ML20040A745
| ML20040A745 | |
| Person / Time | |
|---|---|
| Issue date: | 01/31/1982 |
| From: | Goldberg F, Niyogi P, Vesely W, Weldon P NRC OFFICE OF NUCLEAR REGULATORY RESEARCH (RES) |
| To: | |
| References | |
| NUREG-0873, NUREG-873, NUDOCS 8201220034 | |
| Download: ML20040A745 (48) | |
Text
NUREG-0873 A Bayesian Analysis of Diesel Generator Failure Dat
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NUREG-0873 A Bayesian Analysis of Diesel Generator Failure Data Manuscript Completed: October 1981 Date Published: January 1982 l
W. E. Vesely, P. K. Niyogi, F. F. Goldberg, P. A. Weldon Division of Risk Analysis Office of Nuclear Regulatory Research U.S. Nuclear Regulatory Commission Washington, D.C. 20555 y
- s,
ABSTRACT l
l A simple Bayesian approach has been developed to evaluate failure rate implications from the number of failures and number of successes in a given I
number of diesel tests.
For the Bayesian approach, the diesel is modeled as l
having a constant probability of failure per trial which is unknown and whose possible values are describable by a probability distribution.
The approach utilizes discrete probability distributions (probability mass functions) for ease of implementation.
As a potential tool for the analyst, a computer code has been written to efficiently calculate the diesel posterior failure rate distributions for any input diesel test data and assumed prior distribution.
The code can be used to monitor diesel tests for up-to-date failure rate implications.
In addition, a wide variety of sensitivity analyses can be performed using the code.
The Bayesian approach was applied to determine unacceptable numbers of failures versus. numbers of tests for diesel reliability criteria based on risk consider-ations.
Sensitivity studies are performed to investigate the impact of assuming various prior distributions for the diesel failure rate.
These studies indi-cate an insensitivP;y to the prior assumed in determining unacceptable numbers of failures versus numbers of tests.
The Bayesian results are also compared with classical statistics results.
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iii
1 4
TABLE OF CONTENTS 4
1 P_ age j
ABSTRACT.............................
iii LIST OF TABLES..........................
vii 1
LIST OF FIGURES..........................
ix 1
1 i
1.
INTRODUCTION.........................
1 l
1.1 General Approach - Bayesian Equations..........
2
- 1. 2 Discretizing the Bayesian Approach...........
2
]
t 2.
LER DATA ANALYSIS......................
4 l
2.1 Selection of LER Data..................
4 i
2.2 Distribution Evaluations................
4
- 3.
EXAMINATION OF PRIOR AND POSTERIOR DISTRIBUTIONS.......
10 3.1 Discretized Failure rates................
10 3.2 Candidate Prior Probability Distributions........
10 3.3 Posterior Probabilities for Hypothetical Test Data...
13 4.
CALCULATION OF EXCEEDANCE PROBABILITIES...........
24 4
j 4.1 Definition of Unacceptable Numbers of Failures.....
24 5.
A COMPUTER CODE FOR BAYESIAN ANALYSIS OF BINOMIAL i
SAMPLES (BABS).............,..........
35 l
5.1 Introduction.............
35 j
5.2 BABS Input.
35 i
- 5. 3 BABS Output.......................
36 5.4 Description of Major Subroutines............
36 4
APPENDIX A:
BABS SOURCE CODE LISTING...............
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l LIST OF TABLES i
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Table 1.
Estimated diesel failure rates from NUREG/CR-1362....
5
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Table 2.
Diesel Generator Failure frequency distribution.....
6 i
Table 3.
Selected discrete diesel failure rates..........
10 i
Table 4.
Candidate Prior Probabilities (Probability Mass j
Functions)........................
11 i
Table 5.
Pos te ri o r CCDF s.....................
18 l
Table 6.
Bayesian versus classical bounds for zero failures....
19 1
j Table 7.
Lognormal posteriors versus prior (for test / failures)..
21 i
Table 8.
Exceedance probabilities for A=0.5 Lognormal prior [4-02, 4.5]...............
29 i
)
Table 9.
Exceedance Probabilities for A=0.1 Lognormal prior [4-02, 4. 5]...............
30 1
Table 10. Exceedance Probabilities for A=0.4 Lognormal prior [4-02, 4. 5]...............
31 s
i Table 11. Exceedance Probabilities for A=0.1 1
Lognormal prior [3-02, 3. 0]...............
33 l
i Table 12. Exceedance Probabilities for A=0.1 I
Lognormal prior [1. 6-02, 3. 2]..............
34 i,
l Table 13. Observed Diesel Failure Data Input..
