The chi-square statistic is
chi^2 = (sum)_i [ [ N(i,S) - B(i,x,pB) - S(i,x,pS) ]^2 / sigma(i)^2 ]
where N(i,S) is the total number
of observed counts in bin i of the
on-source region; B(i,x,pB)
is the number of predicted background model counts in
bin i of the on-source region (zero for
background-subtracted data), rescaled from
bin i of the off-source region, and
computed as a function of the model
argument x(i) (e.g., energy or time)
and set of background model parameter
values pB;
S(i,x,pS) is the number of
predicted source model counts in bin i,
as a function of the model argument x(i)
and set of source model parameter
values pS;
and sigma(i) is the error in bin
i.
There are several methods for
assigning sigma(i):
- leastsq
- chi2constvar
- chi2datavar
- chi2gehrels
- chi2modvar
- chi2xspecvar
In each of the
implementations, N(i,B) is the
total number of observed counts in bin i
of the off-source region; A(B) is the
off-source "area", which could be the size of the region from
which the background is extracted, or the length of a
background time segment, or a product of the two, etc.; and
A(S) is the on-source "area".
In the analysis of PHA data, A(B) is
the product of the BACKSCAL
and EXPTIME FITS header keyword values, provided
in the file containing the background
data. A(S) is computed similarly,
from keyword values in the source data file.
Note that it is currently assumed that there is a one-to-one
mapping between a given background region bin and a given
source region bin. For instance, in the analysis of PHA
data, it is assumed that the input background counts spectrum
is binned in exactly the same way as the input source counts
spectrum, and any filter applied to the source spectrum
automatically applied to the background spectrum. This means
that the user cannot, for example, specify arbitrary
background and source regions in two dimensions and get
correct results.
However, this limitation only applies when analyzing
background data that have been entered with the load_bkg
command
("ahelp load_bkg").
One can always enter the background as a separate dataset and
jointly fit the source and background regions.
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