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 Algorithms

GraphEnt is using a modified version of the algorithm described by Gull, S.F. & Daniell, G.J., (1978), Nature, 272, 686-690, some 20 years ago. A rough account of the method is given in the section 17.2 and will not be repeated here. The only thing that I think that I have to mention is the approximations involved in the calculation of a figure-of-merit weighted calculation :

The above mentioned algorithm assumes a Gaussian distribution of error, either on the amplitude alone, or on the real and imaginary parts of the structure factor. The assumed structure factor pdf for the former case would look like this :

\scalebox {0.5}{\includegraphics{/usr/people/glykos/progs/maxent/doc/fig/fig1.ps2}}

This works very well with Patterson function syntheses, or more general, when there is no ambiguity about the values of the phase angles. When we are given a figure of merit for the phases, the structure factor pdf deviates significantly from the above-shown distribution and it looks more like this :

\scalebox {0.5}{\includegraphics{/usr/people/glykos/progs/maxent/doc/fig/fig2.ps2}}

What GraphEnt will do in this case, is to approximate this function with a 2D Gaussian like this one :

\scalebox {0.5}{\includegraphics{/usr/people/glykos/progs/maxent/doc/fig/fig3.ps2}}

As you would imagine, the approximation is good for well-defined (``high FOM, high F/$ \sigma$(F)'') reflections, and as it turns out, it is also acceptable when you have the ``low FOM, low F/$ \sigma$(F)'' and ``high FOM, low F/$ \sigma$(F)'' cases. Things get hairy in the most dangerous case, ie when you have the ``low FOM, high F/$ \sigma$(F)'' case. GraphEnt will still try to approximate the pdf with a 2D Gaussian and because the pdf in this case almost goes around the phase cycle, this Gaussian will end-up centered somewhere near the origin of the complex plane. In other words, in the case of ``low FOM, high F/$ \sigma$(F)'' GraphEnt will artificially reduce the ``expected'' value of the amplitude of the reflection. Although this may sound like a safe thing to do, it is certainly not the best that can be achieved with the given data12.

Finally, I should add that in real life things are even more complex. This is because what you have from the experiment is not a figure-of-merit, but the whole phase probability distribution (usually in the form of Hendrickson-Lattman coefficients). GraphEnt can not tackle this problem.



Footnotes

... data12
To give a semi-philosophical argument : the data tell us that this reflection is strong but its phase is uncertain. If we chose to artificially reduce the amplitude of the reflection (because we do not know its phase), then not only we are ignoring what the data say, but we also introduce unjustified correlations between the amplitudes and phases of the structure factors. What we should be doing is what we set out to do, namely, to determine the value of the phase angle which maximises the configurational entropy of the map, while being consistent with the data.

next up previous contents
Next:  Of F000s, SCALes and Up: GraphEnt Previous:  Implementation specific notes   Contents
NMG, March 2000