# Source code for icet.tools.convex_hull

```
"""
This module provides the ConvexHull class.
"""
import itertools
import numpy as np
from typing import List, Union
from scipy.interpolate import griddata
from scipy.spatial import ConvexHull as ConvexHullSciPy
from scipy.spatial.qhull import QhullError
[docs]class ConvexHull:
"""This class provides functionality for extracting the convex hull
of the (free) energy of mixing. It is based on the `convex hull
calculator in SciPy
<http://docs.scipy.org/doc/scipy-dev/reference/\
generated/scipy.spatial.ConvexHull.html>`_.
Parameters
----------
concentrations : list(float) or list(list(float))
concentrations for each structure listed as ``[[c1, c2], [c1, c2],
...]``; for binaries, in which case there is only one independent
concentration, the format ``[c1, c2, c3, ...]`` works as well.
energies : list(float)
energy (or energy of mixing) for each structure
Attributes
----------
concentrations : np.ndarray
concentrations of the `N` structures on the convex hull
energies : np.ndarray
energies of the `N` structures on the convex hull
dimensions : int
number of independent concentrations needed to specify a point in
concentration space (1 for binaries, 2 for ternaries etc.)
structures : list(int)
indices of structures that constitute the convex hull (indices are
defined by the order of their concentrations and energies are fed when
initializing the ConvexHull object)
Examples
--------
A `ConvexHull` object is easily initialized by providing lists of
concentrations and energies::
>>> data = {'concentration': [0, 0.2, 0.2, 0.3, 0.4, 0.5, 0.8, 1.0],
... 'mixing_energy': [0.1, -0.2, -0.1, -0.2, 0.2, -0.4, -0.2, -0.1]}
>>> hull = ConvexHull(data['concentration'], data['mixing_energy'])
Now one can for example access the points along the convex hull directly::
>>> for c, e in zip(hull.concentrations, hull.energies):
... print(c, e)
0.0 0.1
0.2 -0.2
0.5 -0.4
1.0 -0.1
or plot the convex hull along with the original data using e.g., matplotlib::
>>> import matplotlib.pyplot as plt
>>> plt.scatter(data['concentration'], data['mixing_energy'], color='darkred')
>>> plt.plot(hull.concentrations, hull.energies)
>>> plt.show()
It is also possible to extract structures at or close to the convex hull::
>>> low_energy_structures = hull.extract_low_energy_structures(
... data['concentration'], data['mixing_energy'],
... energy_tolerance=0.005)
A complete example can be found in the :ref:`basic tutorial
<tutorial_enumerate_structures>`.
"""
def __init__(self,
concentrations: Union[List[float], List[List[float]]],
energies: List[float]) -> None:
assert len(concentrations) == len(energies)
# Prepare data in format suitable for SciPy-ConvexHull
concentrations = np.array(concentrations)
energies = np.array(energies)
points = np.column_stack((concentrations, energies))
self.dimensions = len(points[0]) - 1
# Construct convex hull
hull = ConvexHullSciPy(points)
# Collect convex hull points in handy arrays
concentrations = []
energies = []
for vertex in hull.vertices:
if self.dimensions == 1:
concentrations.append(points[vertex][0])
else:
concentrations.append(points[vertex][0:-1])
energies.append(points[vertex][-1])
concentrations = np.array(concentrations)
energies = np.array(energies)
structures = hull.vertices
# If there is just one independent concentration, we'd better sort
# according to it
if self.dimensions == 1:
ces = list(zip(*sorted(zip(concentrations, energies, structures))))
self.concentrations = np.array(ces[0])
self.energies = np.array(ces[1])
self.structures = np.array(ces[2])
else:
self.concentrations = concentrations
self.energies = energies
self.structures = structures
# Remove points that are above the "pure components plane"
self._remove_points_above_tie_plane()
def _remove_points_above_tie_plane(self, tol: float = 1e-6) -> None:
"""
Remove all points on the convex hull that correspond to maximum rather
than minimum energy.
Parameters
----------
tol
Tolerance for what energy constitutes a lower one.
