Abstract

A method is described for estimating the spectra of pure components from the spectra of unknown mixtures with various relative concentrations. This method is based on principal component analysis and a constrained nonlinear optimization technique and is applicable to qualitative analysis of mixtures of more than three components. The method gives two curves as the estimate of a component spectrum: one consists of the set of the maxima and the other consists of the set of the minima for all sampling points subject to a priori information. Experimental results of the estimation of the infrared absorption spectra of xylene–isomer mixtures are shown; the noise problem with this method is also discussed.

© 1983 Optical Society of America

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References

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  1. W. H. Lawton, E. A. Sylvestre, Technometrics 13, 617 (1971).
    [CrossRef]
  2. E. A. Sylvestre, W. H. Lawton, M. S. Maggio, Technometrics 16, 353 (1974).
    [CrossRef]
  3. D. MacNaughtan, L. B. Rogers, G. Wernimont, Anal. Chem. 44, 1421 (1972).
    [CrossRef]
  4. M. Warner, G. D. Christian, E. R. Davidson, Anal. Chem. 49, 564 (1977).
    [CrossRef]
  5. F. J. Knorr, J. H. Futrell, Anal. Chem. 51, 1236 (1979).
    [CrossRef]
  6. M. A. Sharaf, B. R. Kowalski, Anal. Chem. 54, 1921 (1982).
    [CrossRef]
  7. N. Ohta, Anal. Chem. 45, 553 (1973).
    [CrossRef]
  8. T. W. Anderson, An Introduction to Multivariate Statistical Analysis (Wiley, New York, 1958).
  9. J. Kowalik, M. R. Osborne, Methods for Unconstrained Optimization Problems (American Elsevier, New York, 1968).
  10. E. R. Malinowski, Anal. Chem. 49, 606 (1977).
    [CrossRef]
  11. E. R. Malinowski, Anal. Chem. 49, 612 (1977).
    [CrossRef]
  12. P. C. Gillette, J. L. Koenig, Appl. Spectrosc. 36, 535 (1982).
    [CrossRef]

1982 (2)

M. A. Sharaf, B. R. Kowalski, Anal. Chem. 54, 1921 (1982).
[CrossRef]

P. C. Gillette, J. L. Koenig, Appl. Spectrosc. 36, 535 (1982).
[CrossRef]

1979 (1)

F. J. Knorr, J. H. Futrell, Anal. Chem. 51, 1236 (1979).
[CrossRef]

1977 (3)

E. R. Malinowski, Anal. Chem. 49, 606 (1977).
[CrossRef]

E. R. Malinowski, Anal. Chem. 49, 612 (1977).
[CrossRef]

M. Warner, G. D. Christian, E. R. Davidson, Anal. Chem. 49, 564 (1977).
[CrossRef]

1974 (1)

E. A. Sylvestre, W. H. Lawton, M. S. Maggio, Technometrics 16, 353 (1974).
[CrossRef]

1973 (1)

N. Ohta, Anal. Chem. 45, 553 (1973).
[CrossRef]

1972 (1)

D. MacNaughtan, L. B. Rogers, G. Wernimont, Anal. Chem. 44, 1421 (1972).
[CrossRef]

1971 (1)

W. H. Lawton, E. A. Sylvestre, Technometrics 13, 617 (1971).
[CrossRef]

Anderson, T. W.

T. W. Anderson, An Introduction to Multivariate Statistical Analysis (Wiley, New York, 1958).

Christian, G. D.

M. Warner, G. D. Christian, E. R. Davidson, Anal. Chem. 49, 564 (1977).
[CrossRef]

Davidson, E. R.

M. Warner, G. D. Christian, E. R. Davidson, Anal. Chem. 49, 564 (1977).
[CrossRef]

Futrell, J. H.

F. J. Knorr, J. H. Futrell, Anal. Chem. 51, 1236 (1979).
[CrossRef]

Gillette, P. C.

Knorr, F. J.

F. J. Knorr, J. H. Futrell, Anal. Chem. 51, 1236 (1979).
[CrossRef]

Koenig, J. L.

Kowalik, J.

J. Kowalik, M. R. Osborne, Methods for Unconstrained Optimization Problems (American Elsevier, New York, 1968).

Kowalski, B. R.

M. A. Sharaf, B. R. Kowalski, Anal. Chem. 54, 1921 (1982).
[CrossRef]

Lawton, W. H.

E. A. Sylvestre, W. H. Lawton, M. S. Maggio, Technometrics 16, 353 (1974).
[CrossRef]

W. H. Lawton, E. A. Sylvestre, Technometrics 13, 617 (1971).
[CrossRef]

MacNaughtan, D.

D. MacNaughtan, L. B. Rogers, G. Wernimont, Anal. Chem. 44, 1421 (1972).
[CrossRef]

Maggio, M. S.

E. A. Sylvestre, W. H. Lawton, M. S. Maggio, Technometrics 16, 353 (1974).
[CrossRef]

Malinowski, E. R.

E. R. Malinowski, Anal. Chem. 49, 612 (1977).
[CrossRef]

E. R. Malinowski, Anal. Chem. 49, 606 (1977).
[CrossRef]

Ohta, N.

