Optimization Procedure

From a large pool of geometrical and quality variables, a set of analysis variables that exhibited the greatest differences between signal and data distributions (which are assumed to consist solely of background events) were identified. These variables would have the greatest rejection power (R) defined as

$$R= \frac{\epsilon_{signal}}{\epsilon_{background}}$$ (1)

where $\epsilon_{background}= \frac{N_{bg}^{pass}}{N_{bg}^{0}}$ and $\epsilon_{signal}=\frac{N_{sign}^{pass}}{N_{sign}^{0}}$. N_bg⁰ and N_sign⁰ are the number of events prior to the application of the selection variable and N_bg^pass and N_sign^pass are the number of events that are retained after the cut is applied. The set of variables used for version two optimization are discussed in the next section (section 3.2). This section discusses the procedures developed in optimizing the cuts of these set of variables.

After applying the UCI filtering, additional selection criteria were applied with the objective to retain a high percentage of the simulated signal (greater than 90%) while removing a large fraction of the experimental data. The distributions of these variables for both data and simulated signal events were compared. The specific selection value that retained 95% of the simulated signal was applied to same distribution from experimental data. The variable that produced the greatest rejection power, R, was applied, and the evaluation process was iterated for the remaining variables, until the experimental data was reduced to $\sim$5x10⁴<> events. At this point, the analysis was optimized on Signal/Noise (S/N) (Equation 2). The S/N was computed as a function of zenith angle as the value of each variable varied. This insured that stronger cuts continued to improve the S/N ratio at all zenith angles. The goal was to preferentially increase the acceptance of AMANDA-B10 near the horizon. The S/N ratio is computed as:

$$S/N= \frac{ \Phi_{\nu} \cdot \overline{A}_{eff}^{\nu} \cdot \epsilon \cdot t_{live}} { \mu_{s} (N_{0},N_{b})}$$ (2)

where

$$\overline{A}_{eff}^{\nu}= \frac{\int_{E_{\nu}^{min}}^{E_{\nu}... ...{\nu}^{min}}^{E_{\nu}^{max}} E_{\nu}^{-\gamma} \cdot dE_{\nu}}$$ (3)

and $\Phi_{\nu}$ is the total neutrino flux, $\mu_{s}$ is the maximum mean of the signal at a 90% CL which depends on the number of observed events (N₀<>) and the mean number of background events (N_b<>) in a source bin. The neutrino energy is $E_{\nu}$, live-time is t_live<>, and $\epsilon$ is the fraction of the signal in the source bin. Since we are optimizing on high energy neutrinos, the energy limits were $E_{\nu}^{min}$=1 TeV and $E_{\nu}^{max}$=1000 TeV. The total neutrino flux was arbitrarily normalized to 10^-5<> [m^-2<> sec^-1<>]. The average neutrino effective area ( $\overline{A}_{eff}^{\nu}$) is averaged over energy with a power law energy spectra according to Equation 3. The optimization procedure assumed a power law spectral index, $\gamma$, equal to 2. An equivalent definition of the neutrino effective area is found in section 4.2, equation 10.

To compute S/N as function of zenith angle, the average neutrino effective area ( $\overline{A}_{eff}^{\nu}$) is calculated for twenty bins in cos(zenith) from -1 (vertical up-going) to 0 (horizontal) with the Monte Carlo. The calculation averaged over azimuth since the variation is small; conversely, the neutrino effective area depends strongly on energy and zenith. The next step defines how to bin up the sky since the optimal bin size depends on the angular resolution of the signal. One challenge with this calculation is that the angular resolution changes as the cuts are refined. Rather than changing a fixed grid on the sky, we have adopted the following technique to generate the optimal bin size. The idea is to assume a circular bin centered on the neutrino source for each cos(zenith) angle direction. The optimal bin angle ( $\Psi_{opt}$) is estimated by using the median of the space angle error ( $\Delta \Psi_{m}$), which is the angle between the reconstructed track and the true track in the signal MC. Assuming a circular bin with the source at the center, a Gaussian distribution for the space angle error, and a uniform background distribution, the relationship between the optimal bin angle ( $\Psi_{opt}$) and the space angle error ( $\Delta \Psi_{m}$) is $\Psi_{opt}=1.5852 \cdot \Delta \Psi_{m}$. The efficiency ($\epsilon$) is calculated as

