"""Algorithms for determination of Laue group symmetry."""
from __future__ import absolute_import, division, print_function
import json
import logging
import math
import libtbx
import scipy.stats
from cctbx import crystal, sgtbx
from dials.algorithms.symmetry import symmetry_base
from libtbx import table_utils
from scitbx.array_family import flex
from scitbx.math import five_number_summary
logger = logging.getLogger(__name__)
[docs]class determine_space_group(symmetry_base):
"""Determination of Laue group symmetry using algorithms similar to POINTLESS.
See also:
`Evans, P. (2006). Acta Cryst. D62, 72-82
<https://doi.org/10.1107/S0907444905036693>`_ and
`Evans, P. R. (2011). Acta Cryst. D67, 282-292
<https://doi.org/10.1107/S090744491003982X>`_.
"""
def __init__(
self,
intensities,
normalisation="ml_aniso",
lattice_symmetry_max_delta=2.0,
d_min=libtbx.Auto,
min_i_mean_over_sigma_mean=4,
min_cc_half=0.6,
relative_length_tolerance=None,
absolute_angle_tolerance=None,
):
"""Intialise a determine_space_group object.
Args:
intensities (cctbx.miller.array): The intensities on which to perform
symmetry anaylsis.
normalisation (str): The normalisation method to use. Possible choices are
'kernel', 'quasi', 'ml_iso' and 'ml_aniso'. Set to None to switch off
normalisation altogether.
lattice_symmetry_max_delta (float): The maximum value of delta for
determining the lattice symmetry using the algorithm of Le Page (1982).
d_min (float): Optional resolution cutoff to be applied to the input
intensities. If set to :data:`libtbx.Auto` then d_min will be
automatically determined according to the parameters
``min_i_mean_over_sigma_mean`` and ``min_cc_half``.
min_i_mean_over_sigma_mean (float): minimum value of |I|/|sigma(i)| for
automatic determination of resolution cutoff.
min_cc_half (float): minimum value of CC1/2 for automatic determination of
resolution cutoff.
relative_length_tolerance (float): Relative length tolerance in checking
consistency of input unit cells against the median unit cell.
absolute_angle_tolerance (float): Absolute angle tolerance in checking
consistency of input unit cells against the median unit cell.
"""
super(determine_space_group, self).__init__(
intensities,
normalisation=normalisation,
lattice_symmetry_max_delta=lattice_symmetry_max_delta,
d_min=d_min,
min_i_mean_over_sigma_mean=min_i_mean_over_sigma_mean,
min_cc_half=min_cc_half,
relative_length_tolerance=relative_length_tolerance,
absolute_angle_tolerance=absolute_angle_tolerance,
)
self._estimate_cc_sig_fac()
self._estimate_cc_true()
self._score_symmetry_elements()
self._score_laue_groups()
def _estimate_cc_sig_fac(self):
"""Estimation of sigma(CC) as a function of sample size.
Estimate the error in the correlation coefficient, sigma(CC) by using
pairs of reflections at similar resolutions that are not related by
potential symmetry. Using pairs of unrelated reflections at similar
resolutions, calculate sigma(CC) == rms(CC) for groups of size N = 3..200.
