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Fix table format
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adonath committed Mar 24, 2021
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39 changes: 18 additions & 21 deletions docs/estimators/index.rst
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Expand Up @@ -29,37 +29,32 @@ as :math:`-\sqrt{\Delta TS}` by convention.
In general the flux can be estimated using two methods:

#. **Based on model fitting:** given a (global) best fit model with multiple model components,
the flux of the component of interest is re-fitted in the chosen energy, time or spatial
region. The new flux is given as a ``norm`` with respect to the global reference model.
Optionally other component parameters in the global model can be re-optimised. This method
is also named **forward folding**.
the flux of the component of interest is re-fitted in the chosen energy, time or spatial
region. The new flux is given as a ``norm`` with respect to the global reference model.
Optionally other component parameters in the global model can be re-optimised. This method
is also named **forward folding**.

#. ***Based on excess:** in the case of having one energy bin, neglecting the PSF and
not re-optimising other parameters, one can estimate the significance based on the
analytical solution by [LiMa1983]. In this case the "best fit" flux and significance
are given by the excess over the null hypothesis. This method is also named
**backward folding**.
#. **Based on excess:** in the case of having one energy bin, neglecting the PSF and
not re-optimising other parameters, one can estimate the significance based on the
analytical solution by [LiMa1983]. In this case the "best fit" flux and significance
are given by the excess over the null hypothesis. This method is also named
**backward folding**.


Uniformly for both methods most estimators compute the same basic quantities:

================= =================================================
Quantity Definition
================= =================================================
e_ref Reference energy
e_min Minimum energy
e_max Maximum energy
norm Best fit norm with respect to the reference spectral model
norm_err Symmetric error on the norm derived from the Hessian matrix
ts Difference in fit statistics (`stat - stat_null` )
sqrt_ts Square root of ts time sign(norm), in case of one degree of freedom, corresponds to significance (Wilk's theorem)
================= =================================================
stat Fit statistics value of the best fit hypothesis
stat_null Fit statistics value of the null hypothesis
================= =================================================
ts Difference in fit statistics (`stat - stat_null` )
sqrt_ts Square root of ts time sign(norm), in case of one degree of freedom, corresponds to significance (Wilk's theorem)
npred Predicted counts of the best fit hypothesis, equivalent to correlated counts for backward folding
npred_null Predicted counts of the null hypothesis
npred_excess Predicted counts of the excess over `npred_null`, equivalent to (`npred - npred_null`)
npred_null Predicted counts of the null hypothesis, equivalent to correlated null counts for backward folding
npred_excess Predicted counts of the excess over `npred_null`, equivalent to (`npred - npred_null`), equivalent to correlated counts for backward folding
================= =================================================


Expand All @@ -71,7 +66,6 @@ Quantity Definition
norm_errp Positive error of the norm
norm_errn Negative error of the norm
norm_ul Upper limit of the norm
================= =================================================
norm_scan Norm scan
stat_scan Fit statistics scan
================= =================================================
Expand All @@ -80,13 +74,16 @@ To compute the error, assymetric errors as well as upper limits one can
specify the arguments ``n_sigma`` and ``n_sigma_ul``. The ``n_sigma``
arguments are translated into a TS difference assuming ``ts = n_sigma ** 2``.

In addition to the norm values a reference spectral model is given.
Using this reference spectral model the norm values can be converted
In addition to the norm values a reference spectral model and energy ranges
are given. Using this reference spectral model the norm values can be converted
to the following different SED types:

================= =================================================
Quantity Definition
================= =================================================
e_ref Reference energy
e_min Minimum energy
e_max Maximum energy
dnde Differential flux at ``e_ref``
flux Integrated flux between ``e_min`` and ``e_max``
eflux Integrated energy flux between ``e_min`` and ``e_max``
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