# Towards Generalized Noise-Level Dependent Crystallographic Symmetry Classifications of More or Less Periodic Crystal Patterns

## Abstract

**:**

## 1. Introduction and Background

## 2. Kanatani’s Comments on Symmetry as a Continuous and Hierarchic Feature

## 3. Kanatani’s Dictum

^{1}/

_{3}, and x,

^{2}/

_{3}pseudo-mirror lines that intersect the while blobs vertically. In order to avoid overcrowding, these pseudo-mirrors are only given as dotted yellow lines in Figure 2b.

^{1}/

_{6}, x,

^{3}/

_{6}, and x,

^{5}/

_{6}, which intersect the black background areas between the white blobs in the middle vertically. The above-mentioned genuine mirror lines 0,y and ½,y (of Figure 1a and Figure 2a) combine with the perpendicular pseudo-mirror lines as drawn into Figure 2b. This generates pseudo-four-fold and pseudo-two-fold rotation points at the crossings of genuine mirror lines and pseudo-mirror lines so that Fedorov pseudosymmetry group p

_{b/3}4mm ⊃ p1m1 results on the basis of the rectangular Bravais lattice. The pseudo-four-fold rotation points contain in themselves pseudo-two-fold rotation points. There are alternatives to construct the Fedorov pseudosymmetry group that Figure 1a and Figure 2a,b possess, but they all lead to the same end result.

_{b/3}4mm, which carry pseudo-Wyckoff letters a and b, and their locations with respect to the pseudo-square lattice are 0,0 and ½,½, respectively. As a result of the combination of the genuine symmetries and pseudo-symmetries in Figure 1a, the genuine rectangular lattice of Figure 1a and Figure 2a is “truncated” into a pseudo-square lattice as outlined in Figure 2b.

## 4. Geometric Akaike Information Criteria (G-AICs)

#### 4.1. General Considerations

#### 4.2. G-AIC for Plane and Frieze Symmetry Groups

_{more}, to the least-squares residual of a less symmetric (more general) model (S

_{less}) that is non-disjoint:

#### 4.3. G-AIC for 2D Laue Classes

#### 4.4. G-AIC for 2D Bravais Lattice Types

#### 4.5. When is a Noise Level Estimate Mandatory?

## 5. Utilizing Geometric Akaike Information Criteria

#### 5.1. Highlights of the Underlying Information Theory

_{best}represents the best model in the set [10,11,12]. The hat over the sigma means that it is an estimator.

_{i}within a set:

_{i}values of the i models in either a disjoint or a non-disjoint set matter for crystallographic symmetry classifications.

_{best}.

_{i}is the probability that model i is the Kullback-Leibler best model. Akaike weights for a subset of models are additive and can be summed into confidence sets [13]. Obviously, the sum of all Akaike weights in the full set is 100%. While summing into confidence sets is somewhat subjective, there is certainly no arbitrariness in the usage of the equations and inequalities of this review. Note that the alternative name of Akaike weights, i.e. Bayesian posterior model probabilities, does not imply that the information theory approach is to be considered a “Bayesian approach” to statistics. As it can also not be classified as frequentist, it represents something new because it combines the positive features of both approaches while being essentially free of their negative features.

_{i}models’ within a set [13,14,15,16,17]. Model parameters are in the context of this review the values of the unit cell parameters in direct and reciprocal space, the discrete Fourier coefficient amplitude and phase angles of the image intensity that form (in reciprocal space) the ‘Fourier equivalent’ [126] of the asymmetric unit of a plane symmetry group, and the gray level values of the group of individual pixels that collectively form the asymmetric unit in direct space.

_{i}is the ith model and the extended notation ${\widehat{\theta}}_{i}|{g}_{i}$ clarifies that the parameter or prediction estimator is in each of the R cases specific to a model in the set.

