# Copyright (C) 2020 Daniel Finstad
# This program is free software; you can redistribute it and/or modify it
# under the terms of the GNU General Public License as published by the
# Free Software Foundation; either version 3 of the License, or (at your
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# 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.
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# 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.
#
# =============================================================================
#
# Preamble
#
# =============================================================================
#
"""This module provides model classes and functions for implementing
a relative binning likelihood for parameter estimation.
"""
import logging
import numpy
from scipy.interpolate import interp1d
from scipy import special
from pycbc.waveform import get_fd_waveform_sequence
from pycbc.detector import Detector
from pycbc.types import Array
from .gaussian_noise import BaseGaussianNoise
[docs]def setup_bins(f_full, f_lo, f_hi, chi=1.0, eps=0.5):
"""Construct frequency bins for use in a relative likelihood
model. For details, see [Barak, Dai & Venumadhav 2018].
Parameters
----------
f_full : array
The full resolution array of frequencies being used in the analysis.
f_lo : float
The starting frequency used in matched filtering. This will be the
left edge of the first frequency bin.
f_hi : float
The ending frequency used in matched filtering. This will be the right
edge of the last frequency bin.
chi : float, optional
Tunable parameter, see [Barak, Dai & Venumadhav 2018]
eps : float, optional
Tunable parameter, see [Barak, Dai & Venumadhav 2018]. Lower values
result in larger number of bins.
Returns
-------
nbin : int
Number of bins.
fbin : numpy.array of floats
Bin edge frequencies.
fbin_ind : numpy.array of ints
Indices of bin edges in full frequency array.
"""
f = numpy.linspace(f_lo, f_hi, 10000)
# f^ga power law index
ga = numpy.array([-5./3, -2./3, 1., 5./3, 7./3])
dalp = chi * 2.0 * numpy.pi / numpy.absolute((f_lo ** ga) - (f_hi ** ga))
dphi = numpy.sum(numpy.array([numpy.sign(g) * d * (f ** g) for
g, d in zip(ga, dalp)]), axis=0)
dphi_diff = dphi - dphi[0]
# now construct frequency bins
nbin = int(dphi_diff[-1] / eps)
dphi2f = interp1d(dphi_diff, f, kind='slinear', bounds_error=False,
fill_value=0.0)
dphi_grid = numpy.linspace(dphi_diff[0], dphi_diff[-1], nbin+1)
# frequency grid points
fbin = dphi2f(dphi_grid)
# indices of frequency grid points in the FFT array
fbin_ind = numpy.unique([numpy.argmin(numpy.absolute(f_full - ff)) for
ff in fbin])
# make sure grid points are precise
fbin = numpy.array([f_full[i] for i in fbin_ind])
nbin = len(fbin)
return nbin, fbin, fbin_ind
[docs]class Relative(BaseGaussianNoise):
r"""Model that assumes the likelihood in a region around the peak
is slowly varying such that a linear approximation can be made, and
likelihoods can be calculated at a coarser frequency resolution. For
more details on the implementation, see https://arxiv.org/abs/1806.08792.
This model requires the use of a fiducial waveform whose parameters are
near the peak of the likelihood. The fiducial waveform and all template
waveforms used in likelihood calculation are currently generated using
the SPAtmplt approximant.
For more details on initialization parameters and definition of terms, see
:py:class:`BaseGaussianNoise`.
