5.3 Hadronic W
5.3.1 Optimization of the Selection
There are twodierent methodsused tooptimizethe selection.
a cut-based analysis
a multi-variableanalysis
In a cut-based selection, cuts are appliedon aset of variables,which have
sepa-rating power between signal and background. This methodis not an ideal one.
Everytime acut is appliedonanew variable, somefraction of the signalevents
are alsorejected. Sometimes an event is rejected, because it doesn't pass a cut,
eventhoughitfulllsalltheothercriteria. Thusamulti-variableanalysishasthe
advantage thatitdoesn't need many cuts. Thereare manywaysof constructing
amulti-variablemethodincludingthe Fisherdiscriminant[81], aneuralnetwork
[82] or a maximum-likelihood[83]. Common to all these methods is their
con-which couldbeused inthe cut-based selection. The nal variable shouldhavea
goodseparation between signal and background. The selectionis doneapplying
acut onjust this variable. This yields high performance and is normally better
than using a cut-based selection. On the other hand such a multi-variable
me-thoduses highlycomplicatedtechniques and isnot verytransparent. A popular
example isthe use of aneural network.
The cut-based method is transparent as one can directly see what happens.
It alsogives directaccess tothe possiblesystematic sourcesas one can compare
the data and Monte Carlo distributionsstep by step. The set of cuts appliedis
found empirically by looking at the variables using the Monte Carlo signal and
background events. Thus the empirical choice of the selection cuts are not the
optimal cuts. In this thesis an optimization of a cut-based selection is studied.
To assure the optimization and compare the performance with a multi-variable
method, anew multi-variablemethodis alsointroduced.
Cut-based Selection
Themost importantstep intheoptimizationistond the optimizationcriteria.
Theoptimumperformance denedinthis thesisistheset ofcuts thatminimizes
the statistical error on the cross section measurement. The expected statistical
error is minimized if the product of the signal eÆciency and the purity is
maximized(see Appendix A):
sig
= (
sig
L )
1=2
; (5.6)
where
sig
istheexpected statisticalerrorofthe crosssectionand Lstandsfor
the dataluminosity. Toachieve theoptimalset ofcuts, the cutsonthe selection
variables are varied one after the other (sequential optimization). Forexample:
1. arbitrarychoice of the selectioncuts for all the variables(initial cuts)
2. Take the 1. selection variable and vary the cut position tond the
maxi-mum value of the product while keeping the othervariablesat their
initialcut values. If the optimal position is found, replace the initial cut
value of the 1. variable with the new cut value
M inv [GeV]
number of events/2 GeV
● Data qqqq, 1st pairing MC signal
MC background
L3
0 50 100 150
40 60 80 100 120
Abbildung 5.10: Invariant mass distribution of the selected events
4. If the optimization is done for all the variables, the 1. iteration is done.
The new iteration begins with the 2. step.
After some iterations (typically 4-5 iterations), there is no further
improve-mentand the optimalset of cutsare found. Tobesure thatthis set ofcuts does
not correspond to alocalminimumin
sig
, the cuts can be tested usingother
initial cuts (1. step in the example). For the cut-based optimization, all the
selection variables including the variables of the 1. preselection are used. The
optimized cut values of the 1. preselection variables are:
1. E
vis
130 GeV
2. E
k
=E
vis
0:25
3. E max
e;
53GeV
4. N
cl uster 30
5. E =E 0:23
The optimized cut values for the other variablesare:
6. sphericity 0:06
7. thrust0:91
8. sumof the cosines between jets 1:25
9. energy angle relation of the jets0:12 rad
10. two jet mass77GeV
11. minimum cluster multiplicity of the jets4
Figure5.10 shows thedistributionof the invariantmass afterthe
optimizati-on. One can clearly see the enhancement of the W events around 80 GeV. The
eÆciency obtained after the optimization is 87.4% with a purity of 78.1%. A
detailedresult of the selectionis listed inTable 5.3. Note that the high
perfor-mance of the optimization does not result only from the method but also from
the right choice of selectionvariables.
Weighting Method
The motivationfor amulti-variable selectionisdescribed above. A brief
discus-sion of how the weighting method works follows.
Consider avariablex that has a dierent distribution for the signal and the
background as shown in the left plot of the Figure 5.11. Instead of cutting on
this distribution, we could give a weight to the events. The events on the right
side,whichwouldbeaccepted,are assignedaweight ofoneand theother events
theweightofzero. Ifwesum theweightsforallthe variablesanddividethe sum
by the number of variables, we will get a nal distribution as seen in Figure
5.11. This represents a very simpleversion of the weighting method.
Abetterversionwouldbetogiveaweightbetween0and1insteadof0and1.
Forthis wetakereferencehistogramsformedforthesignaland background from
theMCsamplesand calculatethebin weightvaluesforeachofthe variables(see
Figure5.12). Foreachdata eventthe weightinagiven bini ofthe variablexis:
w
i
(sig) =
N
i (sig)
N(sig) + N(bg)
; (5.7)
Sources Accepted SM [pb] EÆciency [%] N
Sources Measured [pb] EÆciency [%] N
expected
qqq q 7:57 0:25 87.4 1163.00
tot N
expected
= 1490.5 EÆciency =87.4%
tot N
observed
=1495.0 Purity = 78.1%
Tabelle 5.3: Number of selected data events, N
observed
, number of
ex-pected events from the MC study and accepted cross sections of the
cut-based selection method. The selection eÆciency for the signal process
e
(sig) is the number of signal events in the ith bin and N
i
(bg) is the
numberof background events in the ith bin of the variable x. This idea can be
extended to more than one variable. In a multi-variable case the joint weight
(
(sig)isthe weightofthe event inthe jthvariableand Nthe numberof
variables. The separation of signal and background is much better, if the joint
weight isnormalised to
=
is the jointweight of anevent tobe background with
w = 1 signal
background
w = 0
Number of events
x Ω
signal background
Number of events
Abbildung 5.11: The left distribution shows two separated classes of
events. A weight of zero and one can be given to these events based on
thecut position. On therightside,the expectednalweightdistributionfor
asimple multi-variableweighting methodis illustrated.
where!
(bg)),iftheeventliesinthe
ithbin for the variable j. Correspondingly, N
ij
stands forthe number of events
intheith binforthevariablej. Fortheselectionofevents,acutisplacedonthe
value of the joint weightasshown in Figure5.13, which minimizesthe expected
statisticalerror.
The event weight procedure is done in three steps. First the 1.
preselecti-on and 2. preselection are applied to the events. In the second step, the same
cuts as used in the cut-based selection are applied for the following variables
E
. In the laststep the weights are
calculatedusing the variablessphericity;thrust,m
jj
;SUMCOS,EANG anda
newvariableY
34 . Y
34
canbeconsideredasaparameterwhichmeasureshowwell
the clusters of an event divide into four jets. Specically, Y
34
comes from the
DURHAMjet algorithmand standsforthe y
ij
value atwhichthe event moves
fromthe 4- tothe 3-jetcategory.
The use ofeventweightsresults inanimprovement ofeÆciency onthe order
of 1% having the same purity as ina cut-based selection. The results are listed
i
signal background
w = N (sig) N (sig) + N (bg)
i i
i
x
Number of events
Abbildung 5.12: This is anillustration of the binned weighting for a
dis-tribution.
From the idea of the weighting method more improvement would have been
expected than a mere 1%. But this indicates that the cut-based selection is
already reallyoptimized to the maximum possible performance. The weighting
method developed here could be of more use in a search analysis or rare decay
mode study, where the ratio of signal to background is much lower.