visualizing the microscopic structure of bilinear data: two components chemical systems
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Visualizing the Microscopic Structure of Bilinear Data: Two components chemical systems. Y. D. R. X. Factorization:. - PowerPoint PPT PresentationTRANSCRIPT
Visualizing the Microscopic Structure of Bilinear Data: Two components
chemical systems
A matrix can be decomposed into the product of two significantly smaller matrices.
Factorization:
In many chemical studies, the measured or calculated properties of the system can be considered to be the linear sum of the term representing the fundamental effects in that system times appropriate weighing factors.
D = X Y + R
D X Y= + R
0 5 10 15 20 25 30 35 40 45 500
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5Concentration Profiles
Retention Time
Conc
entra
tion
400 410 420 430 440 450 460 470 480 490 5000
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1Spectral Profiles
Wavelength (nm)
Abso
rban
ce
400 410 420 430 440 450 460 470 480 490 5000
0.1
0.2
0.3
0.4
0.5
Wavelength
Abso
rban
ce
A simple one component system
435 440 445 450 455 4600
0.1
0.2
0.3
0.4
0.5
wavelength
Abso
rban
ce
00.05
0.10.15
0.20.25
0.30.35
0.40.45
00.05
0.10.15
0.20.25
0.30.35
0.40.45
0.50
0.1
0.2
0.3
0.4
0.5
Absorbance at wavelength #1Absorbance at wavelength #2
Abso
rban
ce a
t wav
elen
gth
#3
Observing the rows of data in wavelength space
00.05
0.10.15
0.20.25
0
0.1
0.2
0.3
0.4
0.50
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
Absorbance at time #1Absorbance at time #2
Abso
rban
ce at
time #
3
19 20 21 22 23 24 25 26 27 28 290
0.1
0.2
0.3
0.4
0.5
Time
Abso
rban
ce
Observing the columns of data in time space
D = USV = u1 s11 v1 + … + ur srr vr
Singular Value Decomposition
=D U S V
d1,:d2,:
dp,:
… =u11
u21
up1…
s11 v1 d1,:= u11 s11 v1d2,:= u21 s11 v1… …
dp,:= up1 s11 v1
For r=1Row vectors:
D = u1 s11 v1
D = USV = u1 s11 v1 + … + ur srr vr
Singular Value Decomposition
[ d:,1 d:,2 … d:,q ] = u1 s11 [v11 v12 … v1q]
d:,1= u1 s11 v11
… …d:,2= u1 s11 v12
d:,q= u1 s11 v1q
Column vectors:
=D U S V
For r=1 D = u1 s11 v1
Rows of measured data matrix in row space:
v1
u11s11v1
up1s11v1
u11s11u21s11…
up1s11
p points (rows of data matrix) in rows space have the following coordinates:
Columns of measured data matrix in column space:
v11s11v12s11…
v1qs11
q points (columns of data matrix) in columnss space have the following coordinates:
u1 v11 s11u1
v1q s11u1
-3 -2.5 -2 -1.5 -1 -0.5 0-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
ui1s11
ui2s
22
Row Space
-1.6 -1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0 0.2-0.1
-0.08
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
0.08
0.1Column space
v1js11
v2js2
2
Visualizing the rows and columns of data matrix
400 410 420 430 440 450 460 470 480 490 5000
0.5
1
1.5
Wavelength
v1js1
1 or A
bsor
banc
e
0 5 10 15 20 25 30 35 40 45 500
0.5
1
1.5
2
2.5
3
Time
ui1s
11 o
r con
cent
ratio
n
Solutions
Pure spectrum
v1js11
Pure conc. profile
ui1s11
400 420 440 460 480 500 520 540 560 580 6000
0.2
0.4
0.6
0.8
1
1.2
1.4Spectral Profiles
Wavelength (nm)
Abso
rban
ce
0 10 20 30 40 50 60 70 80 90 1000
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5Concentration Profiles
Retention Time
Conc
entra
tion
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
