Source code for pycbc.distributions.power_law

# Copyright (C) 2016 Christopher M. Biwer
# This program is free software; you can redistribute it and/or modify it
# Free Software Foundation; either version 3 of the License, or (at your
# option) any later version.
#
# This program is distributed in the hope that it will be useful, but
# WITHOUT ANY WARRANTY; without even the implied warranty of
# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU General
# Public License for more details.
#
# You should have received a copy of the GNU General Public License along
# with this program; if not, write to the Free Software Foundation, Inc.,
# 51 Franklin Street, Fifth Floor, Boston, MA  02110-1301, USA.
"""
This modules provides classes for evaluating distributions where the
probability density function is a power law.
"""

import numpy
from pycbc.distributions import bounded

[docs]class UniformPowerLaw(bounded.BoundedDist):
r"""
For a uniform distribution in power law. The parameters are
independent of each other. Instances of this class can be called like
a function. By default, logpdf will be called, but this can be changed
by setting the class's __call__ method to its pdf method.

The cumulative distribution function (CDF) will be the ratio of volumes:

.. math::

F(r) = \frac{V(r)}{V(R)}

Where :math:R is the radius of the sphere. So we can write our
probability density function (PDF) as:

.. math::

f(r) = c r^n

For generality we use :math:n for the dimension of the volume element,
eg. :math:n=2 for a 3-dimensional sphere. And use
:math:c as a general constant.

So now we calculate the CDF in general for this type of PDF:

.. math::

F(r) = \int f(r) dr = \int c r^n dr = \frac{1}{n + 1} c r^{n + 1} + k

Now with the definition of the CDF at radius :math:r_{l} is equal to 0
and at radius :math:r_{h} is equal to 1 we find that the constant from
integration from this system of equations:

.. math::

1 = \frac{1}{n + 1} c ((r_{h})^{n + 1} - (r_{l})^{n + 1}) + k

Can see that :math:c = (n + 1) / ((r_{h})^{n + 1} - (r_{l})^{n + 1})).
And :math:k is:

.. math::

k = - \frac{r_{l}^{n + 1}}{(r_{h})^{n + 1} - (r_{l})^{n + 1}}

Can see that :math:c= \frac{n + 1}{R^{n + 1}}. So can see that the CDF is:

.. math::

F(r) = \frac{1}{(r_{h})^{n + 1} - (r_{l})^{n + 1}} r^{n + 1} - \frac{r_{l}^{n + 1}}{(r_{h})^{n + 1} - (r_{l})^{n + 1}}

And the PDF is the derivative of the CDF:

.. math::

f(r) = \frac{(n + 1)}{(r_{h})^{n + 1} - (r_{l})^{n + 1}} (r)^n

Now we use the probabilty integral transform method to get sampling on
uniform numbers from a continuous random variable. To do this we find
the inverse of the CDF evaluated for uniform numbers:

.. math::

F(r) = u = \frac{1}{(r_{h})^{n + 1} - (r_{l})^{n + 1}} r^{n + 1} - \frac{r_{l}^{n + 1}}{(r_{h})^{n + 1} - (r_{l})^{n + 1}}

And find :math:F^{-1}(u) gives:

.. math::

u = \frac{1}{n + 1} \frac{(r_{h})^{n + 1} - (r_{l})^{n + 1}} - \frac{r_{l}^{n + 1}}{(r_{h})^{n + 1} - (r_{l})^{n + 1}}

And solving for :math:r gives:

.. math::

r = ( ((r_{h})^{n + 1} - (r_{l})^{n + 1}) u + (r_{l})^{n + 1})^{\frac{1}{n + 1}}

Therefore the radius can be sampled by taking the n-th root of uniform
numbers and multiplying by the radius offset by the lower bound radius.

\**params :
The keyword arguments should provide the names of parameters and their
corresponding bounds, as either tuples or a boundaries.Bounds
instance.

