What do you get for the proton radius using 1963 data?
a.k.a. Horrible Higinbotham’s HUGS Homework
Douglas W. Higinbotham (Jefferson Lab)
EXAMPLE CODES POSTED ON GITHUB:
https://jeffersonlab.github.io/Example-Regression-Codes/
Let’s Make A Deal
• Behind Two of the Doors Are Goats …
• Behind One of the Doors Is The Prize!
Monty Hall Problem
Door #1
Door #2
Door #3
Monty Hall Problem
Door #1
Door #2
Door #3
Wrong
Right
Wrong
Monty Hall Problem
Door #1
Door #2
Door #3
Wrong
Wrong
Right
Monty Hall Problem
Door #1
Door #2
Door #3
Wrong
Wrong
Right!
Probability Says Switch
• Your human reaction is to stay with your
original choice.
• I setup the problem before I even arrived
today (i.e. you have 1/3 chance to pick
correctly at the start)
• I systematically open one of the two
remaining doors to show you a goat, so
the remaining door has a 2/3 chance to be
the winner!
If you are still not convinced: https://en.wikipedia.org/wiki/Monty_Hall_problem
Proton Radius Puzzle
• There are currently only a few ways to
determine the radius of the proton:
– Atomic Hydrogen Lamb Shift ( ~ 0.88 fm )
– Muonic Hydrogen Lamb Shift ( ~ 0.84 fm)
– And of course elastic electron scattering!
• New measurements & checks coming
– PRad: electron scattering
– MUSE: electron and muon scattering
– NIST & other labs: Atomic Hydrogen Lamb Shift
Particle Data Group: 2017 Update
All I know for sure is that the proton only has one radius . . .
Determining the Charge Radius of Carbon
Stanford high Q2 data from I. Sick and J.S. McCarthy, Nucl. Phys. A150 (1970) 631.
National Bureau of Standards (NBS) low Q2 data from L. Cardman et. al., Phys. Lett. B91 (1980) 203.
See the L. Cardman’s paper for details of the carbon radius ( 2.46 fm ) analysis.
Electron Scattering Charge Radii from Nuclei
Fourier Transformation of Ideal Charge Distributions.
Example Plots Made By R. Evan McClellan (Jefferson Lab Postdoc)
e.g. for Carbon: Stanford high Q2 data from I. Sick and J.S. McCarthy, Nucl. Phys. A150 (1970) 631.
National Bureau of Standards low Q2 data from L. Cardman et. al., Phys. Lett. B91 (1980) 203.
Charge Radius of the Proton
• Proton GE has no measured minima and it is too light for the
Fourier transformation to work in a model independent way.
• Thus for the proton we make use of the ideal that as Q2 goes
to zero the charge radius is equal to the slope of GE
We don’t measure to Q2 of zero, so this is going to be an extrapolation problem.
GE and GM Contributions To The Cross Section
Plots by Ethan Buck (Jefferson Lab SULI Student and W&M undergraduate)
Experiments like PRad (Hall B) go to small angle to maximize GE and minimize GM contribution..
Global fits, like typically done with the Mainz 2010 data, need several normalization, GE and GM
Don’t Blindly Fit!
“An ever increasing amount of computational
work is being relegated to computers, and often
we almost blindly assume that the obtained
results are correct.”
- Simon Širca & Martin Horvat
Warning: Danger of Confirmation Bias
In psychology and cognitive science, confirmation bias is a
tendency to search for or interpret information in a way that
confirms one's preconceptions, leading to statistical errors.
i.e. if you start playing with enough data, long enough you can find whatever you radius you want
What did you get from the Hand data?!
<0.82 fm (2 people)
0.82 – 0.86 fm (3 people)
0.86 – 0.90 fm (0)
>0.90 fm (0)
VARIOUS (3 people)
Frequentist or Bayesian
• Did you fix the intercept?!
• Frequentist – The experimental data only
goes to one within errors so I let the endpoint float.
• Bayesian – I know GE(Q2=0)=1 so I fixed
the end-point.
Least Squares Fitting with EXCEL
• Let’s make sure we understand what
exactly our fancy fitting codes are doing,
by first doing something very simple by
calculating sum of square residuals in a
spread sheet.
Fitting with GNUPLOT
• Simon Sirca makes all the plots in the
textbooks with GNUPLOT.
• A simple tool for making nice looking plots
and can very easily fit the Hand data.
Multivariate Errors
As per the particle data handbook, one should
be using a co-variance matrix and calculating the
probably content of the hyper-contour of the
fit. Default setting of Minuit of “up”(often call Δχ2
is one.
Also note standard Errors often underestimate true
uncertainties. (manual of gnuplot fitting has an
explicate warning about this)
The Interpretation of Errors in Minuit (2004 by James)
seal.cern.ch/documents/minuit/mnerror.pdf
In ROOT: SetDefaultErrorDef(real #)
Default is 1 and doesn’t change unless you change it!
R Fits of the Hand Data
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•
•
•
•
Goto R example fits.
Residual
Normal Q-Q
Leverage
Scale-Location
Example plots from: http://data.library.virginia.edu/diagnostic-plots/
Example Residual Plot
Example plots from: http://data.library.virginia.edu/diagnostic-plots/
Normally Distributed Data
Normal Q-Q plots that is not linear usually mean your data have more or less extreme
values than would be expected if they truly came from a Normal distribution.
Example plots from: http://data.library.virginia.edu/diagnostic-plots/
Leverage Plots
( are a few points driving the entire fit?! )
Data points with large residuals (outliners) may distort the outcome and accuracy of a
regression. Cook's distance measures the effect of deleting a given observation.
Example plots from: http://data.library.virginia.edu/diagnostic-plots/
Fitting Is Fun
•
•
•
•
EXCEL
GNUPLUT
Python
R
Please at check the residuals of your fits and don’t just assume that
because reduced chi2 is good you have a good fit.
EXAMPLE CODES POSTED ON GITHUB:
https://jeffersonlab.github.io/Example-Regression-Codes/