The constant CCsigFac is obtained from a linear fit of
sigma(CC) to 1/N^(1/2), i.e.:
sigma(CC) = CCsigFac/N^(1/2)
"""
max_bins = 500
reflections_per_bin = max(
200, int(math.ceil(self.intensities.size() / max_bins))
)
binner = self.intensities.setup_binner_counting_sorted(
reflections_per_bin=reflections_per_bin
)
a = flex.double()
b = flex.double()
ma_tmp = self.intensities.customized_copy(
crystal_symmetry=crystal.symmetry(
space_group=self.lattice_group,
unit_cell=self.intensities.unit_cell(),
assert_is_compatible_unit_cell=False,
)
).map_to_asu()
for i in range(binner.n_bins_all()):
count = binner.counts()[i]
if count == 0:
continue
bin_isel = binner.array_indices(i)
p = flex.random_permutation(count)
p = p[: 2 * (count // 2)] # ensure even count
ma_a = ma_tmp.select(bin_isel.select(p[: count // 2]))
ma_b = ma_tmp.select(bin_isel.select(p[count // 2 :]))
# only choose pairs of reflections that don't have the same indices
# in the asu of the lattice group
sel = ma_a.indices() != ma_b.indices()
a.extend(ma_a.data().select(sel))
b.extend(ma_b.data().select(sel))
perm = flex.random_selection(a.size(), min(20000, a.size()))
a = a.select(perm)
b = b.select(perm)
self.corr_unrelated = CorrelationCoefficientAccumulator(a, b)
n_pairs = a.size()
min_num_groups = 10 # minimum number of groups
max_n_group = int(min(n_pairs / min_num_groups, 200)) # maximum number in group
min_n_group = int(min(5, max_n_group)) # minimum number in group
mean_ccs = flex.double()
rms_ccs = flex.double()
ns = flex.double()
for n in range(min_n_group, max_n_group):
ns.append(n)
ccs = flex.double()
for i in range(200):
isel = flex.random_selection(a.size(), n)
corr = CorrelationCoefficientAccumulator(a.select(isel), b.select(isel))
ccs.append(corr.coefficient())
mean_ccs.append(flex.mean(ccs))
rms_ccs.append(flex.mean(flex.pow2(ccs)) ** 0.5)
x = 1 / flex.pow(ns, 0.5)
y = rms_ccs
fit = flex.linear_regression(x, y)
assert fit.is_well_defined()
self.cc_sig_fac = fit.slope()
def _estimate_cc_true(self):
# A1.2. Estimation of E(CC; S).
# (i)
var_intensities = flex.mean_and_variance(
self.intensities.data()
).unweighted_sample_variance()
var_sigmas = flex.mean_and_variance(flex.pow2(self.intensities.sigmas())).mean()
self.E_cc_true = var_intensities / (var_intensities + var_sigmas)
# (ii)
reindexed_intensities = self.intensities.change_basis(
sgtbx.change_of_basis_op("-x,-y,-z")
).map_to_asu()
x, y = self.intensities.common_sets(
reindexed_intensities, assert_is_similar_symmetry=False
)
self.cc_identity = CorrelationCoefficientAccumulator(x.data(), y.data())
min_sd = 0.05
min_sample = 10
sigma_1 = max(min_sd, self.cc_sig_fac / 200 ** 0.5)
w1 = 0
w2 = 0
if sigma_1 > 0.0001:
w1 = 1 / sigma_1 ** 2
if self.cc_identity.n() > min_sample:
sigma_2 = max(min_sd, self.cc_sig_fac / self.cc_identity.n() ** 0.5)
w2 = 1 / sigma_2 ** 2
assert (w1 + w2) > 0
self.cc_true = (w1 * self.E_cc_true + w2 * self.cc_identity.coefficient()) / (
w1 + w2
)
logger.debug("cc_true = w1 * E_cc_true + w2 * cc_identity)/(w1 + w2)")
logger.debug("w1: %g", w1)
logger.debug("w2: %g", w2)
logger.debug("E_cc_true: %g", self.E_cc_true)
logger.debug("cc_identity: %g", self.cc_identity.coefficient())
logger.debug("cc_true: %g", self.cc_true)
def _score_symmetry_elements(self):
self.sym_op_scores = []
for smx in self.lattice_group.smx():
if smx.r().info().sense() < 0:
continue
self.sym_op_scores.append(
ScoreSymmetryElement(
self.intensities, smx, self.cc_true, self.cc_sig_fac
)
)
def _score_laue_groups(self):
subgroup_scores = [
ScoreSubGroup(subgrp, self.sym_op_scores)
for subgrp in self.subgroups.result_groups
]
total_likelihood = sum(score.likelihood for score in subgroup_scores)
for score in subgroup_scores:
score.likelihood /= total_likelihood
self.subgroup_scores = sorted(
subgroup_scores, key=lambda score: score.likelihood, reverse=True
)
