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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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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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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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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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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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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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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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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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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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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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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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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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Unconditional (predictive) probability

d)()/t(R)t(R

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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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Communication of Epistemic

Uncertainties: The continuous case

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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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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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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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Pumps

Component/Primary

Failure Modes

Assessed Values

Lower Bound Upper Bound

Valves

Failure to start, 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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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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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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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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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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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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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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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.

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