1. Using car4ams, the Bayesian AMS data-analysis code V. Palonen, P. Tikkanen, and J. Keinonen Department of Physics, Division of Materials Physics

2. AMS data In AMS, several measurements are made of each cathode. Each measurement has intrinsic uncertainty from the 14 C counting. Additional (instrumental) error is possible. What is a reliable uncertainty estimate?

3. AMS data analysis Four 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.

4. SDOM: the random scatter Sampling causes random scatter to SDOM. May be too small or too large. 2000 simulated results 10 runs per cathode z-score

5. 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. Combination 1: Max(sampling, counting) Counting statistical uncertainty Overestimates when no instrumental error. May underestimate when instrumental error present.

7. Combination 2: Chi 2 test (NEC) Good when no instrumental error. Underestimates when instrumental error present.

8. The CAR model Known measurement uncertainties from the Poisson distribution of the 14 C 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.

9. CAR results Small scatter and (usually) Gaussian results. Much better control on instrumental error. Uncertainties increase continuously with increasing additional error. Slightly more accurate.

10. 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. δ 13 C 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.

11. δ 13 C correction Prior to CAR analysis, the measured ratios and the stable isotope currents are corrected with where, for 14 C/ 13 C measurements and for 14 C/ 12 C measurements

12. Form of the Ratios.in file Each measured 14 C / 13 C or 14 C / 12 C ratio is given on one line: n(i) t i R i I i τ i with the four columns: n(i) Cathode number t i Time of the measurement of the ratio. In hours from an arbitrary starting point (from 0:00 of the first day convenient). R i The ion current ratio. The number of 14 C counts is converted to ion current. R i = rare isotope current / stable isotope current. I i τ i The product of stable-isotope current and the duration of 14 C counting. Values are given in coulombs (A· s).

13. Run car 4 ams car 4 ams 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 O 1 is the probability density function for the 14 C concentration of cathode 1) car 4 ams outputs the MCMC chain to stdout, which is directed to a file by $ car4ams > c.txt

14. 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 convergenceConvergence OK

15. Output Numerical results in a speadsheet file Plots 15

16. 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.