# The 'confidence' scores are derived from the total probability of the best
# solution p_best and that for the next best solution p_next:
# confidence = [p_best * (p_best - p_next)]^1/2.
for i, score in enumerate(self.subgroup_scores[:-1]):
next_score = self.subgroup_scores[i + 1]
if score.likelihood > 0 and next_score.likelihood > 0:
lgc = score.likelihood * (score.likelihood - next_score.likelihood)
confidence = abs(lgc) ** 0.5
if lgc < 0:
confidence = -confidence
score.confidence = confidence
self.best_solution = self.subgroup_scores[0]
def __str__(self):
"""Return a string representation of the results.
Returns:
str:
"""
output = []
output.append("Input crystal symmetry:")
output.append(str(self.input_intensities[0].space_group_info()))
output.append(str(self.median_unit_cell))
output.append("Change of basis op to minimum cell: %s" % self.cb_op_inp_min)
output.append("Crystal symmetry in minimum cell:")
output.append(str(self.intensities.space_group_info()))
output.append(str(self.intensities.unit_cell()))
output.append("Lattice point group: %s" % self.lattice_group.info())
output.append(
"Overall CC for %i unrelated pairs: %.3f"
% (self.corr_unrelated.n(), self.corr_unrelated.coefficient())
)
output.append(
"Estimated expectation value of true correlation coefficient E(CC) = %.3f"
% self.E_cc_true
)
output.append("Estimated sd(CC) = %.3f / sqrt(N)" % self.cc_sig_fac)
output.append(
"Estimated E(CC) of true correlation coefficient from identity = %.3f"
% self.cc_true
)
header = ("likelihood", "Z-CC", "CC", "N", "", "Operator")
rows = [header]
for score in self.sym_op_scores:
if score.likelihood > 0.9:
stars = "***"
elif score.likelihood > 0.7:
stars = "**"
elif score.likelihood > 0.5:
stars = "*"
else:
stars = ""
rows.append(
(
"%.3f" % score.likelihood,
"%.2f" % score.z_cc,
"%.2f" % score.cc.coefficient(),
"%i" % score.n_refs,
stars,
"%s" % score.sym_op.r().info(),
)
)
output.append("Scoring individual symmetry elements")
output.append(table_utils.format(rows, has_header=True, delim=" "))
header = (
"Patterson group",
"",
"Likelihood",
"NetZcc",
"Zcc+",
"Zcc-",
"CC",
"CC-",
"delta",
"Reindex operator",
)
rows = [header]
for score in self.subgroup_scores:
if score.likelihood > 0.8:
stars = "***"
elif score.likelihood > 0.6:
stars = "**"
elif score.likelihood > 0.4:
stars = "*"
else:
stars = ""
rows.append(
(
"%s" % score.subgroup["best_subsym"].space_group_info(),
stars,
"%.3f" % score.likelihood,
"% .2f" % score.z_cc_net,
"% .2f" % score.z_cc_for,
"% .2f" % score.z_cc_against,
"% .2f" % score.cc_for.coefficient(),
"% .2f" % score.cc_against.coefficient(),
"%.1f" % score.subgroup["max_angular_difference"],
"%s" % (score.subgroup["cb_op_inp_best"]),
)
)
output.append("Scoring all possible sub-groups")
output.append(table_utils.format(rows, has_header=True, delim=" "))
output.append(
"Best solution: %s"
% self.best_solution.subgroup["best_subsym"].space_group_info()
)
output.append(
"Unit cell: %s" % self.best_solution.subgroup["best_subsym"].unit_cell()
)
output.append(
"Reindex operator: %s" % (self.best_solution.subgroup["cb_op_inp_best"])
)
output.append("Laue group probability: %.3f" % self.best_solution.likelihood)
output.append("Laue group confidence: %.3f" % self.best_solution.confidence)
return "\n".join(output)
[docs] def as_dict(self):
"""Return a dictionary representation of the results.
