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Global Analysis with the event-byevent extrapolation
Peter Litchfield
 I will describe how the event-by-event extrapolation method would
work for a global fit including
CC - events
CC + events
NC events
Anti-fiducial events
Rock events
Except for the rock events this is all working now.
Event-by-event Method
 The method is based on the beam MC which relates near detector
events to far detector events from the same parent decay
 Predicted far detector spectra are built up from individual near
detector MC events
 The following stages are carried out
1) Correction of the MC using near detector data
2) Extrapolation to the far detector using beam MC truth events
plus any physics (oscillation parameters etc).
3) Selection of far detector MC truth events with the same physics
as the near detector event and construction of predicted far
detector reconstructed distributions.
4) Parameter fitting and error determination
MC correction
The beam MC is corrected on an event by event basis by weighting
each event by the appropriate element of the reconstructed E v Eshw
matrix of
Number of reconstructed data events with E  -Eshw
Number of reconstructed MC events with E  -Eshw
E is signed by the measured sign of the 
Events with no reconstructed  have their own column
All events are used, with only good beam and
detector cuts (and a minimum of data cleaning cuts)
Global since ,  and nc events are weighted
separately
Run 2 MC cc events
My weights
SKZP weights
Equivalent to the SKZP fit
Does it work?
Event weight
Extrapolation to the far detector
 The extrapolation is done by the beam MC.
 Each near detector MC event has
1) The ratio of the probabilities that the neutrino from this beam
particle decay hit the near detector to that it hit the far detector
(just decay physics and geometry)
2) The energy that a neutrino from this decay would have when it
hit the far detector (decay physics)
 From these one can calculate truth far detector distributions from the
near detector MC
 Any physics that happens on the way, standard oscillations,
separate  oscillations, sterile  oscillations, etc can be added as
weights to the event.
 Since all near MC events are extrapolated the analysis is still global
Far detector predictions
We now have a near detector MC event, weighted by the correction
weight and any physics weights, and a corrected far detector energy.
Events from the far detector MC are selected that have the same
truth physics (CC/NC, QEL/RES/DIS) and the same truth energy and y
The distribution of any reconstructed quantity for these events can
be produced, normalised to one near MC event (unweighted)
Any cuts, selections, etc can be applied to these events yeilding, for
example, CC and NC samples,  and  samples, fiducial and antifiducial events, separation into resolution bins, etc, etc.
These distributions are summed for all the near detector MC events
with relative pot and fiducial volume weights to get the correct final
normalisation
Rock,  and e events
Distributions for event types that are not on the standard far detector
MC can be produced in the same way
Events are selected from Rock MC,  or e far MC files and their
reconstructed parameter distributions calculated, weighted in addition
for cross-section differences.
Again all cuts and selections can be applied and the results summed
with the standard MC distributions
We now have a global prediction for all Far detector reconstructed
events
We can analyse any selection independently or all together as a
global analysis
Run 1,2A Predicted Distributions
No oscillations
Selected NC events
NC with reconstructed track
Selected CC events
NC with no track
Predicted Distributions m2=0.00238, sin22=1.0, 13=0
Selected NC events
NC with reconstructed track
Selected CC events
NC with no track
Comparison with Beam Matrix
Agreement is not bad but there is a systematic difference
Beam peak is narrower in my extrapolation
No definitive explanation of the difference at the moment
My guess is that it is to do with events being mis-reconstructed out of
the beam peak in the near detector
I extrapolate truth so it does not transmit to the far detector prediction
Fitting Procedure
In principle one can use whatever fitting procedure one wants to compare
far data with far prediction
I use methods which fit well with the extrapolation method
“Unbinned” Extended Maximum Likelihood Grid Search
Feldman-Cousins error analysis
Extended maximum likelihood method described in DocDb-5109
Unbinned data is compared with a binned probability density given by the
far detector prediction to produce a likelihood
The low statistics data is unbinned
The probability distribution comes from the MC and near detector data
which have high statistics and can be binned in fine bins.
Likelihoods from many small selections can be summed
Calculation is quick and fits well with the F-C analysis
Error Analysis
I do not like the nuisance parameter fits
Systematic errors are not Gaussian, therefore they do not give correct
coverage
One has to be very careful with MINUIT when near physical boundaries
where derivatives are not continuous
I don’t buy the argument that somehow one is going to improve one’s fit
by fitting for the systematics
They are complicated to do with many systematics
So I use F-C to calculate allowed regions, described in DocDB-5109
Better coverage
Can use my far detector MC event library to produce fake experiments
Systematic effects easy to introduce in the fake experiments
Unbinned likelihood fitting quick
Summary
A system exists for doing a global fit (except for rock events), the only
reason I haven’t done one is that the data was in the closed box
As individual far detector event predictions are available, addition of
rock and anti-fiducial events and any desired separations, e.g. bins of
resolution, are easily added
I use a simple and rigorous fitting procedure which can easily
accommodate different data sets, from a global fit to E v Eshw for all
events, to data divided by event type and/or resolution and/or any other
desired separation