Model Independent Search for Deviations from the Standard Model ...

90 Search Algorithm
oer sucient sensitivity. A binning much ner than the error of the p T measurement does
not make sense.
After dening all relevant combinations of bins, the actual strategy for comparison must
be identied. For a single region, two numbers have to be compared: The amount of data
in the region N data and the sum of all MC contributions in the same region N MC . The
extent of deviation is determined by calculating a p-value:
p-value:
The probability for a given expectation value N MC to measure a uctuation
greater than the one observed with N data when repeating the experiment.
As the number of events observed in data can be too small to assume a Gaussian distribution,
Poisson statistics are applied. Given a Poisson distribution with the expectation
value N MC , the p-value is simply the area of this graph from the point N data onwards,
dened by:
⎧
∞∑
exp(−N MC )(N MC ) i
if N ⎪⎨
data ≥ N MC
i!
i=N
p =
data
N data
. (7.1)
∑ exp(−N MC )(N MC )
⎪⎩
i
if N data < N MC
i!
i=0
This is the probability for N MC to uctuate up to N data and further, given the statistical
error of the measurement.
In this analysis the calculation of the p-value is modied by including the statistical error
of the Monte Carlo mean value N MC itself and the systematic uncertainties of the measurement.
The incorporation of these errors is accomplished by considering a probability
density function made up by a convolution between the Poisson distribution and a Gaussian
of the width δN MC . All errors are interpreted as uncertainties in the expectation
value N MC ; data are xed numbers without any errors. In this way the nal estimator p
for each considered region, based on the procedure of the H1 analysis [19], is dened by:
⎧
∞∑
A ·
⎪⎨
i=N data
p =
⎪⎩
N∑
data
i=0
A ·
∫ ∞
0
∫ ∞
0
db exp ( −(b − N MC ) 2 )
2(δN MC ) 2 · e−b b i
db exp ( −(b − N MC ) 2 )
2(δN MC ) 2 · e−b b i
if N data ≥ N MC
i!
. (7.2)
if N data < N MC
i!
The constant factor A ensures the normalization of the probability density function to
unity.
In this computation N MC is assumed to be the true value, the one nature actually realizes.
Of course simulated Monte Carlo samples are also only the result of generating a variable
numerous times with a given probability density function. Only if the MC-statistics is