Rank Annihilation Factor Analysis

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Institute for Advanced Studies in Basic Sciences,
Zanjan - Iran
Chemometrics workshop, 1384
Rank Annihilation Factor Analysis
Mohsen Kompany-Zareh

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Rank Annihilation Factor Analysis (RAFA)
for two-way (2nd Order) data
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Factor Analysis
Data matrix

• Preparation Variab. Select. / mean cent.
  /scaling

• Reprodution
• svd? signif No. of abstract (principal)
  factors

• Combination gt complete models of real factors

• Prediction

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Transformation
abstract factors into basic factors
1. Abstract rotation varimax, quartimax,
oblimin, ..
2. Target testing of potentially real factors..
r?µT
3. Special Meth.s EFA, PLS, KSFA, RAFA Use of
chemical constraints (No proper target vect.)
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Rankmin(r,c)
Rank 3
Mathem. Full Rank
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Rank
the number of linearly independent columns or
rows of X
Rank 2
lt min(r,c)
Mathem. Rank deficient
2 linearly dependent col.s
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Rank
the number of linearly independent columns or
rows of X
Rank 2
lt min(r,c)
Mathem. Rank deficient
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Evolutionary data
- Chromatography - Reaction kinetics -Titrations
(pH, L, ) -Ex.-Em. Fluorersc. Sp. -
Evolutionary methods
-Modeling meth.s (Hard models) -Self-modeling
meth.s (Soft models)
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One component A
X1 excitation emission fluorescence spectrum
Rank 1
Mathem. Rank defic.
Chemical Full rank
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One component A
X1
f (Emiss.)
e (Excit.)
Conc.
Bilinear data xij?k eik?kfkj
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Two components (A and B)
E
F
Conc.
?
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X2 (Fluoresc.)

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Two components (A and B)
X2
Rank 2
Rank Annihilation
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Rank Annihilation Requires a second set of
data,... the calibration set
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Two components (A and B)
Conc.s 1, 3
X2
Rank 2
Conc. 2
Calibration set (A)
X1
Rank 1
What about subtraction of X1 from X2 ?
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Rank
(X2 2.0? X1)
2
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(X2 1.5? X1)
(X2 - X1)
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(X2 0.5? X1)
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Annihil.
(X2 0.2? X1)
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? ? -1 ?
Iterative
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A GRAY system (A and )
X3
Rank 5
one component (A)
X1
Rank 1
Can RAFA be applied ?
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For the GRAY system (A and )
(X3 0.5? X1) E
Rank 4
Rank 5
? ? -1 ?
Quantification of a component in a complex
mixture without concern for other components.
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Direct Calculation of ?
A. Lorber, Anal Chim Acta 1984,164,293-297
?-1 X2 X1 E
Rankn
Rankn-1
Mathematics Reducing the rank of matrix is
equivalent to saying that the determinant of the
matrix is zero.
?-1 X2 X1 0
Or, as a generalized eigenvalue/eigenvector
problem
X1z ?-1 X2z
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X1z ?-1 X2z
1. Matrices X1 and X2 are usually rectangular. !!
2. X2 is not an Identity matrix.!!
Solution
U1S1V1z ?-1 U2S2V2z
U2U1S1V1 V2S2-1z ?-1 U2U2S2V2 V2S2-1z
U2U1S1V1 V2S2-1z ?-1 z
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U2U1S1V1 V2S2-1z ?-1 z
Eigenvalue problem
a b S2-1z ?-1 z
a
b
S2-1
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a b S2-1
2.000 ?-1
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Rank Annihilation with incomplete information
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Rank Annihilation Requires a second set of
data,... the calibration set
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the calibration set (component A)
X1
Bilinear data
f (Emiss.)
e (Excit.)
Conc.
?
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Two vectors for one component is the complete
information
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Incomplete Information
only one of the vectors is known for a particular
component.
Elution profile
Spectrum
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Two components
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Pure spectrum of component B
The only information (s)
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A soft method
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The pseudo-inverse estimate of c
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Results from pseudoinverse solution
Negative
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p signif. scores and loadings
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Necessary condition for t
1 a1 t1 a2 t2