An Introduction to Model-Free Chemical Analysis

Hamid AbdollahiHamid AbdollahiIASBS, Zanjan

e-mail: abd@iasbs.ac.ir

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