An Introduction to Model-Free Chemical Analysis

Hamid AbdollahiHamid AbdollahiIASBS, Zanjan

e-mail: abd@iasbs.ac.ir

Lecture 1Lecture 1

Model-based vs. model-free analysis

There are no generally applicable tools available to guide the researcher towards finding the model that correctly describes the chemical process under investigation. Model fitting is much easier than model finding.

Information obtained from model-free analysis can guide the researcher toward the correct model

In many instances there is no model or mathematical function at all that could be used to quantitatively describe the process under investigation.

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

Con

cent

ratio

n

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)

Abs

orba

nce

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

Abs

orba

nce

at w

avel

engt

h #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

Abs

orba

nce

at ti

me

#3

19 20 21 22 23 24 25 26 27 28 290

0.1

0.2

0.3

0.4

0.5

Time

Abs

orba

nce

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=1

Row 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

v2js

22

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

v1js

11 o

r Abs

orba

nce

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)

Abs

orba

nce

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

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

Wavelength

Abso

rban

ce

Measured data

Two component systems

00.1

0.20.3

0.40.5

0.60.7

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

Abs

orba

nce

at w

avel

engt

h #3

470 480 490 500 510 520 5300

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Wavelength

Abs

orba

nce

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

Abs

orba

nce

00.1

0.20.3

0.40.5

0.60.7

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

Absorbance at time #1Absorbance at time #2

Abs

orba

nce

at ti

me

#3

Visualizing data in three selected Times

?Is the pattern of points depend on selected variables?

?How is the dependency of pattern to overlapping of concentration and spectral profiles?

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]

+ u1 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 v2

Column 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

-5 -4.5 -4 -3.5 -3 -2.5 -2 -1.5 -1 -0.5 0-1.5

-1

-0.5

0

0.5

1

1.5

ui1s11

ui2s

22

Row Space

-2.5 -2 -1.5 -1 -0.5 0-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6Column Space

v1js11

v2js

22

Visualizing the rows and columns of data matrix

?How is the dependency of pattern in one space to other space?