# Licensed under a 3-clause BSD style license - see LICENSE.rst
"""Conversion functions for test statistic <-> significance <-> probability."""
import numpy as np
import scipy.optimize
import scipy.special
import scipy.stats
__all__ = [
"significance_to_probability_normal",
"probability_to_significance_normal",
"probability_to_significance_normal_limit",
"significance_to_probability_normal_limit",
]
[docs]def significance_to_probability_normal(significance):
"""Convert significance to one-sided tail probability.
Parameters
----------
significance : array_like
Significance
Returns
-------
probability : `numpy.ndarray`
One-sided tail probability
See Also
--------
probability_to_significance_normal,
significance_to_probability_normal_limit
Examples
--------
>>> significance_to_probability_normal(0)
0.5
>>> significance_to_probability_normal(1)
0.15865525393145707
>>> significance_to_probability_normal(3)
0.0013498980316300933
>>> significance_to_probability_normal(5)
2.8665157187919328e-07
>>> significance_to_probability_normal(10)
7.6198530241604696e-24
"""
return scipy.stats.norm.sf(significance)
[docs]def probability_to_significance_normal(probability):
"""Convert one-sided tail probability to significance.
Parameters
----------
probability : array_like
One-sided tail probability
Returns
-------
significance : ndarray
Significance
See Also
--------
significance_to_probability_normal,
probability_to_significance_normal_limit
Examples
--------
>>> probability_to_significance_normal(1e-10)
6.3613409024040557
"""
return scipy.stats.norm.isf(probability)
def _p_to_s_direct(probability, one_sided=True):
"""Direct implementation of p_to_s for checking.
Reference: RooStats User Guide Equations (6,7).
"""
probability = 1 - probability # We want p to be the tail probability
temp = np.where(one_sided, 2 * probability - 1, probability)
return np.sqrt(2) * scipy.special.erfinv(temp)
def _s_to_p_direct(significance, one_sided=True):
"""Direct implementation of s_to_p for checking.
Note: _p_to_s_direct was solved for p.
"""
temp = scipy.special.erf(significance / np.sqrt(2))
probability = np.where(one_sided, (temp + 1) / 2.0, temp)
return 1 - probability # We want p to be the tail probability
[docs]def probability_to_significance_normal_limit(probability):
r"""Tail probability to significance in small probability limit.
Reference: https://ui.adsabs.harvard.edu/abs/2007physics...2156C, Equation (4)
They say it is better than 1% for s > 1.6.
Asymptotically: :math:`s \sim \sqrt{-\log(p)}`
See also: `significance_to_probability_normal_limit`
"""
u = -2 * np.log(probability * np.sqrt(2 * np.pi))
return np.sqrt(u - np.log(u))
[docs]def significance_to_probability_normal_limit(significance, guess=1e-100):
"""Significance to tail probability in large significance limit.
See also: `probability_to_significance_normal_limit`
"""
# Note: s^2 = u - log(u) can't be solved analytically.
def f(probability):
if probability > 0:
return probability_to_significance_normal_limit(probability) - significance
else:
return 1e100
return scipy.optimize.fsolve(f, guess)