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

Output

Numerical results in a speadsheet file

Plots

Thank you for the attention

The program is at

beam.acclab.helsinki.fi/~vpalonen/car4ams/ Future plans:

Include automatic convergence analysis and outlier

detection to the code.

Improve the user interface.