_{q}, exists (in a Bayesian sense [13]) that is best at representing complementary (other) aspects of the finite information in the image data, one is justified to obtain ‘updated’ Bayesian posterior model probabilities by an extension of Equation (8) to:

#### 5.2. Updated Bayesian Posterior Model Probabilities and Confidence Sets for Crystallographic Symmetry Classifications

#### 5.2.1. Illustration of the Updated Bayesian Posterior Model Probabilities Idea

#### 5.2.2. Illustration of the Idea of Rescaled Confidence Sets over Stretches of Crystallographic Symmetry Hierarchy Branches

#### 5.3. Multi-Model Averaging for Better Predictions and Safer Conclusions

_{i}= 0) is the one that does this job best in the sense that it minimizes the expected Kullback-Leibler information loss. Since one ends up with several model descriptions in the information theory inspired approach of this review while following objective criteria, one also has several sets of unit cell parameters and pixel intensities for the asymmetric units, which one may call, collectively, individual model parameters.

_{i}, as defined in Equation (8). The main advantage of multi-model averaging is summed up by Akaike himself in his statement: “If the choice of one single model is not the sole purpose of the analysis of the data the average of the models with respect to the approximate posterior probability C exp {(½) AIC(k)} will provide a better estimate of the true distribution of Y.” [125], where C represents a constant, AIC(k) represents the Δ

_{i}values of a set of models, and Y stands for a set of observations.

## 5.4. Acknowledging Model Selection Uncertainties in Qualitative Ways

_{i}= 0 model is not much stronger than that for the Δ

_{i}> 0.

## 6. Summary and Conclusions

## Acknowledgments

## Conflicts of Interest

## Appendix A. Opportunities and Background in the Context of 1D Periodic 2D Images as Obtained from Plane Edge-On Projected 3D Grain Boundaries

_{3}accommodated significantly different amounts of dopants that substituted for titanium close to the interface. These results are statistically significant as several tens of STEM images with the same frieze symmetry along the same grain boundary containing several hundreds of sectioned CSL unit cells were averaged in order to enhance the image contrast and obtain representative atomic arrangements for the differently-sectioned CSL unit cells. For a related study on the same type of grain boundary with the same types of frieze symmetries in undoped high-purity SrTiO

_{3}, the images of approximately 400 sectioned CSL unit cells in approximately 50 STEM images were averaged in direct space in order to reveal the characteristic atomic arrangements around the interface for each of the different frieze symmetry types [37]. Note that the orientation relationship and crystallographic interface plane, i.e. the five macroscopic parameters, of the grain boundaries in all of these related studies were highly precise because they were prepared with the high-temperature diffusion bonding technique [37,41].

## Appendix B. Pseudo-Symmetries

_{i}

_{,q}pairs’ and ‘w

_{i}

_{,q,s}triples’ of Akaike weights for Bravais lattice types and plane symmetry groups, as well as for Laue classes, will be high in Equations (13) and (14) in Section 5.1 of the main body of this review because each of the individual probability factors will be large. Such matching pairs and triples are defined by mutual crystallographic compatibility conditions.

## Appendix C. Pseudo-Symmetries and Other Problems in Mainstream 3D Crystallography that Have Led to Misclassified Entries in Major Crystallographic Databases

#### Appendix C.1. Space Group Symmetry Misclassifications in Major Crystallographic Databases

_{a}

_{/2}= 0.99, see [141]) combination of translational pseudo-symmetry with motif-based pseudo-symmetry of the electron density of KCl in Fedorov pseudosymmetry group ${P}_{a/2}m\overline{3}m$ ⊂ $Fm\overline{3}m$ (see [123] and Appendix B) and an eight times smaller unit cell. In other words, this Fedorov pseudosymmetry group is broken by only 1% so that Sir William Laurence Bragg’s classification was loosely speaking 99% correct while actually being technically completely incorrect. ‘Ambiguous’ space group assignments account for about one half of the discussed cases in [65] and this appendix will be mainly concerned with analogous misclassifications of crystallographic symmetries.

