Source code for pycbc.distributions.power_law

# Copyright (C) 2016 Christopher M. Biwer
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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 ---------- name : 'uniform_radius' 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 * \[(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 for the distribution are retrieved from the section titled "[`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 of the section header. Returns ------- Uniform A distribution instance from the pycbc.inference.prior module. """ return super(UniformPowerLaw, cls).from_config(cp, section, variable_args, bounds_required=True)
[docs]class UniformRadius(UniformPowerLaw): """ For a uniform distribution in volume using spherical coordinates, this is the distriubtion to use for the radius. For more details see UniformPowerLaw. """ name = "uniform_radius" def __init__(self, dim=3, **params): super(UniformRadius, self).__init__(dim=3, **params)
__all__ = ["UniformPowerLaw", "UniformRadius"]