Anisotropy of the Diffraction Limit
Bayesian Estimation of Structure Amplitudes

GLOSSARY of terminology related to anisotropy:

IsotropyThe property of being independent of direction, implying no variation of specified physical properties with direction.
AnisotropyThe property of being directionally dependent, implying variation of specified physical properties with direction.
Diffraction limitThe 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.
ThresholdNumber 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 shellRegion 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-offCut-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 dataAll 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 dataAll possible reflections with an experimental intensity and standard uncertainty.
Unmeasured dataAll possible reflections without an experimental intensity and standard uncertainty.
Observed dataAll measured reflections inside the diffraction cut-off surface and included in the statistical tables and the reflection output file.
Unobserved dataAll measured reflections outside the diffraction cut-off surface and excluded from the statistical tables and the reflection output file.
Observable dataAll unmeasured reflections inside the diffraction cut-off surface (i.e. expected to be observed).
Unobservable dataAll unmeasured reflections outside the diffraction cut-off surface (i.e. expected to be unobserved).
Fitted ellipsoidEllipsoid 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 factorAttenuation 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 priorExpected 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 factorThe 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.
CompletenessNumber 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).
RedundancyThe 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 redundancyIf 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) multiplicityAs (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 ratioDifference 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 anisotropyThe 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 ratioThe 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 CC½The 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 CC½This 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).