10 30 50
-1.5
-0.5
window size 5
window size 8
EFA
EFA
Local
rank
detection
FSMW-EFA
FSMW-EFA 22

76. FSMW-EFA vs. EFA EFA Displays the evolution of the process.
The compounds are well identified (concentration windows)
Local rank information is not easily interpreted.
FSMW-EFA
Clear definition of local rank.
Sensitive to detection of minor compounds.
The idea of process evolution is not preserved.

77. Getting Local rank information from Evolving Factor Analysis methods
Detection of the selective windows or regions where only one species exists (total selectivity)
Detection of zero concentration windows or regions (no species is present)
Detection of windows or regions where a particular species is not present
Detection of the concentration windows or regions where one species is present (other species can coexist) 18

78. References EFA FSMW-EFA SIMPLISMA
H. Gampp, M. Maeder, C.J. Meyer and A.D. Zuberbühler. Talanta, 32, (1985).
M. Maeder. Anal. Chem. 59, (1987).
FSMW-EFA
H.R. Keller and D.L. Massart. Anal. Chim. Acta, 246, (1991).
SIMPLISMA
W. Windig and J. Guilment. Anal. Chem., 63, (1991).