Rank Annihilation Based Methods

p

n

X

The rank of matrix X is equal to the number of linearly independent vectors from which all p columns of X can be constructed as their linear combination

Geometrically, the rank of pattern of p point can be seen as the minimum number of dimension that is required to represent the p point in the pattern together with origin of space

rank(Pp) = rank(Pn) = rank(X) < min (n, p)

Rank

xP

yP

Variance

P

Q

R

yQ

yR

xQ xR x

y

O

xP yP

xQ yQ

xR yR

OP2= xP2 + yP

2

OQ2= xQ2 + yQ

2

OR2= xR2 + yR

2

OP2 + OQ2 + OR2= xP

2 + yP2 + xQ

2 + yQ

2 + xR2 + yR

2

Sum squared of all elements of a matrix is a criterion for variance in that matrix

Eigenvectors and EigenvaluesFor a symmetric, real matrix, R, an eigenvector v is obtained from:

Rv = v is an unknown scalar-the eigenvalue

Rv – v= 0 (R – Iv= 0The vector v is orthogonal to all of the row

vector of matrix (R-I)

R v = v

0v- R I =

Variance and Eigenvalue

2 43 6

D =1 2

Rank (D) = 1Variance (D) = (1)2 + (2)2 + (3)2 + (2)2 + (4)2 + (6)2 = 70

Eigenvalues (D) = [70 0]

4 23 6

D =1 2

Rank (D) = 2Variance (D) = (1)2 + (4)2 + (3)2 + (2)2 + (2)2 + (6)2 = 70

Eigenvalues (D) = [64.4 5.6]

Free Discussion

The relationship between eigenvalue and variance

10 uni-components samples

Eigenvalues (D) =

204.7

0

0

0

0

0

0

0

0

0

Without noise

204.6

0.0012

0.0012

0.0011

0.0009

0.0009

0.0008

0.0006

0.0006

0.0005

Eigenvalues (D) =with noise

10 bi-components samples

Eigenvalues (D) =

262.6518.9400000000

Without noise

Eigenvalues (D) =with noise

262.6418.930.00110.00090.00080.00080.00070.00070.00060.0005

Bilinearity

As a convenient definition, the matrices of bilinear data can be written as a product of two usually much smaller matrices.

D = C E + R

=

Bilinearity in mono component absorbing systems

=

=

=

=

=

=

=

=

=

=

=

=

=

=

=

Bilinearity in multi-component absorbing systems

A B Ck1

k2

D = C ST

D = cA sAT

+ cB sBT + cC sC

T

sATcA cA sA

T

Bilinearity

sBTcB cB sB

T

Bilinearity

sCTcC cC sC

T

Bilinearity

cA sAT cB sB

T cC sCT+ +

Spectrofluorimetric spectrum

Excitation-Emission Matrix (EEM) is a good example of bilinear data matrix

Exc

ita

tio

n w

ave

len

gth

Emission wavelength

EEM

One component EEM

Two component EEM

Free Discussion

Why the EEM is bilinear?

C=1.0

C=0.8

C=0.6

C=0.4

C=0.2

Trilinearity

Quantitative Determination by Rank Annihilation Factor Analysis

Two components mixture of x and yCx=1.0 and Cy=2.0

Two components mixture of x and y

Two components mixture of x and y

Two components mixture of x and y

Two components mixture of x and y

Two components mixture of x and y

Two components mixture of x and y

Two components mixture of x and y

Two components mixture of x and y

Two components mixture of x and y

Two components mixture of x and y

Two components mixture of x and y

Two components mixture of x and y

Two components mixture of x and y

Two components mixture of x and y

- =

Cx=1.0 Cy=2.0

Cx=0.4 Cy=0.0 Residual

Two components mixture of x and y

Cx=1.0 Cy=2.0

Cx=1.5 Cy=0.0 Residual

- =

Two components mixture of x and y

Cx=1.0 Cy=2.0

Cx=1.0 Cy=0.0 Residual

- =

Two components mixture of x and y

Mixture Standard Residual- =

M – S = R

Rank(M) = n Rank(S) = 1 Rank(R) =n

n-1

Rank Annihilation

The optimal solutions can be reached by decomposing matrix R to the extent that the residual standard deviation (RSD) of the residual matrix obtained after the extraction of n PCs reaches the minimum

j=n+1

c j

n (c– 1)( )RSD(n) =

1/2

RAFA1.m file

Rank Annihilation Factor Analysis

Saving the measured data from unknown and standard

samples

Calling the rank annihilation factor analysis program

Rank estimation of the mixture data matrix

?Use RAFA1.m file for determination one analyte in a ternary mixture using spectrofluorimetry

Simulation of EEM for a sample

?Investigate the effects of extent of spectral overlapping on the results of RAFA