Ranking functions for mass spectra

11 Definition (Linear combinations of spectra) Let e N, n be chemical formulas and o IR. Then the linear combination of with coefficients Oj is the mapping 

How does a mass spectrum arise Let /Sj, e h be the formulas of the fragment ions Mjj, i e n, j e. rij present in the ion mixture in a mass spectrometer, where = / ,. Further we assume that the fragment ions have abundance ajj. Then 

Furthermore, the mass resolution of even recent accurate mass spectrometers is insufficient to identify molecular formulas unambiguously from their theoretical isotope distributions. Kind and Fiehn [159] discuss this in detail, including the influence of various combinations of mass and relative isotopic abundance acciu-acy on formula determination. 

These two facts will be taken into consideration here by treating the abundances as unknowns and the measurement error as the target variable of an optimization problem. We thus write Equation (8.1) as 

12 Proposition Let I tea mass spectrum taken with infinite precision and lien the set of all molecular formulas of fragment ions contributing to the mass spectrum. Then Xj IRg for i n exist, such that 

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If there was a match value that fulfilled the above conditions, the verification step of our structure elucidation problem would be solved. Unfortunately, there is no ranking function for real mass spectra that fulfils the latter two requirements in general. However, if we assume an exact recording of mass spectra intensities we can at least define a ranking function for molecular formulas that satisfies requirements (R) and (T). 

By normalizing the above we can define a ranking function that satisfies requirement (R), and, for mass spectra measured at infinite precision, also (T) ... 

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