$$\epsilon = \frac{1}{\sqrt{2 \pi \sigma^{2}}} \cdot \int_{\Psi... ...i_{opt}} e^{\frac{-(\Psi - \Psi_{0})^{2}}{2 \sigma^{2}}} d\Psi$$ (4)

where the signal has a Gaussian distribution about the source direction ($\Psi_{0}$) with a sigma ($\sigma$) equal to the median of the space angle error ( $\Delta \Psi_{m}$). The value in equation 4 is 0.7156 and was used for efficiency ($\epsilon$) in equation 2.

The observed number of events N₀<> consists of signal events with a mean $\mu_{s}$ and background events with a mean N_b<>. Obtaining N₀<> and N_b<> from the experimental data, the maximum mean of signal events $\mu_{s}$ with 90% CL is determined using tables that give confidence intervals for the signal mean [4]. The maximum mean of the signal events, $\mu_{s}$, is the maximum number of signal events that can be in the source bin and still give the observed number of events (N₀<>) with a given mean of background events (N_b<>) at a certain confidence level (ie. 90% CL). Note these tables give signal means for N₀<<>16. If a bin contains more than 16 counts, then $2\sqrt{N_{0}}$ is used for the maximum mean of the signal with 95% CL assuming Gaussian statistics.

  

Figure 2: of the optimal bin pointing at a source and the cos(zenith) bin. The circle represents the optimal bin pointing at the source for a given cos(zenith) bin. Azimuth angle is arbitrary. The rectangular box represents the cos(zenith) bin where the vertical direction corresponds to zenith direction and horizontal to azimuth direction.

The number of data events, N₀<>, in an idealized circular bin is deduced from the number of data events in each of the 20 slices in cos(zenith) (N_bin<>). The azimuth angle of the data events is not used, but is assumed to be uniform for each cos(zenith) slice. Figure 2 illustrates how the number of data events is determined for the optimal bin. The rectangular box represents a cos(zenith) slice that contains uniformly distributed data events. The circle represents the optimal bin for that particular cos(zenith) slice. The value of N₀<> is determined from the ratio of the solid angle between the optimal bin and the cos(zenith) slice according to

$$N_{0}= \frac{\Omega_{opt}}{\Omega_{bin}} \cdot N_{bin}$$ (5)

where $\Omega_{opt}$ is the solid angle of the optimal bin (circle in figure 2) and $\Omega_{bin}$ is the solid angle of the cos(zenith) bin (rectangular box in figure 2). The relationship between the solid angle of the optimal bin ( $\Omega_{opt}$) and the optimal bin angle ( $\Psi_{opt}$) is

$$\Omega_{opt}=2\pi \cdot (1- \cos \Psi_{opt})$$ (6)

The solid angle in each cos(zenith) ( $\Omega_{bin}$) are $\pi/10$ steradians.

The mean of the background events (N_b<>) is determined by averaging the number of events in each cos(zenith) slice (N_bin<>) per bin. However, assuming the events are uniformly distribution in each cos(zenith) slice then effectively N₀<> determined in equation 5 is equal to N_b<>. This approximation done in calculating S/N enables a relatively simple and CPU time efficient method to investigate signal to noise versus zenith angle during each iteration of cut optimization.

The effectiveness and accuracy of the optimization procedure relies on MC techniques to calculate the average neutrino effective area and angular resolution. This was indirectly checked by examining the distributions of the analysis variables. Background MC and experimental data are compared at several steps in the analysis chain. Good agreement helps to establish confidence in the signal simulation. This comparison is discussed next.