Skip to main content
SpringerLink
Log in

Alternative Algorithm for Simultaneous Determinations of Components Poorly Resolved by Liquid Chromatography with Multiwavelength Detection

  • Original Paper
  • Published:
Analytical Sciences Aims and scope Submit manuscript

Abstract

This paper describes an alternative algorithm for a simultaneous multicomponent analysis with three-mode three-dimensional array data generated by liquid chromatography with diode array detection (LC-DAD). The algorithm is mainly based on a combination of two-step Moore-Penrose pseudoinverse computations based on a truncated singular value decomposition (TSVD) and one-step direct bilinear decomposition. It is modeled with a series of pure or mixed reference samples and then used to analyze several unknown samples. The algorithm was evaluated by Monte-Carlo simulations of two synthetic LC-DAD data sets with different noise levels, and applied to the simultaneous determination of chlorobenzene and toluene by means of HPLC with a photodiode array detector. The obtained results show that the proposed algorithm is suitable for simultaneous determinations of several components, of which the chromatographic peaks and absorption spectra heavily overlap each other, provided that their three-dimensional data have trihnear properties.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Abbreviations

‖ ‖:

the Frobenius or Euclidian norm

m:

a running index for counting reference or standard samples (objects)

M:

the number of the reference samples

h:

a running index for counting selected wavelengths (channels)

H:

the number of selected wavelengths (channels)

k:

a running index for counting selected elution time positions occasions)

K:

the number of selected elution time positions (occasions)

l:

a running index for counting unknown samples

L:

the number of unknown samples

n:

a running index for counting chemical components of interest

N:

the number of chemical components of interest

Rs :

a three-dimensional response array from M pure or mixture reference samples with a two-dimensional response matrix for each sample (size M*H*K)

Ru :

a three-dimensional response array from L unknown samples (size L*H*K)

r mkk :

the element of Rs, the response of the mth reference sample on the hth wavelength at the kth elution time position

r lhk :

the element of Ru, the response of the lth unknown sample on the hth wavelength at the kth elution time position

Rsh:

the kth slice of Rs (size M* H)

Ruk:

the kth slice of Ru (size L*H)

Cs:

a concentration matrix composed of M reference samples (size M*N)

cmn:

the element of Cs, the concentration of the nth chemical component in the mth reference sample

Cu:

a concentration matrix composed of L unknown samples (size L*N)

cln:

the element of Cu, the estimated concentration of the nth chemical component in the lth unknown sample

cln*:

the theorical concentration of the nth chemical component in the lth unknown sample

cn:

the nth column of Cs or Cu, corresponding to the nth chemical component of interest (size M*1 or L*1)

S:

a sensitivity coefficient matrix for N chemical components of interest (size H*N)

s n :

the nth column of S, corresponding to the nth chemical component of interest (size H*1)

shn:

the element of S, the sensitivity coefficient of the nth chemical component on the hth wavelength

T:

a dimensionless time profile matrix of N chemical components of interest (size K*N)

T:

a three-dimensional array of size N*N*K in which the diagonal of the kth slice, Tk, is the kth row of T

T k :

the kth diagonal matrix of T, whose elements are the kth row of T (size N*N)

t n :

the nth column of T, corresponding to the nth chemical component of interest (size K*1)

tkn:

the element of T

I :

a 3-dimensional superdiagonal array with ones on the superdiagonal and zeros elsewhere (size N*N*N)

E :

a three-dimensional residual array for M reference samples in calibration stage (size M*H*K)

E k :

the kth slice of E (size M* H)

emhk:

the element of E

F :

a three-dimensional residual array for L unknown samples in prediction stage (size L*H*K)

F k :

the kth slice of F (size L*H)

fmhk:

the element of F

+:

a notation denoting the Moore-Penrose pseudoinverse

′:

a notation denoting the transpose of a matrix

D :

a three-dimensional array with Dk=(Cs)+Rsk for k=1,...,K(size N*H*K

D k :

the kth slice of D (size N* H)

dnhk:

the element of D

tmax(n):

the maximum value in T corresponding to the nth chemical component

TN(n):

the elution time position corresponding to the tmax(n)

A :

a three-dimensional array with Ak=Ruk(Sr)+ for k=1,..., K (size L*N*K)

A k :

the kth slice of A (size L*N)

alnk:

the element of A

qa:

selected integer used for averaging of shn and cln, depending on the selected elution time interval along the time axis

z:

a relative standard deviation relative to the maximum response value of Rs or Ru as a noise level coefficient used in Monte-Carlo simulation

