rpra5
TRANSCRIPT
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RPRA 5. Data Analysis 1
Engineering Risk Benefit Analysis1.155, 2.943, 3.577, 6.938, 10.816, 13.621, 16.862, 22.82,
ESD.72, ESD.721
RPRA 5. Data Analysis
George E. Apostolakis
Massachusetts Institute of Technology
Spring 2007
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RPRA 5. Data Analysis 2
Statistical Inference
Theoretical Model Evidence
Failure distribution, Sample, e.g.,
e.g., {t1,,tn}
How do we estimate from the evidence?
How confident are we in this estimate? Two methods:
Classical (frequentist) statistics
Bayesian statistics
te)t(f
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RPRA 5. Data Analysis 3
Random Samples
The observed values are independent and the
underlying distribution is constant.
Sample mean:
Sample variance:
n
1it
n
1t
n1 2i2 )tt()1n( 1s
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RPRA 5. Data Analysis 4
The Method of Moments:
Exponential Distribution
Set the theoretical moments equal to the sample
moments and determine the values of the
parameters of the theoretical distribution.
Exponential distribution:
Sample: {10.2, 54.0, 23.3, 41.2, 73.2, 28.0} hrs
t1 =
32.386
9.229)282.732.413.23542.10(
6
1t =
1
hr026.032.38
1 =;hrs32.38MTTF =
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RPRA 5. Data Analysis 5
The Method of Moments:
Normal Distribution
Sample: {5.5, 4.7, 6.7, 5.6, 5.7}
= 68.55
4.28
5
)7.56.57.67.47.5(x
032.2)68.57.5(...)68.55.5()xx(5
1
222i =
=
713.0s
508.0)15(
032.2s
2
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RPRA 5. Data Analysis 6
The Method of Moments:
Poisson Distribution
Sample: {r events in t}
Average number of events: r
{3 eqs in 7 years}
t
rrt =
1yr43.0
7
3 =
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RPRA 5. Data Analysis 7
The Method of Moments:
Binomial Distribution
Sample: {k 1s in n trials}
Average number of 1s: k
qn = k
{3 failures to start in 17 tests}
n
kq =
176.017
3q =
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RPRA 5. Data Analysis 8
Censored Samples and the Exponential
Distribution
Complete sample: All n components fail.
Censored sample: Sampling is terminated at time t0(with k failures observed) or when the rth failureoccurs.
Define thetotal operational time as:
It can be shown that:
Valid for the exponential distribution only (nomemory).
k1
0i t)kn(tT r1
ri t)rn(tT
T
ror
T
k =
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RPRA 5. Data Analysis 9
Example
Sample: 15 components are tested and the testis terminated when the 6th failure occurs.
The observed failure times are:
{10.2, 23.3, 28.0, 41.2, 54.0, 73.2} hrs
The total operational time is:
Therefore
7.8882.73)615(2.73542.41283.232.10T =13 hr10x75.6
7.888
6 =
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RPRA 5. Data Analysis 10
Bayesian Methods
Recall Bayes Theorem (slide 16, RPRA 2):
Prior information can be utilized via the priordistribution.
Evidence other than statistical can be accommodated
via the likelihood function.
Posterior
Probability
Prior
Probability
Likelihood of the
Evidence
( ) ( ) ( )( ) ( )
=N
1 ii
iii
HPHEP
HPHEPEHP
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RPRA 5. Data Analysis 11
The Model of the World
Deterministic, e.g., a mechanistic computer code
Probabilistic (Aleatory), e.g., R(t/ ) = exp(- t)
The MOW deals with observable quantities.
Both deterministic and aleatory models of the world haveassumptions and parameters.
How confident are we about the validity of theseassumptions and the numerical values of theparameters?
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RPRA 5. Data Analysis 12
The Epistemic Model
Uncertainties in assumptions are not handled routinely.If necessary, sensitivity studies are performed.
The epistemic model deals with non-observablequantities.
Parameter uncertainties are reflected on appropriateprobability distributions.
