Standard 1-sigma Gaussian confidence intervals for both SRC_CNTS and BKG_CNTS are computed using an analytical approximation to upper and lower confidence intervals of a Poissonian distribution (Gehrels, 1986, equations 7 and 12).
We propagate the SRC_CNTS and BKG_CNTS 1-sigma confidence intervals through the equation using equation 1.31 in ``A Practical Guide to Data Analysis for Physical Science Students'', L. Lyons, 1991 to get a confidence interval on NET_CNTS, conventionally stated as separate upper and lower 1-sigma ``errors'' on NET_CNTS:
We also compute the ``significance'' of the observed SRC_CNTS as a disproof of the ``null hypothesis'' which is that there is no actual source, i.e. that all the observed counts are background.
Weisskopf et al. (2007, Appendix A2) showed that the probability of observing SRC_CNTS or more counts in the source aperture and observing BKG_CNTS in the background region, conditioned on the assumption that there is no source (i.e. both apertures are sampling the same background population) can be found by integrating a binomial distrubution.
This calculation can be performed with the following call to IDLs binomial function:
Since this expression takes into account the uncertainty one has in estimating the background level (i.e. the Poisson nature of BKG_CNTS), PROB_NO_SOURCE will increase as BKG_CNTS decreases, i.e. a source will be less significant (more consistent with the background) when the background is poorly estimated.
As one might expect, when the background is accurately estimated (BKG_CNTS is large) this expression approaches a simple integral of the Poisson distribution over the interval [SRC_CNTS,] (Weisskopf et al., 2007, § A2), which is equal to:
Important Note: Changing the extraction region will usually change the source significances defined above since both SRC_CNTS and BKG_CNTS will change. The maximal significance is NOT obtained with the largest extraction region (PSF Fraction) because as you enlarge the extraction region you eventually start adding more background than signal and significance goes down.
Some observers choose to use these AE significance statistics in their source detection process--a liberal set of source candidates are nominated by various means, those candidates are extracted by AE, and the catalog is pruned by thresholding on the AE significance statistics. One might choose this approach based on the hope/expectation that AE's significance statistics are more accurate than those computed by source detection codes because AE has a more complex process for estimating the background for each source (critical for these statistics). Or, one might choose this approach because some source candidates are nominated via processes that do not offer significance statistics (e.g. sources detected by-eye, sources detected in reconstructed images, or sources identified from other wavebands).
Note that this sort of procedure has strong analogies to the classic ``cell detection'' algorithm. Viewed in this way, AE can be seen to be evaluating the source significance statistic on only a limited number of cells--the AE extraction regions associated with the list of proposed sources. The procedure for nominating sources can be seen to be merely an efficiency technique--we avoid running AE at the many potential source locations where we have a low expectation that the AE significance statistics would be interesting.
Please note that it is not at all clear whether SRC_SIGNIF or PROB_NO_SOURCE threshold values used to prune the catalog can be used to analytically estimate the number of spurious sources produced by this hybrid procedure. Even though PROB_NO_SOURCE is a classic confidence level, there is no obvious ``number of independent trials'' that can be identified in the hybrid detection process outlined above. Laborious simulations would seem to offer the only hope of quantifying false detection rates for such a detection process. Such simulations may be nearly impossible when the source nomination process is not simple/automated.
Again viewed as a form of cell detection, the hybrid detection process outlined above might be expected to produce spurious sources which are spatially non-uniform because the cell size follows the Chandra 90% PSF size. Since the density of possible cells (extraction regions) is higher on-axis one might expect more spurious sources there (as observed in simulations with CIAOs celldetect). I think observers should be very cautious about performing studies of source spatial distribution that include weak sources near the limits of detection for ANY detection method that is used.
Obviously spectral fitting should produce the most accurate flux estimates.
AE provides estimates for the median (50% quantile), 25% quantile, and 75% quantile of the observed event energies over a variety of bands. These statistics are background-corrected, meaning they seek to characterize the observed spectrum of the astrophysical source if background were not present. AE implements an intuitive and straightforward background correction for standard quartiles, based on the observed cumulative distribution of the net spectrum, as shown in Figure 4. This method appears to be equivalent to that described by Hong et al. (2004, Appendix C), which was developed independently. AE does not yet attempt to estimate individual confidence intervals for these background-corrected quantiles; we expect that resampling techniques would be the best approach to that task.
Similarly, AE provides estimates for the median (50% quantile), 25% quantile, and 75% quantile of the energies of photons incident on Chandra over a variety of bands.