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