Download - Rank Annihilation Based Methods
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