using car4ams, the bayesian ams data-analysis code v. palonen, p. tikkanen, and j. keinonen...
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Using car4ams, the Bayesian AMS data-analysis code
V. Palonen, P. Tikkanen, and J. Keinonen
Department of Physics, Division of Materials Physics
AMS data
In AMS, several measurements are
made of each cathode.
Each measurement has intrinsic
uncertainty from the 14C counting.
Additional (instrumental) error is possible.
What is a reliable uncertainty estimate?
AMS data analysisFour ways
Counting statistical uncertainty Usually the main component of measurement uncertainty
Additional instrumental error possible. Hence, using this only is too
optimistic.
Standard error of the mean (SDOM) Has negative bias.
Has significant random scatter from sampling → >5σ errors.
Not Gaussian.
Combination of the above Better but not optimal.
Bayesian CAR model Small scatter, best detection of instrumental error, accurate.
Applies same kind of instrumental error for all cathodes.
SDOM: the random scatter
Sampling causes random scatter to SDOM. May be too
small or too large.
2000 simulated results10 runs per cathode
z-score
SDOM: deviations not normally distributed
The random scatter of the SDOM leads to a more tailed
distribution for the z-scores (true error/uncertainty).
-6 -4 -2 0 2 4 6
0.0
0.1
0.2
0.3
0.4
0.5
True Error / SDOM
Den
sity
True error / SDOMstd=1 Gaussian
-6 -4 -2 0 2 4 6
1e-0
41e
-03
1e-0
21e
-01
1e+
00
True Error / SDOM
log(
dens
ity)
True error / SDOMstd=1 Gaussian
Combination 1:
Max(sampling, counting)
Counting statisticaluncertainty
Overestimates when no instrumental error. May underestimate when instrumental error present.
Combination 2:
Chi2 test (NEC)
Good when no instrumental error. Underestimates when instrumental error present.
The CAR model
Known measurement uncertainties
from the Poisson distribution of the 14C counts.
Main assumption: Unknown
(instrumental) error is described by a
continuous autoregressive (CAR)
process. The process can describe both white
noise and random walk noise (trend).
Adapts to the most probable
magnitude and type of instrumental
error.
CAR results
Small scatter and (usually) Gaussian results. Much better control on instrumental error. Uncertainties
increase continuously with increasing additional error. Slightly more accurate.
The usage
car4ams, a Linux/Unix/Windows implementation of the
CAR model for AMS is available, along with preprint of an
article on the usage, at:
beam.acclab.helsinki.fi/~vpalonen/car4ams/
The usage:
1. δ13C correct each measured ratio prior to CAR analysis.
2. Make a Ratios.in file of the data.
3. Run car4ams to get a MCMC chain.
4. Summarize car4ams output with cAnalyze.R (an R script).
Summaries are given in a spreadsheet file, all graphs in a
.pdf file.
δ13C correction
Prior to CAR analysis, the measured ratios and the stable
isotope currents are corrected with
where, for 14C/13C measurements
and for 14C/12C measurements
Form of the Ratios.in file
Each measured 14C /13C or 14C /12C ratio is given on one line: n(i) ti Ri Iiτi
with the four columns:
n(i) Cathode number
ti Time of the measurement of the ratio. In hours
from an arbitrary starting point (from 0:00 of the first
day convenient).Ri The ion current ratio. The number of 14C counts
is converted to ion current.
Ri = rare isotope current / stable isotope current.
Iiτi The product of stable-isotope current and the
duration of 14C counting. Values are given in
coulombs (A· s).
Run car4ams
car4ams output an MCMC chain. The chain consists of
parameter-space points distributed as the posterior pdf.
(For example, the histogram of the values of the parameter O1 is
the probability density function for the 14C concentration of
cathode 1)
car4ams outputs the MCMC chain to stdout, which is directed to
a file by
$ car4ams > c.txt
Check run and summarize results
Start R, the free environment for statistical computing.
In R, run the analysis script
> source(’cAnalyze.R’)
Check convergence from the trace plots
If trace plots ok, use the outputs. (If not, run car4ams
again.)
No convergence Convergence OK