Method for exploiting bias in factor analysis using constrained alternating least squares algorithms
View Patent ↗Bias plays an important role in factor analysis and is often implicitly made use of, for example, to constrain solutions to factors that conform to physical reality. However, when components are collinear, a large range of solutions may exist that satisfy the basic constraints and fit the data equally well. In such cases, the introduction of mathematical bias through the application of constraints may select solutions that are less than optimal. The biased alternating least squares algorithm of the present invention can offset mathematical bias introduced by constraints in the standard alternating least squares analysis to achieve factor solutions that are most consistent with physical reality. In addition, these methods can be used to explicitly exploit bias to provide alternative views and provide additional insights into spectral data sets.
1. A method for analyzing a sample from measured spectral data by factoring a spectroscopic data matrix D, describing the measured spectral data, into the product of a factor matrix S, describing estimated spectra of pure components of the sample, and a factor matrix C, describing estimated concentrations of pure components of the sample, according to D=CS T , the method comprising:
a) collecting measured spectral data from the sample with a spectrometer, whereby the measured spectral data is described by the spectroscopic data matrix D,
b) making an initial guess for the factor matrix S,
c) normalizing the factor matrix S,
d) selecting a bias parameter γ C greater than σ min 2 , where σ min 2 is the smallest eigenvalue of S T S,
e) solving
min
C
S
T
D
T
-
(
S
T
S
+
γ
C
I
)
C
T
F
,
subject to constraints on the factor matrix C,
f) selecting a bias parameter γ S greater than −σ min 2 , where σ min 2 in is the smallest eigenvalue of C T C,
g) solving
min
S
C
T
D
T
-
(
C
T
C
+
γ
S
I
)
S
T
F
,
subject to constraints on the factor matrix S,
h) repeating steps c) thorough g) at least once, and
i) computing S=(I+γ S (C T C) −1 )S to provide the factor matrix S.
2. The method of claim 1 , wherein the data matrix D is preprocessed.
3. The method of claim 2 , wherein the data matrix D is weighted.
4. The method of claim 2 , wherein the data matrix D is spatially compressed.
5. The method of claim 2 , wherein the data matrix D is spectrally compressed.
6. The method of claim 1 , wherein the bias parameter γ C is a diagonal matrix comprising individually adjusted diagonal values.
7. The method of claim 1 , wherein the bias parameter γ S is a diagonal matrix comprising individually adjusted diagonal values.
8. The method of claim 1 , wherein the bias parameter γ C is less than σ max 2 , where σ max 2 is the largest eigenvalue of S T S.
9. The method of claim 1 , wherein the bias parameter γ S is less than σ max 2 , where σ max 2 is the largest eigenvalue of C T C.
10. The method of claim 1 , wherein the constraints on the factor matrix C are selected from the group consisting of non-negativity constraints, general linear equality constraints, general linear inequality constraints, combined linear equality and inequality constraints, constraints imposed by the method of weighting, constraints imposed as preferences using the method of weighting, bounded variable constraints, simple variable equality constraints, closure constraints, and combinations of the above constraints.
11. The method of claim 1 , wherein the constraints on the factor matrix S are selected from the group consisting of non-negativity constraints, general linear equality constraints, general linear inequality constraints, combined linear equality and inequality constraints, constraints imposed by the method of weighting, constraints imposed as preferences using the method of weighting, bounded variable constraints, simple variable equality constraints, closure constraints, and combinations of the above constraints.
12. The method of claim 1 , wherein step h) is repeated for a fixed number of iterations.
13. The method of claim 1 , wherein step h) is repeated until the solutions for C in step e) and the solutions for S in step g) converge.
14. A method for analyzing a sample from measured spectral data by factoring a spectroscopic data matrix D, describing the measured spectral data, into the product of a factor matrix S, describing estimated spectra of pure components of the sample, and a factor matrix C, describing estimated concentrations of pure components of the sample, according to D=CS T , the method comprising:
a) collecting measured spectral data from the sample with a spectrometer, whereby the measured spectral data is described by the spectroscopic data matrix D,
b) making an initial guess for the factor matrix C,
c) normalizing the factor matrix C,
d) selecting a bias parameter γ S greater than σ min 2 , where σ min 2 is the smallest eigenvalue of C T C,
e) solving
min
S
C
T
D
T
-
(
C
T
C
+
γ
S
I
)
S
T
F
,
subject to constraints on the factor matrix S,
f) selecting a bias parameter γ C greater than −σ min 2 , where σ min 2 is the smallest eigenvalue of S T S,
g) solving
min
C
S
T
D
T
-
(
S
T
S
+
γ
C
I
)
C
T
F
,
subject to constraints on the factor matrix C,
h) repeating steps c) thorough g) at least once, and
i) computing C=(I+γ C (S T S) −1 )C to provide the factor matrix C.
15. The method of claim 14 , wherein the data matrix D is preprocessed.
16. The method of claim 15 , wherein the data matrix D is weighted.
17. The method of claim 15 , wherein the data matrix D is spatially compressed.
18. The method of claim 15 , wherein the data matrix D is spectrally compressed.
19. The method of claim 14 , wherein the bias parameter γ S is a diagonal matrix comprising individually adjusted diagonal values.
20. The method of claim 14 , wherein the bias parameter γ C is a diagonal matrix comprising individually adjusted diagonal values.
21. The method of claim 14 , wherein the bias parameter γ S is less than σ max 2 , where σ max 2 is the largest eigenvalue of C T C.
22. The method of claim 14 , wherein the bias parameter γ C is less than σ max 2 , where σ max 2 is the largest eigenvalue of S T S.
23. The method of claim 14 , wherein the constraints on the factor matrix S are selected from the group consisting of non-negativity constraints, general linear equality constraints, general linear inequality constraints, combined linear equality and inequality constraints, constraints imposed by the method of weighting, constraints imposed as preferences using the method of weighting, bounded variable constraints, simple variable equality constraints, closure constraints, and combinations of the above constraints.
24. The method of claim 14 , wherein the constraints on the factor matrix C are selected from the group consisting of non-negativity constraints, general linear equality constraints, general linear inequality constraints, combined linear equality and inequality constraints, constraints imposed by the method of weighting, constraints imposed as preferences using the method of weighting, bounded variable constraints, simple variable equality constraints, closure constraints, and combinations of the above constraints.
25. The method of claim 14 , wherein step h) is repeated for a fixed number of iterations.
26. The method of claim 14 , wherein step h) is repeated until the solutions for S in step e) and the solutions for C in step g) converge.