_{free}, the cross validation (CV) statistic introduced in 1992 on which so much reliance is placed today” and (ii) issues a “call to arms to the entire structural biology community so that the important, but entirely correctable problems” which that paper discusses can be resolved as far as this is possible given Kanatani’s dictum [9,10]. While the above-mentioned CV index R

_{free}is described in detail in [56], the review by Jones provides background on the most commonly-used R value and its weighted form (R

_{w}) [90]. Hamilton introduced generalized weighted R values (R

^{G}and R’’) in order to facilitate null hypothesis tests concerning the question if the addition of refinement parameters enhances the validity of a structural model in a statistically significant manner [98,99].

_{3}etc., should remain as they are classified right now, i.e. assigned to one crystallographic symmetry class only as there is no uncertainty to which class they truly belong. Utilizing the combined Akaike weight concept, these structures have been classified with likelihoods exceedingly close to 100% so that there is neither a need nor a basis to spread their entry over several crystallographic classes. All of these structural prototypes are highly symmetric and a thorough review [57] revealed that inorganic materials with very high crystal symmetries are rarely misclassified.

_{1}values may be in the 15% to 30% range depending on the resolution of the data and the amount of solvent that remained within the crystals. Small-molecule crystal structures with up to approximately 200 independent atoms, on the other hand, should be refined to R

_{1}≤ 7% with an ‘allowance’ for disorder of an extra 0.5% [147].

#### Appendix C.2. Reasons Why R Values and Similar “Pure Distance Measures” Are Not Helpful in Crystallographic Symmetry Classifications

_{obs}and

_{cal}stand for “observed” and “calculated”, respectively. Note that using the square root of the reflection intensity introduces non-linearity into otherwise linear least-squares refinements and Hamilton’s test. The normalized sum of Equation (A1) is often multiplied by 100% and the R value that is defined by this equation is also referred to as R

_{1}.

_{2}Si

_{2}O

_{7}, weighted R values (on the basis of the normalized differences of observed and calculated sums of structure factor squares) of 3.25%, 2.83%, and 2.79% for space groups C2/m, Cm, and C2, respectively.

_{I}

_{4}= 19.5%, can be higher than its counterpart, e.g., R

_{I}

_{422}= 18.9%, for that space group as demonstrated by two refinements of a protein structure from the same low-resolution single-crystal X-ray diffraction data [152].

_{1}, wR

_{1}, wR

_{2}, and goodness of fit on F

^{2}values [153].

#### Appendix C.3. Suspected Pseudo-Symmetry Mediated Space Group Symmetry Misclassification of a Highly Topical Material

_{1}value of the single-crystal X-ray crystallography study (on the basis of all hexagonally-indexed reflections) was reported to be 13.17% [101], but was probably so low only because the SQUEEZE function [107] of the well-known OLEX2 [108] program had been utilized to remove a significant amount of experimentally-observed electron density from the mesoscopic channels in NU-1000 during the solving and refining of this structure [101,102].

_{1}≤ 7% as the criterion for a reasonably well-refined small molecule crystal structure in the year 2000 [147]. Participants of a conference on the Critical Evaluation of Chemical and Physical Structure Information considered non-biopolymer crystal structures with R

_{1}> 10% as “suspect” already in 1974 [65]. The goodness of fit on the square of the structure factor amplitudes was for NU-1000 as high as 1.737 [101], while it should ideally have been close to unity [153]. It is, therefore, somewhat doubtful if these two quantitative measures for the alleged ‘model disagreement/correctness’ of the published structure of NU-1000 and the removal of observed electron density by electronic means can lead to “that ‘warm happy feeling’ of confidence in the validity of the scientific work and the results presented” that the participants of the above-mentioned conference were talking about [68,154].

_{α}radiation.

_{88}H

_{44}O

_{32}Zr

_{6}asymmetric unit of this MOF is probably also underreported as far as its chemical composition is concerned. This is because more than one half of the experimentally-observed electron density per unit cell has been removed from the single-crystal X-ray crystallography analysis with the SQUEEZE function [107] of the OLEX2 software [108] (as already mentioned above).

