Transcript Document

ˆ
c =Y Y+c
Y+ = (YTY)-1 YT
Inversion is possible if:
?
YTY

is non-singular and squared
(Full rank)
rank(Y) = 2 =min(#r,#c)
=> Y is full rank
Y’*Y is (33)
but:
rank(Y’*Y)=2 !
1 Y should have:
#rows > #col.s
Column are linearly dependent
2 Y should not be:
Rank deficient
!
3 compon.s, 4 samples
4 wavel.s, 4 samples
rank(x)=min(r(c),r(s))=3
rank(x) < min(# r, #c) =4
=> x is rank deficient
pinv can be performed when
x is rank deficient..
pinv
?X= I (not square and singular X)
svd & estimation of X using
significant factors
?U**V*T=I
V**-1U*TU**V*T=I
pseudo
inverse
U*TU*=I
*-1*=I
V*V*T=I ?
pinv(X)= X+ = V**-1U*T
X = C S classic
Hard
Model
ks
kC
X
ˆ = CC+X
X
1. # samp.s ≥ # compon.s
2. C : full rank
(rank(C)= #compon.s)
(lin indep conc profiles)
ˆ
X
ˆ -X||
|| X
Criterion
for fitting
ks
Projection of X onto space of C
C = X Z inverse
Hard
Model
ks
kC
X
ˆ = XX+C
C
ˆ
C
!
1. # samp.s ≥ # wavel.s
2. X: full rank (rank(X)= # wavel.s)
- variab. Select.
- Factor based methods
ˆ -C||
|| C
Criterion
for fitting
!
ks
Projection of X onto space of C
X is usually near to
singular…
 # samples < # wavel.s
 # wavel.s > # compon.s
XX+
=U**V*TV**-1U*T
=U***-1U*T
=TT+
(signif factors)
C = T RZ inverse
X
Hard
Model
ks
SVD
T
ˆ -C||
|| C
ˆ = TT+C
Criterion
C
for fitting
1. # samp.s ≥ # PCs

2. T: full rank (lin indep col.s)  
ks
kC
ˆ
C
Projection of C onto space of T
T = C R classic
Hard
Model
ks
kC
X
T
SVD
ˆ = CC+T
T
1. # samp.s ≥ # compon.s
2. C : full rank (lin indep.
conc prof.s)
Tˆ
ˆ -T||
|| T
Criterion
for fitting
k
Projection of T onto space of C
ˆ = XX+C
C
ccrX (classical
curve
resolution)
ccrC
ˆ = CC+T
T
pcrT
ˆ = CC+X
X
ˆ = TT+C
C
pcrC (Target
Transform)
T J Thurston, R G Brereton Analyst 127, 2002, 659.
The considered kinetic system:
Second order consecutive
A+B  CD
r(C)=3
1
# indep react.s +1
0.5
2
0
0
5
10
Time (min)
15
20
Spectral meas. In 101 wavel.s
each 30 sec (41 times)
Absorbance
Concentration (microM)
1.5
1.5
1
0.5
0
20
10
Time (min)
420
0
440
460
480
500
Wavelength (nm)
ccrC:
ˆ
C
X(41x101)
r(X)=3
41 samples
101 wavel.s
=X*inv(X‘*X)*X'*C

1
X1=[X(:,50) X(:,70) X(:,90)]
ˆ
C
=X1*inv(X1‘*X1)*X1'*C
ˆ
C
=X*pinv(X)*C
1 # samp.s ≥ # wavel.s
2 rank(X)= # wavel.s


Information
content !
ccrX:
C(41x4)
r(X)=3
41 samples
4 compon.s
ˆ =C*inv(C’*C)*C’*X
X