35 i
Table 14. Prior distribution data..
37 l
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LIST OF FIGURES Page Figure 1.
Histograms of Failure Rate Data (All Plants).
7 Figure 2.
Diesel Failure Rates Plotted on Lognormal Probability Paper.
8 9
Figure 3.
Diesel Failure Rates Plotted on Semi-Log Paper.....
Figure 4.
Comparison of Selected Lognormal and Loguniform 12 Probability Density functions.............
Figure 5.
Lognormal Prior Probability Mass Function..
14 Figure 6.
Loguniform Prior Probability Mass Function.
15 Figure 7.
Uniform Prior Probability Mass Function.
16
~
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Figure 8.
Comparison of the Three Prior Probability Mass 17 Functions.
Figure 9.
Lognormal Posterior versus Prior for 25 Tests with 5 Failures..
22 Figure 10. Lognormal Posterior versus Prior for 100 Tests with 23 Failures.
23 Figure 11. Unacceptable Number of Failures for the Lognormal Prior [4-02, 4.5]......
25 Figure 12. Unacceptable Numbers of Failures Defined from classical statistical considerations 26 Figure 13. Comparison of Bayesian and Classical Results......
27 Figure 14. Further comparison of Bayesian and Classical Results..
28 Figure 15. BABS Sample Input.
38 Figure 16(a) BABS Sample Output.
39 Figure 16(b) BABS Sample Output.
40 Figure 16(c) BABS Sample Output.
41 Figure 16(d) BABS Sample Output.
42 ix l
A BAYESIAN ANALYSIS OF DIESEL GENERATOR FAILURE DATA 1.
INTRODUCIION In order to adequately evaluate the reliability implications of a set of diesel generator test data, a variety of assessments, both qualitative and quantitative, must be performed.
One important assessment is to make statis-tical inferences on the diesel failure rate.
In making statistical inferences, a "best estimate" for the diesel generator failure rate is generally desired together with some type of confidence interval associated with the estimate.
In addition, in various applications the analyst would like to know the confidence he has that the diesel failure rate is above some defined unacceptable value.
Both classical and Bayesian approaches can be used to draw statistical infer-ences from the diesel test data.
The advantage of the Bayesian approach is I
that it can allow more definitive conclusions to be made on small to moderate number of tests.
However, care must be taken in using a Bayesian approach since the initial assessment about the diesel generator failure rate distribu-tion (i.e., the prior distribution) can falsely bias all subsequent results.
For the approaches developed here a diesel generator is modeled as having a constant failure probability per test.
This constant probability will be called the " failure rate per demand" or simply the " failure rate." The out-come of a diesel test (i.e., failure or success) is assumed to be independent of the preceding test outcomes.
Based on these assumptions, for a given diesel failure rate, the number of diesel failures in a given number of tests follows a binomial probability distribution.
In a Bayesian approach, an initial, or prior, probability distribution is assigned for the diesel failure rate.
This prior distribution portrays an assessment of likely diesel failure rate values before any further future diesel tests are performed.
After tests are performed on a specific diesel, the prior distribution is updated according to standard probability laws to i
obtain the posterior distribution.
The posterior distribution then becomes the prior distribution for the next set of diesel test data, and the cycle repeats.
The diesel failure rate distribution is thus continually updated as new test data are received.
The critical part of the Bayesian approach is the selection of the prior probability distribution.
The prior distribution should have features that reasonably represent the assessment of available information on diesel failure rates.
In determining the prior distribution to be used, a process of candi-date selection and diagnostic testing (with simulated data) must be carried out.
The following section, Section 1, discusses the theoretical Bayesian approaches which were used in this report.
Section 2 then discusses the LER data analysis that was performed on diesel data to obtain a candidate set of priors.
Sec-tion 3 discusses the kinds of evaluations that were performed to investigate implications of the candidate priors.
These sections indicate the kind of 1
i evaluations that can be done in selecting a prior (or a set of candidate priors) for a specific problem.
1.1 General Approach - Bayesian Equations For a given diesel failure rate A, the number of diesel failures, x, in a gi*en number of tests, n, is assumed to follow a binomial distribution.
If B(x;n,n) denotes the probability of x failures in n tests for a failure rate A, then B(x;n,A) = (") A* (1 - A)"'*
(1)
Let f(A) be the prior distribution (density function) for the diesel failure rate.
In the Bayesian approach, the failure rate A is treated as a random variable and f(A) describes the assessment of possible failure rate values and their likelihood.
Let f(A;n,x) be the posterior distribution of A given n num-ber of tests and x failures have been recorded.