## Appendix D. Standard Statistical Descriptors and Utility of Contemporary Null Hypothesis Tests in Mainstream 3D Crystallography

^{st}century Statistical Science Decalogue” [159] (with italics in the original for emphasis), they are surely right. In order to make progress, one needs to be pragmatic and accept that one is only testing ‘approximate hypotheses’ [156,157,159] (rather than exact hypotheses that can be specified precisely as, for example, by a certain real number [160] or a single symmetry type, class, or group). One will, in the real world, therefore, never arrive at a definitive conclusion/classification by means of a completely objective procedure when nested models are involved. The good news is that such definiteness is not required for the sake of making progress within the wider scientific (Lakatos) research program to which one is prescribing.

## Appendix E. Comments on the Experimental Studies of Liu and Co-Workers

_{more}> k

_{less}, both variables are positive integers, and k

_{less}> 1.

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**Figure 1.**(

**a**) Image with plane symmetry group p1m1 (pm for short when there is no need to communicate the crystallographic setting) that possesses genuine pseudo-symmetries per design which are in (

**b**) exacerbated by added independent Gaussian noise of mean zero and a standard deviation of 10% of the maximal image intensity. The translation symmetry in (

**a**) is visibly of the rectangular (primitive) Bravais lattice type [7]. In the noisy image (

**b**), the translation symmetry is apparently of the square Bravais lattice type. Both images are in open access [122] and are reproduced here with CC-BY (share—copy and redistribute the material in any medium or format adapt—remix, transform, and build upon the material for any purpose, even commercially) licenses. The labeling of the images with the letters a and b and the outline of one unit cell by a yellow rectangle in (

**a**) are the only modifications that were made. The directions X and Y refer to the edges of both images and are parallel to the unit cell edges x,0 and 0,y (that are defined by the 2D lattice vectors $\overrightarrow{a}$ and $\overrightarrow{b}$).

**Figure 2.**Choices of unit cells that take the prevailing pseudo-symmetries in Figure 1a into account, i.e. which fix the unit cell origins at four-fold pseudo-rotation points. Only the genuine symmetry operations of plane symmetry group p1m1, i.e. the mirror lines 0,y, ½,y, (and 1,y) are highlighted by full yellow lines on the left-hand side in subfigure (

**a**). Subfigure (

**b**), on the right hand side, shows, in addition, pseudo-mirrors as dotted yellow lines that intersect with the genuine mirrors to create two-fold and four-fold pseudo-rotation points as well as a pseudo-square unit cell with three repeats. (Additional pseudo-mirror and pseudo-glide lines are generated by the combination of these genuine pseudo-symmetry operations with the genuine mirror lines, but their locations are not specifically marked.) Fedorov pseudosymmetry [123] group p

_{b/3}4mm ⊃ p1m1 arises as a result of these combinations, as can be straightforwardly seen in (

**b**). The area of one pseudo-unit cell of the pseudo-square type in (

**b**) is just one-third of the genuine rectangular unit cell in (

**a**).

**Figure 3.**Outcomes of the fuzzy plane symmetry classification of three more or less 2D periodic images that possess plane symmetry group pm per design and a combination of translational pseudo-symmetry with a motif-based (four-fold rotation points plus mirror lines) pseudo-symmetry in addition to varying amounts of recording and processing noise. Each of the three individual sketches (at the

**left**, in the

**middle**, and at the

**right**) corresponds to one of three images. Note that the given percentages represent the joint probability that a particular plane symmetry group and Bravais lattice type are the ones which minimize the expected Kullback-Leibler information loss utilizing Equations (13) and (14) rescaled to sum up to 100% in each sketch. For simplicity, these percentages are assumed, rather than derived from experimental data, but this suffices for the illustration of the core idea of updated Bayesian posterior model probabilities confidence set over stretches of plane symmetry hierarchy branches. For added visual appeal, the areas of the displayed unit cells were scaled proportional to the (rescaled) numerical model probability product percentages. The unit cell shapes were chosen in accordance with the prevailing Bravais lattice types. The actual metric of the unit cells and their respective content are utterly irrelevant for the illustration of the confidence set idea.

