10 - H1 - Desy

8.6 Toy Monte Carlo study 121
Since in almost all the cases the calculated χ 2 /ndf is strongly below one 5 , conclusion
can be drawn that both methods extract the correct cross sections within their own
determined uncertainties. As expected, unfolding produces consistently lower values of
χ 2 , which points to it as a more reliable method. In addition, the impact of model
inaccuracy can be observed, as χ 2 steadily rises with the reweight factor. The importance
of proper model description seems to be more important for E γ T , as χ2 rises faster. The
reason for that are much lower bin purities in case of E γ T bins compared to ηγ case.
In figure 8.12 the results of the toy MC study with f toy
E T ,η
reweighting scheme are presented.
One can see that given the amount of the distortion applied to the toy MC, both unfolding
and fitting procedures fail to correctly determine the E γ T
distribution, with unfolding being
more correct in the lowest E γ T bin. In case of ηγ , unfolding manages to determine the
correct distribution, while fitting fails to do so. In addition, the correlation between bins,
taken into account in the unfolding procedure leads to an increase of the evaluated error.
One may observe that fitting leads to systematically higher cross sections, which may be
explained by the harder E γ T
slope in the model used to extract the cross sections (see
figure 8.10). That leads to the overestimation of migration from below E γ T
= 6 GeV cut
value. The unfolding better deals with this problem (see χ 2 /ndf values in table 8.7).
[pb/GeV]
γ
dσ / dE T
20
15
10
Toy MC
Fitting (χ 2 /ndf=3.7)
Unfolding (χ 2 /ndf=1.4)
[pb]
γ
dσ / dη
50
40
30
Toy MC
Fitting (χ 2 /ndf=3.3)
Unfolding (χ 2 /ndf=0.1)
20
5
10
0
6 8 10 12 14
[GeV]
γ
E T
0
-1 0 1 2
γ
η
Figure 8.12: The E γ T and ηγ distribution of the toy Mc compared to the results obtained
with the unfolding and fitting procedures.
The bin-to-bin correction is known to produce unreliable results in case of large model
inaccuracies, while unfolding is expected to produce correct results even with the particularly
wrong model. One should note, that this is fully true only for high (infinite) number
of output bins, where the discrete output shape can be considered as continuous and
unfolding gets the full freedom in its adaptation. In practice though, the unfolding is limited
by the data statistics and is only as good as correct is the model distribution within
5 In addition, number of degrees of freedom is a small number: four for E γ T and five for ηγ .