import numpy
from .ba import BondAngleDescriptor
from .realspace_wrap import compute
[docs]class SmoothedBondAngleDescriptor(BondAngleDescriptor):
"""
Smoothed bond-angle descriptor.
This is a smooth version of the bond-angle descriptor in which the bond angles
:math:`\\theta_{jik}` are multiplied by a weighting function
:math:`f(r_{ij}, r_{ik})` that depends on the radial distances :math:`r_{ij}`
and :math:`r_{ik}` between :math:`(i,j)` and :math:`(i,k)` respectively, where
:math:`j` and :math:`k` can be any particle in the system (*i.e.* not
necessarily a nearest neighbors of :math:`i`).
Essentially, we define the *smoothed* number of bond angles around particle
:math:`i` as
.. math::
N_i^S(\\theta_n) = \\sum_{j=1}^N \\sum_{\\substack{k=1 \\ k \\neq j}}^N f(r_{ij}, r_{ik}) \delta(\\theta_n - \\theta_{jik}) ,
where the superscript :math:`S` indicates the smooth nature of the descriptor,
and :math:`f(r_{ij}, r_{ik})` is the following smoothing function:
.. math::
f(r_{ij}, r_{ik}) = \exp \left[ - \left[ ( r_{ij} / r_{\\alpha\\beta}^c )^\gamma + ( r_{ik} / r_{\\alpha\\beta'}^c )^\gamma \\right] \\right] H( R_\mathrm{max}^c - r_{ij} ) H( R_\mathrm{max}^c - r_{ik}) ,
where:
- :math:`r_{\\alpha\\beta}^c` and :math:`r_{\\alpha\\beta'}^c` are the first minima of the corresponding partial radial distribution functions for the pairs :math:`(i,j)` and :math:`(i,k)`.
- :math:`\gamma` is an integer.
- :math:`R_\mathrm{max}^c = \\xi \\times \max(\{ r_{\\alpha\\beta}^c \})` is the largest nearest neighbor cutoff rescaled by :math:`\\xi > 1`.
- :math:`H` is the `Heavide step function <https://en.wikipedia.org/wiki/Heaviside_step_function>`_, which ensures for efficiency reasons, that the descriptor only has contributions from particles within a distance :math:`R_\mathrm{max}^c`.
We then consider :math:`N_i^S(\\theta_n)` for a set of angles
:math:`\{ \\theta_n \}` that go from :math:`\\theta_0 = 0^\circ` to
:math:`\\theta_{n_\mathrm{max}}=180^\circ` by steps of :math:`\Delta \\theta`.
The resulting feature vector for particle :math:`i` is given by
.. math::
X^\mathrm{SBA}(i) = (\: N_i^S(\\theta_0) \;\; N_i^S(\\theta_1) \;\; \dots \;\; N_i^S(\\theta_{n_\mathrm{max}}) \:) .
See the tutorials for more details.
Attributes
----------
trajectory : Trajectory
Trajectory on which the structural descriptor will be computed.
active_filters : list
All the active filters on both groups prior to the computation of the
descriptor.
dimension : int
Spatial dimension of the descriptor (2 or 3).
grid : numpy.ndarray
Grid of angles :math:`\{ \\theta_n \}`.
features : numpy.ndarray
Array of all the structural features for the particles in group=0 in
accordance with the defined filters (if any). This attribute is
initialized when the method ``compute`` is called (default value is ``None``).
groups : tuple
Composition of the groups: ``groups[0]`` and ``groups[1]`` contain lists of all
the ``Particle`` instances in groups 0 and 1 respectively. Each element of
the tuple is a list of ``Particle`` in ``trajectory``, *e.g.* ``groups[0][0]``
is the list of all the particles in the first frame of ``trajectory`` that
belong to group=0.
verbose : bool
Show progress information and warnings about the computation of the
descriptor when verbose is ``True``, and remain silent when verbose is
``False``.
