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

Bayesian Analysis of Time-intervals for Radiation Monitoring

Peng Luoa

T. A. DeVola and J. L. Sharpb

a. Environmental Engineering and Earth Sciencesb. Applied Economics and Statistics

2011 SRC-HPS Technical Seminar

Time-Interval ?

Timet2t1 t3 t4 t5

…Background


Time-Interval ?

Time

…Presence of

a source

t2’t1’ t3’ t4’ t5’ t6’ t7’ t8’ t9’


Bayesian Statistics Bayes’ theorem• x – observation, counts or

time-interval; r – count rate

)()|( rPrxPP(r|x)

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

PriorLikelihoodPosterior

Bayesian statistics give direct probability statements about the parameter based on prior information and actual data.

Subjective


Conjugate Prior: Gamma distribution Poisson Distribution

( )( )

!

krtrt

P k ek

Poisson Distribution

( ) rtf t re

Time-Interval Distribution

1( , )( )

rGamma r e

Gamma Distribution

Gamma Distribution


Example 1:Background D

ata

0

1

2

3

4

5

6

0 5 10 15 20

coun

ts/o

bser

vatio

n

number of observation

rbkg.=2.0 cpsDetectionLimit= 4.3

Gam

ma

Pri

or Posterior


Example 2: A Source is Present


Data Acquisition & Analysis

Time-interval data acquisition: experiments (~105 pulses) and simulation (~106 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.


Bayesian Analysis Methodology

Ifi=1

Posterior (i) Likelihood Prior

Prior(1)

If r > r0

Prior(i+1)=Posterior(i)

i=i+1

Yes

Yes

No

No

Obs.(i)

……

1

2

3

Obs

erva

tion

s


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)

0

5

10

15

20

25

30

35

402.9 104

3 104

0 2 4 6 8 10

1.65 Bayesian (cnt)Bayesian (ti)

aver

age

ru

n le

ngth

(s)

mean count rate (cps)

0

1

2

3

4

5

6

7

3 4 5 6 7 8 9 10


ave

rag

e r

un

len

gth

(s)


ab

Experimental result

Mean count rate (cps)

Ave

rage

run

leng

th

(s)

0

5

10

15

20

25

900092009400

0 2 4 6 8 10

1.65Bayesian (cnt)Bayesian (ti)

aver

age

ru

n le

ngth

(s)


a

0

1

2

3

4

5

6

3 4 5 6 7 8 9 10


ave

rage

ru

n le

ngth


b

Simulated result

Mean count rate (cps)A

vera

ge r

un le

ngth

(s

)


Detection Probability

Detection probability is defined as 1-

5s bkg.+20s source +5s bkg.

104 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

0

0.2

0.4

0.6

0.8

1

1 2 3 4 5 6

1.65 Bayesian (cnt)Bayesian (ti)de

tect

ion

prob

abi

lity


Det

ectio

n pr

obab

ility


FP ()

FN ()


Detection Probability

5 s20 s

5 s

0.5

0.6

0.7

0.8

0.9

1

1.5 2 2.5 3 3.5 4 4.5 5 5.5


tect

ion

prob

abi

lity


Det

ectio

n pr

obab

ility


95 % 60% DL

ARL – 60% DL

0

5

10

15

20

200

0 2 4 6 8 10


aver

age

ru

n le

ngth


Ave

rage

run

le

ngth



Source Time Effect

2s, 5s, 20s and 50s source time

Bayesian analysis has the ability to reduce both false positive and false negative rates.

a b

c d

0

0.2

0.4

0.6

0.8

1

0 2 4 6 8 10


dete

ctio

n pr

oba

bilit

ymean count rate (cps)

0

0.2

0.4

0.6

0.8

1

0 2 4 6 8 10


tect

ion

prob

abi

lity


0

0.2

0.4

0.6

0.8

1

1 2 3 4 5 6


dete

ctio

n pr

oba

bilit

y


0

0.2

0.4

0.6

0.8

1

1 2 3 4 5 6


tect

ion

prob

abi

lity


2s 5s

20s 50s


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.

0

0.2

0.4

0.6

0.8

1

0 2 4 6 8 10


dete

ctio

n pr

oba

bilit

y


Det

ectio

n pr

obab

ility



Effect of Change Point

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 5 10 15 20


dete

ctio

n pr

oba

bilit

y

change point (s)

0 - 20s5 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.


Enhanced Reset

ifp(r>r0)

discriminator

1st limit

no

start over

yes new data point(s)

incorporated

2nd limit start over


Moving Prior

t

Dat

a

-202468

101214

0 5 10 15 20

coun

ts/o

bser

vatio

n

number of observation

2.0cpsrbkg.=2.0 cps

Critical Level = 4.3

Prior

The moving prior relies on the latest information to calculate the posterior probability by updating the prior probability with each new data point.


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.

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 5 10 15 20

1.65 Bayesian (ti)Enhanced resetmoving prior

dete

ctio

n pr

oba

bilit

y

change point (s)

Time-Interval Data


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.


Acknowledgement

• Funded by DOE Environmental Management Science Program.

(Contract number: DE-FG02-07ER64411)

• Dr. Fjeld and Dr. Powell (EE&ES)

Documents

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