Cash (1979) showed that the minimization
criterion is a very bad one if any of the observed data bins had few counts. A
better criterion is to use a likelihood function:
where
are the observed data
and
the values of the
function. Minimizing C for some model gives the best-fit parameters.
Furthermore, this statistic can be used in the same, familiar way as the
statistic to
find confidence intervals. One finds the parameter values that give
where
N is the same number that gives the required confidence for the number of
interesting parameters as for the case.
Castor (priv. comm.) has pointed out that a better function to use is :
This differs from the first
function by a quantity that depends only upon the data. In the limit of a large
number of counts this second function does provide a goodness-of-fit criterion
similar to that of and it
is now used in XSPEC. It is important to note that the C-statistic assumes
that the error on the counts is pure Poisson, and thus it cannot deal with
data that already has been background subtracted, or has systematic errors.
Arnaud (2003, ApJ submitted) has extended the method of Cash to include the case when a background spectrum is also in use. Note that this requires the source and background spectra to both be available, it does not work on a background-subtracted spectrum.
Suppose we have an
observation which produces events
in the
spectral bins
in an exposure time of
. This
observation includes events from the source of interest along with background
events. Further suppose that we perform a background observation which
generates
events in an exposure
time
. If the source count
rate in bin i is
then
the new fit statistic is
where
and
.
The sign of di in fi is chosen so that fi > 0.
In the limit of large numbers of counts/bin a second-order Taylor expansion shows that W tends to
which
is distributed as with
degrees of freedom,
where the model
has M
parameters (including the normalization).
XSPEC’s
default minimization algorithm is a variant of Marquardt's method (the
Levenberg-Marquadt algorithm) described in §11.5 of Data Reduction and Error
Analysis for the Physical Sciences by Bevington (2002). (The reader is
advised that this description is designed to be read in conjunction with
Bevington.) This method is appropriate for fitting models that are twice
differentiable. We illustrate the implementation of this W statistic using this
method. The algorithm works by finding a matrix and a vector
such
that the equation :
gives
sensible values of the change in parameters, , for the fitting function. Bevington §11.4 gives the
derivation of a and b and shows that b is parallel to the gradient of
.
Now the C-statistic has a gradient with respect to the parameters of the fitting function of :
So, following Bevington,
expand about
:
.
substitute into C and minimize with respect to the changes in the parameters :
setting this to zero we obtain, to first order in the parameter changes,
or :
where:
These
a and b then are substituted
for those used in the case
and the algorithm works as required. Note that
is
to first order in partial derivatives in y,
evaluated at y0.
There is one further
difference in XSPEC between the and
likelihood methods, which is caused by the fact that XSPEC uses an analytic
formula for setting the model normalization. In the
case, this means multiplying the current model by:
where is the error on
. In the likelihood
case the corresponding factor is:
An analogous argument to the above can be followed through for the W statistic. We need the partial derivatives of W which are evaluated as follows.
where
and
Note that in this case there is no analytic formula that can be used to set the model normalization.
Arnaud, K. A. 2003, Astrophysical Journal, submitted
Bevington, P. 2002 Data Reduction and Error Analysis for the Physical Sciences Third Edition (McGraw Hill:Columbus)
Cash, W. 1979 Parameter estimation in astronomy through application of the likelihood ratio Astrophysical Journal 228, 939