https://github.com/gwastro/pycbc
Tip revision: 6fa08f487105a825000dbdd82247c5125f83098d authored by Duncan Brown on 20 February 2017, 14:03:34 UTC
Set for 1.6.6 release (#1461)
Set for 1.6.6 release (#1461)
Tip revision: 6fa08f4
trigger_fits.py
"""
Tools for maximum likelihood fits to single trigger statistic values
For some set of values above a threshold, e.g. trigger SNRs, the functions
in this module perform maximum likelihood fits with 1-sigma uncertainties
to various simple functional forms of PDF, all normalized to 1.
You can also obtain the fitted function and its (inverse) CDF and perform
a Kolmogorov-Smirnov test.
Usage:
# call the fit function directly if the threshold is known
alpha, sigma_alpha = fit_exponential(snrs, 5.5)
# apply a threshold explicitly
alpha, sigma_alpha = fit_above_thresh('exponential', snrs, thresh=6.25)
# let the code work out the threshold from the smallest value via the default thresh=None
alpha, sigma_alpha = fit_above_thresh('exponential', snrs)
# or only fit the largest N values, i.e. tail fitting
thresh = tail_threshold(snrs, N=500)
alpha, sigma_alpha = fit_above_thresh('exponential', snrs, thresh)
# obtain the fitted function directly
xvals = numpy.xrange(5.5, 10.5, 20)
exponential_fit = expfit(xvals, alpha, thresh)
# or access function by name
exponential_fit_1 = fit_fn('exponential', xvals, alpha, thresh)
# get the KS test statistic and p-value - see scipy.stats.kstest
ks_stat, ks_pval = KS_test('exponential', snrs, alpha, thresh)
"""
# Copyright T. Dent 2015 (thomas.dent@aei.mpg.de)
#
# 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 2 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.
from __future__ import division
import numpy
from scipy.stats import kstest
fitalpha_dict = {
'exponential' : lambda vals, thresh : 1. / (numpy.mean(vals) - thresh),
'rayleigh' : lambda vals, thresh : 2. / (numpy.mean(vals**2.) - thresh**2.),
'power' : lambda vals, thresh : numpy.mean(numpy.log(vals/thresh))**-1. + 1.
}
# measurement standard deviation = (-d^2 log L/d alpha^2)^(-1/2)
fitstd_dict = {
'exponential' : lambda vals, alpha : alpha / len(vals)**0.5,
'rayleigh' : lambda vals, alpha : alpha / len(vals)**0.5,
'power' : lambda vals, alpha : (alpha - 1.) / len(vals)**0.5
}
def fit_above_thresh(distr, vals, thresh=None):
"""
Maximum likelihood fit for the coefficient alpha
Fitting a distribution of discrete values above a given threshold.
Exponential p(x) = alpha exp(-alpha (x-x_t))
Rayleigh p(x) = alpha x exp(-alpha (x**2-x_t**2)/2)
Power p(x) = ((alpha-1)/x_t) (x/x_t)**-alpha
Values below threshold will be discarded.
If no threshold is specified the minimum sample value will be used.
Parameters
----------
distr : {'exponential', 'rayleigh', 'power'}
Name of distribution
vals : sequence of floats
Values to fit
thresh : float
Threshold to apply before fitting; if None, use min(vals)
Returns
-------
alpha : float
Fitted value
sigma_alpha : float
Standard error in fitted value
"""
vals = numpy.array(vals)
if thresh is None:
thresh = min(vals)
else:
vals = vals[vals >= thresh]
alpha = fitalpha_dict[distr](vals, thresh)
return alpha, fitstd_dict[distr](vals, alpha)
fitfn_dict = {
'exponential' : lambda x, alpha, t : alpha * numpy.exp(-alpha * (x - t)),
'rayleigh' : lambda x, alpha, t : alpha * x * \
numpy.exp(-alpha * (x**2. - t**2.) / 2.),
'power' : lambda x, alpha, t : (alpha - 1.) * x**(-alpha) * t**(alpha - 1.)
}
def fit_fn(distr, xvals, alpha, thresh):
"""
The fitted function normalized to 1 above threshold
To normalize to a given total count multiply by the count.
Parameters
----------
xvals : sequence of floats
Values where the function is to be evaluated
alpha : float
The fitted parameter
thresh : float
Threshold value applied to fitted values
Returns
-------
fit : array of floats
Fitted function at the requested xvals
"""
xvals = numpy.array(xvals)
fit = fitfn_dict[distr](xvals, alpha, thresh)
# set fitted values below threshold to 0
numpy.putmask(fit, xvals < thresh, 0.)
return fit
cum_fndict = {
'exponential' : lambda x, alpha, t : numpy.exp(-alpha * (x - t)),
'rayleigh' : lambda x, alpha, t : numpy.exp(-alpha * (x**2. - t**2.) / 2.),
'power' : lambda x, alpha, t : x**(1. - alpha) * t**(alpha - 1.)
}
def cum_fit(distr, xvals, alpha, thresh):
"""
Integral of the fitted function above a given value (reverse CDF)
The fitted function is normalized to 1 above threshold
Parameters
----------
xvals : sequence of floats
Values where the function is to be evaluated
alpha : float
The fitted parameter
thresh : float
Threshold value applied to fitted values
Returns
-------
cum_fit : array of floats
Reverse CDF of fitted function at the requested xvals
"""
xvals = numpy.array(xvals)
cum_fit = cum_fndict[distr](xvals, alpha, thresh)
# set fitted values below threshold to 0
numpy.putmask(cum_fit, xvals < thresh, 0.)
return cum_fit
def tail_threshold(vals, N=1000):
"""Determine a threshold above which there are N louder values"""
vals = numpy.array(vals)
if len(vals) < N:
raise RuntimeError('Not enough input values to determine threshold')
vals.sort()
return min(vals[-N:])
def KS_test(distr, vals, alpha, thresh=None):
"""
Perform Kolmogorov-Smirnov test for fitted distribution
Compare the given set of discrete values above a given threshold to the
fitted distribution function.
If no threshold is specified, the minimum sample value will be used.
Returns the KS test statistic and its p-value: lower p means less
probable under the hypothesis of a perfect fit
Parameters
----------
distr : {'exponential', 'rayleigh', 'power'}
Name of distribution
vals : sequence of floats
Values to compare to fit
alpha :
Fitted distribution parameter
thresh : float
Threshold to apply before fitting; if None, use min(vals)
Returns
-------
D : float
KS test statistic
p-value : float
p-value, assumed to be two-tailed
"""
vals = numpy.array(vals)
if thresh is None:
thresh = min(vals)
else:
vals = vals[vals >= thresh]
def cdf_fn(x):
return 1 - cum_fndict[distr](x, alpha, thresh)
return kstest(vals, cdf_fn)