Bayesian Analysis of Time-intervals for Radiation Monitoring Peng Luo a T. A. DeVol a and J. L. Sharp b a.Environmental Engineering and Earth Sciences.

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Presentation transcript:

Bayesian Analysis of Time-intervals for Radiation Monitoring Peng Luo a T. A. DeVol a and J. L. Sharp b a.Environmental Engineering and Earth Sciences b.Applied Economics and Statistics

2011 SRC-HPS Technical Seminar Time-Interval ? Time t2t1t3t4t5 … Background

2011 SRC-HPS Technical Seminar Time-Interval ? Time … Presence of a source t2’t1’t3’t4’t5’t6’t7’t8’t9’

2011 SRC-HPS Technical Seminar Bayesian Statistics Bayes’ theorem x – observation, counts or time-interval; r – count rate  P(r): gives the prior prob. of observing r before data is collected,  P(x|r): gives the prob. to obtain the observation x given r  P(r|x): summarizes the knowledge of r given the prior and data PriorLikelihood Posterior    Bayesian statistics give direct probability statements about the parameter based on prior information and actual data. Subjective

2011 SRC-HPS Technical Seminar Conjugate Prior: Gamma distribution Poisson Distribution Time-Interval Distribution Gamma Distribution

2011 SRC-HPS Technical Seminar Example 1:Background Data Gamma Prior Posterior

2011 SRC-HPS Technical Seminar Example 2: A Source is Present

2011 SRC-HPS Technical Seminar Data Acquisition & Analysis  Time-interval data acquisition: experiments (~10 5 pulses) and simulation (~10 6 pulses).  Three methods are used to analyze the data: classical (1.65σ), Bayesian with counts (cnt), and Bayesian with time-intervals (ti). Fixed count time is 1 second.  Compared the three methods in terms of average run length (time to detect the source) and detection probability (1-  ) where  is false negative (FN) rate.

2011 SRC-HPS Technical Seminar Bayesian Analysis Methodology If i=1 Posterior (i)  Likelihood  Prior Prior(1) If r > r 0 Prior(i+1)= Posterior(i) i=i+1 Yes No Obs.(i) …… Observations

2011 SRC-HPS Technical Seminar Average Run Length  Average Run Length: the average time needed to make a decision whether an alarm is issued.  Time-interval can make a quick decision. (s) a b Experimental result Mean count rate (cps) Average run length (s) a b Simulated result Mean count rate (cps) Average run length (s)

2011 SRC-HPS Technical Seminar Detection Probability  Detection probability is defined as 1-   5s bkg.+20s source +5s bkg.  10 4 trials  95% detection limit (DL)  Bayesian methods give lower false positive (FP) rates  Also lower detection probabilities at low levels 5 s 20 s 5 s Detection probability Mean count rate (cps) FP (  ) FN (  )

2011 SRC-HPS Technical Seminar Detection Probability 5 s 20 s 5 s Detection probability Mean count rate (cps)  95 %  60% DL ARL – 60% DL Average run length Mean count rate (cps)

2011 SRC-HPS Technical Seminar Source Time Effect  2s, 5s, 20s and 50s source time  Bayesian analysis has the ability to reduce both false positive and false negative rates. ab cd 2s 5s 20s 50s

2011 SRC-HPS Technical Seminar If source time < count time  0.5 s source  1 s count time  Time-interval may result in a higher detection probability when the source time is shorter than the fixed count time. Detection probability Mean count rate (cps)

2011 SRC-HPS Technical Seminar Effect of Change Point s 5 s  Change point --- a point at which the radiation level changes to an elevated level.  Change point determines the amount of background data that are included in Bayesian inferences.  The detection of radioactive sources may be delayed.

2011 SRC-HPS Technical Seminar Enhanced Reset

2011 SRC-HPS Technical Seminar Moving Prior t Data Prior  The moving prior relies on the latest information to calculate the posterior probability by updating the prior probability with each new data point.

2011 SRC-HPS Technical Seminar Result of Modified Methods  10 (pulses)/20 (pulses) for ‘Enhanced Reset’.  10 (pulses) is set for ‘moving prior’.  Both modified Bayesian methods result in a higher detection probability than the typical Bayesian analyses.  The performances of two modified methods are independent of the change point. Time-Interval Data

2011 SRC-HPS Technical Seminar Summary  ARL indicates that Bayesian(ti) could more rapidly detect the source than other two methods.  When source time < count time, Bayesian(ti) results in a higher detection probability than Bayesian(cnt), which are both less than the frequentist method.  For no source present, both Bayesian(ti) and Bayesian(cnt) result in lower false positive rates than the frequentist method.  When source time > count time, Bayesian analysis results in a lower detection probability relative to the frequentist method, which increases with source time.  Modified methods can improve the performance of Bayesian analysis by reducing the effect of the background.

2011 SRC-HPS Technical Seminar Acknowledgement Funded by DOE Environmental Management Science Program. (Contract number: DE-FG02-07ER64411) Dr. Fjeld and Dr. Powell (EE&ES)