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:.

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Y

D

R

X

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.

A matrix can be decomposed into the product of two significantly smaller matrices.

D = X Y + R

+

=

Singular Value Decomposition

=

S

V

U

D

d1,:

u11

v1

s11

d2,:

u21

=

up1

dp,:

D = USV = u1s11 v1 + … + ursrr vr

For r=1

D = u1s11 v1

Row vectors:

d1,:

u11 s11v1

=

d2,:

u21 s11v1

=

dp,:

up1 s11v1

=

Singular Value Decomposition

S

V

U

D

d:,1

u1 s11 v11

=

d:,2

u1 s11 v12

=

d:,q

u1 s11 v1q

=

D = USV = u1s11 v1 + … + ursrr vr

=

For r=1

D = u1s11 v1

Column vectors:

[ d:,1 d:,2 … d:,q ] = u1s11 [v11 v12 … v1q]

Rows of measured data matrix in row space:

up1s11v1

u11s11v1

v1

p points (rows of data matrix) in rows space have the following coordinates:

u11s11

u21s11

up1s11

Columns of measured data matrix in column space:

q points (columns of data matrix) in columnss space have the following coordinates:

v11s11

v12s11

v1qs11

u1

v11 s11u1

v1q s11u1

Solutions

v1js11

Pure spectrum

ui1s11

Pure conc. profile

Singular Value Decomposition

For r=2

D = u1s11 v1 + u2s22 v2

d1,:

u11

v1

u12

v2

s11

s22

d2,:

u21

u22

=

+

up1

up2

dp,:

d1,:

u11 s11v1 + u12 s22 v2

=

u21 s11v1 + u22 s22 v2

d2,:

=

up1 s11v1 + up2 s22 v2

dp,:

=

D = USV = u1s11 v1 + … + ursrr vr

Row vectors:

Singular Value Decomposition

For r=2

D = u1s11 v1 + u2s22 v2

[ d:,1 d:,2 … d:,q ] = u1s11 [v11 v12 … v1q]

+ u2s22 [v21 v22 … v2q]

d:,1

s11 v11 u1 + s22 v21 u2

=

d:,2

s11 v12 u1 + s22 v22 u2

=

s11 v1q u1 + s22 v2q u2

d:,q

=

D = USV = u1s11 v1 + … + ursrr vr

Column vectors:

Rows of measured data matrix in row space:

v2

Coordinates of rows

u11s11 u12s22

u21s11 u22s22

up2s22

up1s11 up2s22

u22s22

d2,:

d1,:

v1

up1s11

u21s11

dp,:

u12s22

u11s11

Columns of measured data matrix in column space:

u2

Coordinates of columns

v11s11 v12s11 . . .v1qs11

v21s11 v22s11 . . .v2qs11

v21s22

d:, 1

d:, 2

v22s22

v2qs22

d:, q

u1

v11s11

v12s11

v1qs11

Position of a known profile in corresponding space:

v2

Tv2

dx

Tv1

Coordinates of dx point:

v1

Tv1

Tv2

Tv1 is the length of projection of dx on v1 vector

Tv1 = dx . v1

Tv2 is the length of projection of dx on v1 vector

Tv2 = dx. v2

xj

p

xiT

n

up

vn

xi

xj

u1

v1

uj

vi

Geometrical interpretation of an n x p matrix X

xij

Pn

Sp

Sn

Pp

xij

xij

X

U

Duality based relation between column and row spaces

R= C ST = U D VT = X VT

RT= S CT = VD UT = Y UT

X = U D = RV = U YT V

Y= V D = RT U = V XT U

YT

=

V

Non-negativity constraint and the system of inequalities:

U z ≥ 0

V z ≥ 0

X = U D

U= X D-1

Y = V D

V= Y D-1

U-space Y Points

U z = X D-1z ≥ 0 Hyperplanes

V-space X Points

V z = Y D-1z ≥ 0 Hyperplanes

Duality based relation between column and row spaces

The ith point in V-space: xi

xiD-1= [Ui,1 Ui,2 … Ui,N]

The ithhyperplane in U-space:

Ui,1zi,1 + Ui,2zi,2 … Ui,Nzi,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-space

For two-component systems:

xi = [xi,1 xi,2]

A point in V-space:

A half-plane in V-space:

Ui,1z1 + Ui,2z2 ≥ 0

Half-plane calculation in two-component systems:

Ui,1z1 + Ui,2z2 ≥ 0

z2 ≥ (-Ui,2/Ui,1)z1

General half-plane

General border line can be defined for all points that the ith element of the profile is equal to zero

z2= (-Ui,2/Ui,1)z1

General border line

z2

ith border line

0

ith half-plane

z1

Intensity ambiguity in V space

v2

k2T12

k2a

k1T12

k1a

T12

v1

k1T11

T11

k2T11

a

Normalization to unit length

v2

k2T12

k2a

k1T12

k1a

T12

v1

k1T11

T11

k2T11

an = (1/||a||) a

a

an

Normalization to first eigenvector

v2

k2T12

k2a

k1T12

k1a

T12

v1

k1T11

T11

k2T11

an = (1/(v.a))a

an

a

1

Normalization to unit length

v2

1’

2’

3’

v1

3

4

4’

5

2

1

5’

Normalization to first eigenvector

v2

4’

a’ = v1 + T v2

1’

2’

3’

5’

v1

a = T1 v1 + T2v2

3

4

5

2

1

1

Lawton-Sylvester Plot

• The normalized abstract space of two component systems is one dimensional

One dimensional normalized space

Data points region

• There are 4 critical points in normalized abstract space of two-component systems:

Second inner point

First 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

12

Second feasible region

First feasible region

General Microscopic Structures of Two-Component Systems

Case I)

Second feasible solutions

First feasible solutions

Case II)

Second feasible solutions

Case III)