10 - H1 - Desy

82 Photon signal extraction
n∏
var
p MV S(B) A (i) =
k=1
p S(B),k (x k (i)). (6.12)
This approach assumes statistical independence of all the input variables, which is usually
not fully true. The output of the MVA is a likelihood ratio, denoted further as a
discriminator D. For an event i it is calculated as
D(i) =
p MV A
S
p MV S
A (i)
(i) + p MV B A (i) . (6.13)
It can be shown that in absence of model inaccuracies (such as correlations between input
variables, or an inaccurate probability density model), the ratio given by equation 6.13
provides optimal signal from background separation for the given set of input variables [].
Some other, alternative, mostly more advanced classifier definitions have been also studied
for their possible application in this analysis. The result of this study is presented in
appendix B together with a short discussion of their advantages and disadvantages.
The classifier has been trained independently in double differential bins of transverse
energy and pseudorapidity summarised in appendix in table A-1. The choice of both
dimensions has been motivated by a brief discussion in section 6.1. The discriminator
distribution for both signal and background is presented in figure 6.7 as studied with the
evaluating sample. The discriminator, by definition restricted to the range 0 < D < 1,
tends to peak near one for the signal events and near zero for the background events.
In addition to the separation, defined in 6.1.2, for the peaked distributions as in the case
od D, one can study the significance 〈G〉 of the classifier being equal to the difference
between the classifier means for signal and background divided by the quadratic sum of
their root-meansquares.
∫
pS (x) − p B (x)
〈G〉 =
p 2 S (x) + (6.14)
p2 B
(x)dx,
The transverse energy dependence of both benchmark quantities are shown in the fig 6.8
for each LAr wheel separately. As a reference, in the same plot, the separation of the
best discriminating input variable - R T is shown. One can see the advantage of combining
more than one input variable, as the discriminator separation is higher than the separation
of the single input variable. There is no advantage though in the cases where the
best input variable is highly dominant, as for CB3, FB1 and FB2 LAr wheels, where
separation power of R T is much higher than of all other input variables. Again, one can
see the consistent drop of both, the separation and significance with the transverse energy
following the dependence of the input variables.
The maximum likelihood method chosen in this analysis, is valued for its transparency,
simplicity and speed. At the same time, the method’s limitations are numerous. Particularly,
the incorrect treatment of correlations between input variables is believed to lead
to dimunision of the discrimination performance [116]. In the statistical theory there are
other, more advanced classifier definitions, developed for improvement of the performance.