Wavelength
Abso
rban
ce
Measured data
Two component systems
00.1
0.20.3
0.4 0.50.6
0.7
00.1
0.20.3
0.40.5
0.60.7
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
Absorbance at wavelength #1Absorbance at wavelength #2
Abso
rban
ce at
wav
eleng
th #3
470 480 490 500 510 520 5300
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Wavelength
Abso
rban
ce
Visualizing data in three selected wavelengths
38 40 42 44 46 48 50 52 54 56 580
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Time
Abso
rban
ce
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
00.1
0.20.3
0.40.5
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
Absorbance at time #1Absorbance at time #2
Abso
rban
ce at
time #
3
Visualizing data in three selected Times
Measured Data Matrix
D = USV = u1 s11 v1 + … + ur srr vr
Singular Value Decomposition
Row vectors:d1,:d2,:
dp,:
… =u11
u21
up1
…
s11 v1 u12
u22
up2
…s22 v2
+
d1,:= u11 s11 v1 + u12 s22 v2d2,:=… …
dp,:=
u21 s11 v1 + u22 s22 v2
up1 s11 v1 + up2 s22 v2
…
For r=2 D = u1 s11 v1 + u2 s22 v2
D = USV = u1 s11 v1 + … + ur srr vr
Singular Value Decomposition
[ d:,1 d:,2 … d:,q ] = u1 s11 [v11 v12 … v1q]
+ u2 s22 [v21 v22 … v2q]
d:,1= s11 v11 u1 + s22 v21 u2
… …d:,2=
d:,q=
s11 v12 u1 + s22 v22 u2
s11 v1q u1 + s22 v2q u2
For r=2 D = u1 s11 v1 + u2 s22 v2Column vectors:
Rows of measured data matrix in row space:
u11s11
d1,:
v1
v2
u12s22
d2,:
dp,:
u21s11
u22s22
up2s22
up1s11
…
u11s11 u12s22
…u21s11 u22s22
up1s11 up2s22
…Coordinates of rows
Columns of measured data matrix in column space:
u1
u2
d:, 2
d:, 1
d:, q
…v2qs22
v1qs11v12s11
v22s22
v21s22
v11s11
v11s11 v12s11 . . . v1qs11 Coordinates of columns
v21s11 v22s11 . . . v2qs11
400 450 500 5500
0.2
0.4
0.6
0.8
1
1.2
1.4Spectral Profiles
Wavelength (nm)
Abs
orba
nce
400 450 500 5500
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8Response matrix data
Wavelength (nm)
Abs
orba
nce
0 10 20 30 40 50 60 70 800
0.2
0.4
0.6
0.8
1
1.2
1.4Conc. Profiles
Time
Con
c.
0 1 2 3 4 5 6-1
-0.5
0
0.5
1Visualizing the rows of the data
x1
x2
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-0.5
0
0.5Visualizing the columns of the data
y1
y2
Two component chromatographic system
0 2 4 6 8 10 12 14 16 18 200
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1Concentration Profiles
Time
Con
cent
ratio
n
400 420 440 460 480 500 520 540 560 580 6000.5
1
1.5
2Spectral Profiles
Wavelength (nm)
Inte
nsity
400 420 440 460 480 500 520 540 560 580 6000.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2Simulated Spectra
Wavelength (nm)
Abs
orba
nce
6.6 6.8 7 7.2 7.4 7.6 7.8 8 8.2 8.4-2
-1
0
1
2
3Visualizing the rows of the data
x1
x2
5 6 7 8 9 10 11-2
-1
0
1
2Visualizing the columns of the data
y1
y2
Two component Kinetic system
Two component multivariate calibration
0 5 10 15 20 25 300
1
2
3
4
5
6
7
Sample number
Conc
entra
tion
400 420 440 460 480 500 520 540 560 580 6000.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16Spectral Profiles
Wavelength (nm)
Abs
orba
nce
400 420 440 460 480 500 520 540 560 580 6000
0.2
0.4
0.6
0.8
1
1.2
1.4Response matrix data
Wavelength (nm)
Abs
orba
nce
0 1 2 3 4 5 6 7-1
-0.5
0
0.5
1Visualizing the rows of the data
x1
x2
2 2.5 3 3.5 4 4.5-0.5
0
0.5
1Visualizing the columns of the data
y1
y2
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
