Probability of excess

  

Figure 21: distribution of significance for 319 data bins in declination and right ascension. Significance ($\xi$) defined as -log₁₀(Prob) where Prob, P, is the Poisson probability defined by equation 8. The blacks dots is the expected distribution from random fluctuation of background.

This section discusses the investigation of angular clustering of events in the sky plot (figure 17). The probability of excess in a bin is computed by comparing the distribution of significance ($\xi$) of the data set with random background (figure 21). The significance, $\xi$, defined as -log₁₀(P), is calculated for each bin. P is the Poisson probability defined as

$$P=\sum_{n=N_{0}}^{\infty} \frac{e^{-\mu} \mu^{n}}{n!}$$ (8)

where N₀<> is the observed number of events in a bin, and $\mu$ is the mean of the poisson distribution. The mean, $\mu$, is calculated by summing all events within a declination slice and dividing by the number of bins in that slice. The mean is not computed over the entire sky since the number of background events depends strongly on declination angle (figure 34). Hence a low Poisson probability (P) for random fluctuation implies a large significance ($\xi$).

The black dots in figure 21 is the expected significance distribution for random background. The background is obtained by randomizing the right ascension coordinate of all events within each declination band. The dots were obtained by repeating this procedure for 100 trials and taking the average for a given bin. Now a comparison of the two distributions in figure 21 is done by using equation 8 with N₀<> being the number of bins at a particular significance in question ($\xi_{0}$) and $\mu$ is the expect number of bins from random background at the significance $\xi_{0}$. Equation 8 will then give of the probability of excess (P_e<>) which means the probability for there to be N₀<> bins at $\xi_{0}$ from random fluctuations in background. For example in figure 21 the bin with the largest significance of $\sim$ 2.7, has the number of expected bins from random background (N_e<>) of $\sim$ 0.4. With N₀=1<> and $\mu=0.4$ the probability of excess is 33.0%. There is a 33% chance that the excess in this one bin is from the observed background. This bin doesn't have a significance excess.

The conclusion is that there are no significant clustering in the data set. The events are consistent with random fluctuations expected from Poisson statistics. In case of a source being split by a bin boundary, the same exercise was performed with different bin sizes (ie. 18 declination slices and 36 maximal azimuth bins, and 36 by 72 binning). Again there was no significant excess found above the pre-trial probability distributions, hence no detectable point sources in the data set.