# Licensed under a 3-clause BSD style license - see LICENSE.rst
"""Common fit statistics used in gamma-ray astronomy.
see :ref:`fit-statistics`
"""
from __future__ import absolute_import, division, print_function, unicode_literals
import numpy as np
__all__ = [
"cash",
"cstat",
"wstat",
"get_wstat_mu_bkg",
"get_wstat_gof_terms",
"chi2",
"chi2constvar",
"chi2datavar",
"chi2gehrels",
"chi2modvar",
"chi2xspecvar",
]
N_ON_MIN = 1e-25
[docs]def cash(n_on, mu_on):
r"""Cash statistic, for Poisson data.
The Cash statistic is defined as:
.. math::
C = 2 \left( \mu_{on} - n_{on} \log \mu_{on} \right)
and :math:`C = 0` where :math:`\mu <= 0`.
For more information see :ref:`fit-statistics`
Parameters
----------
n_on : array_like
Observed counts
mu_on : array_like
Expected counts
Returns
-------
stat : ndarray
Statistic per bin
References
----------
* `Sherpa statistics page section on the Cash statistic
<http://cxc.cfa.harvard.edu/sherpa/statistics/#cash>`_
* `Sherpa help page on the Cash statistic
<http://cxc.harvard.edu/sherpa/ahelp/cash.html>`_
* `Cash 1979, ApJ 228, 939
<http://adsabs.harvard.edu/abs/1979ApJ...228..939C>`_
"""
# suppress zero division warnings, they are corrected below
with np.errstate(divide="ignore", invalid="ignore"):
stat = 2 * (mu_on - n_on * np.log(mu_on))
stat = np.where(mu_on > 0, stat, 0)
return stat
[docs]def cstat(n_on, mu_on, n_on_min=N_ON_MIN):
r"""C statistic, for Poisson data.
The C statistic is defined as
.. math::
C = 2 \left[ \mu_{on} - n_{on} + n_{on}
(\log(n_{on}) - log(\mu_{on}) \right]
and :math:`C = 0` where :math:`\mu_{on} <= 0`.
``n_on_min`` handles the case where ``n_on`` is 0 or less and
the log cannot be taken.
For more information see :ref:`fit-statistics`
Parameters
----------
n_on : array_like
Observed counts
mu_on : array_like
Expected counts
n_on_min : array_like
``n_on`` = ``n_on_min`` where ``n_on`` <= ``n_on_min.``
Returns
-------
stat : ndarray
Statistic per bin
References
----------
* `Sherpa stats page section on the C statistic
<http://cxc.cfa.harvard.edu/sherpa/statistics/#cstat>`_
* `Sherpa help page on the C statistic
<http://cxc.harvard.edu/sherpa/ahelp/cash.html>`_
* `Cash 1979, ApJ 228, 939
<http://adsabs.harvard.edu/abs/1979ApJ...228..939C>`_
"""
n_on = np.asanyarray(n_on, dtype=np.float64)
mu_on = np.asanyarray(mu_on, dtype=np.float64)
n_on_min = np.asanyarray(n_on_min, dtype=np.float64)
n_on = np.where(n_on <= n_on_min, n_on_min, n_on)
term1 = np.log(n_on) - np.log(mu_on)
stat = 2 * (mu_on - n_on + n_on * term1)
stat = np.where(mu_on > 0, stat, 0)
return stat
[docs]def wstat(n_on, n_off, alpha, mu_sig, mu_bkg=None, extra_terms=True):
r"""W statistic, for Poisson data with Poisson background.
For a definition of WStat see :ref:`wstat`. If ``mu_bkg`` is not provided
it will be calculated according to the profile likelihood formula.
Parameters
----------
n_on : array_like
Total observed counts
n_off : array_like
Total observed background counts
alpha : array_like
Exposure ratio between on and off region
mu_sig : array_like
Signal expected counts
mu_bkg : array_like, optional
Background expected counts
extra_terms : bool, optional
Add model independent terms to convert stat into goodness-of-fit
parameter, default: True
Returns
-------
stat : ndarray
Statistic per bin
References
----------
* `Habilitation M. de Naurois, p. 141
<http://inspirehep.net/record/1122589/files/these_short.pdf>`_
* `XSPEC page on Poisson data with Poisson background
<https://heasarc.nasa.gov/xanadu/xspec/manual/XSappendixStatistics.html>`_
"""
# Note: This is equivalent to what's defined on the XSPEC page under the
# following assumptions
# t_s * m_i = mu_sig
# t_b * m_b = mu_bkg
# t_s / t_b = alpha
n_on = np.atleast_1d(np.asanyarray(n_on, dtype=np.float64))
n_off = np.atleast_1d(np.asanyarray(n_off, dtype=np.float64))
alpha = np.atleast_1d(np.asanyarray(alpha, dtype=np.float64))
mu_sig = np.atleast_1d(np.asanyarray(mu_sig, dtype=np.float64))
if mu_bkg is None:
mu_bkg = get_wstat_mu_bkg(n_on, n_off, alpha, mu_sig)
term1 = mu_sig + (1 + alpha) * mu_bkg
# suppress zero division warnings, they are corrected below
with np.errstate(divide="ignore", invalid="ignore"):
# This is a false positive error from pylint
# See https://github.com/PyCQA/pylint/issues/2436
term2_ = -n_on * np.log(
mu_sig + alpha * mu_bkg
) # pylint:disable=invalid-unary-operand-type
# Handle n_on == 0
condition = n_on == 0
term2 = np.where(condition, 0, term2_)
# suppress zero division warnings, they are corrected below
with np.errstate(divide="ignore", invalid="ignore"):
# This is a false positive error from pylint
# See https://github.com/PyCQA/pylint/issues/2436
term3_ = -n_off * np.log(mu_bkg) # pylint:disable=invalid-unary-operand-type
# Handle n_off == 0
condition = n_off == 0
term3 = np.where(condition, 0, term3_)
stat = 2 * (term1 + term2 + term3)
if extra_terms:
stat += get_wstat_gof_terms(n_on, n_off)
return stat
[docs]def get_wstat_mu_bkg(n_on, n_off, alpha, mu_sig):
"""Calculate ``mu_bkg`` for wstat
see :ref:`wstat`.
