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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. ApostolakisMassachusetts Institute of Technology
Spring 2007
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 λ−λ=
λ
RPRA 5. Data Analysis 3
Random Samples
• The observed values are independent and the underlying distribution is constant.
• Sample mean:
• Sample variance:
∑=n
1it
n1t
∑ −−
=n
1
2i
2 )tt()1n(
1s
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 =λ
32.386
9.229)282.732.413.23542.10(61t ==+++++=
1hr026.032.38
1 −==λ;hrs32.38MTTF =
RPRA 5. Data Analysis 5
The Method of Moments:Normal Distribution
• Sample: {5.5, 4.7, 6.7, 5.6, 5.7}
μ===++++= 68.55
4.285
)7.56.57.67.47.5(x
032.2)68.57.5(...)68.55.5()xx(5
1
222i =−++−=−∑
σ==
=−
=
713.0s
508.0)15(
032.2s2
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}
trrt =λ⇒=λ
1yr43.073 −==λ⇒
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}
nkq =⇒
176.0173q ==
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 failure occurs.
• Define the total operational time as:
• It can be shown that:
• Valid for the exponential distribution only (no memory).
∑ −+=k
10i t)kn(tT ∑ −+=
r
1ri t)rn(tT
Tror
Tk =λ=λ
RPRA 5. Data Analysis 9
Example
• Sample: 15 components are tested and the test is 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 hr10x75.67.888
6 −−==λ
RPRA 5. Data Analysis 10
Bayesian Methods• Recall Bayes’ Theorem (slide 16, RPRA 2):
• Prior information can be utilized via the prior distribution.
• Evidence other than statistical can be accommodated via the likelihood function.
PosteriorProbability
PriorProbability
Likelihood of theEvidence
( ) ( ) ( )( ) ( )∑
= N
1ii
iii
HPHEP
HPHEPEHP
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 have assumptions and parameters.
– How confident are we about the validity of these assumptions and the numerical values of the parameters?
λ λ
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 appropriate probability distributions.
• For the failure rate: π( ) d = Pr(the failure rate has a value in d about )λ
λλ
λ
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
λλ
RPRA 5. Data Analysis 16
Risk Curves
Figure by MIT OCW.
1,000100Public Acute Fatalities
Prob
abili
ty o
f Exc
eede
nce
1011.0E-09
1.0E-08
1.0E-07
1.0E-06
1.0E-05
1.0E-0495th Percentile
Mean
Median
5th Percentile
RPRA 5. Data Analysis 17
The Quantification of Judgment
• Where does the epistemic distribution π( ) come from?
• Both substantive and normative “goodness” are required.
• Direct assessments of parameters like failure rates should be avoided.
• A reasonable measure of central tendency to estimate is the median.
• Upper and lower percentiles can also be estimated.
λ
RPRA 5. Data Analysis 18
The lognormal distribution
• It is very common to use the lognormal distribution as the epistemic distribution of failure rates.
⎥⎦
⎤⎢⎣
⎡σμ−λ
−σλπ
=λπ 2
2
2)(lnexp
21)(
UB = = exp( + 1.645 )
LB = = exp( - 1.645 )
95λ
05λ μ
μ
σ
σ
RPRA 5. Data Analysis 19
Pumps
Component/PrimaryFailure Modes
Assessed ValuesLower 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, λo:
< 3" diameter, λo:> 3" diameter, λo:
(Normal Environments)
3 x 10-4/d 3 x 10-3/d3 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/dPlug, 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/hr3 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
MechanicalFailure 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.
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), λ -5
o: 3 x 10-6/hr 3 x 10 /hr
Transformers
Open/shorts, λ 3 x 10-7/hr 3 x 10-6o: /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 SwitchesFailure to operate, Q 1 x 10-3
d: 1 x 10-4/d /d
Torque Switches
Failure to operate, Q : 3 x 10-5/d -4d
3 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 SuppliesFailure to provideproper output, λ : 1 x 10-6/hr 1 x 10-5/hr
s
Solid State Devices
Failure to function, λo: 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, λo: 3 x 10-4/hr 3 x 10-2/hr
Instrumentation
Failure to operate, λo: 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 ValuesFailure 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.
RPRA 5. Data Analysis 21
Example
1350 hr10x3)μexp(λ −−==
1295 hr10x3)σ645.1μexp(λ −−=+=
40.1σ,81.5μThen =−=
132
hr10x8)2σμexp(]λ[E −−=+=
1405 hr10x3)σ645.1μexp(λ −−=−=
Lognormal prior distribution with median and 95th percentile given as:
RPRA 5. Data Analysis 22
Updating Epistemic Distributions
• Bayes’ Theorem allows us to incorporate new evidence into the epistemic distribution.
∫=
λd)λ(π)λ/E(L)λ(π)λ/E(L)E/λ('π
RPRA 5. Data Analysis 23
Example of Bayesian updating of epistemic distributions
• Five components were tested for 100 hours each and no failures 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 not fail AND… comp. 5 did not fail) = exp(-100 ) x exp(-100 ) x…x exp(-100 ) = exp(-500 )
• L(E/ ) = exp(-500 )– Note: The classical statistics point estimate is zero since no
failures were observed.
λλ
λ
λ
λλ
λ
λ
RPRA 5. Data Analysis 24
Prior ( ) and posterior ( ) distributions
λ
1e-4 1e-3 1e-2 1e-1
Pro
babi
lity
0.000
0.001
0.002
0.003
0.004
0.005
0.006
0.007
0.008
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
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 for Reliability 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 Aging Effects 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.