Fourier Studies: Looking at Data A. Cerri
Jan 13, 2016
Fourier Studies:Looking at Data
A. Cerri
2
Outline
• Introduction• Data Sample• Toy Montecarlo
– Expected Sensitivity– Expected Resolution
• Frequency Scans:– Fourier– Amplitude Significance– Amplitude Scan– Likelihood Profile
• Conclusions
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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
4Data 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
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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
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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
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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
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Toy Montecarlo: confidence bandsSignal or background depth of deepest minimum in toys
•Tail integral of distribution gives detection & false alarm probabilities
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Toy Montecarlo: m resolutionTwo approaches:
•Fit pulls distributions and measure width
•Fit two parabolic branches to L minimum in a toy by toy basis
Negative Error
Positive Error
RMS~0.5
DataAll the plots you are going to see are
based on Fourier transform & toy montecarlo distributions, unless
explicitely mentioned
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Data: Fourier and Amplitude
53.2~A
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Compare with standard A-scan
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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
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Data Results
• Peak in L ratio is: -2.84 (A/=2.53)– Detection (signal) probability: 53%– False Alarm (background fake) probability: 25%
• Likelihood profile:
141.055.023.17
psms
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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
17Tool Structure
BootstrapToy MC
Ct Histograms
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)2e-t/]),
Functions:
(Re,Im) (+,-,0, tags)(S,B)
Ascii Flat File
(ct, ct, Dexp, tag dec., Kfactor)
Data
Fourier Transform Amplitude Scan
Re(~[ms=])()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
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Ingredients in Fourier space
3/)1(
3
1
i
e bi
/2)()( tebxbx
2
22
1 x
e
Resolution Curve (e.g. single gaussian)
Ct efficiency curve, random example
Ct (ps)
Ct (ps)
m (ps-1)
m (ps-1) m (ps-1)
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1 e 222 1
1
im
iD
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Toy
Data
Toy Montecarlo
• 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
Data+Toy
Realistic MC+Toy
Ct (ps)
Ct (ps)
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Flavor-neutral checks
Re(+)+Re(-)+Re(0) Analogous to a lifetime fit:
•Unbiased WRT mixing
•Sensitive to:
•Eff. Curve
•Resolution
Ct efficiency
Resolution
…when things go wrong
Realistic MC+Model Realistic MC+Toy
m (ps-1)
m (ps-1) Realistic MC+Wrong Model Ct (ps)
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“Lifetime Fit” on Data
Ct (ps) m (ps-1)
Data vs Toy Data vs Prediction
Comparison in ct and m spaces of data and toy MC distributions
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“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
Mea
nR
MS
m (ps-1)
m (ps-1)
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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!
Next plots are based on data skimmed, using the OST only in the winter blessing style. No box has been open.
M (GeV)
25From Fourier to Amplitude
•Recipe is straightforward:
1)Compute (freq)
2)Compute expected N(freq)=(freq | m=freq)
3)Obtain A= (freq)/N(freq)•No more data driven [N(freq)]•Uses all ingredients of A-scan•Still no minimization involved though!
•Here looking at Ds() only (350 pb-1, ~500 evts)
•Compatible with blessed results
m (ps-1)
m (ps-1)
Fourier Transform+Error+Normalization
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Toy MC
• Same configuration as Ds() but ~1000 events• Realistic toy of sensitivity at higher effective
statistics (more modes/taggers)
Able to run on data (ascii file) and even generate toy MC off of it
m (ps-1) m (ps-1)
Fourier Transform+Error+Normalization
Confidence Bands
28Peak Search
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
Minuit-based search of maxima/minima in the chosen parameter vs m
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“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
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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
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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
32Integral Distributions of Maxima heights
Linear scale
Logarith. scale
Determining the Peak Position
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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
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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…
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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
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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)