LBOZ

Let ${\displaystyle \mathbf {X} }$ 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:

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

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

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 ${\displaystyle 1/\xi }$ 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.

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