For the failure rate: ( )d = Pr(the failure rate has avalue in d about )
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RPRA 5. Data Analysis 13
Unconditional (predictive) probability
d)()/t(R)t(R
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RPRA 5. Data Analysis 14
Communication of Epistemic
Uncertainties: The discrete case
Suppose that P( = 10-2) = 0.4 and P( = 10-3) = 0.6
Then, P(e-0.001t) = 0.6 and P(e-0.01t) = 0.4
R(t) = 0.6 e-0.001t + 0.4 e-0.01t
t
1.0
exp(-0.001t)
exp(-0.01t)
0.6
0.4
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RPRA 5. Data Analysis 15
Communication of Epistemic
Uncertainties: The continuous case
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RPRA 5. Data Analysis 16
Risk Curves
Figure by MIT OCW.
1,000100
Public Acute Fatalities
Probability
ofExceedence
101
1.0E-09
1.0E-08
1.0E-07
1.0E-06
1.0E-05
1.0E-04
95th Percentile
Mean
Median
5th Percentile
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RPRA 5. Data Analysis 17
The Quantification of Judgment
Where does the epistemic distribution ( )come from?
Both substantive and normative goodness arerequired.
Direct assessments of parameters like failurerates should be avoided.
A reasonable measure of central tendency to
estimate is the median. Upper and lower percentiles can also be
estimated.
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RPRA 5. Data Analysis 18
The lognormal distribution
It is very common to use the lognormal distribution as
the epistemic distribution of failure rates.
2
2
2
)(lnexp
2
1)(
UB = = exp( + 1.645 )
LB = = exp( - 1.645 )
95
05
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RPRA 5. Data Analysis 19
Pumps
Component/Primary
Failure Modes
Assessed Values
Lower Bound Upper Bound
Valves
Failure to start, Qd:
Failure to operate, Qd:
Failure to operate, Qd:
Failure to operate, Qd:
Failure to open, Qd:
Failure to open, Qd:
Plug, Qd:
Plug, Qd:
Plug, Qd
:
Failure to run, o:
< 3" diameter, o:
> 3" diameter, o:
(Normal Environments)
3 x 10-4/d 3 x 10-3/d
3 x 10-6/hr 3 x 10-4/hr
3 x 10-4/d 3 x 10-3/d
3 x 10-5/d 3 x 10-4/d
3 x 10-4/d 3 x 10-3/d
Plug, Qd:
3 x 10-5/d 3 x 10-4/d
1 x 10-4/d 1 x 10-3/d
3 x 10-5/d 3 x 10-4/d
3 x 10-5/d 3 x 10-4/d
3 x 10-6/d 3 x 10-5/d
3 x 10-5
/d 3 x 10-4
/d
3 x 10-11/hr 3 x 10-8/hr
3 x 10-12/hr 3 x 10-9/hr
1 x 10-4/d 1 x 10-3/d
Motor Operated
Solenoid Operated
Check
Relief
Manual
Plug/rupture
Mechanical
Failure to engage/disengage
Pipe
Clutches
Mechanical Hardware
Air Operated
Table by MIT OCW.
Adapted from Rasmussen, et al."The Reactor Safety Study."WASH-1400, US Nuclear RegulatoryCommission, 1975.
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RPRA 5. Data Analysis 20
Table by MIT OCW.
Adapted from Rasmussen, et al.The Reactor Safety Study."
WASH-1400, US Nuclear RegulatoryCommission, 1975.
