In the name of GOD

In the name of GOD

Some about model-based Analysis

6th Iranian Chemometrics workshop

Institute for Advances studies in Basic Sciences (IASBS), Zajan, Iran

Feb 2007

6th Iranian Chemometrics workshop

Institute for Advances studies in Basic Sciences (IASBS), Zajan, Iran

Feb 2007

By: Mohsen Kompany-ZarehBy: Mohsen Kompany-Zareh

Three optically active components in a sample, absorbance spectrum in three wavelengths

Three optically active components in a sample, absorbance spectrum in three wavelengths

a1= c1s11 + c2 s12 + c3 s13a1= c1s11 + c2 s12 + c3 s13

a2= c1s21 + c2 s22 + c3 s23a2= c1s21 + c2 s22 + c3 s23

a3= c1s31 + c2 s32 + c3 s33a3= c1s31 + c2 s32 + c3 s33

oror a = S ca = S c

S-1 a = S-1S cS-1 a = S-1S cSolution:Solution:

SquareNon-singular

SquareNon-singular

S-1 a = cS-1 a = c S-1S=IS-1S=I

a = S ca = S c

What if S is not square?What if S is not square?

S-1S=I

(for square nonsing. S)

S-1S=I

(for square nonsing. S)S-1 a = S-1S cS-1 a = S-1S c

?S=I (for not square S)?S=I (for not square S)

STS is squareSTS is square

(STS)-1 STS =I (STS)-1 STS =I(STS)-1 ST (STS)-1 ST

pseudo inversepseudo inverse

S+ S=IS+ S=I

S+ a = cS+ a = c

S-1 a = cS-1 a = c

S+ a = cS+ a = c

a = S ca = S c= S S+ aa

Projection of a in space of S

Projection of a in space of S

aa

r = || - a|| 0aif a is in space of S

if a is in space of S

X = C SX = C S

C = X ZC = X Z

420 440 460 480 500

0

10

200

0.5

1

1.5

2

Wavelength (nm)Time (min)

Abs

orba

nce

XX

0 5 10 15 200

0.5

1

1.5

Time (min)

Conce

ntr

ati

on (

mic

roM

)

CCclassicalclassical

inverseinverseTo fit the parameters that form C

= X X+ C= X X+ CC

= C C+ XX

C

CC

rr XX

r = || - C|| = 0.08

C= f(K)C= f(K)

K=2K=2

= X X+ C= X X+ CC

C

C

CC

rr XX

r = || - C|| = 0.04

C= f(K)C= f(K)

K=3K=3

= X X+ C= X X+ CC

C

C

CC

rr XX

r = || - C|| = 0.01

C= f(K)C= f(K)

K=4K=4

= X X+ C= X X+ CC

C

C

CC

rr XX

r = || - C|| = 0.0001

C= f(K)C= f(K)

K=5K=5

= X X+ C= X X+ CC

C

XX

rr CC

r = || - X|| = 0.15

C= f(K)C= f(K)

K=2K=2

X

X

= C C+ X= C C+ XX

XX

rr CC

r = || - X|| = 0.08

C= f(K)C= f(K)

K=3K=3

= C C+ X= C C+ XX

X

X

XXrr

CC

r = || - X|| = 0.011

C= f(K)C= f(K)

K=4K=4

= C C+ X= C C+ XX

X

X

XXrr

CC

r = || - X|| = 0.001

C= f(K)C= f(K)

K=5K=5

= C C+ X= C C+ XX

X

X

M + L MLML

[M] [L][ML

K f= [M] [L][ML]

K f

CL = [L] + [ ML]

CM = [M] + [ ML]

One-step complex formation equilibrum

CL = [L] + Kf [M] [L]

CM = [M] + Kf [M] [L]

Kf [L]2 + (KfCM-KfCL+1)[L] –CL =0

Estimation of [L] at any Kf, CM and CL Estimation of [L] at any Kf, CM and CL

MM

LL

300 350 400 450 5000

0.2

0.4

0.6

0.8

1

1.2x 10

-3 Response matrix data

Wavelength (nm)

Ab

sorb

ance

Spectroph.Spectroph.

= X X+ [L]= X X+ [L]]L[ r = || - [L]|| 0]L[

log(Kf)=4log(Kf)=4