"""
n_on = np.atleast_1d(np.asanyarray(n_on, dtype=np.float64))
n_off = np.atleast_1d(np.asanyarray(n_off, dtype=np.float64))
alpha = np.atleast_1d(np.asanyarray(alpha, dtype=np.float64))
mu_sig = np.atleast_1d(np.asanyarray(mu_sig, dtype=np.float64))
# NOTE: Corner cases in the docs are all handled correcty by this formula
C = alpha * (n_on + n_off) - (1 + alpha) * mu_sig
D = np.sqrt(C ** 2 + 4 * alpha * (alpha + 1) * n_off * mu_sig)
mu_bkg = (C + D) / (2 * alpha * (alpha + 1))
return mu_bkg
[docs]def get_wstat_gof_terms(n_on, n_off):
"""Calculate goodness of fit terms for wstat
see :ref:`wstat`.
"""
term = np.zeros(len(n_on))
# suppress zero division warnings, they are corrected below
with np.errstate(divide="ignore", invalid="ignore"):
term1 = -n_on * (1 - np.log(n_on))
term2 = -n_off * (1 - np.log(n_off))
term += np.where(n_on == 0, 0, term1)
term += np.where(n_off == 0, 0, term2)
return 2 * term
[docs]def chi2(N_S, B, S, sigma2):
r"""Chi-square statistic with user-specified variance.
.. math::
\chi^2 = \frac{(N_S - B - S) ^ 2}{\sigma ^ 2}
Parameters
----------
N_S : array_like
Number of observed counts
B : array_like
Model background
S : array_like
Model signal
sigma2 : array_like
Variance
Returns
-------
stat : ndarray
Statistic per bin
References
----------
* Sherpa stats page (http://cxc.cfa.harvard.edu/sherpa/statistics/#chisq)
"""
N_S = np.asanyarray(N_S, dtype=np.float64)
B = np.asanyarray(B, dtype=np.float64)
S = np.asanyarray(S, dtype=np.float64)
sigma2 = np.asanyarray(sigma2, dtype=np.float64)
stat = (N_S - B - S) ** 2 / sigma2
return stat
[docs]def chi2constvar(N_S, N_B, A_S, A_B):
r"""Chi-square statistic with constant variance.
"""
N_S = np.asanyarray(N_S, dtype=np.float64)
N_B = np.asanyarray(N_B, dtype=np.float64)
A_S = np.asanyarray(A_S, dtype=np.float64)
A_B = np.asanyarray(A_B, dtype=np.float64)
alpha2 = (A_S / A_B) ** 2
# Need to mulitply with np.ones_like(N_S) here?
sigma2 = (N_S + alpha2 * N_B).mean()
stat = chi2(N_S, A_B, A_S, sigma2)
return stat
[docs]def chi2datavar(N_S, N_B, A_S, A_B):
r"""Chi-square statistic with data variance.
"""
N_S = np.asanyarray(N_S, dtype=np.float64)
N_B = np.asanyarray(N_B, dtype=np.float64)
A_S = np.asanyarray(A_S, dtype=np.float64)
A_B = np.asanyarray(A_B, dtype=np.float64)
alpha2 = (A_S / A_B) ** 2
sigma2 = N_S + alpha2 * N_B
stat = chi2(N_S, A_B, A_S, sigma2)
return stat
[docs]def chi2gehrels(N_S, N_B, A_S, A_B):
r"""Chi-square statistic with Gehrel's variance.
"""
N_S = np.asanyarray(N_S, dtype=np.float64)
N_B = np.asanyarray(N_B, dtype=np.float64)
A_S = np.asanyarray(A_S, dtype=np.float64)
A_B = np.asanyarray(A_B, dtype=np.float64)
alpha2 = (A_S / A_B) ** 2
sigma_S = 1 + np.sqrt(N_S + 0.75)
sigma_B = 1 + np.sqrt(N_B + 0.75)
sigma2 = sigma_S ** 2 + alpha2 * sigma_B ** 2
stat = chi2(N_S, A_B, A_S, sigma2)
return stat
[docs]def chi2modvar(S, B, A_S, A_B):
r"""Chi-square statistic with model variance.
"""
S = np.asanyarray(S, dtype=np.float64)
B = np.asanyarray(B, dtype=np.float64)
A_S = np.asanyarray(A_S, dtype=np.float64)
A_B = np.asanyarray(A_B, dtype=np.float64)
stat = chi2datavar(S, B, A_S, A_B)
return stat
[docs]def chi2xspecvar(N_S, N_B, A_S, A_B):
r"""Chi-square statistic with XSPEC variance.
"""
N_S = np.asanyarray(N_S, dtype=np.float64)
N_B = np.asanyarray(N_B, dtype=np.float64)
A_S = np.asanyarray(A_S, dtype=np.float64)
A_B = np.asanyarray(A_B, dtype=np.float64)
# TODO: is this correct?
mask = (N_S < 1) | (N_B < 1)
# _stat = np.empty_like(mask, dtype='float')
# _stat[mask] = 1
stat = np.where(mask, 1, chi2datavar(N_S, N_B, A_S, A_B))
return stat