Generalized Rank Annihilation Factor Analysis

Anal Chem 58(1986)496.E Sanchez B R Kowalski

Bilinear data

2

1

3

2 .1 .2 .3 .2

e(Excit.)

f(Emiss.)

X1 (Fluoresc.)

=

One component Rank =1

Conc.

0.4 0.8 1.2 0.8

0.2 0.4 0.6 0.4

0.6 1.2 1.8 1.2

2 1

1 2

3 1

1 0

0 3

.1 .2 .3 .2

.2 .4 .4 .3

21.1+13.2

21.1+13.2

11.1+23.2

31.2+13.3

Two components

E FConc.

=

X2 (Fluoresc.)

0.2 +0.6 0.4 +1.2 0.6 +1.2 0.4 + .9

0.1 +1.2 0.2 +2.4 0.3 +2.4 0.2 +1.8

0.3 +0.6 0.6 +1.2 0.9 +1.2 0.6 +0.9

Two components

X2 =

Rank = 2

0.4 0.8 1.2 0.8

0.2 0.4 0.6 0.4

0.6 1.2 1.8 1.2

one component (calibration matrix)

Rank = 1

X1 =

Lorber, 1984 X2- 0.5 X1 = ERank=2 Rank=1

Quantification of one component.

0.2 +0.6 0.4 +1.2 0.6 +1.2 0.4 +0.9

0.1 +1.2 0.2 +2.4 0.3 +2.4 0.2 +1.8

0.3 +0.6 0.6 +1.2 0.9 +1.2 0.6 +0.9

Two components(sample)

X2 =

Rank = 2

0.4+0.2 0.8+0.4 1.2+0.4 0.8+0.3

0.2+0.4 0.4+0.8 0.6+0.8 0.4+0.6

0.6+0.2 1.2+0.4 1.8+0.4 1.2+0.3

Two components(calibration)

Rank = 2

X3 =

What about quantific. of more than one component?

Generalized RAFA [Anal Chem 1986, 58, 496-499. B.R. Kowalski]

1. Non-iterative [Lorber, 1984].

2. Simultaneous detn. of analytes using

Just one bilinear calibration spectrum

from one mixture of standards.

a. Bilinear spectrum of each analyte

b. Relative conc.s

Theory

E FT = X2

E FT = X1

sample :

Calibration :

E = X2(FT)+ -1

E = X1(FT)+ -1

X1(FT)+ -1 = X2(FT)+ -1

X1 Z = U S VT Z -1

Z = V S-1 Z* (definition)

Common F and E

Trilinearity

X1 V S-1 Z* = U S VT V S-1 Z* -1

I

I

UT X1 V S-1 Z* = Z* -1

R V = V (eigenvector analysis)

FT = (V S-1 Z*)+

E = U Z* -1

-1 =

?

Simult. detn. of two acids in a sample

H2A HA A

H2B HB B

using pH-metric titration

H2A HA A

H2B HB Asample

C0A ?

C0B ?

H2A HA A

H2B HB Acalibr.

C0A =0.02 M

C0B =0.04 M

Data matrices

sample

calibration

Only HA-

and HB- are optically active.

sample

calibration

[Zstar,λ]=eig(Usm‘ * Xcl‘ * Vsm* inv(Ssm))

[Usm,Ssm,Vsm] = svd(Xsm')

0.6669

1.9998

λ=

(CoA)cl

(CoA)sm

(CoB)cl

(CoB)sm

, (CoA)cl=0.02 M

=> (CoA)sm=0.03 M

, (CoB)cl=0.04 M

=> (CoB)sm=0.02 M

0.03

0.02

β =

15

F = pinv( Vsm * inv( Ssm ) * Zstar)

Conc. profiles

E = Usm * Zstar * inv(β)

spectral profiles

What if:

The calibration sample includes some components that are not present in unknown sample,

And there be some components in unknown sample not present in the calibration sample.

HPLC-DAD chromatogram for A,B, and C (as CL),

for ?,?,and ? (as SM)

Example:The General C

ondition

Xcl

CAcl= 1 mM

CBcl= 3 mM

CCcl= 2 mM

Xsm

?, ?, and ?, ..