neighbors_boost : float, default: 1.5
Scaling factor to estimate the number of neighbors relative to a
an ideal gas with the same density. This is used internally to set
the dimensions of lists of neighbors. A too small number creates a
risk of overfilling the lists of neighbors, and a too large number
increases memory usage. This only works if the associated ``Trajectory``
has valid cutoffs in the ``Trajectory.nearest_neighbors_cutoffs`` list
attribute. This sets the value of the ``max_num_neighbors`` attribute
during the computation of the descriptor.
max_num_neighbors : int, default: 100
Maximum number of neighbors. This is used internally to set the dimensions
of lists of neighbors. This number is automatically adjusted to limit
memory usage if the associated ``Trajectory`` has valid cutoffs in the
``Trajectory.nearest_neighbors_cutoffs`` list attribute. The
default value ``100`` is used if no cutoffs can be used to estimate a
better value. The default value is sufficient in most cases, otherwise
this number can manually be increased **before** computing the descriptor.
"""
name = 'smoothed-bond-angle'
symbol = 'sba'
[docs] def __init__(self, trajectory, dtheta=3.0, cutoff_enlargement=1.3, exponent=8,
accept_nans=True, verbose=False):
"""
Parameters
----------
trajectory : Trajectory
Trajectory on which the structural descriptor will be computed.
dtheta : float
Bin width :math:`\Delta \\theta` in degrees.
cutoff_enlargement : float, default: 1.3
Scaling factor :math:`\\xi` for the largest nearest neighbors cutoff
:math:`\max(\{ r_\mathrm{\\alpha\\beta}^c \})` to consider neighbors :math:`j`
a distance :math:`R_\mathrm{max}^c = \\xi \\times \max(\{r_{\\alpha\\beta}^c\})`
away from the central particle :math:`i`.
exponent : int, default: 8
Exponent :math:`\gamma` in the smoothing function
:math:`f(r_{ij},r_{ik})`.
accept_nans: bool, default: True
If ``False``, discard any row from the array of features that contains a
`NaN` element. If ``True``, keep `NaN` elements in the array of features.
verbose : bool, default: False
Show progress information and warnings about the computation of the
descriptor when verbose is ``True``, and remain silent when verbose
is ``False``.
"""
BondAngleDescriptor.__init__(self, trajectory,
dtheta=dtheta,
accept_nans=accept_nans,
verbose=verbose)
self.cutoff_enlargement = cutoff_enlargement
self.exponent = exponent
[docs] def compute(self):
"""
Compute the smoothed bond-angle correlations for the particles in group=0
for the grid of angles :math:`\{ \\theta_n \}`. Returns the data matrix and
also overwrites the ``features`` attribute.
Returns
-------
features : numpy.ndarray
Data matrix with smoothed bond-angle correlations.
"""
# set up
self._set_up(dtype=numpy.float64)
self._manage_nearest_neighbors_cutoffs()
n_frames = len(self.trajectory)
row = 0
# all relevant arrays
pos_0 = self.dump('position', group=0)
pos_all = self.trajectory.dump('position')
idx_0 = self.dump('_index', group=0)
spe_0_id = self.dump('species_id', group=0)
spe_all_id = self.trajectory.dump('species_id')
box = self.trajectory.dump('cell.side')
n_species = len(self.trajectory[0].distinct_species)
# compute extended neighbors with extended cutoffs
standard_cutoffs = numpy.asarray(self.trajectory.nearest_neighbors_cutoffs)
extended_cutoffs = self.cutoff_enlargement * standard_cutoffs
self._compute_extended_neighbors(extended_cutoffs)
# computation
standard_cutoffs = standard_cutoffs.reshape(n_species, n_species).T
start = 0
for n in self._trange(n_frames):
pos_0_n = pos_0[n].T
pos_all_n = pos_all[n].T
npart = len(self.groups[0][n])
feat_n = compute.smoothed_angular_histogram_all(idx_0[n],
pos_0_n, pos_all_n,
spe_0_id[n], spe_all_id[n],
self._extended_neighbors[n],
self._extended_neighbors_number[n],
standard_cutoffs,
self.exponent, box[n],
self.n_features,
self.dtheta)
self.features[start: start+npart, :] = feat_n
start += npart
self._handle_nans()
return self.features