Multivariate Resolution in Chemistry

1
Multivariate Resolution in Chemistry

• Lecture 3

Roma Tauler IIQAB-CSIC, Spain e-mail
rtaqam_at_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
Examples of Three-way data in Chemistry
Luminiscence excitacion /emission
spectra/sample Process/Reaction spectroscopic
monitoring time/pH/temperature wavelength sampl
e/system/run Analytical Hyphenated
Methods LC/DAD LC/FTIR GC/MS
LC/MS time/wavelength/sample time/m/z
ratios/sample Environmental monitoring samples/co
ncentrations/time or conditions Spectroscopic
imaging multiple spectroscopic images from
different samples
4
Three-way data in Chemistry
Example Multiple excitacion emission spectra
(standards and unknown samples)
5
Three-way data in Chemistry
Example Multiple HPLC-DAD-MS runs of a system
(standards and unknown samples)
6
Three-way data in Chemistry
Example A chemical reaction or proces monitored
spectrsocopically
7
Three-way data Unfolding / Matricizing Matrix
Augmentation
NC
Multiple data matrices in a cube (NR,NC,NM)
NR
NM
Row-wise data matrix augmentation (NR,NCxNM)
NR
NC x NM
Column-wise data matrix augmentation (NC,NRxNM)
NR x NM
Tube-wise data matrix augmentation (NM,NRxNC)
NC
NR x NC
NM
3
8
Multiple data sets (e.g. environmental data)
9
Extension of Bilinear Models (PCA or MCR)
Matrix Augmentation
S1T
S2T
S3T
Row-wise
C



D
D
D
D
D
D
1
2
3
1
2
3
ST
C
D
D
D
Column-wise
Row and column-wise
ST
S1T
S2T
S3T
D
D
C1
1
1
C1
ST
D
D
D
D
D
D
1
2
3
1
2
C2
D
D




2
2
C2
D
D
D
D
D
D
4
5
6
4
5
6
C3
D
D
3
3
D
D
C
C
D
Several experiments monitored with the same
technique
10
Ex. Hyphenated Chromatography
D1 D2 D3
B
DT1 DT2 DT3
A
C
??
A
B
D1 . Mixture matrix formed by A, B (analytes) and
C (interferent). D2 . Standard of A. D3 .
Standard of B.
Column-wise data matrix augmentation
11
Ex. CD-UV absorption monitoring of a protein
folding process
12
Bilinear models to describe augmented matrices
Extension of Bilinear models for simultaneous
analysis of multiple two way data sets
ST
Caug
Dk
Ck
?
(n,J)
ST
(I x J)
(I,n)
Ck
?
Dk
PCA orthogonality max. variance
MCR non-negativity, nat. constraints
Matrix augmentation strategy
Daug
ST
Dk
Ck
?
Stretched/unfolded representation ? Dk
Ck ST C tk ST
(n,J)
(I x J)
(I,n)
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
PARAFAC (trilinear model)
The same number of components In the three modes
Ni Nj Nk N No interactions between
components Different slices Dk are decomposed In
bilinear profiles having the same shape!
15
PARAFAC trilinear model
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
Chemometric models to describe chemical
measurements
Trilinear models for three-way data
Dk
dijk is the concentration of chemical contaminant
j in sample I at time (condition) k n1,...,N are
a reduced number of independent environmental
sources cin is the amount of source n in sample
i fnj is the amount of contaminant j in source
n Dk is the data matrix of the measured
concentrations of j1,...,J contaminants
in i1,...,I samples at time k1,,K C is the
factor matrix describing the row (sample)
profiles. Scores. Map of the samples ST is the
factor matrix describing the column (spectra)
profiles. Loadings. Map of variables T is the
factor matrix describing the third mode
(conditions, situations,) TTk
18
Three Way data models
In general Np, Nq and Nr may be different,
19
Three-way data models
variables
T-mode
Nq
Nr
Np
D
T
C
ST
samples
C-mode
conditions
S-mode
20
Three-way data general model Tucker3 model
Data cube decomposition
G (Np,Nq,Nr) is a cube of reduced dimensions,
giving the interaction between the factors in
the different modes/orders
Decomposition gives different number of
components in the three modes/orders
21
Tucker3 models
In PARAFAC Np Nq Nr N and core array G is a
superdiagonal identity cube
22
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
Multivariate Curve resolution Alternating Least
Squares MCR-ALS
NC
D1
C1
ST
NR1
column-, spectra profiles
D2
C2
NR2

Different row sizes
D3
C3
NR3
row-, concentration profiles
NM 3
D
C
column-wise augmented data matrix
quantitative information
T
26
Bilinear Model MCR-ALS of column-wise augmented
data matrices
27
Unconstrained Alternating Least Squares solution
matrix pseudoinverse calculation
Optional constraints are applied at each ALS
iteration!!!
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
MCR-ALS constraints for three-way data

• 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 .....
• .........

