Transcript Document 7581432

An Introduction to Model-Free
Chemical Analysis
Lecture 2
Hamid Abdollahi
IASBS, Zanjan
e-mail: [email protected]
Position of a known profile in corresponding space:
v2
Tv2
dx
v1
Tv1
Tv1 is the length of projection of dx on v1 vector
Tv1 = dx . v1
Tv2 is the length of projection of dx on v1 vector
Tv2 = dx . v2
Coordinates of dx point:
Tv1
Tv2
Position of real spectral profiles in V space
Row Space
4
Real spectrum 1
3
2
Tv2
1
0
-1
-2
-3
-4
-8
Real spectrum 2
-7
-6
-5
-4
Tv1
-3
-2
-1
0
Position of real spectral profiles in V space
Row Space
4
Real spectrum 1
3
2
Tv2
1
0
-1
-2
-3
-4
-8
Real spectrum 2
-7
-6
-5
-4
Tv1
-3
-2
-1
0
Position of real spectral profiles in V space
Row Space
4
Real spectrum 1
3
2
Tv2
1
0
-1
-2
-3
-4
-8
Real spectrum 2
-7
-6
-5
-4
Tv1
-3
-2
-1
0
Heuristic Evolving Latent Projection (HELP)
The main contributions of HELP have been offering a
sophisticated graphical tool to visually detect potential
selective zones in the score plot of the data matrix and
a statistical method to confirm the presence of the
selectivity in the concentration and/or spectral
windows graphically chosen.
Heuristic Evolving
FreeLatent Projection
Discussion
(HELP)
Vspace.m file
Visualizing the points in V space
Use the Vspace.m file and find the points
which define the similar spectral shapes.
?
Solution of a soft-modeling method
D = USV = CA
C ≠ US
A≠V
D = US (T-1 T) V = CA
C = US T-1
Two component systems:
t11 t12
-1=
T= t
T
21 t22
A =T V
ti11 ti12
ti21 ti22
Solution of a soft-modeling method
a1 = t11 v1 + t12 v2
A=
a2 = t21 v1 + t22 v2
C = [c1 = ti11 s11u1 + ti21 s22u2
c2 = ti12 s11u1 + ti22 s22u2]
The elements of T matrix are the coordinates of
real spectral profiles in V space
The elements of ST-1 matrix are the coordinates
of real concentration profiles in U space
V_U_space.m file
Visualizing the points in V and U
spaces
Real spectrum 1
Real spectrum 2
Intensity ambiguity in V space
V Space
2
1.5
1
ui2s22
0.5
0
-0.5
-1
-1.5
-2
-9
-8
-7
-6
-5
-4
ui1s11
-3
-2
-1
0
Intensity ambiguity in U space
U Space
2.5
2
1.5
1
vj2s22
0.5
0
-0.5
-1
-1.5
-2
-2.5
-12
-10
-8
-6
vj1s11
-4
-2
0
Rotational ambiguity in V space
V Space
2
1.5
1
ui2s22
0.5
0
-0.5
-1
-1.5
-2
-12
-10
-8
-6
ui1s11
-4
-2
0
Rotational ambiguity in U space
U Space
0.6
0.4
0.2
vj2s22
0
-0.2
-0.4
-0.6
-0.8
-5
-4.5
-4
-3.5
-3
-2.5
vj1s11
-2
-1.5
-1
-0.5
0
Rotational ambiguity
The one major problem with all model-free
methods is the fact that often there is no unique
solution for the task of decomposing the data
matrix into the product of two physically
meaningful matrices.
Where there is rotational ambiguity, the solution of
soft-modeling methods is one particular point
within the range of possibilities.
Use the V_U_space.m file and investigate the
effect of overlapping in concentration and
spectral profiles on the possible solutions
?
A first order kinetic as a closed system
Spectral Profiles
0.8
0.7
0.7
0.6
0.6
0.5
Intensity
0.8
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
2
4
6
8
10
12
14
16
18
0
400
20
420
440
460
Time
480
500
520
Wavelength (nm)
Simulated Spectra
0.8
0.7
0.6
0.5
Absorbance
Concentration
Concentration Profiles
0.9
0.4
0.3
0.2
0.1
0
400
420
440
460
480
500
520
Wavelength (nm)
540
560
580
600
540
560
580
600
V Space
4
3
2
ui2s22
1
0
-1
-2
-3
-4
-5
-6
-5
-4
-3
ui1s11
-2
-1
0
V Space
4
3
2
ui2s22
1
0
-1
-2
-3
-4
-5
-6
-5
-4
-3
ui1s11
-2
-1
0
Spectral Profiles
0.8
0.7
0.6
Absorbance
0.5
0.4
0.3
0.2
0.1
0
0
50
100
Wavelength
150
200
250
U Space
1
0.5
vj2s22
0
-0.5
-1
-1.5
-3
-2.5
-2
-1.5
vj1s11
-1
-0.5
0
Concentration Profiles
1
0.9
0.8
Concentration
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
2
4
6
8
10
Time
12
14
16
18
20
Use the V_U_space.m file and investigate the
possible solutions for a first order kinetic data
?
Intensity ambiguity in V space
v2
k2T12
k1T12
T12
k2a
k1a
a
T11
k1T11 k2T11
v1
Normalization
a=
T1 v1 + T2v2
…
k1a = k1T1 v1 + k1T2v2
k2a = k2T1 v1 + k2T2v2
kna = knT1 v1 + knT2v2
a’ = v1 + T v2
Normalization
a = T1 v1 + T2v2
v2
4’
a’ = v1 + T v2
3
4
1’ 2’ 3’
5
2
1
5’
1
v1
n_V_U_space.m file
Visualizing the normalized points
in V and U spaces