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- poisson_conf_interval

# poisson_conf_interval

`astropy.stats.``poisson_conf_interval`(n, interval=u'root-n', sigma=1)

  Poisson parameter confidence interval given observed counts

  Parameters:

  n : int or numpy.ndarray

  > Number of counts (0 <= `n`).

  interval : {‘root-n’,’root-n-0’,’pearson’,’sherpagehrels’,’frequentist-confidence’}, optional

  > Formula used for confidence interval. See notes for details. Default is `'root-n'`.

  sigma : float

  > Number of sigma for confidence interval; only supported for the ‘frequentist-confidence’ mode.

  Returns:

  conf_interval : numpy.ndarray

  > `conf_interval[0]` and `conf_interval[1]` correspond to the lower and upper limits, respectively, for each element in `n`.

  Notes

  The “right” confidence interval to use for Poisson data is a matter of debate. The CDF working group [recommends][pois_eb] using root-n throughout, largely in the interest of comprehensibility, but discusses other possibilities. The ATLAS group also [discusses][ErrorBars] several possibilities but concludes that no single representation is suitable for all cases. The suggestion has also been [floated][ac12] that error bars should be attached to theoretical predictions instead of observed data, which this function will not help with (but it’s easy; then you really should use the square root of the theoretical prediction).

  The intervals implemented here are:

  1. ‘root-n’ This is a very widely used standard rule derived from the maximum-likelihood estimator for the mean of the Poisson process. While it produces questionable results for small n and outright wrong results for n=0, it is standard enough that people are (supposedly) used to interpreting these wonky values. The interval is

  CI = (n-\sqrt{n}, n+\sqrt{n})

  2. ‘root-n-0’ This is identical to the above except that where n is zero the interval returned is (0,1).

  3. ‘pearson’ This is an only-slightly-more-complicated rule based on Pearson’s chi-squared rule (as [explained][pois_eb] by the CDF working group). It also has the nice feature that if your theory curve touches an endpoint of the interval, then your data point is indeed one sigma away. The interval is

  CI = (n+0.5-\sqrt{n+0.25}, n+0.5+\sqrt{n+0.25})

  4. ‘sherpagehrels’ This rule is used by default in the fitting package ‘sherpa’. The [documentation][sherpa_gehrels] claims it is based on a numerical approximation published in [Gehrels 1986][gehrels86] but it does not actually appear there. It is symmetrical, and while the upper limits are within about 1% of those given by ‘frequentist-confidence’, the lower limits can be badly wrong. The interval is

  CI = (n-1-\sqrt{n+0.75}, n+1+\sqrt{n+0.75})

  5. ‘frequentist-confidence’ These are frequentist central confidence intervals:

  CI = (0.5 F_{\chi^2}^{-1}(\alpha;2n), 0.5 F_{\chi^2}^{-1}(1-\alpha;2(n+1)))

  where F_{\chi^2}^{-1} is the quantile of the chi-square distribution with the indicated number of degrees of freedom and \alpha is the one-tailed probability of the normal distribution (at the point given by the parameter ‘sigma’). See [Maxwell 2011][maxw11] for further details.

  Examples

      >>> poisson_conf_interval(np.arange(10), interval='root-n').T
      array([[  0.        ,   0.        ],
             [  0.        ,   2.        ],
             [  0.58578644,   3.41421356],
             [  1.26794919,   4.73205081],
             [  2.        ,   6.        ],
             [  2.76393202,   7.23606798],
             [  3.55051026,   8.44948974],
             [  4.35424869,   9.64575131],
             [  5.17157288,  10.82842712],
             [  6.        ,  12.        ]])

      >>> poisson_conf_interval(np.arange(10), interval='root-n-0').T
      array([[  0.        ,   1.        ],
             [  0.        ,   2.        ],
             [  0.58578644,   3.41421356],
             [  1.26794919,   4.73205081],
             [  2.        ,   6.        ],
             [  2.76393202,   7.23606798],
             [  3.55051026,   8.44948974],
             [  4.35424869,   9.64575131],
             [  5.17157288,  10.82842712],
             [  6.        ,  12.        ]])

      >>> poisson_conf_interval(np.arange(10), interval='pearson').T
      array([[  0.        ,   1.        ],
             [  0.38196601,   2.61803399],
             [  1.        ,   4.        ],
             [  1.69722436,   5.30277564],
             [  2.43844719,   6.56155281],
             [  3.20871215,   7.79128785],
             [  4.        ,   9.        ],
             [  4.8074176 ,  10.1925824 ],
             [  5.62771868,  11.37228132],
             [  6.45861873,  12.54138127]])

      >>> poisson_conf_interval(np.arange(10),
      ...                       interval='frequentist-confidence').T
      array([[  0.        ,   1.84102165],
             [  0.17275378,   3.29952656],
             [  0.70818544,   4.63785962],
             [  1.36729531,   5.91818583],
             [  2.08566081,   7.16275317],
             [  2.84030886,   8.38247265],
             [  3.62006862,   9.58364155],
             [  4.41852954,  10.77028072],
             [  5.23161394,  11.94514152],
             [  6.05653896,  13.11020414]])

      >>> poisson_conf_interval(7,
      ...                       interval='frequentist-confidence').T
      array([  4.41852954,  10.77028072])

  [pois_eb]: http://www-cdf.fnal.gov/physics/statistics/notes/pois_eb.txt

  [ErrorBars]: http://www.pp.rhul.ac.uk/~cowan/atlas/ErrorBars.pdf

  [ac12]: http://adsabs.harvard.edu/abs/2012EPJP..127...24A

  [maxw11]: http://adsabs.harvard.edu/abs/2011arXiv1102.0822M

  [gehrels86]: http://adsabs.harvard.edu/abs/1986ApJ...303..336G

  [sherpa_gehrels]: http://cxc.harvard.edu/sherpa4.4/statistics/#chigehrels