average-minimum-distance

average-minimum-distance: geometry based crystal descriptors

• PyPI project: https://pypi.org/project/average-minimum-distance
• Documentation: https://average-minimum-distance.readthedocs.io
• Source code: https://github.com/dwiddo/average-minimum-distance
• References (bib references at the bottom of this page):
  • Average minimum distances of periodic point sets - foundational invariants for mapping periodic crystals. MATCH Communications in Mathematical and in Computer Chemistry, 87(3):529-559 (2022). https://doi.org/10.46793/match.87-3.529W
  • Resolving the data ambiguity for periodic crystals. Advances in Neural Information Processing Systems (NeurIPS 2022), v.35. https://openreview.net/forum?id=4wrB7Mo9_OQ.

What's amd?

Average minimum distances (AMD) and pointwise distance distributions (PDD) are geometry-based crystal descriptors designed to have desirable properties such as independence from choice of a unit cell and continuity under perturbations of points.

The typical representation of a crystal as a motif and unit cell is ambiguous, because many choices of cell and motif define the same crystal. This package implements descriptors which are isometry invariants, meaning they are always the same for any two crystals which are geometrically equivalent, independent of a choice of unit cell and motif. AMDs/PDDs can be compared to give a distance between crystals, which is 0 for identical crystals and close to 0 for similar crystals (a continuous metric).

The pointwise distance distribution (PDD) records the environment of each atom in a unit cell by listing distances to neighbouring atoms in order. Two PDDs are compared using an optimal matching algorithm (Earth Mover's distance). Taking the average of a PDD gives a vector called the average minimum distance (AMD), which are simpler and faster to compare but still identify crystals with similar geometry. Both have one parameter k, equal to the number of neighbouring atoms considered for each atom in the unit cell.

Getting started

Use pip to install average-minimum-distance:

pip install average-minimum-distance

Then import average-minimum-distance with import amd.

amd.compare() compares crystals in cif files by AMD or PDD descriptors:

import amd

# compare items in file.cif pairwise by AMD, k=100
dm = amd.compare('file.cif', by='AMD', k=100)
# compare items in file1.cif vs file2.cif by PDD, k=100
dm = amd.compare('file1.cif', 'file2.cif', by='PDD', k=100)

The distance matrix returned is a pandas DataFrame. amd.compare() can also accept paths to folders or lists of paths.

amd.compare() reads crystals from CIFs, calculates their descriptors and compares them, but these steps can be done separately if needed (see below). amd.compare() accepts several optional parameters, see the documentation for a full list.

CSD Python API only: amd.compare() interfaces with csd-python-api. It can accept one or more CSD refcodes if passed refcode_families=True or other file formats instead of cifs if passed reader='ccdc'.

Choosing a value of k

The parameter k is the number of neighbouring atoms considered for each atom in a unit cell. Two crystals with the same unit molecule will have a small AMD/PDD distance for small enough k (e.g. k = 5), and a larger k means the geometry must be similar up to a larger radius for the distance to be small. The default for amd.compare() is k = 100, but if this is significantly less than the number of atoms in the unit molecule, it may be better to choose a larger value. It is usually not useful to choose k too large (many times larger than the number of atoms in a unit cell).

Reading crystals, calculating AMDs/PDDs

This code reads a cif file and computes the list of AMDs (k = 100):

import amd

reader = amd.CifReader('file.cif')
amds = [amd.AMD(crystal, 100) for crystal in reader]
# # To calculate the PDDs:
# pdds = [amd.PDD(crystal, 100) for crystal in reader]

CifReader accepts some optional parameters, e.g. for removing Hydrogen and handling disorder, see here for a full list.

CSD Python API only: CSD entries can be accessed via the CSD Python API with amd.CSDReader, see the documentation for details. CifReader can accept file formats other than .cif by passing reader='ccdc'.

Comparing by AMD or PDD

To compare all crystals in one collection with each other, use amd.AMD_pdist() or amd.PDD_pdist(), which accept a list of AMDs/PDDs and return a condensed distance matrix like SciPy's pdist(). Here's a full example of reading crystals from a .cif, calculating the descriptors and comparing them:

import amd

# read and calculate AMDs and PDDs (k = 100)
crystals = list(amd.CifReader('path/to/file.cif'))
amds = [amd.AMD(crystal, 100) for crystal in reader]
pdds = [amd.PDD(crystal, 100) for crystal in reader]

amd_cdm = amd.AMD_pdist(amds) # compare AMDs pairwise
pdd_cdm = amd.PDD_pdist(pdds) # compare PDDs pairwise

# Use squareform for a symmetric 2D distance matrix
from scipy.distance.spatial import squareform
amd_dm = squareform(amd_cdm)

Note: AMDs can be quickly computed from PDDs with amd.PDD_to_AMD().

The default metric for comparison is chebyshev (L-infinity), though it can be changed to anything accepted by SciPy's pdist, e.g. euclidean.

To compare crystals in one set with those in another set, use amd.AMD_cdist or amd.PDD_cdist:

import amd

amds1 = [amd.AMD(c, 100) for c in amd.CifReader('set1.cif')]
amds2 = [amd.AMD(c, 100) for c in amd.CifReader('set2.cif')]
# dm[i][j] = AMD distance between amds1[i] & amds2[j]
dm = amd.AMD_cdist(amds)

Example: AMD-based dendrogram

This example compares some crystals in a cif by AMD (k = 100) and plots a single linkage dendrogram:

import amd
import matplotlib.pyplot as plt
from scipy.cluster import hierarchy

crystals = list(amd.CifReader('crystals.cif'))
names = [crystal.name for crystal in crystals]
amds = [amd.AMD(crystal, 100) for crystal in crystals]
cdm = amd.AMD_pdist(amds)
Z = hierarchy.linkage(cdm, 'single')
dn = hierarchy.dendrogram(Z, labels=names)
plt.show()

Cite us

Use the following bib references to cite AMD or PDD.

Average minimum distances of periodic point sets - foundational invariants for mapping periodic crystals. MATCH Communications in Mathematical and in Computer Chemistry, 87(3), 529-559 (2022). https://doi.org/10.46793/match.87-3.529W.

@article{widdowson2022average,
  title = {Average Minimum Distances of periodic point sets - foundational invariants for mapping periodic crystals},
  author = {Widdowson, Daniel and Mosca, Marco M and Pulido, Angeles and Kurlin, Vitaliy and Cooper, Andrew I},
  journal = {MATCH Communications in Mathematical and in Computer Chemistry},
  doi = {10.46793/match.87-3.529W},
  volume = {87},
  number = {3},
  pages = {529-559},
  year = {2022}
}

Resolving the data ambiguity for periodic crystals. Advances in Neural Information Processing Systems (NeurIPS 2022), v.35. https://openreview.net/forum?id=4wrB7Mo9_OQ.

@inproceedings{widdowson2022resolving,
  title = {Resolving the data ambiguity for periodic crystals},
  author = {Widdowson, Daniel and Kurlin, Vitaliy},
  booktitle = {Advances in Neural Information Processing Systems},
  year = {2022},
  url = {https://openreview.net/forum?id=4wrB7Mo9_OQ}
}