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Astronomical Data Analysis Software and Systems VI

ASP Conference Series, Vol. 125, 1997

Editors: Gareth Hunt and H. E. Payne

**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

© Copyright 1997 Astronomical Society of the Pacific, 390 Ashton Avenue, San Francisco, California 94112, USA

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