Tiles Method

1. Tiles Method • Divide variable plane into nxn tiles with (n-1) tile boundaries per variable • n2 observables: Nij (number of observed events in each tile) • expected number of events in tile (ij) given by: • assume variable independence for signal: • 2(n-1) unknown signal fractions fSi, fSj and 2 unknown normalizations NSM, NS • n = 2: analytically solvable (4 observables + 4 unknowns) • n > 2: problem is overconstrained (more observables than unknowns) • minimize negative log-likelihood estimator: • Background corr. taken into account • Signal contamination no problem • Signal shape unconstrained • Assumesbkgfractions well described by MC • Signal correlation not taken into account

2. Tiles Method • Use 2 (or more) discriminating variables to split data sample into bins (tiles). • Observed numbers of events in each tile are described by one binned signal model, and one (or more) binned background model(s).

3. Tiles Method • Use 2 (or more) discriminating variables to split data sample into bins (tiles). • Observed numbers of events in each tile are described by one binned signal model, and one (or more) binned background model(s). • Simulated data are used to derive the fractions of background events in each tile, while the overall background abundance is determined by the fit. • Assumptions about the signal events are kept to a minimum: In the case of 2 discriminating variables, it is sufficient to assume that these variables are uncorrelated for signal events. • No other assumption about the shape of the signal events in the two variables is made. • In particular, no regions with signal or background dominance are assumed in the Tiles Method. • The expected background and signal event yields, as well as the binned signal shape are obtained by the fit.

4. Tiles Method: Systematics The Tiles method has systematic errors due to • Uncertainties in the Monte Carlo shape of individual background processes. Some examples • Uncertainties due to detector effects:  (jet, lepton, ETmiss) energy scale, resolution  • Theoretical uncertainties(PDFs, generator uncertainties in ttbar+Jets, W+Jets etc...) • Uncertainties in the Monte Carlo cross-sections for the individual background components. This leads to an uncertainty in the inclusive background shape. In the Tiles Method with separate background shapes, this systematic uncertainty is (partly) absorbed in the statistical error. • Theoretical uncertainties in the production cross-sections • Any efficiency uncertainty (e.g. in the lepton trigger and PID efficiencies) • The Tiles Method has a bias if signal events are correlated between the two discriminating variables.  We estimated and minimized this effect using various SUSY benchmark models. 

5. Example from SUSY toy studies • Left plot : (Fitted – MC) signal yield, depending on number of tiles used. • Right plot: Fit error (signal yield), depending on number of tiles used. • Slight bias (left plot) due to neglect of signal correlations (open circles)test fits where signal corr. are switched off by means of event mixing return the true value on average. • Statistical error decreases with higher number of tiles due to additional (bkg) information.

6. Tiles Method: References • Tiles Method talks: • Sep  3, 2009, SUSY WG mtg : http://indico.cern.ch/contributionDisplay.py?contribId=5&confId=66331 • Jan 23, 2009, SUSY WG mtg : http://indico.cern.ch/getFile.py/access?contribId=4&resId=0&materialId=slides&confId=49330 • Dec  4, 2008, SUSY WG mtg : http://indico.cern.ch/getFile.py/access?contribId=7&resId=0&materialId=slides&confId=45545 • Aug 28, 2008, SUSY WG mtg : http://indico.cern.ch/getFile.py/access?contribId=1&resId=0&materialId=slides&confId=38999 • Tiles Method notes: • Atlas Int note: ATL-PHYS-INT-2009-010http://cdsweb.cern.ch/record/1154123/ • Atlas Pub note: ATL-PHYS-PUB-2009-077http://cdsweb.cern.ch/record/1177160/ • We are about to submit a new INT note: "Improving the Tiles Method”

7. 4-Tiles Method • Expected number of event in each tile: • NA=fSMANSM+ (1-fS1)(1-fS2)NS • NB= fSMBNSM+ fS1 (1-fS2)NS • NC= fSMCNSM+ (1-fS1) fS2 NS • ND= fSMDNSM+ fS1 fS2 NS • Take SM fractions (fSM) for each tile (A,B,C,D) from MC: • fSMA + fSMB + fSMC + fSMD = 1 • Measure NA,NB,NC,ND from data: • NA+ NB+ NC+ ND= Nobs • Signal fractions for 2 variables: fS1,fS2 • solve equations for NS • SM correlation is taken into account • Signal contamination no problem • No assumption on signal • Assumes SM eff. well described by MC • Signal correlation not taken into account System of 4 independent linear equations B D A C

8. Splitting background contributions • For tiles configurations with n ≥ 3: extend number of fitted event yields to fit bkg. contributions • Example (3 x 3 tiles): 9 – 6 = 3 degrees of freedom -> up to 4 bkg. contributions • expected number of events in tile (ij) with K bkg. Contributions: • Example: tt->bblnln + SM w/o tt->bblnln on MC sample MC sample: 768 SU3 events 256tt->bblnln events 1612SM w/o tt->bblnln events • bias in NS is larger (sig. corr.) • NSM (tt->bblnln) overestimated • due to shape similarity to signal • (highly anticorrelated!) • stat. error is higher compared to std. Tiles Method • Method requires distinct signal and background shapes • otherwise signal migrates to background • could add penalties on relative background fractions