C1=C(:,2:4)
ˆ =C1*inv(C1’*C1)*C1’*X
X
ˆ =C*pinv(C)*X
X
1. # samp.s ≥ # compon.s
2. rank(C)= #compon.s


pcrT:
ˆ =C*inv(C’*C)*C’*T
T

C1=C(:,2:4)
ˆ =C1*inv(C1’*C1)*C1’*T
T
ˆ =C*pinv(C)*T
T


pcrC:
ˆ =T*T’*C
C

1. # samp.s ≥ # PCs
2. rank(T)= # col.s (always it is so…)
Overlap effect
1.5
Concentration (microM)
0.8
0.6
0.4
0.2
0
420
440
460
480
wavelength (nm)
500
1
0.5
0
0
5
10
Time (min)
15
20
2
Absorbance
Absorbance
1
1.5
1
0.5
+Rand
noise
0
20
10
Time (min)
420
0
440
460
480
500
Wavelength (nm)
0.11
ccrX
noise 0.02
0.12
0.11
0.1
k2
k2
0.105
ccrC
noise 0.02
0.1
0.09
0.095
0.08
0.09
0.36
0.38
0.11
0.4
k1
0.42
0.44
0.3
pcrT
noise 0.02
0.4
k1
0.45
0.5
pcrC
noise 0.02
0.12
0.105
0.11
0.1
k2
k2
0.35
0.1
0.09
0.095
0.08
0.09
0.36
0.38
0.4
k1
0.42
0.44
0.3
0.35
0.4
k1
0.45
0.5
Spectral overlap (in the
presence of some noise) results
in some deviation in the results
from
***C
methods
Results from application
of ***C and ***X methods
are different …
One way to obtain more similar
results from ***C and ***X
methods are application of
constraints
Presence of
heteroscedastic noise
41 reaction times
&101 wavelengths
1.5
1
0.5
0
0
40
50
Variable No.
20
100
0
reaction time
0.015
0.01
+ a heterosced.
noise
Absorbance
Absorbance
2
0.005
0
-0.005
-0.01
-0.015
0
20
40
60
Variable No.
80
100
0.106
0.104
k2
0.102
0.1
0.098
0.096
0.094
0.36
0.38
0.4
0.42
0.44
k1
Inaccurate results from ccrX !
weighted regression…
ˆ -X)||
||W ( X
Absorbance
0.05
weights
0
-0.05
0
20
40
60
Variable No.
80
100
W=
1/SD1
1/SD2
…
1/SDn
0.1005
k2
n=50
0.1
0.0995
0.3995
0.4
k1
0.4005
Accurate results from
weighted ccrX !
FSMWFA in daset 6
FSMWFA
log (eigvalue)
2.5
0.5
-1.5
-3.5
-5.5
-7.5
300 320 340 360 380 400 420 440 460 480 500
wavelength
Recognition of the presence
of heterosc. noise
Sampling error coefficient
1.01
A more
serious source
of error
1.005
1
0.995
0.99
0
1.015
10
20
30
Sample number
40
1.01
1.005
1
Non-random
sampling error
0.995
0.99
0
5
0
10
10
20
30
40
50
Square, symmetric,
But not diagonal W matrix:
J Chemometr 2002, 16, 378.
R. Bro, N.D. Sidiropoulos, A.K. Smilde
Maximum likelihood fitting
0.105
0.105
0.1
0.1
0.095
0.095
0.09
0.09
k2
k2
Presence of non-random
sampling error nS=0.005
0.085
0.08
0.3
0.08
ccrX
0.075
0.4
k1
0.5
0.085
0.075
0.6
ˆ -X||
|| X
J Chemom 2002, 16,387. R.Bro et al
0.3
0.4
k1
0.5
0.6
ˆ -X)||
||W ( X
Weighted
regression
Presence of unknown
interference
Changing interference, drift , or shift
1.5
1.4
concentration (microM)
absorbance of pure components
1
0.8
0.6
0.4
0.2
0
400
concentration profiles
1
0.5
2.5
420
440
460
480
0
0
500
wavelength 2
(nm)
Absorbance
absorbance
1.2
Data
5
10
15
reaction time(min)
1.5
1
0.5
0
400
rank(Data)=4
420
440
460
Wavelength (nm)
480
500
20
0.103
0.103
ccrX
k2
k2
0.102
0.102
0.101
0.1
0.101
0.099
0.1
0.099
0.385
0.098
0.39
0.395
0.4
0.405
0.097
0.39
ccrC
0.395
k1
0.4
0.405
k1
0.13
0.125
0.102
pcrC
0.101
0.115
k2
k2
0.12
pcrT
0.11
0.099
0.105
0.098
0.1
0.095
0.38
0.1
0.385
0.39
0.395
k1
0.4
0.405
0.097
0.39
0.395
0.4
k1
0.405
Presence of shift or drift (a
changing interference) results in
serious deviations in
***X
Methods
(but not in ***C methods)
Why?
In the presence of shift, drift or changing interferences:
T or X space includes 1. the concentration changes
according to the model 2. variations from shift, drift or
changing interference
C space includes only the concentration changes
according to the model
ˆ = CC+T
T
ˆ = CC+X
X

