CStat

class sherpa.stats.CStat(name='cstat')

Bases: sherpa.stats.Likelihood

Maximum likelihood function (XSPEC style).

This is equivalent to the XSpec implementation of the Cash statistic [1] except that it requires a model to be fit to the background. To handle the background in the same manner as XSpec, use the WStat statistic.

Counts are sampled from the Poisson distribution, and so the best way to assess the quality of model fits is to use the product of individual Poisson probabilities computed in each bin i, or the likelihood L:

L = (product)_i [ M(i)^D(i)/D(i)! ] * exp[-M(i)]

where M(i) = S(i) + B(i) is the sum of source and background model amplitudes, and D(i) is the number of observed counts, in bin i.

The cstat statistic is derived by (1) taking the logarithm of the likelihood function, (2) changing its sign, (3) dropping the factorial term (which remains constant during fits to the same dataset), (4) adding an extra data-dependent term (this is what makes it different to Cash, and (5) multiplying by two:

C = 2 * (sum)_i [ M(i) - D(i) + D(i)*[log D(i) - log M(i)] ]

The factor of two exists so that the change in the cstat statistic from one model fit to the next, (Delta)C, is distributed approximately as (Delta)chi-square when the number of counts in each bin is high. One can then in principle use (Delta)C instead of (Delta)chi-square in certain model comparison tests. However, unlike chi-square, the cstat statistic may be used regardless of the number of counts in each bin.

The inclusion of the data term in the expression means that, unlike the Cash statistic, one can assign an approximate goodness-of-fit measure to a given value of the cstat statistic, i.e. the observed statistic, divided by the number of degrees of freedom, should be of order 1 for good fits.

Notes

The background should not be subtracted from the data when this statistic is used. It should be modeled simultaneously with the source.

The cstat statistic function evaluates the logarithm of each data point. If the number of counts is zero or negative, it’s not possible to take the log of that number. The behavior in this case is controlled by the truncate and trunc_value settings in the .sherpa.rc file:

• if truncate is True (the default value), then log(trunc_value) is used whenever the data value is <= 0. The default is trunc_value=1.0e-25.
• when truncate is False an error is raised.

References

[1]	The description of the Cash statistic (cstat) in https://heasarc.gsfc.nasa.gov/xanadu/xspec/manual/XSappendixStatistics.html

Methods Summary

calc_stat(data, model)
calc_staterror(data)

Methods Documentation

calc_stat(data, model)

calc_staterror(data)