10 - H1 - Desy

8.2 Unfolding application 105
8.2.1.3 Background bins
Every single Reconstructed bin and Side bin in real data contains an a priori unknown
fraction of background (see chapter 6). Migration matrix definition includes one
Background bin defined on the Output direction for each of the Input bins. In the
example of figure 8.2, there are eight Background bins corresponding to six Reconstructed
Bins and two Side bins. Due to special usage of the discriminator, there is
a high correlation between Background bins and the lower discriminator bins of the
corresponding Input bins. Signal bins and Migration bins couple to the high discriminator
values, so adding Background bins does not introduce any ambiguity to the
unfolding problem.
8.2.2 Selection efficiency correction
Events generated in the studied phase space, but not fulfilling the selection criteria are
accommodated in a special underflow bin of the migration matrix. For every Signal bin
the efficiency correction factor is defined as a ratio between the sum of weights of all
the events generated in the particular Signal bin to the sum of weights of the events
reconstructed either in one of the Reconstructed bins or in one of the Side bins. After
the calculation of the output vector, the solution is appropriately corrected.
The trigger efficiency correction needs a special treatment, as the MC simulation differs
in this respect significantly from the measured one in data. The correction factor w Trigger ,
defined with the equation 4.4 is used to artificially decrease the simulated trigger efficiency.
For each MC event the original event weight w is modified according to the formula:
w ′ = w · w Trigger . Such modified w ′ weight is used to fill the migration matrix. In the
same time, the corresponding underflow bin, responsible for the detector inefficiency is
filled using the weight w ′′ = w ·(1 −w Trigger ). Since w ′ +w ′′ = w, and taking into account
that the detector inefficiency bin does not participate directly in the unfolding procedure,
the second paradigm holds and the unambiguity of the solution is preserved.
The final migration matrix A is normalised in a following way:
∑
A ij = 1, (8.15)
i
where the sum runs over every Input bin i and the independent normalisation is performed
for every Output bin j.
8.2.3 Regularisation
The multidimensional output binning prevents the usage of the derivative or curvature
regularisation schemes, as neighboring bins should not always remain connected. For
example the highest E γ T bin stays next to the lowest Eγ T bin of the next ηγ bin. Therefore,
this analysis is using the regularisation on the size of the output. As explained earlier,
the only applicable regularisation parameter τ choice method is the L-curve method (see