Projection of a
larger space to
a smaller one
ˆ = TT+C
C
ˆ = XX+C
C

Projection of a
smaller space
to a larger one
ˆ = TT+C
C
in the presence of unknown
interference, drift or shift.
Target Transform (pcrC) is the most
preferred method
Constant interference
1.5
1.4
concentration (microM)
absorbance of pure components
1
0.8
0.6
2.5
0.4
0.2
0
400
2
420
Absorbance
absorbance
1.2
440
460
480
concentration profiles
A+B  CD
1
0.5
Data
0
0
500
5
10
15
reaction time(min)
wavelength (nm)
1.5
1
0.5
0
400
rank(Data)=3 !
420
440
460
Wavelength (nm)
480
500
20
0.101
0.101
ccrX
0.1005
k2
k2
0.1005
0.1
0.1
0.0995
0.0995
0.099
0.399
ccrC
0.3995
0.4
0.4005
0.099
0.399
0.401
0.3995
0.101
pcrT
0.1005
k2
k2
0.401
0.4005
0.401
0.101
0.1
0.0995
0.099
0.399
0.4005
k1
k1
0.1005
0.4
pcrC
0.1
0.0995
0.3995
0.4
k1
0.4005
0.401
0.099
0.399
0.3995
0.4
k1
A constant interference
does not show any
significant effect the
accuracy of ***X and ***C
methods.
Target test fitting
From:
J Chemometr. 2001, 15, 511.
P.Jandanklang, M. Maeder, A. C. whitson
Differential
pulse
Voltammetry
35
voltammograms
Unitary current
30
25
20
Each voltammog. depends
only on its own E1/2
15
10
5
0
-0.1
-0.05
0
E
0.05
0.1
Successive complexation:
M L
 ML
M  2L 
 ML2
M  3L 
 ML3
[ ML]
 ML 
[ M ][L]
[ ML2 ]
 ML2 
[ M ][L]2
[ ML3 ]
 ML3 
[ M ][L]3
1 
[ MLn ]
[ ML]
,....., n 
[ M ][L]
[ M ][L]n
CTM  [ M ]  [ ML]  .... [ MLn ]
CTL  [ L]  [ ML]  .... n[ MLn ]
[L]n1 n  [L]nn(CTM  CTL)  n1   [L]n1n1((n  1)CTM  CTL)  n2  ....  CTL  0
[M ] 
1
(1  1[ L]   2 [ L]2  ....  n [ L]n )
[ MLn ] 
 n [ L]n
(1  1[ L]   2 [ L]2  ....  n [ L]n )
Analyst , 2001 , 126 , 371-377
Each concn. profile
includes 1,…, n
35
1
voltammograms
Unitary current
0.8
0.6
0.4
0.2
0
0
25
20
15
10
5
20
40
60
0
-0.1
80
-0.05
0
E
Ctotal L
35
X
Data
30
25
current
Current microA
30
20
15
10
5
0
-0.1
-0.05
0
E
0.05
0.1
0.05
0.1
X=CS
X=UVT=TV
ˆ
s = VVT s
voltammogr
ˆ
c = UUT c = TTT c
concn.
For estimation of concn.
profiles 1,…,n
(n parameters) should
be optimized
simultaneously
1,…,n are
dependent
parameters
Simultaneous optimization of
