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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: [email protected]

Goal: Given an I x J matrix, D, of N species, determine N and the pure spectra of each specie.

Model: DIxJ = CIxNSNxJ

Common assumptions:

Non-negative spectra and concentrations

Unimodal concentrations

Kinetic profiles

1

0.5

=

S

N

x

J

0

30

60

20

C

40

10

20

I

x

N

0

0

samples

sensors

D

I

x

J

Multicomponent Curve ResolutionBasic 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

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

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

MCR-ALS Theory

- In the case of spectroscopic data
- Beer-Lambert Law for a mixture
- D(mn) absorbance data of k absorbing species

D = CS

- C(mk) concentration profile
- S(kn) pure spectra

MCR-ALS Theory

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

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

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)

Constraints in MCR-ALS

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

Rotational Ambiguity

- Rank Deficiency

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

Basic EFA ExampleCalculate Forward Singular Values

1

___ 1st Singular Value

1

0.9

----- 2nd Singular Value

SVD

...… 3rd Singular Value

0.8

S

0.7

i

R

0.6

i

0.5

0.4

0.3

0.2

0.1

I

0

0

5

10

15

20

25

I samples

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

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

0

0

5

10

15

20

25

I

samples

Example data

- Spectrophotometric monitoring of the kinetic of a consecutive first order reaction of the form of

A B C

k1

k2

Pseudo first-order reaction with respect to A

- A + R B C
- [R]1 k1=0.20 k2=0.02
- [R]2 k1=0.30 k2=0.08
- [R]3 k1=0.45 k2=0.32

k1

k2

EFA Analysis

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

http://www.ub.edu/mcr/welcome.html

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.

Object (i.e. time or sample)

Data matrix

- 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.

2

20

Pure spectra

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

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

2) Normalisation of the data matrix: Each spectrum xiis normalised by dividing each element of a row xijby the length of the row ||xi||:

When an offset is added, the same offset is also included in the normalisation of the spectra.

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 ziof 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

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]]

*

*

*

Example data

HPLC-DAD data of a binary mixture

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