Returns:
dict
"""
d = {
"input_symmetry": {
"hall_symbol": self.input_intensities[0]
.space_group()
.type()
.hall_symbol(),
"unit_cell": self.median_unit_cell.parameters(),
},
"cb_op_inp_min": self.cb_op_inp_min.as_xyz(),
"min_cell_symmetry": {
"hall_symbol": self.intensities.space_group().type().hall_symbol(),
"unit_cell": self.intensities.unit_cell().parameters(),
},
"lattice_point_group": self.lattice_group.type().hall_symbol(),
"cc_unrelated_pairs": self.corr_unrelated.coefficient(),
"n_unrelated_pairs": self.corr_unrelated.n(),
"E_cc_true": self.E_cc_true,
"cc_sig_fac": self.cc_sig_fac,
"cc_true": self.cc_true,
}
d["sym_op_scores"] = [score.as_dict() for score in self.sym_op_scores]
d["subgroup_scores"] = [score.as_dict() for score in self.subgroup_scores]
return d
[docs] def as_json(self, filename=None, indent=2):
"""Return a json representation of the results.
Args:
filename (str): Optional filename to export the json representation of
the results.
indent (int): The indent level for pretty-printing of the json. If ``None``
is the most compact representation.
Returns:
str:
"""
json_str = json.dumps(self.as_dict(), indent=indent)
if filename:
with open(filename, "w") as f:
f.write(json_str)
return json_str
[docs]class ScoreCorrelationCoefficient(object):
def __init__(self, cc, sigma_cc, expected_cc, lower_bound=-1, upper_bound=1, k=2):
self.cc = cc
self.sigma_cc = sigma_cc
self.expected_cc = expected_cc
self._lower_bound = lower_bound
self._upper_bound = upper_bound
self._k = k
self._compute_p_cc_given_s()
self._compute_p_cc_given_not_s()
def _compute_p_cc_given_s(self):
self._p_cc_given_s = trunccauchy_pdf(
self.cc,
self._lower_bound,
self._upper_bound,
self.expected_cc,
self.sigma_cc,
)
def _compute_p_cc_given_not_s(self):
sump = scipy.integrate.quad(self._numerator, 0, 1)[0]
sumw = scipy.integrate.quad(self._denominator, 0, 1)[0]
self._p_cc_given_not_s = sump / sumw
@property
def p_cc_given_s(self):
"""Probability of observing this CC if the sym op is present, p(CC; S).
Modelled by a Cauchy distribution centred on cc_true and width gamma = sigma_cc
"""
return self._p_cc_given_s
@property
def p_cc_given_not_s(self):
"""Probability of observing this CC if the sym op is NOT present, p(CC; !S).
"""
return self._p_cc_given_not_s
@property
def p_s_given_cc(self):
"""The likelihood of this symmetry element being present.
p(S; CC) = p(CC; S) / (p(CC; S) + p(CC; !S))
"""
return self._p_cc_given_s / (self._p_cc_given_s + self._p_cc_given_not_s)
def _p_mu_power_pdf(self, m):
return (1.0 - pow(m, self._k)) ** (1.0 / self._k)
def _numerator(self, x):
return trunccauchy_pdf(
self.cc, self._lower_bound, self._upper_bound, loc=x, scale=self.sigma_cc
) * self._p_mu_power_pdf(x)
def _denominator(self, x):
return self._p_mu_power_pdf(x)
[docs]class ScoreSymmetryElement(object):
"""Analyse intensities for presence of a given symmetry operation.
1) Calculate the correlation coefficient, CC, for the given sym op.
2) Calculate the probability of observing this CC if the sym op is present,
p(CC; S), modelled by a Cauchy distribution centred on cc_true and width
gamma = sigma_cc.
