Isotropy  The property of being
independent of direction, implying no variation of specified physical
properties with direction.

Anisotropy  The property of being
directionally dependent, implying variation of specified physical
properties with direction.

Diffraction limit  The isotropic or
anisotropic limit on average of statistically significant data (also
known as the 'resolution limit'), depending on both the
properties of the crystal and the datacollection strategy.

Threshold  Number used with a test variable
and a relational operator to construct a decision criterion for
rejection of data. Example: if the test variable is
'mean I / σ(I)', the relational operator
is 'greater than' and the threshold value is 2, then the criterion
is 'mean I / σ(I) > 2'.

Spherical shell  Region of reciprocal space
lying between two defined reciprocalspace radii (d*). The
statistics of both isotropic and anisotropic properties are analyzed in
spherical shells.

Diffraction cutoff  Cutoff applied to all
diffraction data outside a single closed isotropic or anisotropic
surface in reciprocal space defining the assumed boundary between, on
average, statistically significant and insignificant data. In the
case of an isotropic surface only one parameter is needed to define it,
i.e. the radius of the limiting sphere. In the general case
the surface is anisotropic and one parameter (or even a small number of
parameters) is not sufficient.

Possible data  All reflections inside the
greatest value of d* for diffraction cutoffs in all directions,
regardless of whether or not an intensity was measured (systematic
absences are excluded everywhere).

Measured data  All possible reflections with
an experimental intensity and standard uncertainty.

Unmeasured data  All possible reflections
without an experimental intensity and standard uncertainty.

Observed data  All measured reflections
inside the diffraction cutoff surface and included in the statistical
tables and the reflection output file.

Unobserved data  All measured reflections
outside the diffraction cutoff surface and excluded from the
statistical tables and the reflection output file.

Observable data  All unmeasured reflections
inside the diffraction cutoff surface (i.e. expected to be
observed).

Unobservable data  All unmeasured
reflections outside the diffraction cutoff surface (i.e.
expected to be unobserved).

Fitted ellipsoid  Ellipsoid fitted by a
multidimensional search and weighted least squares algorithm to the
reflection data immediately neighboring the diffraction cutoff
surface. Used to reduce the number of parameters required to define
the anisotropic diffraction cutoff surface and thereby estimate the
anisotropic diffraction limits and predict observable data.

DebyeWaller factor  Attenuation factor of
each measured intensity and its standard uncertainty due to thermal
motion and/or static disorder:
exp(4π^{2} s_{h}^{T} U s_{h}),
where s_{h} is the reciprocallattice (column)
vector, s_{h}^{T} is the same as a row
vector and U is the overall anisotropy tensor (=
B / 8π^{2}).

Intensity prior  Expected intensity,
estimated from a spherical intensity distribution obtained from
experimental data or from a model, combined with the symmetry
enhancement factor and the isotropic or anisotropic DebyeWaller factor.

Anisotropy correction factor  The inverse
of the square root of the anisotropic DebyeWaller factor applied as a
multiplicative correction of the amplitudes after Bayesian estimation
from the intensities:
exp(2π^{2} s_{h}^{T} ΔU s_{h})
where ΔU is the overall anisotropy U tensor after
subtraction of U_{eq} from its diagonal elements, and
U_{eq} is the isotropic equivalent U (mean of the
eigenvalues), so that the trace of ΔU is zero.

Completeness  Number of data as a fraction
of the number of possible data in any defined volume of reciprocal
space (e.g. within the intersection of a spherical shell with the
fitted ellipsoid).

Redundancy  The count of the measurements of
a reflection that contribute to the final average, either as exact
repeats of the Laue indices, or as
equivalent Laue indices in the Laue class
of the crystal. The purpose of redundancy here is faulttolerance,
i.e. improvement of the reliability of the averaged intensity, in
whatever way that averaging is performed, possibly with outlier
rejection.

Anomalous redundancy  If anomalous
scattering is present, the counts of contributing equivalent reflections
whose Laue indices are related only by a proper rotation to the
arbitrarilychosen Laue indices in the asymmetric unit of the Laue class
may be counted separately from reflections whose Laue indices are
related by a combination of proper rotation with inversion ('Bijvoet
equivalents'). For example, these are the columns labelled 'N(+)'
and 'N()' respectively in a merged MTZ file, i.e. total
redundancy = N(+) + N().

(Anomalous) multiplicity  As (anomalous)
redundancy. Beware that it may also refer to other ways in
which multiple instances of a reflection's Laue indices may be counted,
depending on the context.

Anisotropy ratio  Difference between the
largest and smallest eigenvalues of the overall anisotropy U
tensor divided by U_{eq}:
(E_{max}  E_{min}) / U_{eq}.
This metric lies between zero (isotropic) and 3 (maximally anisotropic).

Fractional anisotropy  The RMS deviation of
the eigenvalues from the mean, scaled to lie between zero and one:
√(1.5 Σ_{i} (E_{i}  U_{eq})^{2} / Σ_{i} E_{i}^{2}).
Unlike the anisotropy ratio this metric takes into account all 3
eigenvalues.

Anisotropic S/N ratio  The maximum value
over all observed reflections of the absolute deviation of the squared
anisotropy correction factor from 1, multiplied by the local mean
intensity / standard uncertainty ratio:
max_{h} ( exp(4π^{2} s_{h}^{T} ΔU s_{h})  1  <I_{h} / σ(I_{h})>).
Unlike the anisotropy ratio and the fraction anisotropy this takes into
account both the fact that large anisotropies represent larger
differences in intensity at high d* and that there will be a
greater contribution from reflections with a high value of the local
mean I / σ(I). This metric is zero
in the isotropic case, with no limit in the anisotropic case.

Weighted CC_{½}  The
reciprocal distanceweighted halfdataset Pearson productmoment
correlation coefficient of
I / σ(I). Note that the definition of
CC_{½} employed here differs materially from that currently
implemented in most other software in that (1) it is reciprocal
distanceweighted and (2) I / σ(I) is used
in place of I. This is because the significance test for the
Pearson correlation coefficient is appropriate for use only in very
tightly defined situations and consequently its use in the current
context as generally implemented is inconsistent with some of the assumptions used in its derivation. In particular
the assumptions of normally distributed variables and homoscedascity
(uniform variance: the variance of I in a typical dataset spans
several orders of magnitude whereas the variance of
I / σ(I) is always 1 by definition) are not
satisfied. In any case, CC_{½} is inappropriate for
anisotropic data because the anisotropy itself will result in
overestimation of CC_{½} by introducing a false correlation
between the halfdatasets. The normality assumption is not strictly
required since that is a property of the probability model and
significance tests applicable to all datasets can in principle be
derived for the Wilson distributions. However, the heteroscedascity
and anisotropy of the Is are obviously datadependent, which
means that the significance thresholds for the tests will be different
for every dataset and cannot be known in advance. If the
halfdataset inversevariance weighted mean intensities are available
(e.g. from MRFANA), CC_{½} is computed using the
standard formula for the weighted correlation coefficient. Otherwise it is
computed using a modification (i.e. using reciprocaldistance
weighting and I / σ(I) instead of I
as above) of the 'στ' method of Assmann et al.

Fisher transformation of the weighted
CC_{½}  This uses the standard formula.

KullbackLeibler divergence (a.k.a. 'relative
entropy')  This uses the standard formula with base 2 logarithms, so its unit
is the binary digit (bit).
