| 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 data-collection 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 reciprocal-space radii (d*). The
statistics of both isotropic and anisotropic properties are analyzed in
spherical shells.
|
| Diffraction cut-off | Cut-off 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 cut-offs 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 cut-off surface and included in the statistical
tables and the reflection output file.
|
| Unobserved data | All measured reflections
outside the diffraction cut-off surface and excluded from the
statistical tables and the reflection output file.
|
| Observable data | All unmeasured reflections
inside the diffraction cut-off surface (i.e. expected to be
observed).
|
| Unobservable data | All unmeasured
reflections outside the diffraction cut-off surface (i.e.
expected to be unobserved).
|
| Fitted ellipsoid | Ellipsoid fitted by a
multi-dimensional search and weighted least squares algorithm to the
reflection data immediately neighboring the diffraction cut-off
surface. Used to reduce the number of parameters required to define
the anisotropic diffraction cut-off surface and thereby estimate the
anisotropic diffraction limits and predict observable data.
|
| Debye-Waller factor | Attenuation factor of
each measured intensity and its standard uncertainty due to thermal
motion and/or static disorder:
exp(-4π2 shT U sh),
where sh is the reciprocal-lattice (column)
vector, shT is the same as a row
vector and U is the overall anisotropy tensor (=
B / 8π2).
|
| 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 Debye-Waller factor.
|
| Anisotropy correction factor | The inverse
of the square root of the anisotropic Debye-Waller factor applied as a
multiplicative correction of the amplitudes after Bayesian estimation
from the intensities:
exp(2π2 shT ΔU sh)
where ΔU is the overall anisotropy U tensor after
subtraction of Ueq from its diagonal elements, and
Ueq 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 fault-tolerance,
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
arbitrarily-chosen 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 Ueq:
(Emax - Emin) / Ueq.
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 Σi (Ei - Ueq)2 / Σi Ei2).
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:
maxh (| exp(4π2 shT ΔU sh) - 1 | <Ih / σ(Ih)>).
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 CC½ | The
reciprocal distance-weighted half-dataset Pearson product-moment
correlation coefficient of
I / σ(I). Note that the definition of
CC½ employed here differs materially from that currently
implemented in most other software in that (1) it is reciprocal
distance-weighted 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, CC½ is inappropriate for
anisotropic data because the anisotropy itself will result in
overestimation of CC½ by introducing a false correlation
between the half-datasets. 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 data-dependent, which
means that the significance thresholds for the tests will be different
for every dataset and cannot be known in advance. If the
half-dataset inverse-variance weighted mean intensities are available
(e.g. from MRFANA), CC½ is computed using the
standard formula for the weighted correlation coefficient. Otherwise it is
computed using a modification (i.e. using reciprocal-distance
weighting and I / σ(I) instead of I
as above) of the 'σ-τ' method of Assmann et al.
|
| Fisher transformation of the weighted
CC½ | This uses the standard formula.
|
| Kullback-Leibler divergence (a.k.a. 'relative
entropy') | This uses the standard formula with base 2 logarithms, so its unit
is the binary digit (bit).
|