"
Motors
Failure to start, Q 1 x 10-3d: 1 x 10-4/d /d
Failure to run
(Normal Environments), -5o: 3 x 10-6/hr 3 x 10 /hr
Transformers
Open/shorts,
3 x 10-7
/hr 3 x 10-6
o: /hr
Relays
Failure to energize, Q -5 -4d: 3 x 10 /d 3 x 10 /d
Circuit Breaker
Failure to transfer, Qd: 3 x 10-4/d 3 x 10-3/d
Limit Switches
Failure to operate, Q 1 x 10-3d: 1 x 10-4/d /d
Torque SwitchesFailure to operate, Q : 3 x 10-5/d -4
d3 x 10 /d
Pressure Switches
Failure to operate, Qd: 3 x 10-5/d 3 x 10-4/d
Manual Switches
Failure to operate, Q : 3 x 10-6 -5d
/d 3 x 10 /d
Battery Power Supplies
Failure to provideproper output, : 1 x 10-6/hr 1 x 10-5/hr
s
Solid State Devices
Failure to function, o: 3 x 10-7/hr 3 x 10-5/hr
Diesels (complete plant)
Failure to start, Q -2d: 1 x 10 /d 1 x 10-1/d
Failure to run, o: 3 x 10-4/hr 3 x 10-2/hr
Instrumentation
Failure to operate, o: 1 x 10-7/hr 1 x 10-5/hr
Electrical Hardware
Electrical Clutches
Failure to operate, Qd: 1 x 10-4/d 1 x 10-3/d
Component/Primary Assessed Values
Failure Modes Lower Bound Upper Bound
a. All values are rounded to the nearest
half order of magnitude on the exponent.
b. Derived from averaged data on pumps,
combining standby and operate time.
c. Approximated from plugging that was
detected.
d. Derived from combined standby and
operate data.
e. Derived from standby test on batteries,
which does not include load.
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RPRA 5. Data Analysis 21
Example
1350 hr10x3)exp(
=12
95 hr10x3)645.1exp(=
40.1,81.5Then =132 hr10x8)
2exp(][E =
14
05 hr10x3)645.1exp(
=
Lognormal prior distribution with median and
95th percentile given as:
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RPRA 5. Data Analysis 22
Updating Epistemic Distributions
Bayes Theorem allows us to incorporate newevidence into the epistemic distribution.
d)()/E(L )()/E(L)E/('
E l f B i d ti f i t i
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RPRA 5. Data Analysis 23
Example of Bayesian updating of epistemic
distributions
Five components were tested for 100 hours each and nofailures were observed.
Since the reliability of each component is exp(-100 ),the likelihood function is:
L(E/ ) = P(comp. 1 did not fail AND comp. 2 did notfail AND comp. 5 did not fail) = exp(-100 ) x exp(-100 ) xx exp(-100 ) = exp(-500 )
L(E/ ) = exp(-500 ) Note: The classical statistics point estimate is zero since no
failures were observed.
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RPRA 5. Data Analysis 24
Prior ( ) and posterior ( )
distributions
1e-4 1e-3 1e-2 1e-1
Probability
0.000
0.001
0.002
0.003
0.004
0.005
0.006
0.007
0.008
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RPRA 5. Data Analysis 25
Impact of the evidence
Mean
(hr-1)
95th
(hr-1)
Median
(hr-1)
5th
(hr-1)
Prior
distr.
8.0x10-3 3.0x10-2 3x10-3 3.0x10-3
Posterior
distr.
1.3x10-3 3.7x10-3 9x10-4 1.5x10-4
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RPRA 5. Data Analysis 26
Selected References
Proceedings of Workshop on Model Uncertainty: Its Characterization and
Quantification, A. Mosleh, N. Siu, C. Smidts, and C. Lui, Eds., Center forReliability Engineering, University of Maryland, College Park, MD, 1995.
Reliability Engineering and System Safety, Special Issue on the Treatment
of Aleatory and Epistemic Uncertainty, J.C. Helton and D.E. Burmaster,Guest Editors., vol. 54, Nos. 2-3, Elsevier Science, 1996.
Apostolakis, G., The Distinction between Aleatory and Epistemic
Uncertainties is Important: An Example from the Inclusion of AgingEffects into PSA,Proceedings of PSA 99, International Topical Meeting
on Probabilistic Safety Assessment, pp. 135-142, Washington, DC, August
22 - 26, 1999, American Nuclear Society, La Grange Park, Illinois.