Exposure Ratio

The analysis of H.E.S.S. data with a background subtraction algorithm relies in gen-eral on the measurement of the number of events N_ON in the signal and N_OFF in the background region. An excess, ∆ =NON−αNOFF, is calculated whereα is the ratio of the exposures of the signal and background region. Typically, a measurement of a γ-ray excess is performed with a large dataset that consist of multiple smaller subdatasets -usually (f.i. in the case of the rotated pixel and driftscan method) multiple observation runs. In general, the method employed for the background subtraction gives a prediction for the value of α individually for every subdataset. In the following,α_i, i.e. the expo-sure ratio for a specific subdataset, is discussed in detail. The combination of multiple subdataset exposure ratios is discussed afterwards.

Exposure Ratio for One Subdataset

To understand the significance and the precise meaning of αi, consider the case where N_i^ON and N_i^OFF are the number of events that pass standard γ-ray event selection criteria in the signal and background region of a subdataset i, respectively. In general, N_i^ON =N_i^ON,sig+N_i^ON,bkg and N_i^OFF =N_i^OFF,sig+N_i^OFF,bkg, i.e. the number of events in the signal and background region is the sum of the number of background events N_i^ON/OFF,bkg that are selected as γ-ray event candidates but in reality are background events and the number of real γ-ray events N_i^ON/OFF,sig.

The excess,

∆_i =N_i^ON−α_iN_i^OFF=N_i^ON,sig+ N_i^ON,bkg−α_iN_i^OFF, (4.3) is in an ideal case intended to be equal to the number of signal events in the signal region, i.e.

∆_i=N_i^ON,sig. (4.4)

This condition defines the exposure ratio as αi= N_i^ON,bkg

N_i^OFF . (4.5)

The given definition of the exposure ratio is precise but in practice only useful as a starting point for further considerations because the number of background events that passγ-ray event selection criteria in the signal region, N_i^ON,bkg, is not measurable inde-pendently from N_i^ON,sig. As a first step, the exposure ratio is defined such that eq. 4.4 is fulfilled only on average, i.e. h∆_ii=hN_i^ON,sigi. This is possible with

αi = hN_i^ON,bkgi

N_i^OFF (4.6)

which leads to statistical fluctuations of ∆_i =N_i^ON,sig+ N_i^ON,bkg− hN_i^ON,bkgi around h∆_ii=hN_i^ON,sigi.

A further step is to approximate N_i^OFF in the exposure ratio by hN_i^OFF,bkgi which in general changes theγ-ray excess to be

h∆_ii=hN_i^ON,sigi −α_ihN_i^OFF,sigi (4.7) on average with the exposure ratio

α_i= hN_i^ON,bkgi

hN_i^OFF,bkgi. (4.8)

A meaningful measurement is thus only possible if hN_i^ON,sigi > α_ihN_i^OFF,sigi, i.e. typi-cally if theγ-ray flux per solid angle in the signal region is larger than in the background region.

In the following, the background acceptances A^ON_i and A^OFF_i for the signal and back-ground region are introduced by

hN_i^ON,bkgi=A^ON_i Ω^ON_i T_i^ON (4.9) and

hN_i^OFF,bkgi=A^OFF_i Ω^OFF_i T_i^OFF (4.10) where

4.1 Introduction

• Ω^ON_i and Ω^OFF_i are the field of views of the signal and background region observa-tions,

• T_i^ON and T_i^OFF are the livetimes, i.e. the dead time corrected observation times, of the signal and background region observations.

This leads to the preliminary expression for the exposure ratio αi= Ω^ON_i T_i^ON

Ω^OFF_i T_i^OFF A^ON_i

A^OFF_i (4.11)

in which all quantities except the acceptances for the signal and background region ob-servation are determinable. A background subtraction algorithm will typically construct the signal and background regions such that the acceptance in the signal and background region are equal and the acceptance ratio cancels thus out in the exposure ratio. The equality of the acceptance in the signal and the background region will, however, in general only hold up to a systematic error which in turn introduces a systematic error σ_α(i) in the exposure ratio that has to be estimated from case to case. Statistical fluctu-ations of the number of signal and background events lead to statistical fluctufluctu-ations of the excess around its average value (eq. 4.7). The statistical fluctuations of the excess around its average value are of the order of magnitude of

σ_∆(i)∼^qN_i^ON+α²_iN_i^OFF (4.12) if the exposure ratio αi is known to have a negligible error σ_α(i). The error on the exposure ratio is in general negligible if

σ_α(i)N_i^OFFσ_∆(i), (4.13)

i.e. if the statistical error on the excess due to fluctuations of the number of signal and background events is much larger than the error on the excess introduced due to the finite precision with which the exposure ratio is known. Note that the systematic error on the excess, σ_α(i)N_i^OFF, scales linearly with the number of events in the background region but the statistical error on the excess due to Poisson fluctuations of the number of signal and background events scales only with the square root of the number of back-ground events.

