Transcript A bin-free Extended Maximum Likelihood Fit

A bin-free Extended Maximum
Likelihood Fit
Peter Litchfield
 A bin free Extended Maximum Likelihood method of fitting
oscillation parameters is described
A Feldman-Cousins style error analysis has been developed
Systematic errors are incorporated into the MC experiments
comprising the F-C analysis giving error contours with statistical
and systematic components
Extended Maximum Likelihood
 Described by Roger Barlow; NIM A297,496
 Maximum Likelihood with a normalisation condition
 The standard maximum likelihood method maximises the likelihood
function
M
L   p( xi ;a1 ...an )
i 1
 where p is the probability density and is normalised to 1, M is the
number of events, x is a measured quantity and the ai are parameters
to be determined.
The fit thus only fits the shape and says nothing about the number of
events
Extended Maximum Likelihood
 In Extended Maximum Likelihood p is replaced by the unnormalised quantity P where
 P( x ;a ...a
i
1
n
)dxi  N( a 1 ....an )
The predicted number of events, N, is a function of the fitted
parameters.
M
It can then be shown that
L  e  N  P( x i ;a1 ....an )
i 1
M
ln L   P( xi ;a1 ....an )  N
i 1
It can also be shown that lnL is maximised for N=M
Extended Maximum Likelihood
In our case the function P is just the extrapolated predicted neutrino
energy distribution for the given set of parameters.
Strictly P should be a continuous function but with a high statistics MC
we can approximate it by the finely binned MC.
So we just sum over the number of predicted MC events Ni(Em) in the
bin corresponding to the measured energy Em of each data event
M
ln L   Ni ( Em )  N
i 1
In the plots that follow I use 125 200 MeV MC bins between 0 and 50
GeV. The bins can be as narrow as the MC and extrapolation
procedure warrents.
Results
Binned likelihood has the standard 500 MeV bins below 10GeV
Unbinned gains at high m2 because of the improved resolution on the
falling edge of the oscillation dip
Little gain at low m2 where there is no data
Feldman-Cousins error analysis
Following the F-C prescription, for each m2-sin22 bin I generate fake
experiments with numbers of events Poisson fluctuated about the number
predicted by my extrapolation.
For each experiment I select events at random from the full Far detector
MC sample, up to the fluctuated number and according to the predicted
energy spectrum.
The lnL distribution is calculated on the m2-sin22 grid for each
experiment and the 2 difference between the best fit point and the
generated point determined.
If say 1000 experiments are generated and fitted, the 2 are sorted
and the 900th 2 from the minimum gives the 90% 2 (290) for that grid
point.
If the data 2 for that grid point is less than 290, that grid point is within
the 90% confidence allowed region
Systematics Analysis
For each MC experiment the parameters of the experiment are varied
according to a set of systematic errors.
The errors for a given experiment are taken randomly from a uniform
distribution between + and – the estimated systematic error.
Notice that CPU time forbids repeating the extrapolation for the > 2.5
million FC experiments required, so all errors are simulated by varying the
selected far MC events.
Systematic parameters can be varied individually or all together.
Correlations between systematic parameters are accounted for.
All identified systematics can be included without significant time or
complication penalty
Systematics Included
1) Normalisation
 The generated event distribution is scaled by a factor randomly
selected between 10.04.
2) Relative hadronic energy scale
 The hadronic energy of the selected far detector events is scaled by
10.033 for each experiment
3) Muon energy scale
 The muon energy is scaled by 10.036
4) Absolute energy scale
 I cannot change the energy in the predicted distribution but a change
in the absolute scale is equivalent to shifting the predicted oscillation
dip in the far detector. The far detector truth energy is shifted by a
random amount between 100MeV in calculating the oscillation
probability
Systematics included
5) PID cut
 The far MC events available at this point in the program have
been selected by the PID. At the moment I can only make a one
sided cut in the selection. Events with PID with a value randomly
selected between 0 and 0.05 above the standard cut are
removed from the fake experiments
6) Extrapolation error
 To try to allow for the extrapolation error I have taken the ratio of
the SKZP extrapolation to my extrapolation and scaled the
predicted distribution by a random fraction between 0 and 1 of
that difference for each experiment