References

  1. T. Hirschfeld, Anal. Chem., 52, 297A (1980).

    Article  CAS  Google Scholar 

  2. K. S. Booksh and B. R. Kowalski, Anal. Chem., 66, 782A (1994).

    Article  CAS  Google Scholar 

  3. P. Geladi, Chemom. Intell. Lab. Syst., 7, 11 (1989).

    Article  CAS  Google Scholar 

  4. A. K. Smilde, Chemom. bitell Lab. Syst., 15, 143 (1992).

    Article  CAS  Google Scholar 

  5. E. Sanchez and B. R. Kowalski, Anal. Chem., 58, 496 (1986).

    Article  CAS  Google Scholar 

  6. B. Wilson, E. Sanchez, and B. R. Kowalski, J. Chemom., 3, 493 (1989).

    Article  CAS  Google Scholar 

  7. R. A. Harshman and M. E. Lundy, Computational Statistics & Data Analysis, 18, 15 (1994).

    Article  Google Scholar 

  8. S. Wold, P. Geladi, K. Esbensen and J. Ohman, J. Chemom., 1, 41 (1987).

    Article  CAS  Google Scholar 

  9. J. Ohman, P. Geladi, and S. Wold, J. Chemom., 4, 135 (1990).

    Article  CAS  Google Scholar 

  10. A. K. Smilde and D. A. Doornbos, J. Chemom., 6, 11 (1992).

    Article  CAS  Google Scholar 

  11. E. Sanchez and B. R. Kowalski, J. Chemom., 4, 29 (1990).

    Article  CAS  Google Scholar 

  12. K. S. Booksh, Z. Lin, Z. Wang and B. R. Kowalski, Anal. Chem., 66, 2561 (1994).

    Article  CAS  Google Scholar 

  13. A. K. Smilde, Y. Wang, and B. R. Kowalski, J. Chemom., 8, 21 (1994).

    Article  CAS  Google Scholar 

  14. A. K. Smilde, R. Tauler, J. M. Henshaw, L. W. Burgess and B. R. Kowalski, Anal. Chem., 66, 3345 (1994).

    Article  CAS  Google Scholar 

  15. E. Sanchez and B. R. Kowalski, J. Chemom., 2, 265 (1988).

    Article  CAS  Google Scholar 

  16. S. Li and P. Gemperline, J. Chemom., 7, 77 (1993).

    Article  Google Scholar 

  17. T. Hoshino, M. Senda, T. Hondo, M. Saito and S. Tohei, J. Chromatogr., 316, 473 (1984).

    Article  CAS  Google Scholar 

  18. T. Hoshino, T. Hondo, M. Senda, M. Saito and S. Tohei, J. Chromatogr., 332, 139 (1985).

    Article  CAS  Google Scholar 

  19. E. R. Malinowski, “Factor Analysis in Chemistry”, 2nd ed., John Wiley & Sons, Inc., New York, 1991.

    Google Scholar 

  20. D. S. Burdick, X. M. Tu, L. B. McGown and D. W. Millican, J. Chemom., 4, 15 (1990).

    Article  CAS  Google Scholar 

  21. S. Leurgans and R. T. Ross, Statistical Science, 7, 289 (1992).

    Article  Google Scholar 

  22. R. A. Harshman, UCLA Working Papers in Phonetics, 16, 1 (1970).

    Google Scholar 

  23. J. D. Carroll and J. Chang, Psychometrika, 35, 283 (1970).

    Article  Google Scholar 

  24. H. Law, C. Snyder, Jr., J. Hattie and R. McDonald (ed.), “Research Methods for Multimode Data Analysis”, Praeger, New York, 1984.

    Google Scholar 

  25. R. Coppi and S. Bolasco (ed.), “Multiway Data Analysis”, Elsevier Science Publishers B.V., North-Holland, 1989.

    Google Scholar 

  26. J. B. Kruskal, Linear Algebra Appl., 18, 95 (1977).

    Article  Google Scholar 

  27. L. R. Tucker, Psychometrika, 31, 279 (1966).

    Article  CAS  Google Scholar 

  28. P. M. Kroonenberg, “Three-Mode Principal Component Analysis”, DSMO Press, Leiden, 1983.

    Google Scholar 

  29. H. Martens and T. Næs, “Multivariate Calibration”, Wiley, New York, 1989.

    Google Scholar 

  30. G. Golub and C. Vanloan, “Matrix Computations”, 2nd ed., The John Hopkins University Press, Baltimore, 1989.

    Google Scholar 

  31. H.-L. Wu, K. Oguma, and R.-Q. Yu, Anal. Sci., 10, 875 (1994).

    Article  CAS  Google Scholar 

  32. C. L. Lawson and R. J. Hanson, “Solving Least Squares Problems”, Prentice-Hall, Englewood Cliffs, 1974.

    Google Scholar 

  33. R. I. Shrager, Chemom. Intell. Lab. Syst., 1, 59 (1986).

    Article  CAS  Google Scholar 

  34. G. J. Hahn and S. S. Shapiro, “Statistical Models in Engineering”, Section 7-3, John Wiley and Sons, New York, 1967.

    Google Scholar 

  35. L. M. Schwartz, Anal. Chem., 47, 963 (1975).

    Article  CAS  Google Scholar 

  36. J. W. Harbaugh and G. Bonham-Carter, “Computer Simulation in Geology”, R. E. Krieger Pub. Co., Malabar, 1981.

    Google Scholar 

  37. H. R. Keller and D. L. Massart, Anal. Chim. Acta, 246, 379 (1991).

    Article  CAS  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Faculty of Engineering, Chiba University, Yayoi-cho, Inage-ku, Chiba, 263, Japan

    Hai-Long Wu, Yu-Qi Feng, Masami Shibukawa & Koichi Oguma

Authors
  1. Hai-Long Wu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yu-Qi Feng
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Masami Shibukawa
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Koichi Oguma
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Wu, HL., Feng, YQ., Shibukawa, M. et al. Alternative Algorithm for Simultaneous Determinations of Components Poorly Resolved by Liquid Chromatography with Multiwavelength Detection. ANAL. SCI. 13, 99–108 (1997). https://doi.org/10.2116/analsci.13.99

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.2116/analsci.13.99

Keywords

Search

Navigation