"""
# Identify the "complex concentration hull", i.e. the extremal
# concentrations. In the simplest case, these should simply be the
# pure components.
if self.dimensions == 1:
# Then the ConvexHullScipy function doesn't work, so we just pick
# the indices of the lowest and highest concentrations.
vertices = []
vertices.append(np.argmin(self.concentrations))
vertices.append(np.argmax(self.concentrations))
vertices = np.array(vertices)
else:
concentration_hull = ConvexHullSciPy(self.concentrations)
vertices = concentration_hull.vertices
# Remove all points of the convex energy hull that have an energy that
# is higher than what would be gotten with pure components at the same
# concentration. These points are mathematically on the convex hull,
# but in the physically uninteresting upper part, i.e. they maximize
# rather than minimize energy.
to_delete = []
for i, concentration in enumerate(self.concentrations):
# The points on the convex concentration hull should always be
# included, so skip them.
if i in vertices:
continue
# The energy obtained as a linear combination of concentrations on
# the convex hull is the "z coordinate" of the position on a
# (hyper)plane in the (number of independent concentrations +
# 1)-dimensional (N-D) space. This plane is spanned by N points.
# If there are more vertices on the convex hull, we need to loop
# over all combinations of N vertices.
for plane in itertools.combinations(vertices,
min(len(vertices),
self.dimensions + 1)):
# Calculate energy that would be gotten with pure components
# with ascribed concentration.
energy_pure = griddata(self.concentrations[np.array(plane)],
self.energies[np.array(plane)],
concentration,
method='linear')
# Prepare to delete if the energy was lowered. `griddata` gives
# NaN if the concentration is outside the triangle formed by
# the three vertices. The result of the below comparison is
# then False, which is what we want.
if energy_pure < self.energies[i] - tol:
to_delete.append(i)
break
# Finally remove all points
self.concentrations = np.delete(self.concentrations, to_delete, 0)
self.energies = np.delete(self.energies, to_delete, 0)
self.structures = list(np.delete(self.structures, to_delete, 0))
[docs] def get_energy_at_convex_hull(self, target_concentrations:
Union[List[float],
List[List[float]]]) -> np.ndarray:
"""Returns the energy of the convex hull at specified concentrations.
If any concentration is outside the allowed range, NaN is
returned.
Parameters
----------
target_concentrations
concentrations at target points
If there is one independent concentration, a list of
floats is sufficient. Otherwise, the concentrations ought
to be provided as a list of lists, such as ``[[0.1, 0.2],
[0.3, 0.1], ...]``.
"""
if self.dimensions > 1:
assert len(target_concentrations[0]) == self.dimensions
# Loop over all complexes of N+1 points to make sure that the lowest
# energy plane is used in the end. This is needed in two dimensions
# but in higher.
hull_candidate_energies = []
for plane in itertools.combinations(range(len(self.energies)),
min(len(self.energies),
self.dimensions + 1)):
try:
plane_energies = griddata(self.concentrations[list(plane)],
self.energies[list(plane)],
np.array(target_concentrations),
method='linear')
except QhullError:
# If the points lie on a line, the convex hull will fail, but
# we do not need to care about these "planes" anyway
continue
hull_candidate_energies.append(plane_energies)
# Pick out the lowest energies found
hull_energies = np.nanmin(hull_candidate_energies, axis=0)
return hull_energies
[docs] def extract_low_energy_structures(self, concentrations:
Union[List[float],
List[List[float]]],
energies: List[float],
energy_tolerance: float) -> List[int]:
"""Returns the indices of energies that lie within a certain
tolerance of the convex hull.
Parameters
----------
concentrations
concentrations of candidate structures
If there is one independent concentration, a list of
floats is sufficient. Otherwise, the concentrations must
be provided as a list of lists, such as ``[[0.1, 0.2],
[0.3, 0.1], ...]``.
energies
energies of candidate structures
energy_tolerance
include structures with an energy that is at most this far
from the convex hull
"""
# Convert to numpy arrays, can be necessary if, for example,
# they are Pandas Series with "gaps"
concentrations = np.array(concentrations)
energies = np.array(energies)
n_points = len(concentrations)
if len(energies) != n_points:
raise ValueError('concentrations and energies must have '
'the same length')
# Calculate energy at convex hull for specified concentrations
hull_energies = self.get_energy_at_convex_hull(concentrations)
# Extract those that are close enough
close_to_hull = [i for i in range(n_points)
if energies[i] <= hull_energies[i] + energy_tolerance]
return close_to_hull
```