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{{Multiple issues|no footnotes = December 2010|orphan = February 2009}} | |||
'''LBOZ''' is a coefficient used in [[spectrophotometry]] to estimate [[selectivity]]{{dn|date=October 2012}} (amount of [[overlapping]]{{dn|date=October 2012}} of [[spectrum|spectra]]) in quantitative manner. It is named after its creators: Lorber, Bergmann, von Oepen, and Zinn. | |||
== Definition == | |||
[[Image:lboz-spectra.png|thumb|right 300 px|Sample synthetic spectra of four components]] | |||
Let <math>\mathbf{X}</math> 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: | |||
:<math>\xi_k = \frac{1}{\|\mathbf{X}_{k-row}\| \|\mathbf{X}^{+}_{k-col}\|}</math> | |||
where <math>\mathbf{X}^{+}</math> means a [[pseudoinverse]] of the matrix and <math>\| \cdots \|</math> means an [[Euclidean space|euclidean]] length of a vector. | |||
== Properties == | |||
The image above show synthetic [[Gaussian function|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 <math>1/\xi</math> 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. | |||
== 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. | |||
[[Category:Spectroscopy]] | |||
Revision as of 02:29, 12 December 2013
LBOZ is a coefficient used in spectrophotometry to estimate selectivityTemplate:Dn (amount of overlappingTemplate:Dn of spectra) in quantitative manner. It is named after its creators: Lorber, Bergmann, von Oepen, and Zinn.
Definition
Let 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:
where means a pseudoinverse of the matrix and 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 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.
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.