3) Calculate the probability of observing this CC if the sym op is
NOT present, p(CC; !S).
4) Calculate the likelihood of symmetry element being present,
p(S; CC) = p(CC; S) / (p(CC; S) + p(CC; !S))
See appendix A1 of `Evans, P. R. (2011). Acta Cryst. D67, 282-292.
<https://doi.org/10.1107/S090744491003982X>`_
"""
def __init__(self, intensities, sym_op, cc_true, cc_sig_fac):
"""Initialise a ScoreSymmetryElement object.
Args:
intensities (cctbx.miller.array): The intensities on which to perform
symmetry anaylsis.
sym_op (cctbx.sgtbx.rt_mx): The symmetry operation for analysis.
cc_true (float): the expected value of CC if the symmetry element is present,
E(CC; S)
cc_sig_fac (float): Estimation of sigma(CC) as a function of sample size.
"""
self.sym_op = sym_op
assert self.sym_op.r().info().sense() >= 0
self.cc = CorrelationCoefficientAccumulator()
cb_op = sgtbx.change_of_basis_op(self.sym_op)
cb_ops = [cb_op]
if self.sym_op.r().order() > 2:
# include inverse symmetry operation
cb_ops.append(cb_op.inverse())
for cb_op in cb_ops:
if cb_op.is_identity_op():
cb_op = sgtbx.change_of_basis_op("-x,-y,-z")
reindexed_intensities = intensities.change_basis(cb_op).map_to_asu()
x, y = intensities.common_sets(
reindexed_intensities, assert_is_similar_symmetry=False
)
sel = sgtbx.space_group().expand_smx(self.sym_op).epsilon(x.indices()) == 1
x = x.select(sel)
y = y.select(sel)
outliers = flex.bool(len(x.data()), False)
iqr_multiplier = 20 # very generous tolerance
for col in (x.data(), y.data()):
if col.size():
min_x, q1_x, med_x, q3_x, max_x = five_number_summary(col)
iqr_x = q3_x - q1_x
cut_x = iqr_multiplier * iqr_x
outliers.set_selected(col > q3_x + cut_x, True)
outliers.set_selected(col < q1_x - cut_x, True)
if outliers.count(True):
logger.debug(
"Rejecting %s outlier value%s"
% (libtbx.utils.plural_s(outliers.count(True)))
)
x = x.select(~outliers)
y = y.select(~outliers)
self.cc += CorrelationCoefficientAccumulator(x.data(), y.data())
self.n_refs = self.cc.n()
if self.n_refs <= 0:
self.likelihood = 0
self.z_cc = 0
return
self.sigma_cc = max(0.1, cc_sig_fac / self.n_refs ** 0.5)
self.z_cc = self.cc.coefficient() / self.sigma_cc
score_cc = ScoreCorrelationCoefficient(
self.cc.coefficient(), self.sigma_cc, cc_true
)
self.p_cc_given_s = score_cc.p_cc_given_s
self.p_cc_given_not_s = score_cc.p_cc_given_not_s
self.likelihood = score_cc.p_s_given_cc
def __str__(self):
"""Return a string representation of the symmetry element scoring.
Returns:
str:
"""
return "%.3f %.2f %.2f %i %s" % (
self.likelihood,
self.z_cc,
self.cc.coefficient(),
self.n_refs,
self.sym_op.r().info(),
)
[docs] def as_dict(self):
"""Return a dictionary representation of the symmetry element scoring.
The dictionary will contain the following keys:
- likelihood: The likelihood of the symmetry element being present
- z_cc: The Z-score for the correlation coefficent
- cc: The correlation coefficient for the symmetry element
- n_ref: The number of reflections contributing to the correlation
coefficient
- operator: The xyz representation of the symmetry element
Returns:
dict:
"""
return {
"likelihood": self.likelihood,
"z_cc": self.z_cc,
"cc": self.cc.coefficient(),
"n_ref": self.n_refs,
"operator": self.sym_op.as_xyz(),
}
[docs]class ScoreSubGroup(object):
"""Score the probability of a given subgroup being the true subgroup.