n dependent nonlinear
parameters:
• Simplex method.
• Levenberg-Marquardt
•…
estimation of
(E1/2)1, …, (E1/2 )n
values for
voltammograms
(E1/2)1, …, (E1/2 )n
are independent
parameters
35
voltammograms
25
20
15
10
5
0
-0.1
2
-0.05
0
0.05
0.1
E
1
log10(norm(r))
Unitary current
30
0
-1
-2
-3
-0.1
-0.05
0
E
0.05
0.1
35
voltammograms
25
20
15
10
5
0
-0.1
2
-0.05
0
0.05
0.1
E
1
log10(norm(r))
Unitary current
30
0
-1
-2
-3
-0.1
-0.05
0
E
0.05
0.1
35
voltammograms
25
20
15
10
5
0
-0.1
2
-0.05
0
0.05
0.1
E
1
log10(norm(r))
Unitary current
30
0
-1
-2
-3
-0.1
-0.05
0
E
0.05
0.1
35
voltammograms
25
20
15
10
5
0
-0.1
2
-0.05
0
0.05
0.1
E
1
log10(norm(r))
Unitary current
30
0
-1
-2
-3
-0.1
-0.05
0
E
0.05
0.1
35
voltammograms
25
20
15
10
5
0
-0.1
2
-0.05
0
0.05
0.1
E
1
log10(norm(r))
Unitary current
30
0
-1
-2
(E1/2)M
-3
-0.1
-0.05
0
r = || ˆ
s - s||  0
E
0.05
0.1
35
voltammograms
25
20
15
10
5
0
-0.1
2
-0.05
0
0.05
0.1
E
1
log10(norm(r))
Unitary current
30
0
-1
-2
-3
-0.1
-0.05
0
E
0.05
0.1
35
voltammograms
25
20
15
10
5
0
-0.1
2
-0.05
0
0.05
0.1
E
1
log10(norm(r))
Unitary current
30
0
-1
-2
-3
-0.1
-0.05
0
E
0.05
0.1
35
voltammograms
25
20
15
10
5
0
-0.1
2
-0.05
0
0.05
0.1
E
1
log10(norm(r))
Unitary current
30
0
-1
-2
-3
-0.1
-0.05
0
E
0.05
0.1
35
voltammograms
25
20
15
10
5
0
-0.1
2
-0.05
0
0.05
0.1
E
1
log10(norm(r))
Unitary current
30
0
-1
-2
-3
-0.1
-0.05
0
E
0.05
0.1
35
voltammograms
25
20
15
10
5
0
-0.1
2
-0.05
0
0.05
0.1
E
1
log10(norm(r))
Unitary current
30
0
-1
-2
(E1/2)ML
(E1/2)M
-3
-0.1
-0.05
0
r = || ˆ
s - s||  0
E
0.05
0.1
35
voltammograms
25
20
15
10
5
0
-0.1
2
-0.05
0
0.05
0.1
E
1
log10(norm(r))
Unitary current
30
0
-1
-2
-3
-0.1
-0.05
0
E
0.05
0.1
35
voltammograms
25
20
15
10
5
0
-0.1
2
-0.05
0
0.05
0.1
E
1
log10(norm(r))
Unitary current
30
0
-1
-2
-3
-0.1
-0.05
0
E
0.05
0.1
35
voltammograms
25
20
15
10
5
0
-0.1
2
-0.05
0
0.05
0.1
E
1
log10(norm(r))
Unitary current
30
0
-1
-2
-3
-0.1
-0.05
0
E
0.05
0.1
35
voltammograms
25
20
15
10
5
0
-0.1
2
-0.05
0
0.05
0.1
E
1
log10(norm(r))
Unitary current
30
0
-1
-2
-3
-0.1
-0.05
0
E
0.05