Tv2Coordinates of dx point:
Position of real profiles in V and U spaces
0 1 2 3 4 5 6 7-2
-1
0
1
2Visualizing the rows of the data
x1
x2
0 5 10 15 20 25-10
-5
0
5
10Visualizing the columns of the data
y1
y2
Position of real profiles in V and U spaces
0 1 2 3 4 5 6-1.5
-1
-0.5
0
0.5
1
1.5Visualizing the rows of the data
x1
x2
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-1
-0.5
0
0.5
1Visualizing the columns of the data
y1
y2
Position of real profiles in V and U spaces
0 1 2 3 4 5 6 7 8-2
-1
0
1
2
3Visualizing the rows of the data
x1
x2
0 1 2 3 4 5 6 7 8 9 10 11-2
-1
0
1
2
3Visualizing the columns of the data
y1
y2
Duality in Measured Data Matrix
p
n
xiT
xj
xij
Pp
Sn
vn
v1
vi
PnSp
up
u1
uj
xi xj
xijxij
Geometrical interpretation of an n x p matrix X
R= C ST = U D VT = X VT
RT= S CT = VD UT = Y UT
X = U D = RV = U YT VY= V D = RT U = V XT U
X YT
U V
Duality based relation between column and row spaces
=
Non-negativity constraint and the system of inequalities:
U z ≥ 0V z ≥ 0
X = U D U= X D-1
Y = V D V= Y D-1
U z = X D-1 z ≥ 0 Hyperplanes
V z = Y D-1 z ≥ 0 Hyperplanes
U-space Y Points
V-space X Points
Duality based relation between column and row spaces
xi = [xi,1 xi,2]
Ui,1 zi,1 + Ui,2 zi,2 … Ui,N zi,N ≥ 0 The coordinates of each point in one space defines the coefficient of related hyper plane in dual space
Point x in V-space Hyper plane (D-1x) z in U-spaceFor two-component systems:
The ith point in V-space: xi
xiD-1= [Ui,1 Ui,2 … Ui,N] The ith hyperplane in U-space:
A point in V-space:
Ui,1 z1 + Ui,2 z2 ≥ 0 A half-plane in V-space:
Half-plane calculation in two-component systems:
General half-planeGeneral border line can be defined for all points that the ith element of the profile is equal to zero
z1
z2
0ith half-plane
ith border line
General border line
Ui,1 z1 + Ui,2 z2 ≥ 0
z2 ≥ (-Ui,2/Ui,1)z1
z2 = (-Ui,2/Ui,1)z1
0 1 2 3 4 5 6-1
-0.5
0
0.5
1Visualizing the rows of the data
x1
x2
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-1
-0.5
0
0.5
1Visualizing the columns of the data
y1
y2
Chromatographic data
0 1 2 3 4 5 6 7
-2
-1
0
1
2
Visualizing the rows of the data
x1
x2
0 1 2 3 4 5 6 7-2
-1
0
1
2Visualizing the columns of the data
y1
y2
Chromatographic data
0 1 2 3 4 5 6 7 8 9-5
0
5Visualizing the rows of the data
x1
x2
0 2 4 6 8 10 12 14 16-10
-5
0
5
10
15Visualizing the columns of the data
y1
y2
Kinetics data
0 1 2 3 4 5 6 7 8 9-5
0
5Visualizing the rows of the data
x1
x2
0 2 4 6 8 10 12 14-4
-2
0
2
4
6
8
10Visualizing the columns of the data
y1
y2
Kinetics data
Calibration data
0 1 2 3 4 5 6 7 8 9-6
-4
-2
0
2
4Visualizing the rows of the data
x1
x2
0 1 2 3 4 5 6 7-4
-2
0
2
4Visualizing the columns of the data
y1
y2
Calibration data
0 1 2 3 4 5 6 7 8 9-6
-4
-2
0
2
4Visualizing the rows of the data
x1
x2
0 1 2 3 4 5 6-2
-1
0
1
2Visualizing the columns of the data
y1
y2
Intensity ambiguity in V space
v1
v2
a
k1ak2a
T11
T12
k1T11
k1T12
k2T11
k2T12
Normalization to unit length
v1
v2
a
k1ak2a
T11
T12
k1T11
k1T12
k2T11
k2T12
an
an = (1/||a||) a
Normalization to first eigenvector
v1
v2
a
k1ak2a
T11
T12
k1T11
k1T12
k2T11
k2T12
1
an = (1/(v.a))a an
v1
v2
12
4
3
51’ 2’ 3’
4’
5’
Normalization to unit length
v1
v2
1
12
4
3
5
a = T1 v1 + T2v2
a’ = v1 + T v2
1’ 2’ 3’
4’
5’
Normalization to first eigenvector
Chromatographic data- Normalized to unit length
