Fourier Studies: Looking at Data

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# Fourier Studies: Looking at Data - PowerPoint PPT Presentation

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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### 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
• 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
• 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
• Exercise the whole procedure on a realistic case (see BML 7/18)
• Toy simulation configured to emulate sample from previous page
• 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
• 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
• 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
• Signal or background depth of deepest minimum in toys
• Tail integral of distribution gives detection & false alarm probabilities
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

### 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
• Automated code looks for –log(Lratio) minimum
• Depth of minimum compared to toy MC distributions gives signal/background probabilities

Background

Signal

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
• 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

### Backup

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:

• Toy MC Models
• “Fitter” Response
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 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

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)

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”
• 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
• 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

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)

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

### Confidence Bands

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):
• 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
• 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 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
• 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

Linear scale

Logarith. scale

### Determining 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
• “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
• 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
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)