0.1
35
voltammograms
25
20
15
10
5
0
-0.1
2
-0.05
0
0.05
0.1
E
1
log10(norm(r))
Unitary current
30
0
-1
-2
-3
-0.1
-0.05
0
E
0.05
0.1
35
voltammograms
25
20
15
10
5
0
-0.1
2
-0.05
0
0.05
0.1
E
1
log10(norm(r))
Unitary current
30
0
-1
-2
(E1/2)ML
(E1/2)ML2
(E1/2)M
-3
-0.1
-0.05
0
r = || ˆ
s - s||  0
E
0.05
0.1
35
voltammograms
25
20
15
10
5
0
-0.1
2
-0.05
0
0.05
0.1
E
1
log10(norm(r))
Unitary current
30
0
-1
-2
-3
-0.1
-0.05
0
E
0.05
0.1
35
voltammograms
25
20
15
10
5
0
-0.1
2
-0.05
0
0.05
0.1
E
1
log10(norm(r))
Unitary current
30
0
-1
-2
-3
-0.1
-0.05
0
E
0.05
0.1
35
voltammograms
25
20
15
10
5
0
-0.1
2
-0.05
0
0.05
0.1
E
1
log10(norm(r))
Unitary current
30
0
-1
-2
(E1/2)ML3
(E1/2)ML
(E1/2)ML2
(E1/2)M
-3
-0.1
-0.05
0
r = || ˆ
s - s||  0
E
0.05
0.1
35
voltammograms
25
20
15
10
5
0
-0.1
2
-0.05
0
0.05
0.1
E
1
log10(norm(r))
Unitary current
30
0
-1
-2
-3
-0.1
-0.05
0
E
0.05
0.1
35
voltammograms
25
20
15
10
5
0
-0.1
2
-0.05
0
0.05
0.1
E
1
log10(norm(r))
Unitary current
30
0
-1
-2
-3
-0.1
-0.05
0
E
0.05
0.1
35
voltammograms
25
20
15
10
5
0
-0.1
2
-0.05
0
0.05
0.1
E
1
log10(norm(r))
Unitary current
30
0
-1
-2
(E1/2)ML3
(E1/2)ML
(E1/2)ML2
(E1/2)L
(E1/2)M
-3
-0.1
-0.05
0
r = || ˆ
s - s||  0
E
0.05
0.1
35
voltammograms
25
20
15
10
5
0
-0.1
2
-0.05
0
0.05
0.1
E
1
log10(norm(r))
Unitary current
30
0
-1
-2
-3
-0.1
-0.05
0
E
0.05
0.1
35
voltammograms
25
20
15
10
5
0
-0.1
2
-0.05
0
0.05
0.1
E
1
log10(norm(r))
Unitary current
30
0
-1
-2
-3
-0.1
-0.05
0
E
0.05
0.1
35
voltammograms
25
20
15
10
5
0
-0.1
2
-0.05
0
0.05
0.1
E
1
log10(norm(r))
Unitary current
30
0
-1
-2
(E1/2)ML3
(E1/2)I
(E1/2)ML
(E1/2)ML2
(E1/2)L
(E1/2)M
-3
-0.1
-0.05
0
r = || ˆ
s - s||  0
E
0.05
0.1
35
voltammograms
25
20
15
10
5
0
-0.1
2
-0.05
0
0.05
0.1
E
1
log10(norm(r))
Unitary current
30
0
-1
-2
-3
-0.1
-0.05
0
E
0.05
0.1
Optimum values for
n independent parameters
can be estimated
by
grid search of one parameter.
A difficult aspect of hard modeling is
determination of correct model
Thanks.
Thanks to:
Miss Maryam Khoshkam
and
Mr Yaser Beyad
for a number of m-files and slides.