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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