LBOZ

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LBOZ is a coefficient used in spectrophotometry to estimate selectivity (amount of overlapping of spectra) in quantitative manner. It is named after its creators: Lorber, Bergmann, von Oepen, and Zinn.

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[edit] Definition

Let \mathbf{X} be a matrix of the spectra (absorbances), where the k rows correspond to the components in mixture and n columns correspond to the sequence of wavelengths. The LBOZ criterion for kth component is calculated from the following formula:

\xi_k = \frac{1}{\|\mathbf{X}_{k-row}\| \|\mathbf{X}^{+}_{k-col}\|}

where \mathbf{X}^{+} means a pseudoinverse of the matrix and \| \cdots \| means an euclidean length of a vector.

[edit] Properties

Sample synthetic spectra of four components

The image above show synthetic gaussian spectra. The LBOZ criteria are: 0.561 for black compound, 0.402 for red compound, 0.899 for green and 0.549 for blue. LBOZ always lie in range <0,1> and has strong mathematical sense - it presents the amount of spectral signal which is not overlapped by the others. Hence, the uncertainty of a compound quantity increases by 1 / ΞΎ in presence of the other compounds. In this case, the highest uncertainty is expected during determination of red compound - theoretically 2.38 times greater than during determination of its compound alone.

[edit] Implementation

The following function in GNU Octave/Matlab can be useful to calculate LBOZ:

function ksi=LBOZ(X)
X = X';
ksi=1./sqrt(sum(X.^2).*sum((pinv(X).^2)'));
endfunction

[edit] References

  • A. Lorber, Anal. Chem. 58 (1986) 1167.
  • A. Lorber, A. Harel, Z. Goldbart, I.B. Brenner, Anal. Chem. 59 (1987) 1260.
  • G. Bergmann, B. von Oepen, P. Zinn, Anal. Chem. 59 (1987) 2522.
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