Parameters
----------
variable_params : (tuple of) string(s)
A tuple of parameter names that will be varied.
data : dict
A dictionary of data, in which the keys are the detector names and the
values are the data (assumed to be unwhitened). All data must have the
same frequency resolution.
low_frequency_cutoff : dict
A dictionary of starting frequencies, in which the keys are the
detector names and the values are the starting frequencies for the
respective detectors to be used for computing inner products.
mass1_ref : float
The primary mass in solar masses used for generating the fiducial
waveform.
mass2_ref : float
The secondary mass in solar masses used for generating the fiducial
waveform.
spin1z_ref : float
The component of primary dimensionless spin along the orbital angular
momentum used for generating the fiducial waveform.
spin2z_ref : float
The component of secondary dimensionless spin along the orbital angular
momentum used for generating the fiducial waveform.
ra_ref : float
The right ascension in radians used for generating the fiducial
waveform.
dec_ref : float
The declination in radians used for generating the fiducial waveform.
tc_ref : float
The GPS time of coalescence used for generating the fiducial waveform.
epsilon : float, optional
Tuning parameter used in calculating the frequency bins. Lower values
will result in higher resolution and more bins.
\**kwargs :
All other keyword arguments are passed to
:py:class:`BaseGaussianNoise`.
"""
name = "relative"
def __init__(self, variable_params, data, low_frequency_cutoff,
fiducial_params=None, epsilon=0.5, **kwargs):
super(Relative, self).__init__(
variable_params, data, low_frequency_cutoff, **kwargs)
# check that all of the frequency cutoffs are the same
# FIXME: this can probably be loosened at some point
kmins = list(self.kmin.values())
kmaxs = list(self.kmax.values())
if any(kk != kmins[0] for kk in kmins):
raise ValueError("All lower frequency cutoffs must be the same")
if any(kk != kmaxs[0] for kk in kmaxs):
raise ValueError("All high frequency cutoffs must be the same")
# store data and frequencies
d0 = list(self.data.values())[0]
self.f = numpy.array(d0.sample_frequencies)
self.df = d0.delta_f
self.end_time = float(d0.end_time)
self.det = {ifo: Detector(ifo) for ifo in self.data}
self.epsilon = float(epsilon)
# store data and psds as arrays for faster computation
self.comp_data = {ifo: d.numpy() for ifo, d in self.data.items()}
self.comp_psds = {ifo: p.numpy() for ifo, p in self.psds.items()}
# store fiducial waveform params
self.fid_params = fiducial_params
# get detector-specific arrival times relative to end of data
dt = {ifo:
self.det[ifo].time_delay_from_earth_center(
self.fid_params['ra'], self.fid_params['dec'],
self.fid_params['tc'])
for ifo in self.data}
self.ta = {ifo: self.fid_params['tc'] + dt[ifo] - self.end_time
for ifo in self.data}
# generate fiducial waveform
f_lo = kmins[0] * self.df
f_hi = kmaxs[0] * self.df
logging.info("Generating fiducial waveform from %s to %s Hz",
f_lo, f_hi)
# prune low frequency samples to avoid waveform errors
nbelow = sum(self.f < 10)
fpoints = Array(self.f.astype(numpy.float64))[nbelow:]
approx = self.static_params['approximant']
fid_hp, fid_hc = get_fd_waveform_sequence(approximant=approx,
sample_points=fpoints,
**self.fid_params)
self.h00 = {}
for ifo in self.data:
# make copy of fiducial wfs, adding back in low frequencies
hp0 = numpy.concatenate([[0j] * nbelow, fid_hp.copy()])
hc0 = numpy.concatenate([[0j] * nbelow, fid_hc.copy()])
fp, fc = self.det[ifo].antenna_pattern(
self.fid_params['ra'], self.fid_params['dec'],
self.fid_params['polarization'], self.fid_params['tc'])
tshift = numpy.exp(-2.0j * numpy.pi * self.f * self.ta[ifo])
self.h00[ifo] = numpy.array(hp0 * fp + hc0 * fc) * tshift
# compute frequency bins
logging.info("Computing frequency bins")
nbin, fbin, fbin_ind = setup_bins(f_full=self.f, f_lo=kmins[0]*self.df,
f_hi=kmaxs[0]*self.df,
eps=self.epsilon)
logging.info("Using %s bins for this model", nbin)
# store bins and edges in sample and frequency space
self.edges = fbin_ind
self.fedges = numpy.array(fbin).astype(numpy.float64)
self.bins = numpy.array([(self.edges[i], self.edges[i+1]) for
i in range(len(self.edges) - 1)])
self.fbins = numpy.array([(fbin[i], fbin[i+1]) for
i in range(len(fbin) - 1)])
# store low res copy of fiducial waveform
self.h00_sparse = {ifo: self.h00[ifo].copy().take(self.edges) for ifo
in self.h00}
# compute summary data
logging.info("Calculating summary data at frequency resolution %s Hz",
self.df)
self.sdat = self.summary_data()
[docs] def summary_data(self):
"""Compute summary data bin coefficients encoding linear
approximation to full resolution likelihood.