1) Calculates the combined correlation coefficients for symmetry operations
present/absent from the subgroup.
2) Calculates overall Zcc scores for symmetry elements present/absent from
the subgroup.
3) Calculates the overall likelihood for this subgroup.
See appendix A2 of `Evans, P. R. (2011). Acta Cryst. D67, 282-292.
<https://doi.org/10.1107/S090744491003982X>`_
"""
def __init__(self, subgroup, sym_op_scores):
"""Initialise a ScoreSubGroup object.
Args:
subgroup (dict): A dictionary describing the subgroup as generated by
:class:`cctbx.sgtbx.lattice_symmetry.metric_subgroups`.
sym_op_scores (list): A list of :class:`ScoreSymmetryElement` objects for each
symmetry element possibly in the lattice symmetry.
"""
# Combined correlation coefficients for symmetry operations
# present/absent from subgroup
self.subgroup = subgroup
patterson_group = subgroup["subsym"].space_group()
self.cc_for = CorrelationCoefficientAccumulator()
self.cc_against = CorrelationCoefficientAccumulator()
for score in sym_op_scores:
if score.sym_op in patterson_group:
self.cc_for += score.cc
else:
self.cc_against += score.cc
# Overall Zcc scores for symmetry elements present/absent from subgroup
self.z_cc_for = 0
self.z_cc_against = 0
n_for = 0
n_against = 0
PL_for = 0
PL_against = 0
power = 2
for score in sym_op_scores:
if score.n_refs <= 2:
continue
if score.sym_op in patterson_group:
self.z_cc_for += score.z_cc ** power
n_for += 1
PL_for += math.log(score.p_cc_given_s)
else:
self.z_cc_against += score.z_cc ** power
n_against += 1
PL_against += math.log(score.p_cc_given_not_s)
# Overall likelihood for this subgroup
self.likelihood = math.exp(PL_for + PL_against)
if n_against > 0:
self.z_cc_against = (self.z_cc_against / n_against) ** (1 / power)
if n_for > 0:
self.z_cc_for = (self.z_cc_for / n_for) ** (1 / power)
self.z_cc_net = self.z_cc_for - self.z_cc_against
self.confidence = 0
def __str__(self):
"""Return a string representation of the subgroup scores.
Returns:
str:
"""
return "%s %.3f %.2f %.2f %.2f %.2f %.2f" % (
self.subgroup["best_subsym"].space_group_info(),
self.likelihood,
self.z_cc_net,
self.z_cc_for,
self.z_cc_against,
self.cc_for.coefficient(),
self.cc_against.coefficient(),
)
[docs] def as_dict(self):
"""Return a dictionary representation of the subgroup scoring.
The dictionary will contain the following keys:
- patterson_group: The current subgroup
- likelihood: The likelihood of the subgroup being correct
- confidence: The confidence of the subgroup being correct
- z_cc_for: The combined Z-scores for all symmetry elements present in the
subgroup
- z_cc_against: The combined Z-scores for all symmetry elements present in
the lattice group but not in the subgroup
- z_cc_net: The net Z-score, i.e. z_cc_for - z_cc_against
- cc_for: The overall correlation coefficient for all symmetry elements
present in the subgroup
- cc_against: The overall correlation coefficient for all symmetry
elements present in the lattice group but not in the subgroup
- max_angular_difference: The maximum angular difference between the
symmetrised unit cell and the P1 unit cell.
- cb_op: The change of basis operation from the input unit cell to the
'best' unit cell.