N. Ohta, Anal. Chem. 45, 553 (1973).
[CrossRef]

Osborne, M. R.

J. Kowalik, M. R. Osborne, Methods for Unconstrained Optimization Problems (American Elsevier, New York, 1968).

Rogers, L. B.

D. MacNaughtan, L. B. Rogers, G. Wernimont, Anal. Chem. 44, 1421 (1972).
[CrossRef]

Sharaf, M. A.

M. A. Sharaf, B. R. Kowalski, Anal. Chem. 54, 1921 (1982).
[CrossRef]

Sylvestre, E. A.

E. A. Sylvestre, W. H. Lawton, M. S. Maggio, Technometrics 16, 353 (1974).
[CrossRef]

W. H. Lawton, E. A. Sylvestre, Technometrics 13, 617 (1971).
[CrossRef]

Warner, M.

M. Warner, G. D. Christian, E. R. Davidson, Anal. Chem. 49, 564 (1977).
[CrossRef]

Wernimont, G.

D. MacNaughtan, L. B. Rogers, G. Wernimont, Anal. Chem. 44, 1421 (1972).
[CrossRef]

Anal. Chem. (7)

D. MacNaughtan, L. B. Rogers, G. Wernimont, Anal. Chem. 44, 1421 (1972).
[CrossRef]

M. Warner, G. D. Christian, E. R. Davidson, Anal. Chem. 49, 564 (1977).
[CrossRef]

F. J. Knorr, J. H. Futrell, Anal. Chem. 51, 1236 (1979).
[CrossRef]

M. A. Sharaf, B. R. Kowalski, Anal. Chem. 54, 1921 (1982).
[CrossRef]

N. Ohta, Anal. Chem. 45, 553 (1973).
[CrossRef]

E. R. Malinowski, Anal. Chem. 49, 606 (1977).
[CrossRef]

E. R. Malinowski, Anal. Chem. 49, 612 (1977).
[CrossRef]

Appl. Spectrosc. (1)

Technometrics (2)

W. H. Lawton, E. A. Sylvestre, Technometrics 13, 617 (1971).
[CrossRef]

E. A. Sylvestre, W. H. Lawton, M. S. Maggio, Technometrics 16, 353 (1974).
[CrossRef]

Other (2)

T. W. Anderson, An Introduction to Multivariate Statistical Analysis (Wiley, New York, 1958).

J. Kowalik, M. R. Osborne, Methods for Unconstrained Optimization Problems (American Elsevier, New York, 1968).

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Figures (7)

Fig. 1
Fig. 1

Explanatory illustration of the constraints and the restriction on the coordinate system of the coefficients {tj1,tj2,tj3} for all j, in the case of a three-component system. The a priori information Eqs. (4) and (5) limits the possible coefficient set within three subspaces and the normalization Eq. (6) cuts these subspaces by plane A, resulting in three subplanes (shaded areas in the figure) enclosed by solid line B and broken lines C1, C2, and C3.

Fig. 2
Fig. 2

Absorbance spectra of nineteen mixtures of xylene isomers with various relative concentrations.

Fig. 3
Fig. 3

Nineteen eigenvalues of the second moment matrix of the mixture spectra (Fig. 2) in order of magnitude.

Fig. 4
Fig. 4

Normalized eigenvectors corresponding to (a) the largest, (b) the second, and (c) the third eigenvalues in magnitude.

Fig. 5
Fig. 5

Resultant solution bands of three-component spectra. The solutions are limited by two curves in the individual graphs.

Fig. 6
Fig. 6

Pure component spectra of (a) o-xylene, (b) m -xylene, and (c) p -xylene.

Fig. 7
Fig. 7

Computer simulation of noise influence: (a) the set of computer-generated mixture spectra composed of two Gaussians at distinct concentration ratios with SNR = 140; (b) same as (a) but with SNR = 28; (c) and (d) resultant solution bands of two-component spectra by our proposed method from the data sets of (a) and (b), respectively (shaded regions); (e) and (f) same as (c) and (d) but by Lawton-Sylvestre method.

Equations (11)

Equations on this page are rendered with MathJax. Learn more.

x i = j = 1 N c ij s j i = 1 , , M ,
[ R ] = i = 1 M x i x i t / M .
s j = k = 1 N t jk v k j = 1 , , N ,
s jl 0 for all j , l .
c ij 0 for all i , j .
l = 1 L s jl = 1 for all j .
U jl ( + ) = + s jl + P min , j = 1 , , N , U jl ( ) = s jl + P min , l = 1 , , L .
P ( t 11 , , t NN , γ ) = γ { j = 1 N l = 1 L H ( s jl ) s jl 2 + i = 1 M j = 1 N H ( c ij ) c ij 2 } ,
H ( y ) = { 0 ( y 0 ) , 1 ( y < 0 ) .
k = 1 N ( l = 1 L υ kl ) t jk = 1 j = 1 , , N ,
t j N = 1 l = 1 L υ N l { 1 k = 1 N 1 ( l = 1 L υ k l ) t jk } j = 1 , , N .

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