an introduction to model-free chemical analysis hamid abdollahi iasbs, zanjan e-mail:...
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An Introduction to Model-Free Chemical Analysis
Hamid AbdollahiHamid AbdollahiIASBS, Zanjan
e-mail: [email protected]
Lecture 2Lecture 2
Position of a known profile in corresponding space:
dx
Tv1
Tv2
v1
v2
Tv1 is the length of projection of dx on v1 vectorTv1 = dx . v1
Tv2 is the length of projection of dx on v1 vectorTv2 = dx . v2 Tv1
Tv2
Coordinates of dx point:
-8 -7 -6 -5 -4 -3 -2 -1 0-4
-3
-2
-1
0
1
2
3
4
Tv1
Tv2
Row Space
Position of real spectral profiles in V space
Real spectrum 1
Real spectrum 2
-8 -7 -6 -5 -4 -3 -2 -1 0-4
-3
-2
-1
0
1
2
3
4
Tv1
Tv2
Row Space
Position of real spectral profiles in V space
Real spectrum 1
Real spectrum 2
-8 -7 -6 -5 -4 -3 -2 -1 0-4
-3
-2
-1
0
1
2
3
4
Tv1
Tv2
Row Space
Position of real spectral profiles in V space
Real spectrum 1
Real spectrum 2
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.
Free Discussion
Heuristic Evolving Latent Projection (HELP)
Vspace.m fileVisualizing 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 A =T V
t11 t12
t21 t22
T=ti11 ti12
ti21 ti22
T-1=
Two component systems:
a1 = t11 v1 + t12 v2
a2 = t21 v1 + t22 v2
A =
C = [c1 = ti11 s11u1 + ti21 s22u2 c2 = ti12 s11u1 + ti22 s22u2]
Solution of a soft-modeling method
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 fileVisualizing the points in V and U
spaces
Real spectrum 1
Real spectrum 2
-9 -8 -7 -6 -5 -4 -3 -2 -1 0-2
-1.5
-1
-0.5
0
0.5
1
1.5
2V Space
ui1s11
ui2s
22
Intensity ambiguity in V space
Intensity ambiguity in U space
-12 -10 -8 -6 -4 -2 0-2.5
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5U Space
vj1s11
vj2s
22
Rotational ambiguity in V space
-12 -10 -8 -6 -4 -2 0-2
-1.5
-1
-0.5
0
0.5
1
1.5
2V Space
ui1s11
ui2s
22
Rotational ambiguity in U space
-5 -4.5 -4 -3.5 -3 -2.5 -2 -1.5 -1 -0.5 0-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6U Space
vj1s11
vj2s
22
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
2 4 6 8 10 12 14 16 18 200.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9Concentration Profiles
Time
Con
cent
ratio
n
400 420 440 460 480 500 520 540 560 580 6000
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8Spectral Profiles
Wavelength (nm)
Inte
nsity
400 420 440 460 480 500 520 540 560 580 6000
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8Simulated Spectra
Wavelength (nm)
Abs
orba
nce
A first order kinetic as a closed system
-6 -5 -4 -3 -2 -1 0-5
-4
-3
-2
-1
0
1
2
3
4V Space
ui1s11
ui2s
22
-6 -5 -4 -3 -2 -1 0-5
-4
-3
-2
-1
0
1
2
3
4V Space
ui1s11
ui2s
22
0 50 100 150 200 2500
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8Spectral Profiles
Wavelength
Abs
orba
nce
-3 -2.5 -2 -1.5 -1 -0.5 0-1.5
-1
-0.5
0
0.5
1U Space
vj1s11
vj2s
22
0 2 4 6 8 10 12 14 16 18 200.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1Concentration Profiles
Time
Con
cent
ratio
n
?Use the V_U_space.m file and investigate the possible solutions for a first order kinetic data
Intensity ambiguity in V space
v1
v2
a
k1a
k2a
T11
T12
k1T11
k1T12
k2T11
k2T12
Normalization
a = T1 v1 + T2v2
k1a = k1T1 v1 + k1T2v2
k2a = k2T1 v1 + k2T2v2
kna = knT1 v1 + knT2v2
… a’ = v1 + T v2
v1
v2
Normalization
1
1
2
4
3
5
a = T1 v1 + T2v2
a’ = v1 + T v2
1’ 2’ 3’
4’
5’
n_V_U_space.m fileVisualizing the normalized points
in V and U spaces