Returns:
dict:
"""
return {
"patterson_group": self.subgroup["best_subsym"]
.space_group()
.type()
.hall_symbol(),
"likelihood": self.likelihood,
"confidence": self.confidence,
"z_cc_net": "% .2f" % self.z_cc_net,
"z_cc_for": "% .2f" % self.z_cc_for,
"z_cc_against": "% .2f" % self.z_cc_against,
"cc_for": "% .2f" % self.cc_for.coefficient(),
"cc_against": "% .2f" % self.cc_against.coefficient(),
"max_angular_difference": "%.1f" % self.subgroup["max_angular_difference"],
"cb_op": "%s" % (self.subgroup["cb_op_inp_best"]),
}
[docs]class CorrelationCoefficientAccumulator(object):
"""Class for incremental computation of correlation coefficients.
Uses the single-pass formula for Pearson correlation coefficient:
https://en.wikipedia.org/wiki/Pearson_correlation_coefficient#For_a_sample
"""
def __init__(self, x=None, y=None):
"""Initialise a CorrelationCoefficientAccumulator object.
Args:
x (list): Optional list of `x` values to initialise the accumulator.
y (list): Optional list of `y` values to initialise the accumulator.
"""
self._n = 0
self._sum_x = 0
self._sum_y = 0
self._sum_xy = 0
self._sum_x_sq = 0
self._sum_y_sq = 0
if x is not None and y is not None:
self.accumulate(x, y)
[docs] def accumulate(self, x, y):
"""Accumulate the `x` and `y` values provided.
Args:
x (list): The list of `x` values to accumulate.
y (list): The list of `y` values to accumulate.
"""
assert x.size() == y.size()
self._n += x.size()
self._sum_x += flex.sum(x)
self._sum_y += flex.sum(y)
self._sum_xy += flex.sum(x * y)
self._sum_x_sq += flex.sum(flex.pow2(x))
self._sum_y_sq += flex.sum(flex.pow2(y))
[docs] def coefficient(self):
"""Calculate the correlation coefficient.
Returns:
float: The correlation coefficient.
"""
if self._n == 0:
return 0
return self.numerator() / self.denominator()
[docs] def n(self):
"""Return the number of values contributing to the correlation coefficient.
Returns:
n (int)
"""
return self._n
[docs] def numerator(self):
r"""Calculate the numerator of the correlation coefficient formula.
.. math:: n \sum{x y} - \sum{x} \sum{y}
Returns:
float: The value of the numerator.
"""
return self._n * self._sum_xy - self._sum_x * self._sum_y
[docs] def denominator(self):
r"""Calculate the denominator of the correlation coefficient formula.
.. math:: \sqrt{n \sum{x^2} - \sum{x}^2} \sqrt{n \sum{y^2} - \sum{y}^2}
Returns:
float: The value of the denominator.
"""
return math.sqrt(self._n * self._sum_x_sq - self._sum_x ** 2) * math.sqrt(
self._n * self._sum_y_sq - self._sum_y ** 2
)
def __iadd__(self, other):
"""Add together two instances of :class:`CorrelationCoefficientAccumulator`.
Args:
other (CorrelationCoefficientAccumulator):
The :class:`CorrelationCoefficientAccumualator` to add to the current object.
Returns:
self (CorrelationCoefficientAccumulator): The current object.
"""
self._n += other._n
self._sum_x += other._sum_x
self._sum_y += other._sum_y
self._sum_xy += other._sum_xy
self._sum_x_sq += other._sum_x_sq
self._sum_y_sq += other._sum_y_sq
return self
[docs]def trunccauchy_pdf(x, a, b, loc=0, scale=1):
"""Calculate a truncated Cauchy probability density function.
Args:
x (float): The point at which to calculate the PDF.
a (float): The lower bound of the truncated distribution.
b (float): The upper bound of the truncated distribution.
loc (float): The location parameter for the Cauchy distribution.
scale (float): The scale parameter for the Cauchy distribution.
Returns:
float: The value of the probability density function.
"""
assert b > a
rv = scipy.stats.cauchy(loc=loc, scale=scale)
return rv.pdf(x) / (rv.cdf(b) - rv.cdf(a))