**Table 1.**Letter key for references to the three algorithms/computer programs for which we will quote quantitative results [114] in this review.

Algorithm’s Number in the Final Reference Section | Algorithm’s Letter Reference in this Review |
---|---|

[115] | A |

[116] | B |

[117] | C |

**Table 2.**Extracted lattice parameters from the noise-free image in Figure 1a and derived unit cell areas utilizing the default settings of three computer programs. The qualitatively correct result is marked in bold font.

Algorithm’s Reference Letter | a/b | γ in ° | Unit Cell Area in Square Pixels |
---|---|---|---|

A | 0.333 ± 0.06 | 90.0 ± 0.4 | 300.0 ± 2.0 |

B | 1.004 ± 0.01 | 90.0 ± 0.05 | 99.4 ± 1.0 |

C | 0.998 ± 0.01 | 90.0 ± 0.05 | 99.8 ± 1.0 |

**Table 3.**Extracted lattice parameters from the noise-free image in Figure 1a and derived unit cell areas after a re-interpretation of the results from Algorithm B and as obtained in a non-default setting of Algorithm C. Both results are qualitatively correct and, therefore, marked in bold font.

Algorithm’s Reference Letter | a/b | γ in ° | Unit Cell Area in Square Pixels |
---|---|---|---|

B | 0.335 ± 0.01 | 90.0 ± 0.05 | 298.2 ± 3.0 |

C | 0.333 ± 0.02 | 90.0 ± 0.05 | 300.0 ± 2.0 |

**Table 4.**Extracted lattice parameters from the noisy image in Figure 1b and derived unit cell areas utilizing the default settings of three different computer programs. There is no qualitatively correct result to be marked in bold font.

Algorithm’s Reference Letter | a/b | γ in ° | Unit Cell Area in Square Pixels |
---|---|---|---|

A | 1.414 ± 0.07 | 135.8 ± 0.6 | 98.6 ± 1.5 |

B | 0.995 ± 0.01 | 90.2 ± 0.05 | 99.1 ± 1.0 |

C | 1.000 ± 0.01 | 90.0 ± 0.05 | 100.0 ± 1.0 |

**Table 5.**Extracted lattice parameters from the noisy image in Figure 1b and derived unit cell areas after a re-interpretation of the results from Algorithm B and as obtained in a non-default setting of Algorithm C. Both results are qualitatively correct and, therefore, marked in bold font.

Algorithm’s Reference Letter | a/b | γ in ° | Unit Cell Area in Square Pixels |
---|---|---|---|

B | 0.332 ± 0.01 | 90.2 ± 0.05 | 297.3 ± 3.0 |

C | 0.333 ± 0.02 | 90.0 ± 0.05 | 300.0 ± 2.0 |

**Table 6.**Number of constraints that enter Inequality (2) in a G-AIC for the fuzzy classification into Bravais lattice types.

Parallelogram | Rectangle | General Rhombus | Square | Hexagonal Rhombus |
---|---|---|---|---|

2 | 3 | 3 | 4 | 4 |

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Moeck, P.
Towards Generalized Noise-Level Dependent Crystallographic Symmetry Classifications of More or Less Periodic Crystal Patterns. *Symmetry* **2018**, *10*, 133.
https://doi.org/10.3390/sym10050133

**AMA Style**

Moeck P.
Towards Generalized Noise-Level Dependent Crystallographic Symmetry Classifications of More or Less Periodic Crystal Patterns. *Symmetry*. 2018; 10(5):133.
https://doi.org/10.3390/sym10050133

**Chicago/Turabian Style**

Moeck, Peter.
2018. "Towards Generalized Noise-Level Dependent Crystallographic Symmetry Classifications of More or Less Periodic Crystal Patterns" *Symmetry* 10, no. 5: 133.
https://doi.org/10.3390/sym10050133