Sensitivity Analysis
7.3 Event Classification
7.3.3 Kernel-PDE and Classification Results
The likelihood ratio defined in equation (7.3) provides a very powerful discriminant and is evaluated using the estimated PDF functions for the remaining events. Fig-ure 7.7 shows the logarithm of the ratio for signal and background, normalized to the minimal value of both distributions. Events with a negativeL= log10(fS/fB) are more likely to be background. Most background events are classified correctly with the method described here. Only a small contribution extends into the sig-nal regime with positive values. The sigsig-nal samples are partly misidentified as background, however, the peak of the distribution is just above zero and a large fraction is clearly classified to be signal. The distribution extends to larger val-ues in the signal region due to the normalization to the minimum value obtained, which compresses the background regime from -1 to 0.
There are discontinuities in the distribution of L>0.1 for background events, which is visible in the magnified inset plot. The problem arises due to too small statistics of the background sample, the peaks are build by single events with rel-atively large event weights. In the plot shown, this has been eliminated to some extend by applying an additional cut on the energy deposit, which exploits the different energy spectra of signal and background. Also previous analyses used the reconstructed energy in the final event selection to remove a large fraction of remaining background [98, 131]. Since no energy reconstruction was performed, the total charge parameter Σcharge is used as an approximate measure of the en-ergy deposit . The lower plot in Figure 7.7 shows the signal-to-background ratio in this two dimensional parameter space. In particular the background events with low energy deposits with Σcharge below ∼1·103 cause discontinuities because of large event weights. To remove these events, a cut of Σcharge >1·103 is applied.
Although the plot suggests a two dimensional cut in the energy deposit and likeli-hood ratio parameter space, this path is not followed. Clearly the contribution of background events decreases to high energy deposits, however, the limited statis-tics do not motivate a optimization of the final event selection. Time limitations dictated the use of the available samples.
For increasing L-cut values the signal and background efficienciesNL>Lcut/N are plotted in Figure 7.8 (left plot). The background rejection is very good, a reduction of ∼105 is reached, while still more than 60% of the signal remains.
Beyond that the interpretation becomes difficult due to the fluctuations of the surviving background. It is not clear, whether the gradient is still exponential or flattens out. At the point where the background vanishes, the signal efficiency is still above 20%.
Also shown in the right plot of the same Figure are the event rates within one year and the signal-to-background ratio as a function of the cut value of L.
Applying a cut at 0.05 slightly above the transition point at 0, where events are classified to be signal, the signal-to-noise ratio is∼0.1. Again, the behavior beyond that point is difficult to assess, the background contribution has discontinuities when the remaining events are removed one after the other. At a cut ofL>0.28 the last background event is removed and the signal-to-noise ratio becomes infinite.
0 5 10 15
Figure 7.7: Top: The likelihood ratio distribution for signal and background sam-ples. The ratio is defined by log10(fS/fB) with probability density functions ob-tained from kernel PDE of signal and background training samples. The inset plot shows a magnified version with finer binning at the cross-over region around 0, where events become classified as being signal rather than background. Bottom: Signal-to-background ratio as a function of total charge andL. Note the spots in the upper right to center area, these are the events causing discontinuities due to high event weights.
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
-0.10 -0.05 0.00 0.05 0.10 0.15 0.20 0.25
log10(fS/fB)
Figure 7.8: Left: Background and signal efficiencies with increasing L-cut. The signal efficiency falls on a linear scale, while the background drops exponentially. Too few statistics of the background sample lead to discontinuities when the last remain-ing event samples are removed which have large event weights. Right: The event rate expected in one year calculated from background and signal event samples as a function of the cut value ofL. The signal-to-background ratio is plotted on a different scale. Again, fluctuations in the background sample introduce large background rate variations.
A cut ofL>0.36 is chosen in order to be unaffected by statistical fluctuation.
The value is more than three standard deviations above the maximumL value of the background distribution. The standard deviation is estimated using only values withL >0. An extrapolation of the likelihood ratio distribution for background events yields a background expectation of 4.1·10−4 events per year, which can be treated in the sensitivity calculation as an expectation of 0 background events.
The final passing rates are summarized in Table 7.2. The following Chapter gives the results obtained from this analysis, namely the electron-neutrino effective area of the detector and the sensitivity to electron neutrino fluxes.
Table 7.2: Event rate and filter efficiency for the E−2 signal flux and background sample after application of the final cutL >0.36. The efficiency is calculated with respect to the level one selection. The expected background rate for L > 0.36 is extrapolated from the likelihood ratio plot (Figure 7.7). The background sample yields an event count of 0.
Signal Background
Full E ≤ 1 PeV E > 1 PeV
Events/year 18.8 12.5 6.3 4.1·10−4
Efficiency 15.3% 15.1% 15.8% < 10−11%
Results
This chapter summarizes the results obtained from the analysis described in the previous Chapter and discusses the sensitivity of the IceCubedetector for ultra-high energy electron-neutrino events.