Attributes
----------
The name of this distribution.
dim : int
The dimension of volume space. In the notation above dim
is :math:n+1. For a 3-dimensional sphere this is 3.

Attributes
----------
params : list of strings
The list of parameter names.
bounds : dict
A dictionary of the parameter names and their bounds.
norm : float
The normalization of the multi-dimensional pdf.
lognorm : float
The log of the normalization.
"""
name = "uniform_power_law"
def __init__(self, dim=None, **params):
super(UniformPowerLaw, self).__init__(**params)
self.dim = dim
self._norm = 1.0
self._lognorm = 0.0
for p in self._params:
self._norm *= self.dim  / \
(self._bounds[p][1]**(self.dim) -
self._bounds[p][0]**(self.dim))
self._lognorm = numpy.log(self._norm)

@property
def norm(self):
return self._norm

@property
def lognorm(self):
return self._lognorm

[docs]    def rvs(self, size=1, param=None):
"""Gives a set of random values drawn from this distribution.

Parameters
----------
size : {1, int}
The number of values to generate; default is 1.
param : {None, string}
If provided, will just return values for the given parameter.
Otherwise, returns random values for each parameter.

Returns
-------
structured array
The random values in a numpy structured array. If a param was
specified, the array will only have an element corresponding to the
given parameter. Otherwise, the array will have an element for each
parameter in self's params.
"""
if param is not None:
dtype = [(param, float)]
else:
dtype = [(p, float) for p in self.params]
arr = numpy.zeros(size, dtype=dtype)
for (p,_) in dtype:
offset = numpy.power(self._bounds[p][0], self.dim)
factor = numpy.power(self._bounds[p][1], self.dim) - \
numpy.power(self._bounds[p][0], self.dim)
arr[p] = numpy.random.uniform(0.0, 1.0, size=size)
arr[p] = numpy.power(factor * arr[p] + offset, 1.0 / self.dim)
return arr

def _cdfinv_param(self, param, value):
"""Return inverse of cdf to map unit interval to parameter bounds.
"""
n = self.dim - 1
r_l = self._bounds[param][0]
r_h = self._bounds[param][1]
new_value = ((r_h**(n+1) - r_l**(n+1))*value + r_l**(n+1))**(1./(n+1))
return new_value

def _pdf(self, **kwargs):
"""Returns the pdf at the given values. The keyword arguments must
contain all of parameters in self's params. Unrecognized arguments are
ignored.
"""
for p in self._params:
if p not in kwargs.keys():
raise ValueError(
'Missing parameter {} to construct pdf.'.format(p))
if kwargs in self:
pdf = self._norm * \
numpy.prod([(kwargs[p])**(self.dim - 1)
for p in self._params])
return float(pdf)
else:
return 0.0

def _logpdf(self, **kwargs):
"""Returns the log of the pdf at the given values. The keyword
arguments must contain all of parameters in self's params. Unrecognized
arguments are ignored.
"""
for p in self._params:
if p not in kwargs.keys():
raise ValueError(
'Missing parameter {} to construct pdf.'.format(p))
if kwargs in self:
log_pdf = self._lognorm + \
(self.dim - 1) * \
numpy.log([kwargs[p] for p in self._params]).sum()
return log_pdf
else:
return -numpy.inf

[docs]    @classmethod
def from_config(cls, cp, section, variable_args):
"""Returns a distribution based on a configuration file. The parameters
"[section-variable_args]" in the config file.

Parameters
----------
cp : pycbc.workflow.WorkflowConfigParser
A parsed configuration file that contains the distribution
options.
section : str
Name of the section in the configuration file.
variable_args : str
The names of the parameters for this distribution, separated by
prior.VARARGS_DELIM. These must appear in the "tag" part

Returns
-------
Uniform
A distribution instance from the pycbc.inference.prior module.
"""
return super(UniformPowerLaw, cls).from_config(cp, section,
variable_args,
bounds_required=True)