Traditionally the significance of an excess measured with a Cherenkov telescope is calcu-lated with the method described in Li and Ma [1983]. This method is, however, not able to treat a systematic error on the exposure ratio. A modification that is capable of treat-ing systematic errors is described in appendix D. If no significant excess is measured, an upper limit on theγ-ray event excess is usually derived. The method described in Feld-man and Cousins [1998] is known to have very good frequentist coverage properties but can on the other hand not treat a systematic error on the exposure ratio. Alternatively, the method described in Rolke et al. [2005] is known to be in general over-covering but can consider a systematic error on the excess.

Combination of Multiple Subdatasets

Consider now the situation that multiple subdatasetsi= 1..K are to be combined. The combination of the dataset has to state the significance of a putative signal measurement.

This would in principle be possible with a likelihood ratio method similar to the one applied in Li and Ma [1983] when a likelihood function for the combined measurement is defined as the product of the likelihood functions for the individual subdataset. This is possible because all measurements are assumed to be independent but it leads to a log-likelihood ratio which is asymptotically behaving like a χ² distribution with K degrees of freedom if no signal is present. In contrast, the same method for only one subdataset leads in the same situation (no signal present) to a log-likelihood ratio which is asymptotically behaving like aχ²distribution with only one degree of freedom, i.e. like the absolute value of a standard normal random variable. This is obviously much easier to interpret. Apart from that there are other practical problems, the most problematic being that it is in general intended to determine a γ-ray flux in case of a significant signal or, in case that no significant signal is measured, an upper limit on theγ-ray flux.

This is in practice only possible if aγ-ray excess is measured and leads to the necessity to combine the individually measured γ-ray excesses ∆i for every subdataset into one combinedγ-ray excess ∆. In practice, it is argued that the individual subdatasets are combined by summing up the number of signal and background events,N^ON =^P_iN_i^ON andN^OFF =^P_iN_i^OFF. The combined excess is then given by

∆ =N^ON−αN^OFF (4.14)

with a new exposure ratio α for the combined dataset. A consideration similar to the one given above for only one subdataset leads to the expression for the exposure ratio

α= P

iΩ^ON_i T_i^ONA^ON_i P

iΩ^OFF_i T_i^OFFA^OFF_i . (4.15) One interesting point to note here is that the average excess,h∆i, is given by

h∆i=hN^ONi −αhN^OFFi=hN^ONi −^X

i

αihN_i^OFFi=h^X

i

∆_ii (4.16) by means of the definitions 4.9 and 4.10 and the rewriting of eq. 4.15 in the form

α= P

iαiT_i^OFFΩ^OFF_i A^OFF_i P

iT_i^OFFΩ^OFF_i A^OFF_i . (4.17) Equation 4.16 means that the average combined excess is equal to the average of the sum of the excess for every subdataset, a result that is supported by intuition. However, the exposure ratio defined in eq. 4.15 is not a simple quantity. The primary reason for the complexity of the general exposure ratio is that even if a suitably chosen back-ground subtraction algorithm manages to gain perfectly equal acceptances in the signal and background region for every single subdataset,A^ON_i =A^OFF_i , the subdataset

accep-4.1 Introduction tances do not cancel out of the exposure ratio. The cancellation would only hold if the acceptances were additionally equal for every subdataset which is highly unrealistic³. In practice, there are only two ways out of this situation.

• One can try to construct the signal and background regions such thatαi∼const.

In that case, the average ¯α = 1/K^P_iα_i can be used and the exposure ratio for the combined dataset becomes trivially α= ¯α. This method can be applied if the RMS of the αi, RMS(αi), fulfills

RMS(αi)N^OFFpN^ON+α²N^OFF. (4.18) Alternatively, a systematic error on the exposure ratio must be considered.

• The other option is to argue that the acceptances A_i do not vary too much from subdataset to subdataset. In that case, the average ¯A is used and the exposure ratio becomes

For the case where the exposure ratio, α_i, for every subdataset is only known up to a non-negligible systematic error,σ_α(i), it is of central importance that the systematic error σ_α(i) is of random nature and not biasing towards a preferred direction. For instance for the first of the two options considered above with almost equal exposure ratios for every subdataset, αi = ¯α, and almost equal systematic error for the exposure ratio for every subdataset, σ_α(i)=σα_¯, the random nature of the systematic error leads to an on average smaller systematic error on the exposure ratioα= ¯α for the combined dataset, i.e. σα = 1/√

Kσα¯. This means that the systematic error on the excess introduced by the systematic error on the exposure ratio scales with the number of subdataset K like

√K, i.e. like the statistical error on the excess. If the systematic error were in contrast

not random, the systematic error on the excess introduced by the systematic error on the exposure ratio would scale directly with K and at some point limit the potential to improve the sensitivity by means of increasing the number of subdataset.