30
MCR-ALS constraints for three-way data
Correspondence between common species in the
different data matrices
Zero values give selectivity and local rank
resolution conditions!!!! Appropriate design of
experiments will help for total resolution
and remove of rotational ambiguities!!
31
Bilinear modelling of three-way data (Matrix
Augmentation, matricizing, stretching, unfolding )
Chemometrics and Intelligent Laboratory Systems,
2007, 88, 69-83
Xaug
contaminants
YT
sites
F
1
4
F
S
PCA MCR-ALS
W
?
S
2
5
sites
sites
W
3
6
D
32
MA-MCR-ALS Trilinearity constraint
contaminants
sites
F
F
compartments
S
sites
W
S
sites
contaminants
sites
W
D
33
Trilinearity can be implemented independently for
each component (chemical species) in MCR-ALS!

ST
C
34
Effect of application of the trilinearity
constraint
one profile in C augmented data matrix
Profiles with different shape
Trilinearity constraint
Profiles with equal shape
35
MA-MCR-ALS component interaction constraint
This is analogous to a restricted Tucker3 model
Xaug
metals
Y
sites
F
1
4
F
S
W
S

MCR-ALS
2
5
sites
sites
W
3
6
D
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
MCR-ALS constraints for three-way data

• 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!)

38
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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
Solving intensity ambiguities in MCR-ALS
k is arbitrary. How to find the right one?
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
41
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42
Recovery of quantitative information

• Relative Quantitation
• Unknown reference concn. Cr
• C1/Cr A1 / Ar
• C2/Cr A2 / Ar
• Absolute Quantitation
• Known reference concn. Cr
• C1 (A1 / Ar) Cr
• C2 (A2 / Ar) Cr

43
Quantitative MCR-ALS for three-way data
44
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45
Quantitative information in iterative three-way
methods (PARAFAC-ALS and Tucker-ALS)
Quantitative information is available from
matrix Tk (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 ST rank(D)
min(rank (C,ST)) rank(D) min (R1, Q) R num.
of reactions, Q num. of species Rank augmentation
can be obtained by matrix augmentation! A ? B 2
species, 1 reaction, rank is 2 A B ?gt C 3
species, 1 reaction, rank is only 2 (rank
deficiency) A ?gt B C 3 species, 1
reaction, rank is only 2 (rank deficiency) A B
?gt C D 4 species, 1 reaction, rank is only 2
(rank deficiency) A ? B C ? D 4 species, 2
reactions, rank is 3 (rank deficiency) ...........
..................................................
...............................
48
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
ACUA
R1
R2
ACU
?
pH 13.3
pH 9.4
MCR- ALS
A
pH 10.5
R3
Rank deficiency is broken By means of matrix
augmentation Quantitative determinations with
errors lt 3
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
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Advantages of MCR-ALS ofThree-way Data

• 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
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
T
C
ST
D

PARAFAC
55
T
ST
G
D
(r x c x t)

C
TUCKER
Dk
Nk
C
ST

(c x n)
(r x c)
(m x n)
(m x r)
56
ST

D
C
T
MCR-ALS
Ck
Dk
ST

(c x n)
(m x n)
(m x c)
Ck Ck Tk
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
Resolution methods for trilinear data

• Non-iterative (Eigenvector Decomposition)
• GRAM (Generalized Rank Annihilation)
• TLD (Trilinear Data Decomposition)
• Iterative (Alternating Least Squares, ALS)
• PARAFAC-ALS
• Tucker-ALS
• MCR-ALS

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
• ST 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
Find the minimum of

• Determination of the number of chemical compounds
  (N) in the original three-way array.
• Calculation of initial estimates for C and ST.
• Estimation of T, given DT, C and ST.
• Estimation of C, given DR, ST and T.
• Estimation of ST, given DC, C and T.
• Go to 3 until convergence is achieved.
• This data decomposition gives the same
  number of components in the different
  modes/orders!!

61

• PARAFAC-ALS
• R.Bro, Chemolab, (1997) 149-171
• Step 4 of the algorithm (example)
• 4. Estimation of C, given DR, ST and T.

