In SHELXL. a refinement program sometimes used by small molecule
crystallographers, all Fourier map for at least the last 20 years were
weighted by Ic^2/(Ic^2+sigma^2(I)), where Ic is the calculated squared
structure factor and sigma(I) is the square root of 1/w. w is the weight
assigned to a reflection in the refinement (e.g. w=1/(sig(I)^2+(gI)^2),
where sig(I) is the esd of the measured intensity I and g is a small
constant. This purely empirical scheme appears to result in a
significant reduction in the noise level of the map, at least for
typical small molecule structures. Such schemes have been called
'maximum likelihood by intuition', a proper maximum likelihood treatment
taking the esds of the intensities into account would of course do much
better.
George
On 05/23/2012 06:59 PM, Dale Tronrud wrote:
On 05/23/12 08:06, Nicholas M Glykos wrote:
Hi Ed,
I may be wrong here (and please by all means correct me), but I think
it's not entirely true that experimental errors are not used in modern
map calculation algorithm. At the very least, the 2mFo-DFc maps are
calibrated to the model error (which can be ideologically seen as the
"error of experiment" if you include model inaccuracies into that).
This is an amplitude modification. It does not change the fact that the
sigmas are not being used in the inversion procedure [and also does not
change the (non) treatment of missing data]. A more direct and relevant
example to discuss (with respect to Francisco's question) would be the
calculation of a Patterson synthesis (where the phases are known and
fixed).
I have not done extensive (or any for that matter) testing, but my
evidence-devoid gut feeling is that maps not using experimental errors
(which in REFAMC can be done either via gui button or by excluding SIGFP
from LABIN in a script) will for a practicing crystallographer be
essentially indistinguishable.
It seems that although you are not doubting the importance of maximum
likelihood for refinement, you do seem to doubt the importance of closely
related probabilistic methods (such as maximum entropy methods) for map
calculation. I think you can't have it both ways ... :-)
The reason for this is that "model errors" as estimated by various
maximum likelihood algorithms tend to exceed experimental errors. It
may be that these estimates are inflated (heretical thought but when you
think about it uniform inflation of the SIGMA_wc may have only
proportional impact on the log-likelihood or even less so when they
correlate with experimental errors). Or it may be that the experimental
errors are underestimated (another heretical thought).
My experience from comparing conventional (FFT-based) and maximum-entropy-
related maps is that the main source of differences between the two maps
has more to do with missing data (especially low resolution overloaded
reflections) and putative outliers (for difference Patterson maps), but in
certain cases (with very accurate or inaccurate data) standard deviations
do matter.
In a continuation of this torturous diversion from the original question...
Since your concern is not how the sigma(Fo) plays out in refinement but
how uncertainties are dealt with in the map calculation itself (where an
FFT calculates the most probable density values and maximum entropy would
calculate the "best", or centroid, density values) I believe the most
relevant measure of the uncertainty of the Fourier coefficients would be
sigma(2mFo-DFc). This would be estimated from a complex calculation of
sigma(sigmaA), sigma(Fo), sigma(Fc) and sigma(Phic). I expect that the
contribution of sigma(Fo) would be one of the smallest contributors to this
calculation, as long as Fo is "observed". I wouldn't expect the loss of
sigma(Fo) to be catastrophic.
Wouldn't sigma(sigmaA) be the largest component since sigmaA is a function
of resolution and based only on the test set?
Dale Tronrud
All the best,
Nicholas
--
Prof. George M. Sheldrick FRS
Dept. Structural Chemistry,
University of Goettingen,
Tammannstr. 4,
D37077 Goettingen, Germany
Tel. +49-551-39-3021 or -3068
Fax. +49-551-39-22582
