Initial estimates for MCR-ALS method: EFA and SIMPLISMA

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Initial estimates for MCR-ALS method EFA and
SIMPLISMA
7th Iranian Workshop on Chemometrics 3-5 February
2008                                        

• Bahram Hemmateenejad
• Chemistry Department, Shiraz University, Shiraz,
  Iran
• E-mail hemmatb_at_sums.ac.ir

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Chemical modeling

• Fitting data to model (Hard model)
• Fitting model to data (soft model)

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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

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Basic Principles of MCR methods

• PCA DTP
• Beer-Lambert DCS
• In MCR we want to reach from PCA to Beer-Lambert
• D TP TRR-1P, R rotation matrix
• D (TR)(R-1P)
• CTR, SR-1P
• The critical step is calculation of R

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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) 2040-2047, 1993.
• de Juan A, Tauler R, CRITICAL REVIEWS IN
  ANALYTICAL CHEMISTRY 36 (3-4) 163-176 2006

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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

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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

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MCR-ALS Theory

• Initial estimate of C or S
• Evolving Factor Analysis (EFA) C
• Simple-to-use Interactive Self-Modeling Mixture
  Analysis (SIMPLISMA) S

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MCR-ALS Theory

• Consider we have initial estimate of C (Cint)
• Determination of the chemical rank
• Least square solution for S SCint D
• Least square solution for C CDS
• Reproducing of Dc DcCS
• Calculating lack of fit error (LOF)
• Go to step 2

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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)

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Constraints in MCR-ALS

• Peak shape constraint
• Hard model constraint (combined hard model
  MCR-ALS)

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• Rotational Ambiguity
• Rank Deficiency

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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) 315-317 1985
• Maeder M, Zuberbuhler AD, ANALYTICA CHIMICA ACTA
  181 287-291, 1986
• Gampp H, Maeder M, Meyer CJ, Zuberbuhler AD,
  TALANTA 33 (12) 943-951, 1986

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Basic EFA ExampleCalculate Forward Singular
Values
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___ 1st Singular Value
1
0.9
----- 2nd Singular Value
SVD
... 3rd Singular Value
0.8
S
0.7
i
R
i
0.6
0.5
0.4
0.3
0.2
0.1
I
0
0
5
10
15
20
25
I samples
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Basic EFA ExampleCalculate Backward Singular
Values
1
1
___ 1st Singular Value
0.9
----- 2nd Singular Value
0.8
... 3rd Singular Value
0.7
R
0.6
0.5
0.4
i
SVD
0.3
S
0.2
i
0.1
I
0
0
5
10
15
20
25
I
samples
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Basic EFA

• Use forward and backward singular values to
  estimate initial concentration profiles
• Area under both nth forward and (K-n1)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
0
0
5
10
15
20
25
I
samples
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Basic EFA
First estimated spectra Area under 1st forward
and 3rd backward singular value plot.
(Blue) Compare to true component (Black)
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Basic EFA
First estimated spectra Area under 2nd forward
and 2nd backward singular value plot.
(Red) Compare to true component (Black)
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Basic EFA
First estimated spectra Area under 3rd forward
and 1st backward singular value plot.
(Green) Compare to true component (Black)
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Example data

• Spectrophotometric monitoring of the kinetic of a
  consecutive first order reaction of the form of
• A B C

k1
k2
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• Pseudo first-order reaction with respect to A
• A R B C
• R1 k10.20 k20.02
• R2 k10.30 k20.08
• R3 k10.45 k20.32

k1
k2
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K10.2 K20.02
K10.3 K20.08
K10.45 K20.32
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K10.30 K20.08
K10.20 K20.02
K10.45 K20.32
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Noisy data
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EFA Analysis

• The m.file is downloadable from the MCR-ALS home
  page
• http//www.ub.edu/mcr/welcome.html

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Simple-to-use Interactive Self-Modeling Mixture
Analysis (SIMPLISMA)
W. Windigm J. Guilment, Anal. Chem. 1991, 63,
1425-1432. F.C. Sanchez, D.L. Massart, Anal.
Chim. Acta 1994, 298, 331-339.
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• SIMPLISMA is based on the selection of what are
  called pure variables or pure objects.

• 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.

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Absorbance spectra
Chromatographic profile
Pure object
Pure variable
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?1
?2
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35
20
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Mean vector
Standard deviation vector
?
t 0
t m
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. . .
Mean vector
. . .
Standard deviation vector
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chromatogram
Pure spectra
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Pure spectra
Standard deviation
Mean
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chromatogram
Mean
Standard deviation
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SIMPLISMA steps
1) The ratio between the standard deviation, si,
and the mean, µi, of each spectrum is determined
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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
0ltoffsetlt3
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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.
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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 (H1), and
the weight of each spectrum is equal to the
square of the length of the normalized spectrum
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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
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i1 HI i2
Hp1 i3 Hp1
p2 i4 Hp1 p2 p3
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Offset0
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Offset1
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• Example data
• HPLC-DAD data of a binary mixture

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chromatogram
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Pure spectra
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