Augmented classical least squares multivariate spectral analysis
First Claim
1. A method of multivariate spectral analysis, comprising the steps of:
- a) obtaining an estimate of spectral error covariance EA for measured set of multivariate spectral data A;
b) decomposing the spectral error covariance EA according to EA=TP+E, where T is a set of n×
r scores and P is a set of r×
p loading vectors obtained from factor analysis of the spectral error covariance EA, and E is a set of n×
p random errors and spectral variations not useful for prediction;
c) guessing pure-component spectra K for the set of multivariate spectral data A;
d) predicting a set of component values Ĉ
according to
Ĉ
AKT(KKT)−
1=A(KT)+;
e) augmenting the set of predicted component values Ĉ
with at least one vector of the T scores to obtain a first set of augmented component values f) estimating augmented pure-component spectra according to g) testing for convergence according to h) predicting a second set of augmented component values
according to i) replacing the augmented portion of the second set of augmented component values
with the at least one vector of the T scores to obtain a third set of augmented component values
and j) repeating steps f) through i) at least once.
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Abstract
A method of multivariate spectral analysis, termed augmented classical least squares (ACLS), provides an improved CLS calibration model when unmodeled sources of spectral variation are contained in a calibration sample set. The ACLS methods use information derived from component or spectral residuals during the CLS calibration to provide an improved calibration-augmented CLS model. The ACLS methods are based on CLS so that they retain the qualitative benefits of CLS, yet they have the flexibility of PLS and other hybrid techniques in that they can define a prediction model even with unmodeled sources of spectral variation that are not explicitly included in the calibration model. The unmodeled sources of spectral variation may be unknown constituents, constituents with unknown concentrations, nonlinear responses, non-uniform and correlated errors, or other sources of spectral variation that are present in the calibration sample spectra. Also, since the various ACLS methods are based on CLS, they can incorporate the new prediction-augmented CLS (PACLS) method of updating the prediction model for new sources of spectral variation contained in the prediction sample set without having to return to the calibration process. The ACLS methods can also be applied to alternating least squares models. The ACLS methods can be applied to all types of multivariate data.
110 Citations
48 Claims
-
1. A method of multivariate spectral analysis, comprising the steps of:
-
a) obtaining an estimate of spectral error covariance EA for measured set of multivariate spectral data A;
b) decomposing the spectral error covariance EA according to EA=TP+E, where T is a set of n×
r scores and P is a set of r×
p loading vectors obtained from factor analysis of the spectral error covariance EA, and E is a set of n×
p random errors and spectral variations not useful for prediction;
c) guessing pure-component spectra K for the set of multivariate spectral data A;
d) predicting a set of component values Ĉ
according to
Ĉ
AKT(KKT)−
1=A(KT)+;
e) augmenting the set of predicted component values Ĉ
with at least one vector of the T scores to obtain a first set of augmented component valuesf) estimating augmented pure-component spectra according to g) testing for convergence according to h) predicting a second set of augmented component values
according toi) replacing the augmented portion of the second set of augmented component values
with the at least one vector of the T scores to obtain a third set of augmented component values
andj) repeating steps f) through i) at least once. - View Dependent Claims (2, 3, 4, 5, 6, 7, 8, 9)
-
- 10. The method of claim wherein the measured set of multivariate spectral data A comprises image data.
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13. A method of multivariate spectral analysis, comprising the steps of:
-
a) obtaining an estimate of spectral error covariance EA for measured set of multivariate spectral data A;
b) decomposing the spectral error covariance EA according to EA=TP+E, where T is a set of n×
r scores and P is a set of r×
p loading vectors obtained from factor analysis of the spectral error covariance EA, and E is a set of n×
p random errors and spectral variations not useful for prediction;
c) guessing pure-component spectra K for the set of multivariate spectral data A;
d) augmenting the pure-component spectra K with at least one vector of the P loading vectors to obtain first augmented pure-component spectra {tilde over (K)};
e) predicting a first set of augmented component values
according tof) estimating second augmented pure-component spectra
according tog) testing for convergence according to h) replacing the augmented portion of the second augmented pure-component spectra
with the at least one vector of the P loading vectors to obtain third augmented pure-component spectra
andi) predicting a second set of augmented component values
according toj) repeating steps f) through i) at least once. - View Dependent Claims (14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27)
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28. A method of multivariate spectral analysis, comprising the steps of:
-
a) obtaining an estimate of the spectral error covariance EA for measured set of multivariate spectral data A;
b) decomposing the spectral error covariance EA according to EA=TP+E, where T is a set of n×
r scores and P is a set of r×
p loading vectors obtained from factor analysis of the spectral error covariance EA, and E is a set of n×
p random errors and spectral variations not useful for prediction;
c) guessing a set of component values C for the set of multivariate spectral data A;
d) estimating pure-component spectra {circumflex over (K)} according to {circumflex over (K)}=(CTC)−
1CTA=C+A;
e) augmenting the pure-component spectra {circumflex over (K)} with at least one vector of the P loading vectors to obtain first augmented pure-component spectra f) predicting a first set of augmented component values
according tog) testing for convergence according to h) estimating second augmented pure-component spectra
according toi) replacing the augmented portion of the second augmented pure-component spectra
with the at least one vector of the P loading vectors to obtain a third augmented pure-component spectra
andj) repeating steps f) through i) at least once. - View Dependent Claims (29, 30, 31, 32, 33, 34, 35, 36)
-
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37. A method of multivariate spectral analysis, comprising the steps of:
-
a) obtaining an estimate of the spectral error covariance EA for measured set of multivariate spectral data A;
b) decomposing the spectral error covariance EA according to EA=TP+E, where T is a set of n×
r scores and P is a set of r×
p loading vectors obtained from factor analysis of the spectral error covariance EA, and E is a set of n×
p random errors and spectral variations not useful for prediction;
c) guessing a set of component values C for the set of multivariate spectral data A;
d) augmenting the set of component values C with at least one vector of the T scores to obtain a first set of augmented component values {tilde over (C)};
e) estimating augmented pure-component spectra
according tof) testing for convergence according to g) predicting a second set of augmented component values
according toh) replacing the augmented portion of the second set of augmented component values
with the at least one vector of the T scores to obtain a third set of augmented component values
andi) repeating steps e) through h) at least once, using the augmented component values
in step f). - View Dependent Claims (38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48)
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Specification