Returns
-------
dict of dicts
Dictionary keyed by detector name, whose values are dictionaries
containing bin coefficients a0, b0, a1, b1, for each frequency
bin.
"""
# calculate coefficients
sdat = {}
for ifo in self.data:
hd = numpy.conjugate(self.comp_data[ifo]) * self.h00[ifo]
hd /= self.comp_psds[ifo]
hh = (numpy.absolute(self.h00[ifo]) ** 2.0) / self.comp_psds[ifo]
# constant terms
a0 = numpy.array([4. * self.df * numpy.sum(hd[l:h]) for
l, h in self.bins])
b0 = numpy.array([4. * self.df * numpy.sum(hh[l:h]) for
l, h in self.bins])
# linear terms
bin_lefts = [fl for fl, fh in self.fbins]
a1 = numpy.array([4. * self.df
* numpy.sum(hd[l:h] * (self.f[l:h] - bl)) for
(l, h), bl in zip(self.bins, bin_lefts)])
b1 = numpy.array([4. * self.df
* numpy.sum(hh[l:h] * (self.f[l:h] - bl)) for
(l, h), bl in zip(self.bins, bin_lefts)])
sdat[ifo] = {'a0': a0, 'a1': a1,
'b0': b0, 'b1': b1}
return sdat
def _loglr(self):
r"""Computes the log likelihood ratio,
.. math::
\log \mathcal{L}(\Theta) = \sum_i
\left<h_i(\Theta)|d_i\right> -
\frac{1}{2}\left<h_i(\Theta)|h_i(\Theta)\right>,
at the current parameter values :math:`\Theta`.
Returns
-------
float
The value of the log likelihood ratio.
"""
# get model params
p = self.current_params.copy()
p.update(self.static_params)
hh = 0.
hd = 0j
for ifo in self.data:
# get detector antenna pattern
fp, fc = self.det[ifo].antenna_pattern(p['ra'], p['dec'],
p['polarization'],
p['tc'])
# get timeshift relative to end of data
dt = self.det[ifo].time_delay_from_earth_center(p['ra'], p['dec'],
p['tc'])
dtc = p['tc'] + dt - self.end_time
tshift = numpy.exp(-2.0j * numpy.pi * self.fedges * dtc)
# generate template and calculate waveform ratio
hp, hc = get_fd_waveform_sequence(sample_points=Array(self.fedges),
**p)
htilde = numpy.array(fp * hp + fc * hc) * tshift
r = (htilde / self.h00_sparse[ifo]).astype(numpy.complex128)
r0 = r[:-1]
r1 = (r[1:] - r[:-1]) / (self.fedges[1:] - self.fedges[:-1])
# <h, d> is sum over bins of A0r0 + A1r1
hd += numpy.sum(self.sdat[ifo]['a0'] * r0
+ self.sdat[ifo]['a1'] * r1)
# <h, h> is sum over bins of B0|r0|^2 + 2B1Re(r1r0*)
hh += numpy.sum(self.sdat[ifo]['b0'] * numpy.absolute(r0) ** 2.
+ 2. * self.sdat[ifo]['b1']
* (r1 * numpy.conjugate(r0)).real)
hd = abs(hd)
llr = numpy.log(special.i0e(hd)) + hd - 0.5 * hh
return float(llr)