A weighted mean is only unbiased while the weights are independent of the values
being averaged. The IUCr's own nomenclature report (Schwarzenbach et al., Acta
Cryst A45 (1989) 63-75) puts it directly: weights in averaging "should not be
based on the counting statistics of the individual observations whose estimated
variances are biased and result in larger weights for accidentally low
intensities". Two places in the rotation pipeline were doing exactly that, and
between them they drove whole resolution shells of merged intensity negative.
1. The profile fit computed its non-signal variance as
var_bkg = max(0, 1/den - max(0, I) + bkg-estimate term)
The point of a separate var_bkg is that it does NOT move with the
reflection's own fluctuation, and 1/den - I is the quantity that does not:
1/den is the fit variance taken at the fitted intensity and grows with it
roughly one for one. Clamping the subtrahend at zero left a down-fluctuated
reflection's own deflated variance standing as its background variance.
Measured over 6.9 M partials of one weak rotation dataset, var_bkg/bkg came
out at 3.7-5.4 for observations with I < 0 against 11.4-13.7 for I > 0 - the
down-fluctuated half of every reflection carried a variance ~2.7x too small
and was weighted up by the same factor, first in the 3D combine and then
again in the merge. Removing the clamp makes var_bkg flat in I (~13 x bkg
across the whole range).
2. The merge then weighted each combined full by 1/sigma_full^2, and sigma_full
is by construction a function of the full's own answer: the combine's
variance carries a corr*max(0, F) signal term, so every full with F <= 0 got
the smallest variance the model allows while the strongest quartile got
2.26x more. The merge now rebuilds that variance at the reflection's mean
instead, from a linear model var(I) = var_bkg + var_per_I * I that the
combine measures and stores on the full. This mirrors
MergeOnTheFly::CorrectedSigma, whose comment already claimed to mirror the
rotation combine.
Verified against an estimator that cannot see the fluctuation - summing the
partials and dividing by the summed partiality, the classical construction every
other program uses (Greenhough & Suddath, J. Appl. Cryst. 19 (1986) 400-409, via
Leslie, Acta Cryst D55 (1999) 1696-1702: profile fitting biases the individual
partials but not their sum). Reproducing the merge on dumped observations, the
shipped weighting sat ~1.9 sigma below that reference in the noise shells; the
two changes recover most of it, and every intensity-independent weighting
scheme agrees with the reference once (1) is in.
Four-crystal probe, XDS resolution limits, branch fingerprint identical on all
four (so none of these is a two-pass branch flip):
weak cubic case last shell <I/sig> -1.6 -> +0.2 (XDS +0.10), last shell
R_meas 478% -> 250% (XDS 246%), overall <I/sig> 6.1 -> 7.5
(XDS 7.18), R_meas 18.3% -> 18.1%, CC1/2_hi 38.2% -> 43.7%
tetragonal case outer shells <I/sig> -0.4/-0.8/-0.9/-1.0 -> +1.8/+1.2/
+0.9/+0.4, R_meas 184%/595%/7614%/nan -> 95%/119%/135%/232%
(the nan was the shell mean crossing zero), R_meas 33.3% ->
32.9%, CC1/2_hi 38.3% -> 56.5%
trigonal case R_meas 13.0% -> 12.5%, CC1/2_hi 14.4% -> 16.5%
strong control unchanged to every printed digit but ISa
Cost: ISa falls (17.2 -> 14.0 and 16.7 -> 14.9 on the two mid-strength cases,
28.3 -> 27.8 on the control). Strong reflections are untouched by (1) - their
partials are all positive, so var_bkg is bit-identical - but the joint a/b fit
redistributes: honest weak sigmas lower a, and b rises to keep the strong bins
fitted. The median reduced chi^2 improves (1.25 -> 1.14, 1.35 -> 1.28) so the
new split describes the scatter better, but ISa is the one headline metric that
moves the wrong way and it should be watched over the full battery.
The integrator change is shared, so the stills merge sees it too; there it feeds
GetExpectedVarianceMerge, which had been handed the same contaminated var_bkg.
That path is untested here.
Co-Authored-By: Claude Opus 5 (1M context) <noreply@anthropic.com>
leonarski_fandClaude Opus 5
d8029524e7