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James Theiler and Jeff Bloch
Astrophysics and Radiation Measurements Group, MS-D436
Los Alamos National Laboratory, Los Alamos, NM 87545
e-mail:
jt@lanl.gov,
jbloch@lanl.gov
We test the null hypothesis of no point source (assuming a
spatially uniform background) at a given location by enclosing that
location with a source kernel (whose area 
is generally matched
to the point-spread-function of the telescope) and then
enclosing the source kernel with a relatively large background annulus
(area 
). Given 
photons in the source kernel, and
photons in the background annulus, the problem is to determine
whether the number of source photons is significantly larger
than expected under the null.
More sensitive point source detection is obtained by weighting the photons to match the point-spread function of the telescope more precisely. Further enhancements are obtained for ALEXIS data by weighting also according to instantaneous scalar background rate, pulse height, and position on the detector. In this case, we ask whether the weighted sum of photons in the source region is significantly larger than expected under the null.
If counts are unweighted (i.e., all weights are equal), then it
is possible to write down an exact, explicit expression for the
probability of seeing 
or more photons in the source kernel,
assuming 
is fixed. This is a binomial
distribution, and Lampton (1994) showed that the p-value associated
with this observation can be expressed in terms of the incomplete beta
function: 
, where 
. See
also Alexandreas et al. (1994), for an alternative derivation of
an equivalent expression (the assumption that 
is fixed is
replaced by a Bayesian argument).
If the count rate is high (or the exposure long), so that 
and
are large, then an appropriate Gaussian approximation can be used.
In general, this involves finding a ``signal'' and dividing it by
the square root of its variance.
Case 1u. The most straightforward approach uses
the signal 
, where 
.
Under the null hypothesis, this signal has an expected value of zero, and
a variance-if 
and 
are treated as independent Poisson
sources-of 
. To get a p-value, use
where 
converts
``sigmas'' of significance into a one-tailed p-value.
Case 2u. An alternative approach, suggested by Li & Ma (1983),
treats the sum 
,
as fixed, so that 
and 
are binomially distributed. In
particular, choose the signal 
, and note that the
variance of 
is given by 
, while the variance of
is by definition zero. In that case
Case 3u. By looking at a ratio of Poisson likelihoods, Li & Ma (1983) also derived a more complicated equation
where 
and 
. This is
considerably more accurate than Eqs. (14,15) when
and 
are not large, but is still just an approximation to
Lampton's exact formula. Abramowitz & Stegun (1972) provide several
approximations to the incomplete beta function, one of which (25.5.19) is an
asymptotic series whose first term looks very much like the Li & Ma
formula.
The left panel of Figure 1 compares these cases, along
with the Lampton (1994) formula, using a Monte-Carlo simulation.
Figure: Results of Monte-Carlo experiments with N=100 photons, with
, and with 
trials. For the weighted experiment,
N weights were uniformly chosen from zero to one,
and assigned to the N photons. The photons
were randomly assigned to the source kernel or background annulus with
probabilities f and 1-f respectively. Values of 
, 
,
, and 
were computed, and a p-value was computed using
the formulas for the three cases.
As the p-values were computed, a cumulative histogram 
was
built indicating the number of times a p-value less than p was observed.
Since we expect 
, we plotted 
as the frequency of
``overoccurrence'' of that p-value. The plot is this overoccurrence
as a function of ``significance,'' defined by 
.
Original PostScript figure (87kB).
Define 
and
, where 
is the weight of the
i-th photon. Notice
that when all the weights are equal to one, we have
and 
. Note
also that 
, and that
. We do not
make any assumptons about weights averaging or summing to unity. (We
define 
and 
similarly.)
Generalizing Case 1u, we define the signal as 
and
then treating source and background as independent, we can write the
variance as 
. We can similarly generalize Case 2u
and obtain:
Case 3w: It is not as straightforward to generalize Eq. (16), but we have tried the following heuristic:
where 
and 
.
The Monte-Carlo results shown in Figure 1 indicate that this
heuristic provides reasonably accurate p-values even for very small
values of p.
An interesting limit occurs as the background annulus becomes large. Here,
, and the expected
backgrounds 
, 
, etc. are all precisely known.
For the unweighted counts, the exact p-value can be expressed
in terms of the incomplete gamma function:
. The
Gaussian estimate of significance is straightforward
both for the unweighted case, 
, and for the weighted
case: 
.
In this limit, Eq. (19) becomes
Marshall (1994) has suggested an empirical formula
,
where 
, which produced reasonable results
in his simulations, but does not appear
well suited for p-values at the far tail
of the distribution.
This work was supported by the United States Department of Energy.
Abramowitz, M., & Stegun, I. A. 1972, Handbook of Mathematical Functions (Dover, New York), 945
Alexandreas, D. E., et al. 1993, Nucl. Instr. Meth. Phys. Res. A328, 570
Babu, G. J., & Feigelson, E. D. 1996, Astrostatistics (Chapman & Hall, London), 113
Lampton, M. 1994, ApJ, 436, 784
Li, T.-P., & Ma, Y.-Q. 1983, ApJ, 272, 317
Marshall, H. L. 1994, in Astronomical Data Analysis Software and Systems III, ASP Conf. Ser., Vol. 61, eds. D. R. Crabtree, R. J. Hanisch, & J. Barnes (San Francisco, ASP), 403
Priedhorsky, W. C., Bloch, J. J., Cordova, F., Smith, B. W., Ulibarri, M., Chavez, J., Evans, E., Seigmund, O., H. W., Marshall, H., & Vallerga, J. 1989, in Berkeley Colloquium on Extreme Ultraviolet Astronomy, Berkeley, CA, vol 2873, 464
Roussel-Dupré, D., Bloch, J. J., Theiler, J., Pfafman, T., & Beauchesne, B. 1996, in Astronomical Data Analysis Software and Systems V, ASP Conf. Ser., Vol. 101, eds. G. H. Jacoby and J. Barnes (San Francisco, ASP), 112
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