Thanks Joe and others.
Bits and pieces of this story appear in 11.4.8 of International
Tables volume F, Borek et. al. Acta Cryst D59 (2003) and the
Scalepack manual, but none are complete or have enough detail to
follow easily. None of them give the expression for Chi-square for
this problem.
I found a presentation by Jay Ponder online (for his Bio5325 course)
that gives:
chi-2 = 1/N sum (I_avg - I_meas)^2/(sigma_avg^2 + sigma_meas^2)
where the sum probably runs over all reflections and the I_avg is the
average of the appropriate group of symmetry-related reflections.
Sigma_avg^2 should be the sigma computed from the error model below
(not given in the presentation) I think and sigma_meas is the sigma^2
from the actual symmetry-related reflections.
One would then adjust the error parameters below to give chi-square
approx unity and this leads to the proper scaling factors for
intensities and sigmas.
One confusing hitch seems to be that (according to the International
Tables F Eqs.(11.4.8.5) and (11.4.8.6)), the error model is also
implicitly defined and must be solved iteratively ... though it's
hard to see that from the text.
Does this sound right?
Richard
On Jul 17, 2009, at 4:12 PM, [email protected] wrote:
Dear Richard,
I *think* it works like this, don't know if it's detailed or rigourous
enough for you!
If I(l,h,i) is the intensity of the ith observation of reflection h on
frame l, with error sig(l,h,i), and S(l) is the (inverse) scale
factor to
be applied to frame l, then the error in the scaled intensity
I(l,h,l)/S(l) is parameterised in terms of the error scale factor
(E1) and
estimated error (E2) as
sqrt( (E1*sig(l,h,i))**2 + (E2*I(h)*S(l))**2 )
where I(h) is the weighted mean of the scaled intensity values for
index h
(i.e. the merged, scaled intensity).
The reasoning behind this that the errors in the intensities of
strong and
weak reflections generally arise from different sources. Weak data are
noisy, whilst very strong data can often be systematically badly
measured,
especially in DENZO, which assumes that all your spots are the same
size
and shape, or overloaded. E1 and E2 thus tend to dominate the error
model
at high and low resolutions, respectively.
Hope that helps,
Joe
Does anyone know of a detailed rigorous discussion of how the
scalepack error model/Bayesian reasoning works? The scalepack manual
has no equations for this.
Richard Gillilan
MacCHESS
