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Fourier Studies: Looking at Data. A. Cerri. Outline. Introduction Data Sample Toy Montecarlo Expected Sensitivity Expected Resolution Frequency Scans: Fourier Amplitude Significance Amplitude Scan Likelihood Profile Conclusions. Introduction.

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outline
Outline
  • Introduction
  • Data Sample
  • Toy Montecarlo
    • Expected Sensitivity
    • Expected Resolution
  • Frequency Scans:
    • Fourier
    • Amplitude Significance
    • Amplitude Scan
    • Likelihood Profile
  • Conclusions
introduction
Introduction
  • Principles of Fourier based method presented on 12/6/2005, 12/16/2005, 1/31/2006, 3/21/2006
  • Methods documented in CDF7962 & CDF8054
  • Full implementation described on 7/18/2006 at BLM
  • Aims:
    • settle on a completely fourier-transform based procedure
    • Provide a tool for possible analyses, e.g.:
      • J/ direct CP terms
      • DsK direct CP terms
    • Perform the complete exercise on the main mode ()
    • All you will see is restricted to . Focusing on this mode alone for the time being
  • Not our Aim: bless a mixing result on the full sample
data sample
Data Sample
  • Full 1fb-1
  • Ds, main Bs peak only
  • ~1400 events in [5.33,5.41] consistent with baseline analysis
  • S/B ~ 8:1
  • Background modeled from [5.7,6.4]
  • Efficiency curve measured on MC
  • Taggers modeled after winter ’05 (cut based) + OSKT
toy montecarlo
Toy Montecarlo
  • Exercise the whole procedure on a realistic case (see BML 7/18)
  • Toy simulation configured to emulate sample from previous page
  • Access to MC truth:
    • Study of pulls (see BML 7/18)
    • Projected sensitivity
    • Construction of confidence bands to measure false alarm/detection probability
    • Projected m resolution
toy montecarlo sensitivity
Toy Montecarlo: sensitivity
  • Rem: Golden sample only
  • Reduced sensitivity, but in line with what expected for the statistics
  • All this obtained without t-dependend fit
  • Iterating we can build confidence bands
distribution of maxima
Distribution of Maxima
  • Run toy montecarlo several times
    • “Signal”default toy
    • “Background”toy with scrambled taggers
  • Apply peak-fitting machinery
  • Derive distribution of maxima (position,height)

Max A/: limited separation and uniform peak distribution for background, but not model (&tagger parameter.) dependent

Min log Lratio: improved separation and localized peak distribution for background

toy montecarlo confidence bands
Toy Montecarlo: confidence bands
  • Signal or background depth of deepest minimum in toys
  • Tail integral of distribution gives detection & false alarm probabilities
toy montecarlo m resolution
Toy Montecarlo: m resolution
  • Two approaches:
  • Fit pulls distributions and measure width
  • Fit two parabolic branches to L minimum in a toy by toy basis

RMS~0.5

Negative Error

Positive Error

slide10

Data

All the plots you are going to see are based on Fourier transform & toy montecarlo distributions, unless explicitely mentioned

data where we look for a peak
Data: Where we look for a Peak
  • Automated code looks for –log(Lratio) minimum
  • Depth of minimum compared to toy MC distributions gives signal/background probabilities

Background

Signal

data results
Data Results
  • Peak in L ratio is: -2.84 (A/=2.53)
    • Detection (signal) probability: 53%
    • False Alarm (background fake) probability: 25%
  • Likelihood profile:
conclusions
Conclusions
  • Worked the exercise all the way through
  • Method:
    • Assessed
    • Viable
    • Power equivalent to standard technique
  • Completely independent set of tools/code from standard analysis, consistent with it!
  • Tool is ready and mature for full blown study
  • Next: document and bless result as proof-of-principle
tool structure

Data

Configuration Parameters

Signal

(ms,,ct,Dtag,tag,Kfactor),

Background

(S/B,A,Dtag,tag, fprompt, ct, prompt, longliv,),

curves (4x[fi(t-b)(t-b)2e-t/]),

Toy MC

Bootstrap

Ascii Flat File

(ct, ct, Dexp, tag dec., Kfactor)

  • Functions:

(Re,Im) (+,-,0, tags)(S,B)

Re(~[ms=])()

Fourier Transform

Amplitude Scan

Tool Structure

Ct Histograms

Same ingredients as standard L-based A-scan

  • Consistent framework for:
  • Data analysis
  • Toy MC generation/Analysis
  • Bootstrap Studies
  • Construction of CL bands
validation

Validation:

  • Toy MC Models
  • “Fitter” Response
ingredients in fourier space
Ingredients in Fourier space

Resolution Curve (e.g. single gaussian)

m (ps-1)

Ct (ps)

Ct (ps)

m (ps-1)

m (ps-1)

Ct efficiency curve, random example

toy montecarlo1
Toy Montecarlo

Data+Toy

  • As realistic as it can get:
    • Use histogrammed ct, Dtag, Kfactor
    • Fully parameterized curves
    • Signal:
      • m, , 
    • Background:
      • Prompt+long-lived
      • Separate resolutions
      • Independent curves

Toy

Data

Ct (ps)

