stats - Statistics

Introduction

gammapy.stats holds statistical estimators, fit statistics and algorithms commonly used in gamma-ray astronomy.

It is mostly concerned with the evaluation of one or several observations that count events in a given region and time window, i.e. with Poisson-distributed counts measurements.

For on-off methods we will use the following variable names following the notation in [Cousins2007]:

Variable	Dataset attribute name	Definition
n_on	counts	Total observed counts in the on region
n_off	counts_off	Total observed counts in the off region
mu_on	npred	Total expected counts in the on region
mu_off	npred_off	Total expected counts in the off region
mu_sig	npred_sig	Signal expected counts in the on region
mu_bkg	npred_bkg	Background expected counts in the on region
a_on	acceptance	Relative background exposure in the on region
a_off	acceptance_off	Relative background exposure in the off region
alpha	alpha	Background efficiency ratio a_on / a_off
n_bkg	background	Background estimate in the on region

The following formulae show how an on-off measurement \((n_{on}, n_{off})\) is related to the quantities in the above table:

\[n_{on} \sim Pois(\mu_{on})\text{ with }\mu_{on} = \mu_s + \mu_b n_{off} \sim Pois(\mu_{off})\text{ with }\mu_{off} = \mu_b / \alpha\text{ with }\alpha = a_{on} / a_{off}\]

With the background estimate in the on region

\[n_{bkg} = \alpha\ n_{off},\]

the maximum likelihood estimate of a signal excess is

\[n_{excess} = n_{on} - n_{bkg}.\]

When the background is known and there is only an “on” region (sometimes also called “source region”), we use the variable names n_on, mu_on, mu_sig and mu_bkg.

These are references describing the available methods: [LiMa1983], [Cash1979], [Stewart2009], [Rolke2005], [Feldman1998], [Cousins2007].

Getting Started

Li & Ma Significance

[LiMa1983] (see equation 17)

As an example, assume you measured \(n_{on} = 18\) counts in a region where you suspect a source might be present and \(n_{off} = 97\) counts in a background control region where you assume no source is present and that is \(a_{off}/a_{on}=10\) times larger than the on-region.

Here’s how you compute the statistical significance of your detection with the Li & Ma formula:

>>> from gammapy.stats import significance_on_off
>>> significance_on_off(n_on=18, n_off=97, alpha=1. / 10, method='lima')
2.2421704424844875

Confidence Intervals

Assume you measured 6 counts in a Poissonian counting experiment with an expected background \(b = 3\). Here’s how you compute the 90% upper limit on the signal strength \(\mu\):

import numpy as np
from scipy import stats
import gammapy.stats as gstats

x_bins = np.arange(0, 100)
mu_bins = np.linspace(0, 50, 50 / 0.005 + 1, endpoint=True)

matrix = [stats.poisson(mu + 3).pmf(x_bins) for mu in mu_bins]
acceptance_intervals = gstats.fc_construct_acceptance_intervals_pdfs(matrix, 0.9)
LowerLimitNum, UpperLimitNum, _ = gstats.fc_get_limits(mu_bins, x_bins, acceptance_intervals)
mu_upper_limit = gstats.fc_find_limit(6, UpperLimitNum, mu_bins)

The result is mu_upper_limit == 8.465.

Using gammapy.stats

• Feldman and Cousins Confidence Intervals
• Fit statistics
• Derivation of the WStat formula

Reference/API

gammapy.stats Package

Statistics.

Functions

background(n_off, alpha)	Estimate background in the on-region from an off-region observation.
background_error(n_off, alpha)	Estimate standard error on background in the on region from an off-region observation.
cash(n_on, mu_on)	Cash statistic, for Poisson data.
cash_sum_cython()	Summed cash fit statistics.
cstat(n_on, mu_on[, n_on_min])	C statistic, for Poisson data.
excess(n_on, n_off, alpha)	Estimate excess in the on region for an on-off observation.
excess_error(n_on, n_off, alpha)	Estimate error on excess for an on-off measurement.
excess_matching_significance(mu_bkg, …[, …])	Compute excess matching a given significance.
excess_matching_significance_on_off(n_off, …)	Compute sensitivity of an on-off observation.
excess_ul_helene(excess, excess_error, …)	Compute excess upper limit using the Helene method.
fc_construct_acceptance_intervals(…)	Convenience function that calculates the PDF for the user.
fc_construct_acceptance_intervals_pdfs(…)	Numerically choose bins a la Feldman Cousins ordering principle.
fc_find_acceptance_interval_gauss(mu, sigma, …)	Analytical acceptance interval for Gaussian with boundary at the origin.
fc_find_acceptance_interval_poisson(mu, …)	Analytical acceptance interval for Poisson process with background.
fc_find_average_upper_limit(x_bins, matrix, …)	Calculate the average upper limit for a confidence belt.
fc_find_limit(x_value, x_values, y_values)	Find the limit for a given x measurement.
fc_fix_limits(lower_limit, upper_limit)	Push limits outwards as described in the FC paper.
fc_get_limits(mu_bins, x_bins, …)	Find lower and upper limit from acceptance intervals.
get_wstat_gof_terms(n_on, n_off)	Goodness of fit terms for WSTAT.
get_wstat_mu_bkg(n_on, n_off, alpha, mu_sig)	Background estimate mu_bkg for WSTAT.
probability_to_significance_normal(probability)	Convert one-sided tail probability to significance.
probability_to_significance_normal_limit(…)	Tail probability to significance in small probability limit.
significance(n_on, mu_bkg[, method, n_on_min])	Compute significance for an observed number of counts and known background.
significance_on_off(n_on, n_off, alpha[, method])	Compute significance of an on-off observation.
significance_to_probability_normal(significance)	Convert significance to one-sided tail probability.
significance_to_probability_normal_limit(…)	Significance to tail probability in large significance limit.
wstat(n_on, n_off, alpha, mu_sig[, mu_bkg, …])	W statistic, for Poisson data with Poisson background.