Equilibrium systems Chromatography systems Number of PCs original Mean centered Number of PCs...
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Transcript of Equilibrium systems Chromatography systems Number of PCs original Mean centered Number of PCs...
Equilibrium systems
Chromatography systems
Number of PCs
original Mean centered
Number of PCs
originalMean
centered
2 1
2 1
2 1
2 1
2 2
3 2
The rank of a product of two matrices X and Y is equal or smaller to the smallest of the rank of X and Y:
Rank (X Y) ≤ min (rank (X) , rank (Y))
A = CS
HA A- + H+
[HA] =
Ct [H+][H+] + Ka
[A-] =
Ct Ka
[H+] + Ka
[Int] =p
[HA] + [A-] = Ct
Ct = p = [Int]
[HA] + [A-] = [Int]
[HA] + [A-] - [Int] = 0
?
Determine the rank of data matrix of following hypothesis system (rank.mat file)
A Rank Deficiency problem and Its Solution
Rank deficient system
=
=
Full rank system
Augmentation
=
Ard C E=
As CE
=
E=Ard
As
C
C
Rank (Ard) = 2
Rank (As) = 1
Rank (Ard ; As) = 3
Augmentation
1
2
3
3
2
1
0.5
0.5
0.5
1
2
3
3
2
1
0.5
0.5
0.5
+ + (-8)
0
0
0
=
Rank deficiency in C matrix
1
2
3
3
2
1
0.5
0.5
0.5
2
4
6
6
4
2
0
0
0
Augmentation
1
2
3
3
2
1
0.5
0.5
0.5
2
4
6
6
4
2
0
0
0
Rank deficiency in a system with two independent chemical processes
HA A- + H+
HB B- + H+
[HA] [A-]
[HB] [B-] HA
A-
HBB-
HAM.m file
Spectrophotometric monitoring of pH-metric titration of mixture of two acids
HA A- + H+
[HA] =
Ct1 [H+][H+] + Ka1
[A-] =
Ct1 Ka1
[H+] + Ka1
[HA] + [A-] = ([HB] + [B-])
Ct1 = Ct2
HB B- + H+
[H+] + Ka2
[HB] =
Ct2 [H+][A-] =
Ct2 Ka2
[H+] + Ka2
[HB] + [B-] = Ct2
1
2
3
3
2
1
2
4
6
6
4
2
1
2
3
3
2
1
+ +
0
0
0
=
2
4
6
6
4
2
+
1
2
3
3
2
1
+
4
4
4
=
2
4
6
6
4
2
+
8
8
8
=
4
4
4
8
8
8
= (1/2)
(-1/2) (-1/2)
Augmentation
=
1
2
3
3
2
1
2
4
6
6
4
21
2
3
3
2
1
0
0
0
0
0
0
1
2
3
3
2
1
2
4
6
6
4
2
1
2
3
3
2
1
0
0
0
0
0
0
1
2
3
3
2
1
1
2
3
3
2
1
+
4
4
4
4
4
4
=
2
4
6
6
4
2
0
0
0
0
0
0
+
8
8
8
0
0
0
=
?
Is it possible to solve rank deficiency in systems by exact same pKa values?
Non-linear data
Principal Component Analysis
Samples in two dimensional wavelength space
Non-homogeneous deviation in wavelength space
Principal Component Analysis
Samples in two dimensional wavelength space
?
Use NLC.m file and create data matrix for two component system and investigate the effect of non-linearity on numbers of PCs
NLC.m file
Non-linear calibration absorbance data matrix
Number of significant eigenvalues and evolutionary chemical processes
1.5915 0.0023 0.0000 0.0000
Singular values (0-6 sec)
2.4518 0.0069 0.0000 0.0000
Singular values (2-8 sec)
3.4849 0.0193 0.0001 0.0000
Singular values (4-10 sec)
4.5878 0.0488 0.0002 0.0000
Singular values (6-12 sec)
5.6211 0.1120 0.0008 0.0000
Singular values (8-14 sec)
6.4493 0.2323 0.0023 0.0000
Singular values (10-16 sec)
6.9869 0.4338 0.0060 0.0000
Singular values (12-18 sec)
7.2375 0.7233 0.0138 0.0000
Singular values (14-20 sec)
7.3157 1.0627 0.0279 0.0000
Singular values (16-22 sec)
7.4354 1.3499 0.0495 0.0000
Singular values (18-24 sec)
7.8152 1.4677 0.0752 0.0000
Singular values (20-26 sec)
8.5105 1.4042 0.0951 0.0000
Singular values (22-28 sec)
9.3713 1.2588 0.0963 0.0000
Singular values (24-30 sec)
10.1608 1.1141 0.0761 0.0000
Singular values (26-32 sec)
10.6552 0.9779 0.0476 0.0000
Singular values (28-34 sec)
10.6909 0.8265 0.0245 0.0000
Singular values (30-36 sec)
10.1926 0.6522 0.0108 0.0000
Singular values (32-38 sec)
9.1870 0.4713 0.0042 0.0000
Singular values (34-40 sec)
7.7941 0.3089 0.0014 0.0000
Singular values (36-42 sec)
6.1977 0.1830 0.0004 0.0000
Singular values (38-44 sec)
4.5999 0.0980 0.0001 0.0000
Singular values (40-46 sec)
3.1735 0.0475 0.0000 0.0000
Singular values (42-148 sec)
2.0275 0.0209 0.0000 0.0000
Singular values (44-50 sec)
FSW.m file
Eigen analysis on moving fixed size window
?