0 0.2 0.4 0.6 0.8 1-0.5
0
0.5Normalization to unit length-Row Space
x1
x2
0 0.2 0.4 0.6 0.8 1 1.2-0.4
-0.2
0
0.2
0.4
Normalization to unit length-Column Space
y1
y2
Chromatographic data- Normalized to 1th eigenvector
0 0.2 0.4 0.6 0.8 1 1.2-0.5
0
0.5Normalization to first eigenvector-Row Space
x1
x2
0 0.2 0.4 0.6 0.8 1 1.2-0.4
-0.2
0
0.2
0.4
Normalization to first eigenvector-Column Space
y1
y2
Kinetics data- Normalized to unit length
0 0.2 0.4 0.6 0.8 1
-0.4
-0.2
0
0.2
0.4
0.6
Normalization to unit length-Row Space
x1
x2
0 0.2 0.4 0.6 0.8 1-0.4
-0.2
0
0.2
0.4
0.6
0.8
Normalization to unit length-Column Space
y1
y2
Kinetics data- Normalized to 1th eigenvector
0 0.2 0.4 0.6 0.8 1
-0.4
-0.2
0
0.2
0.4
0.6
0.8
Normalization to first eigenvector-Row Space
x1
x2
0 0.2 0.4 0.6 0.8 1
0
0.5
1
Normalization to first eigenvector-Column Space
y1
y2
Multivariate calibration data- Normalized to unit length
0 0.2 0.4 0.6 0.8 1 1.2-0.5
0
0.5
1Normalization to unit length-Row Space
x1
x2
0 0.2 0.4 0.6 0.8 1 1.2 1.4
-0.2
0
0.2
0.4
Normalization to unit length-Column Space
y1
y2
Multivariate calibration data- Normalized to 1th eigenvector
0 0.2 0.4 0.6 0.8 1 1.2-0.5
0
0.5
1Normalization to first eigenvector-Row Space
x1
x2
0 0.2 0.4 0.6 0.8 1 1.2
-0.2
0
0.2
0.4
Normalization to first eigenvector-Column Space
y1
y2
• The normalized abstract space of two component systems is one dimensional
Data points regionOne dimensional normalized space
• There are 4 critical points in normalized abstract space of two-component systems:
First inner point Second inner point
First outer point Second outer point• The 4 critical points can be calculated very easily
and so the complete resolving of two component systems is very simple
First feasible region Second feasible region
Lawton-Sylvester Plot
Microscopic Observation of Two Component systems using Lawton-Sylvester Plot
First feasible solutions Second feasible solutions
General Microscopic Structures of Two-Component Systems
Second feasible solutions
Case I)
Case II)
Case III)
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.60
0.5
1
1.5
2Visualizing the rows of the data
x2
x1
-0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.40
0.5
1
1.5
2Visualizing the columns of the data
y2
y1
Lawton-Sylvester Plot- Multivariate calibration
Feasible Bands- Multivariate calibration
Feasible Bands- Chromatographic Data
Feasible Bands- Kinetics Data
400 420 440 460 480 500 520 540 560 580 6000.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2Simulated Spectra
Wavelength (nm)
Abs
orba
nce
Kinetics Data (I)
-0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 10
0.5
1
1.5
2Visualizing the rows of the data
x2
x1
-0.4 -0.2 0 0.2 0.4 0.6 0.8 1 1.20
0.5
1
1.5
2Visualizing the columns of the data
y2
y1
LS plot as a microscope for kinetics data (I)
Feasible bands for kinetics data (I)
400 420 440 460 480 500 520 540 560 580 6000
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8Simulated Spectra
Wavelength (nm)
Abs
orba
nce
Kinetics Data (II)
-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.80
0.5
1
1.5
2Visualizing the rows of the data
x2
x1
-0.4 -0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.40
0.5
1
1.5
2Visualizing the columns of the data
y2
y1
LS plot as a microscope for kinetics data (II)
Feasible bands for kinetics data (II)