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Astronomical Data Analysis Software and Systems VI
ASP Conference Series, Vol. 125, 1997
Editors: Gareth Hunt and H. E. Payne

Heuristic Estimates of Weighted Binomial Statistics for Use in Detecting Rare Point Source Transients

James Theiler and Jeff Bloch
Astrophysics and Radiation Measurements Group, MS-D436 Los Alamos National Laboratory, Los Alamos, NM 87545 e-mail:,



The ALEXIS (Array of Low Energy X-ray Imaging Sensors) (Priedhorsky et al. 1989) satellite scans nearly half the sky every fifty seconds, and downlinks time-tagged photon data twice a day. The standard science quicklook processing produces over a dozen sky maps at each downlink, and these maps are automatically searched for potential transient point sources. We are interested only in highly significant point source detections, and, based on earlier Monte-Carlo studies (Roussel-Dupré et al. 1996), only consider p<10-7, which is about 5.2 ``sigmas.'' Our algorithms are therefore required to operate on the far tail of the distribution, where many traditional approximations break down. Although an exact solution is available for the case of unweighted counts (Lampton 1994), the problem is more difficult in the case of weighted counts. We have found that a heuristic modification of a formula derived by Li & Ma (1983) provides reasonably accurate estimates of p-values for point source detections even for very low p-value detections.


1. Introduction

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.

2. Unweighted Counts

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).

3. Weighted Counts

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.

4. Limit of Precisely Known Background

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 straightforwardgif 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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