How a seemingly innocent question can explode ...
I actually thought I understood this but little of what has been
discussed matches my "mental picture" of the truncate process.
Truncate can do multiple things, but the truncate part I believe really
just deals with converting I to F and the inherent problems due to
experimental error and mathematical problems in deriving SigF from SigI
when I is near zero. This only depends on how close I is to zero
(relative to SigI), and not on the Wilson distribution itself.
My mental picture is as follows:
Visualize a gaussian distribution representing I and its standard
deviation, with I being close to zero (either positive or negative).
Part of the gaussian will stretch into negative-I territory, which is
fine for the experimental I (because of experimental error) but not the
true I. Given this prior knowledge you can re-estimate I by TRUNCATEing
the negative tail of the gaussian and integrating just the positive part
to find the new mean and standard deviation. As a result any reflection
will become positive (including those starting out with negative I). The
extend to which the method affects the intensity depends on how much of
a negative tail it has, so nearly no effect on I/SigI>=2 reflections and
not really that much on even I/SigI=2 reflections.
I actually think this is a very elegant solution. The only thing better,
is to use I directly and avoid the entire issue. I personally think you
want to use the experimental I without correcting it as explained above
since it will introduce bias and the refinement procedure should take
proper care of random experimental error, unless you mess around with
it. However, when you need amplitudes, truncate is the way to go.
Bart
Ian Tickle wrote:
But there's a fundamental difference in approach, the authors here
assume the apparently simpler prior distribution P(I) = 0 for I < 0 &
P(I) = const for I >= 0. As users of Bayesian priors well know this is
an improper prior since it integrates to infinity instead of unity.
This means that, unlike the case I described for the French & Wilson
formula based on the Wilson distribution which gives unbiased estimates
of the true I's and their average, the effect on the corrected
intensities of using this prior really will be to increase all
intensities (since the mean I for this prior PDF is also infinite!),
hence the intensities and their average must be biased (& I'm sure the
same goes for the corresponding F's). But as you say in practice the
errors introduced may well not be significant compared with those
introduced by (for example) deconvoluting the overlapping peaks in the
powder pattern. Also I'm not sure the F vs I argument can be carried
over from the powder to the single crystal case because the kinds of
errors encountered in each case are quite different.
-- Ian
-----Original Message-----
From: [EMAIL PROTECTED]
[mailto:[EMAIL PROTECTED] On Behalf Of [EMAIL PROTECTED]
Sent: 08 September 2008 22:20
To: Jacob Keller
Cc: CCP4BB@JISCMAIL.AC.UK
Subject: Re: [ccp4bb] truncate ignorance
I would also recommend reading of the following paper:
D.S. Sivia & W.I.F. David (1994), Acta Cryst. A50, 703-714. A
Bayesian
Approach to Extracting Structure-Factor Amplitudes from Powder
Diffraction Data.
Despite of the title, most of the analysis presented in this paper
applies equally well to single-crystal data (see especially
sections 3
and 5). If you are not interested in the specific powder-diffraction
problems (i.e. overlapping peaks), you can simply skip
sections 4 and 6.
A few interesting points from this paper :
(1) The conversion from I's to F's can be done (in a Bayesian
way) by
applying two simple formula (equations 11 and 12 in the
paper), which,
for all practical purposes, are as valid as the more complicated
French & Wilson procedure (see discussion in section 5).
(2) Re. the use of I's rather than F's : this is discussed on
page 710
(final part of section 5). The authors seem to be more in favor of
using F's.
Marc Schiltz
Quoting Jacob Keller <[EMAIL PROTECTED]>:
Does somebody have a .pdf of that French and Wilson paper?
Thanks in advance,
Jacob
*******************************************
Jacob Pearson Keller
Northwestern University
Medical Scientist Training Program
Dallos Laboratory
F. Searle 1-240
2240 Campus Drive
Evanston IL 60208
lab: 847.491.2438
cel: 773.608.9185
email: [EMAIL PROTECTED]
*******************************************
----- Original Message -----
From: "Ethan Merritt" <[EMAIL PROTECTED]>
To: <CCP4BB@JISCMAIL.AC.UK>
Sent: Monday, September 08, 2008 3:03 PM
Subject: Re: [ccp4bb] truncate ignorance
On Monday 08 September 2008 12:30:29 Phoebe Rice wrote:
Dear Experts,
At the risk of exposing excess ignorance, truncate makes me
very nervous because I don't quite get exactly what it is
doing with my data and what its assumptions are.
From the documentation:
========================================================
... the "truncate" procedure (keyword TRUNCATE YES, the
default) calculates a best estimate of F from I, sd(I), and
the distribution of intensities in resolution shells (see
below). This has the effect of forcing all negative
observations to be positive, and inflating the weakest
reflections (less than about 3 sd), because an observation
significantly smaller than the average intensity is likely
to be underestimated.
=========================================================
But is it really true, with data from nice modern detectors,
that the weaklings are underestimated?
It isn't really an issue of the detector per se, although in
principle you could worry about non-linear response to the
input rate of arriving photons.
In practice the issue, now as it was in 1977 (French&Wilson),
arises from the background estimation, profile fitting, and
rescaling that are applied to the individual pixel contents
before they are bundled up into a nice "Iobs".
I will try to restate the original French & Wilson argument,
avoiding the terminology of maximum likelihood and
Bayesian statistics.
1) We know the true intensity cannot be negative.
2) The existence of Iobs<0 reflections in the data set means
that whatever we are doing is producing some values of
Iobs that are too low.
3) Assuming that all weak-ish reflections are being processed
equivalently, then whatever we doing wrong for reflections with
Iobs near zero on the negative side surely is also going wrong
for their neighbors that happen to be near Iobs=0 on the positive
side.
4) So if we "correct" the values of Iobs that went negative, for
consistency we should also correct the values that are nearly
the same but didn't quite tip over into the negative range.
Do I really want to inflate them?
Yes.
Exactly what assumptions is it making about the expected
distributions?
Primarily that
1) The histogram of true Iobs is smooth
2) No true Iobs are negative
How compatible are those assumptions with serious anisotropy
and the wierd Wilson plots that nucleic acids give?
Not relevant
Note the original 1978 French and Wilson paper says:
"It is nevertheless important to validate this agreement for
each set of data independently, as the presence of atoms in
special positions or the existence of noncrystallographic
elements of symmetry (or pseudosymmetry) may abrogate the
application of these prior beliefs for some crystal
structures."
It is true that such things matter when you get down to the
nitty-gritty details of what to use as the "expected distribution".
But *all* plausible expected distributions will be non-negative
and smooth.
Please help truncate my ignorance ...
Phoebe
==========================================================
Phoebe A. Rice
Assoc. Prof., Dept. of Biochemistry & Molecular Biology
The University of Chicago
phone 773 834 1723
http://bmb.bsd.uchicago.edu/Faculty_and_Research/01_Faculty/01
_Faculty_Alphabetically.php?faculty_id=123
RNA is really nifty
DNA is over fifty
We have put them
both in one book
Please do take a
really good look
http://www.rsc.org/shop/books/2008/9780854042722.asp
--
Ethan A Merritt
Biomolecular Structure Center
University of Washington, Seattle 98195-7742
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