LBOZ

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.

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 "k"th 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.

Properties

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 $1/xi$ 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.

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

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.