Realistic MC+Toy

Toy

Data

Ct (ps)

flavor neutral checks
Flavor-neutral checks

Realistic MC+Model

Realistic MC+Toy

Ct efficiency

Resolution

Ct (ps)

m (ps-1)

Realistic MC+Wrong Model

  • Re(+)+Re(-)+Re(0) Analogous to a lifetime fit:
  • Unbiased WRT mixing
  • Sensitive to:
    • Eff. Curve
    • Resolution

…when things go wrong

m (ps-1)

lifetime fit on data
“Lifetime Fit” on Data

Data vs Prediction

Data vs Toy

m (ps-1)

Ct (ps)

Comparison in ct and m spaces of data and toy MC distributions

fitter validation pulls
“Fitter” Validation“pulls”
  • Re(x) or =Re(+)-Re(-) predicted (value,) vs simulated.
  • Analogous to Likelihood based fit pulls
  • Checks:
    • Fitter response
    • Toy MC
  • Pull width/RMS vs ms shows perfect agreement
  • Toy MC and Analytical models perfectly consistent
  • Same reliability and consistency you get for L-based fits

Mean

m (ps-1)

RMS

m (ps-1)

unblinded data
Unblinded Data
  • Cross-check against available blessed results
  • No bias since it’s all unblinded already
  • Using OSTags only
  • Red: our sample, blessed selection
  • Black: blessed event list
  • This serves mostly as a proof of principle to show the status of this tool!

M (GeV)

Next plots are based on data skimmed, using the OST only in the winter blessing style. No box has been open.

from fourier to amplitude
From Fourier to Amplitude

Fourier Transform+Error+Normalization

  • Recipe is straightforward:
    • Compute (freq)
    • Compute expected N(freq)=(freq | m=freq)
    • Obtain A= (freq)/N(freq)
  • No more data driven [N(freq)]
  • Uses all ingredients of A-scan
  • Still no minimization involved though!
  • Here looking atDs() only (350 pb-1, ~500 evts)
  • Compatible with blessed results

m (ps-1)

m (ps-1)

toy mc
Same configuration as Ds() but ~1000 events

Realistic toy of sensitivity at higher effective statistics (more modes/taggers)

Toy MC

Fourier Transform+Error+Normalization

m (ps-1)

m (ps-1)

Able to run on data (ascii file) and even generate toy MC off of it

peak search
Peak Search

Minuit-based search of maxima/minima in the chosen parameter vs m

Two approaches:

  • Mostly Data driven: use A/
    • Less systematic prone
    • Less sensitive
  • Use the full information (L ratio):
    • More information needed
    • Better sensitivity

(REM here sensitivity is defined as ‘discovery potential’ rather than the formal sensitivity defined in the mixing context)

  • We will follow both approaches in parallel
toy study
“Toy” Study
  • Based on full-fledged toy montecarlo
    • Same efficiency and ct as in the first toy
    • Higher statistics (~1500 events)
    • Full tagger set used to derive D distribution
  • Take with a grain of salt: optimistic assumptions in the toy parameters
  • The idea behind this: going all the way through with our studies before playing with data
distribution of maxima1
Distribution of Maxima
  • Run toy montecarlo several times
    • “Signal”default toy
    • “Background”toy with scrambled taggers
  • Apply peak-fitting machinery
  • Derive distribution of maxima (position,height)

Max A/: limited separation and uniform peak distribution for background, but not model (&tagger parameter.) dependent

Min log Lratio: improved separation and localized peak distribution for background

maxima heights
Maxima Heights
  • Separation gets better when more information is added to the “fit”
  • Both methods viable “with a grain of salt”. Not advocating one over the other at this point: comparison of them in a real case will be an additional cross check
  • ‘False Alarm’ and ‘Discovery’ probabilities can be derived, by integration
integral distributions of maxima heights
Integral Distributions of Maxima heights

Linear scale

Logarith. scale

measuring the peak position
Measuring the Peak Position
  • Two ways of evaluating the stat. uncertainty on the peak position:
    • Bootstrap off data sample
    • Generate toy MC with the same statistics
  • At some point will have to decide which one to pick as ‘baseline’ but a cross check is a good thing!
  • Example: ms=17 ps-1
error on peak position
Error on Peak Position
  • “Peak width” is our goal (ms)
  • Several definitions: histogram RMS, core gaussian, positive+negative fits
  • Fit strongly favors two gaussian components
  • No evidence for different +/- widths
  • The rest, is a matter of taste…
next steps
Next Steps
  • Measure accurately for the whole fb-1 the ‘fitter ingredients’:
    • Efficiency curves
    • Background shape
    • D and ct distributions
  • Re-generate toy montecarlos and repeat above study all the way through
  • Apply same study with blinded data sample
  • Be ready to provide result for comparison to main analysis
  • Freeze and document the tool, bless as procedure
conclusions1
Conclusions
  • Full-fledged implementation of the Fourier “fitter”
  • Accurate toy simulation
  • Code scrutinized and mature
  • The exercise has been carried all the way through
    • Extensively validated
    • All ingredients are settled
    • Ready for more realistic parameters
    • After that look at data (blinded first)