PHYSTAT 05 - Oxford 12th - 15th September 2005

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Sifting data in the real world Martin
BlockNorthwestern University
PHYSTAT 05 - Oxford 12th - 15th September
2005 Statistical problems in Particle Physics,
Astrophysics and Cosmology
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Sifting Data in the Real World, M. Block,
arXivphysics/0506010 (2005).
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Generalization of the Maximum Likelihood Function
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Hence,minimize Si b(z), or equivalently, we
minimize c2 º Si Dc2i
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Problem with Gaussian Fit when there are Outliers
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Lorentzian Fit used in Sieve Algorithm
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Why choose normalization constant g0.179 in
Lorentzian L02?
Computer simulations show that the choice of
g0.179 tunes the Lorentzian so that minimizing
L02, using data that are gaussianly distributed,
gives the same central values and approximately
the same errors for parameters obtained by
minimizing these data using a conventional c2
fit.
If there are no outliers, it gives the same
answers as a c2 fit. Hence, using the tuned
Lorentzian L02 , much like using the Hippocratic
oath, does no harm.
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Sieve Algorithm SUMMARY
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All cross section data for Ecms gt 6 GeV, pp and
pbar p, from Particle Data Group
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All r data (Real/Imaginary of forward scattering
amplitude), for Ecms gt 6 GeV, pp and pbar p,
from Particle Data Group
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Fitting the Sieved pp and pp data with analytic
amplitudes
We use real analytical amplitudes that saturate
the Froissart bound with the term ln2(n/m), where
n is the laboratory energy and m is the proton
(pion) mass. We simultaneously fit the cross
section s and r (the ratio of the real to the
imaginary portion of the forward scattering
amplitude), where
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Only 3 Free Parameters
However, only 2, c1 and c2, are needed in cross
section fits !
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Cross section model fits for Ecms gt 6 GeV,
anchored at 4 GeV, pp and pbar p, after applying
Sieve algorithm to Real World data
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r-value fits for Ecms gt 6 GeV, anchored at 4
GeV, pp and pbar p, after applying Sieve
algorithm
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What the Sieve algorithm accomplished for the
pp and pbar p data
Before imposing the Sieve algorithm
c2/d.f.5.7 for 209 degrees of freedom Total
c21182.3.
After imposing the Sieve algorithm
Renormalized c2/d.f.1.09 for 184 degrees of
freedom, for Dc2i gt 6 cut Total
c2201.4. Probability of fit 0.2. The 25
rejected points contributed 981 to the total c2
, an average Dc2i of 39 per point.


Similar results were found when fitting pp and
p-p data from the Particle Data Group (not shown
due to lack of time!)
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Cross section and r-value predictions for
pp and pbar-p
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100 data points, gaussianly distributed on the
straight line y1-2x 20 noise points, randomly
distributed, with Dc2igt6.
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100 data points, gaussianly distributed about the
constant y10 40 noise points, randomly
distributed, with Dc2igt4.
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Lessons learned from computer studies of a
straight line and a constant model
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What happens when we try to separate two similar
distributions?
BONUS Seems to also work reasonably well in
separating two similar distributions!
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log2(n/mp) fit compared to log(n/mp) fit All
known n-n data
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gp log2(n/m) fit, compared to the pp even
amplitude fit
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All cross section data for Ecms gt 6 GeV, pp and
p-p, from Particle Data Group
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All r data (Real/Imaginary of forward scattering
amplitude), for Ecms gt 6 GeV, pp and p-p, from
Particle Data Group
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Cross section model fits for Ecms gt 6 GeV,
anchored at 2.6 GeV, pp and p-p, after
applying Sieve algorithm to Real World data
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r-value fits for Ecms gt 6 GeV, anchored at 2.6
GeV, pp and p-p, after applying Sieve algorithm