Row-wise augmented data matrix DR
DR
ALS
C
C DR Z Z T ? ST ? Kronecker product
62
Tucker-ALS P.M.Kroonenberg and J.DeLeeuw,
Psychometrika, 45 (1980) 9
Find the minimum of

• Determination of the number of components
• in each order.
• Calculation of initial estimates for C, S and T.
• Estimation of G, given C, S and T.
• Estimation of C, given G, S and T.
• Estimation of ST, given G, C and T.
• Estimation of T, given G, C and ST
• Go to 3 until convergence is achieved.
• This data decomposition allows different umber
  of
• components in the different orders!!

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)
• it is very flexible and easy to use and interpret
• only needs one order/mode/direction in common
• different number of rows are allowed in differnt
  matrices
• constraints can be applied for each individual
  species and matrix
• 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
Guidelines for selection of resolution
method Journal of Chemometrics, 2001, 15, 749-771
Deviations from trilinearity Mild Medium
Strong Array size PARAFAC
Small PARAFAC2 Medium TUCKER
Large MCR, PCA, SVD,..
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 4
Run 3
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 R2 a)
Theoretical 1.634 0.99973
(added noise) b) MA-MCR-ALS-tril
1.624 0.99974 c) PARAFAC 1.613 0.99974
(small overfitting)
O PARAFAC MA-MCR-ALS tril - theoretical
O PARAFAC MA-MCR-ALS tril - theoretical
70
Three-way trilinear data spectra recovery
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 norma
lization and selectivity constraints Totally
unique solutions are not achieved in this case!
74
Feasible bands for the 4th spectrum obtained
under selectivity constraints after the
simultanous analysis of the 4 runs (this is the
profile with more rotational ambiguity)
75
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
Trilinearity gives unique solutions!
76
Non-trilinear data
4
77
Non-trilinear data
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
• 2.0524e01 2.0593e001 1.8918e01
  1.9148e001
• 3.8184e00 3.4987e000 3.1731e00
  2.5268e000
• 1.2735e00 8.7933e-001 2.2716e00
  9.0939e-001
• 5.0908e-001 7.7666e-001 1.0068e00
  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 4
Run 3
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 R2 a)
Theoretical 0.9754 0.99990 (added noise) b)
MA-MCR-ALS-tril 17.096 0.97077 (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 R2 a)
Theoretical 0.9754 0.99990 (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 R2 a)
Theoretical 0.9754 0.99995 (added noise) b)
MA-MCR-ALS-non-tril 0.9959 0.99990 (good MA
and local rank conditions for total resolution
without ambiguities)
MA-MCR-ALS non tril - theoretical
MA-MCR-ALS non tril - theoretical
86
Three-way non-trilinear data spectra recovery
87
Non-trilinear data quantitative recovery
88
Example of Quantiative determinations Determinatio
n of triphenyltin in sea-water by excitation-emiss
ion 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
• Reaction with a fluorogenic reagent (flavonol in
  a micellar medium)
• Excitation-emission fluorescence measurements
  (giving an EEM data matrix)
• MCR-ALS analysis of EEM data matrices
• 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
MCR-ALS resolution of EEM data
92
MCR-ALS resolution of EEM data Model USRB
Daug YaugXT Eaug Resolution
(emission) Yaug Daug (XT) Constraints
- non-negativity (excitation) XT (Yaug)Daug -
trilinearity Quantitation cU Area(y1,U)
/ Area(y1,S) cS
93
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94
MCR-ALS resolution/quantitation of EEM data
Plot of the emission profiles areas for TPhT
species in standards, synthetic and sea-water
samples respect the analyte concentration
95
Comparison between 'true' and MCR-ALS calculated
TPhT concentrations in sea-water samples
Quantitation cU Area(yU) / Area(yS) cS
overall prediction errors were always below 13!
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
Building the Calibration Curve and Sensitivity
3.5
3
2.5
2
Relative Area
1.5
ri ai / astd f(cstd)
1
0.5
0
0
5
10
15
TPhT concentration (µg / L)
98
(No Transcript)
99
Accuracy of the method in the prediction of TPhT
in real samples
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
matrices identification A4S2RB gt A4
unknown sample A4SA1S2RB gt A4SA1 A4
0.20 µg l-1 TPhT A4SA2S2RB gt A4SA2 A4
0.75 µg l-1 TPhT A4SA3S2RB gt A4SA3 A4
1.05 µg l-1 TPhT A4SA4S2RB gt A4SA4 A4
1.87 µg l-1 TPhT A4SA5S2RB gt A4SA5 A4
3.30 µg l-1 TPhT A4SA6S2RB gt A4SA6 A4
4.52 µg l-1 TPhT A4SA7S2RB gt 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
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