[Zstar,λ]=eig(Utot‘ * Xsm‘ * Vtot* inv(Stot))

[Utot,Stot,Vtot] = svd(Xtot')

Xtot = Xcl + Xsm The total space, rank =4

(includes A, B, C ,and D)

0.9999 0 0 0

0 0.0003 0 0

0 0 0.5000 0

0 0 0 0.3334

λ= β / ( β + ξ )

C?sm

C?sm+C?cl=0.9999 0.0003 0.5000 0.3334

C?cl=0

Only in sm

C?sm=0

Only in cl

C?sm= C?cl

2C?sm= C?clC

BAD

CBsm= 3 mM

CCsm= 1 mM

F = pinv( Vtot * inv( Stot ) * Zstar)

Conc. profiles

E = Utot * Zstar

spectral profiles

Non-bilinear RA

Analyte detn.

..in the presence of unaccounted spectral interference..

Rank for the pure component >1

H2A HA A

One compon, but Rank=…3

Xcl

H2A and H2B

Rank(Xsm)=5H2A and H2B

Interference

Conc. Prof.s

Spect. Prof.s

0.9415 0 0 0 0

0 0.0003 0 0 0

0 0 -0.003 0 0

0 0 0 2.0044 0

0 0 0 0 2.0010

λ of H2B

DirectExponentialCurveResolutionAlgorithm

J. Chemom. 14 (2000) 213-227.

DECRA

Model base: an exponential decay

162

54

18

6

2

162

54

18

6

2

x2 x1

162/54=

54/18=

18/6=

6/2=

3

3

3

3

shift

x

C1 = e –k t

C2 = e –k (t+S)

C1

C2 ===e –kt +k(t+S)= e –k S

e –k t

e –k (t+S)

k = ln() / S

x2 :

x1 :

Shift

Shift=7

x2

x1

1st Ord Data

From 1 sample

k = ln() / 7

=0.1

cP

sPT

cQ

sQT

cR

sRT

= + +

XExpon.Decay

2st Ord Data

From 1 sample

Trilinear structureN

X1

X2

= +

E

Gives k1 and k2X

2-way

(MN)

X

3-way

((M-S) N 2)

Stacking

E

F

λ

1

M-S

1+S

M

Decomposition of a number of colorants to colorless products..

A A’

B B’

C C’

…

1st order reactions

svd(X)=

6279.5

294.0

34.4

0.7

0.6

0.6

…

Three components

Shift = 10 min

Estimated F

estmated E

k = ln(λ) / shift

3.3201 0 0

0 2.2255 0

0 0 1.4918

λ =

0.12 0 0

0 0.08 0

0 0 0.04

A consecutive reaction:

No Expon. Decaying concn.

A B Dk1 k2

Reaction model

First order, consecutive

CA,i = CA,0 e –k1 ti

CB,i = (e –k1 ti - e –k2 ti )k1 CB,0

k2- k1

CD,i = CA,0 - CA,i - CB,i

A B Dk1 k2

Columns of C matrix

cA

cB

cD

X* = cA sAT + cB sB

T + cD sD

T + cL sL

T

= (e-k1t) sAT

+ k(e-k1t) sBT - k(e-k2t) sB

T

- (e-k1t)sD

T + e0tsD

T - k(e-k1t)sDT + k(e-k2t)sD

T

+ e0t sL

T

= e-k1t ( sA + k sB

- sD – k sD) T

+ e-k2t (- k sB + ksD)T

+ e0t (sD + sL )

T

Sum of exponentially decaying functions

Unique decomp.

But not result into actual spectra and concn. profiles

= e-k1t ( sP + k sQ

- sR – k sR) T

+ e-k2t (- k sQ + ksR)T

+ e0t (sR + sL )

T

e1 e2 e3

Trilinear structure

1

M-s

1+s

M

NX1

X2

= +

X

2-way

X

3-way

E

Gives k1 and k2

(MN)

((M-S) N 2)

Stacking

E

F

λ

Expon.Decaying

X*

fA

eAT

fB

eBT

fD

eDT

= +

+ +

1..1

N+1

eLT =[0 0 .. 0 1]

fL =[e0 e0 .. e0]

What if :

Not applying the

ones column?

An Examplefor consecutive reaction

0.99997 0 0

0 5.77 0

0 0 19.848

λ=

k = ln() / shift

0.0000 0 0

0 0.0701 0

0 0 0.1195

Not proper pure spectra !

Not proper pure conc. Prof.s !

What about estimation of spectral and concentration profiles?

An NMR example

PGSE NMR

Pulsed Gradient Spin Echo NMR

A Mixture,

with exponential decay of the contribution of each component

A series of spectra

A function of diffusion coefficient of component

Low-MW

Poly(dimethylsiloxane) PDMS

MRI14 images

(echo times (TEs) from 15 to 210ms)

Exponential decay of signal from each component

=f(sp.-sp. relax. Time of compon.)

Thanks