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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
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