Use FSW.m file and investigate on pH-window of each component for H2A system (H2A.m file)
?
Investigate the effects of selected window size on accuracy of determining the concentration window for each species
1.4327 0.0024
Singular values (0-2 sec)
2.2664 0.0083 0.0000
Singular values (0-4 sec)
3.2730 0.0231 0.0000 0.0000
Singular values (0-6 sec)
4.4044 0.0563 0.0001 0.0000
Singular values (0-8 sec)
5.5834 0.1245 0.0004 0.0000
Singular values (0-10 sec)
6.7299 0.2517 0.0012 0.0000
Singular values (0-12 sec)
7.7864 0.4668 0.0036 0.0000
Singular values (0-14 sec)
8.7323 0.7956 0.0099 0.0000
Singular values (0-16 sec)
9.5808 1.2484 0.0244 0.0000
Singular values (0-18 sec)
10.3637 1.8119 0.0552 0.0000
Singular values (0-20 sec)
11.1136 2.4512 0.1133 0.0000
Singular values (0-22 sec)
11.8506 3.1232 0.2110 0.0000
Singular values (0-24 sec)
12.5772 3.7923 0.3561 0.0000
Singular values (0-26 sec)
13.2808 4.4360 0.5455 0.0000
Singular values (0-28 sec)
13.9413 5.0402 0.7623 0.0000
Singular values (0-30 sec)
14.5360 5.5893 0.9812 0.0000
Singular values (0-32 sec)
15.0430 6.0633 1.1776 0.0000
Singular values (0-34 sec)
15.4449 6.4435 1.3359 0.0000
Singular values (0- 36 sec)
15.7348 6.7216 1.4512 0.0000
Singular values (0- 38 sec)
15.9215 6.9040 1.5268 0.0000
Singular values (0- 40 sec)
16.0273 7.0098 1.5713 0.0000
Singular values (0- 42 sec)
16.0794 7.0634 1.5942 0.0000
Singular values (0- 44 sec)
16.1015 7.0868 1.6044 0.0000
Singular values (0- 46 sec)
16.1096 7.0955 1.6083 0.0000
Singular values (0-48 sec)
16.1122 7.0983 1.6096 0.0000
Singular values (0-50 sec)
0.7231 0.0017 0.000 0.000
Singular values (50-48 sec)
1.2648 0.0060 0.0000 0
Singular values (50-46 sec)
2.0245 0.0164 0.0000 0.0000
Singular values (50-44 sec)
3.0143 0.0395 0.0000 0.0000
Singular values (50-42 sec)
4.2074 0.0865 0.0001 0.0000
Singular values (50-40 sec)
5.5408 0.1738 0.0003 0.0000
Singular values (50-38 sec)
6.9305 0.3215 0.0009 0.0000
Singular values (50-36 sec)
8.2934 0.5483 0.0029 0.0000
Singular values (50-34 sec)
9.5650 0.8639 0.0082 0.0000
Singular values (50-32 sec)
10.7064 1.2627 0.0213 0.0000
Singular values (50-30 sec)
11.7008 1.7245 0.0504 0.0000
Singular values (50-28 sec)
12.5455 2.2228 0.1080 0.0000
Singular values (50-26 sec)
13.2478 2.7381 0.2091 0.0000
Singular values (50-24 sec)
13.8235 3.2656 0.3639 0.0000
Singular values (50-22 sec)
14.2956 3.8130 0.5684 0.0000
Singular values (50-20 sec)
14.6900 4.3880 0.8003 0.0000
Singular values (50-18 sec)
15.0288 4.9811 1.0266 0.0000
Singular values (50-16 sec)
15.3247 5.5579 1.2200 0.0000
Singular values (50-14 sec)
15.5782 6.0693 1.3680 0.0000
Singular values (50-12 sec)
15.7824 6.4753 1.4711 0.0000
Singular values (50-10 sec)
15.9307 6.7613 1.5372 0.0000
Singular values (50-8 sec)
16.0254 6.9387 1.5759 0.0000
Singular values (50-6 sec)
16.0776 7.0349 1.5963 0.0000
Singular values (50- 4 sec)
16.1022 7.0801 1.6058 0.0000
Singular values (50-2 sec)
16.1122 7.0983 1.6096 0.0000
Singular values (50-0 sec)
EFA.m file
Forward and backward eigen analysis
?
Investigate the variation of eigenvalues in an evolutionary chemical process in the presence of noise
?
Show that the eigenvalue analysis can be used for estimating the selective wavelength range for each chemical species
Principal Component Regression (PCR)PCR is simply PCA followed by a regression step
A= C E = S L
A C E= S L=
A= C E = (S R) (R-1 L)
C = S R
C S R=
S r=c1
A data matrix can be represented by its score matrixA regression of score matrix against one or several dependent variables is possible, provided that scores corresponding to small eigenvalues are omittedThis regression gives no matrix inversion problemPCR has the full-spectrum advantages of the CLS methodPCR has the ILS advantage of being able to perform the analysis one chemical components at a time while avoiding the ILS wavelength selection problem
c = S b
Calibration and Prediction Steps in PCR
=c1
n
1
Sn
r
br
1
b = ( STS)-1 ST c
Calibration Step
Axm
p
L
p
rr
mSx =
Prediction StepSx = Ax L
cx = Sx b