Figures of merit.

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Figures of merit.

The quality of the multiple solutions produced after application of the tangent formula are assessed by three figures of merit. ABSFOM is a measure of the internal consistency of the triplet phase relationships. We can define as the expectation value of for a random set of phases and is given by [37]

ABSFOM is defined as


For a poor solution with near random phases ABSFOM while for a good solution where approaches its estimated value , ABSFOM.

The second FOM is defined as

and tests how far the triplets deviate from their statistically predicted behaviour. The lowest values of RESID will generally indicate the best solutions.

The third FOM is

The summations w.r.t are taken over the weak reflection which are output after normalisation. The summation term over in the numerator is equivalent to the right hand side of Sayre's equation (Eq. gif) and is taken over all the triplet phase relationships linking the weaker structure factors to those structure factors for which phases have been evaluated. The numerator will therefore tend to be small if the phases of and calculated by the Tangent formula are correct, thus giving a low value of PSIZERO.

A final combined FOM , CFOM is calculated using,


where the subscripts and refer to the maximum and minimum values of the FOM's of all the solutions given by the Tangent formula and are weights with , and . CFOM has a maximum value of 3.0 and a minimum value of 0. The weights have been determined from experience. ABSFOM is generally considered to be the least reliable of the three FOM's and is given a lower weight.

Gwyndaf Evans
Fri Oct 7 15:42:16 MET 1994