The posterior distribution f(A;n,x) represents the updated assessment of possible diesel failure rates and their likelihood which accounts for the new knowledge of the number of tests and failures which have been recorded.
The posterior distribution f(A;n,x) is determined using standard probability laws:
8(x;n,A) f(A) f(A;n,x) =
(2) f B(x;n,A) f(A) dA o
The posterior distribution represents complete information on the diesel failure rate.
The median, the average value, and the most probable value (if it exists) are directly obtained.
The posterior distribution can be integrated over various ranges to calculate the probability tnat the diesel failure rate lies in these ranges.
The probability F (A;n,x) that the diesel failure rate is greater than some value A is
=
t F (A;n,x) = f f(A ;n,x) dA1 (3)
A f(A;n,x) is termed the complimentary cumulative, or exceedance probability.
l The quantity 1 - F (A;n,x) is the cumulative probability, the probability that the failure rate is less than or equal to A.
The posterior distribution f(A;n,x) becomes the prior distribution for the future test and failure data; Equations (1) and (2) are used with f(A;n,x) replacing f(A).
The distribution of diesel failure rates is thus updated after every new set of test and failure data is received.
i 1.2 Discretizing the Bayesian Approach In the computer code BABS developed for this study, the diesel failure rate, A, was discretized into n values A, i = 1,
...n.
For discrete A, equations (1) j and (2) become, respectively, 2
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t B(x;n,A ) = (") A * (1 A )"~*
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B (x;n,A ) p(A )
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g 4
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where p(A ) is th9 probability (mass function) for A and p(A ;n,x) is the 4
z z
corresponding posteisvr probability.
Equation (3) b4comes a sum of p(A ;n,x) 9 for all A9 greater than or equal to A.
The prior probabilities p(A ) can be selected directly or can be obtained from acontinuousdistribution,h.g.,
j A +4 4
p(A ) =
f f(A) dA (6) 4 Ai As in any discretization, a certain coarseness and impreciseness will enter
.i the evaluations.
The computer code allows various kinds of discretizations to i
be performed so that the effects of discretization can be sttidied via sensitivity studies.
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2.
LER DATA ANALYSIS 2.1 Selection of LER Data Diesel Generator Failure data was obtained and analyzed to determine probability distributions which appeared to adequately describe the plant-to plant variation in diesel failure rates.
These distributions, and other distributions obtained by varying these distributions, were then used as the candidate set of priors for the Bayesian approach.
The source of the data is the publication " Data Summaries of Licensee Event Report of Diesel Generators at U.S. Commercial Nuclear Power Plants," NUREG/CR-1362, March 1980.
The model and assumptions used for the data analysis were as follows:
1.
Each plant diesel failure rate was a random variable with the same probability distribution.
and 2.
The plant diesel failure rate estimates in NUREG/CR-1362 had associated sampling errors which were small compared to the plant to plant variations.
The specific data used in this study was the " Demand Rate-Does Not Start (Monthly Testing)" data for the period 1976-1978 (pp 339-341 of NUREG/CR-1362),
and is presented in Table 1.
The data on Bia Rock Point 1 (BP1), which has a failure rate of 0.530, was not taken into account because of its nontypical testing schedule.
Plants which had no diesel generator failure during this period were also eliminated from consideration.
The data in Table 1 are presented in the form of frequency distributions in Table 2 for two different class intervals.
Figure 1 illustrates the corres-ponding histograms.
A lognormal and a loguniform distribution were also fitted to the data.
2.2 Distribution Evaluations It is apparent from Figure 2 that the estimated diesel failure rates fit a lognormal distribution reasonably well; the fit was also not unreasonable for loguniform distribution shown in Figure 3, particularly when the upper and lower 5% data (circled in Figure 3) were ignored.
The objective of this data analysis was to evaluate general features of the probability distributions which could be implied from the diesel failure rate estimates.
From this point of view, the lognormal and loguniform are not unreasonable distributions.
To provide a better fit three separate loguniform segments could also be used to describe as the middle part and the two tails.
No formal statistical analysis was performed since the intent was to find candidate priors and not one "best fitting" prior distribution.
(Out of the forty-two plants considered here, twenty-three had diesel generators manufactured by General Motors.
For the rest, the diesel generators were supplied by Fairbanks Morse (11), Alco (2), Cooper-Bessemer (2), Worthington (2), De Laval (1), and Nordburg Manufacturing (1).
In Table 1, failure rates of plants with diesel generators manufactured by General Motors are identified with an asterisk. When separate probability plots were made, these estimated failure rates appeared to have the same distribution pattern as that of the composite data.)
4
1 l
l Table 1 Estimated diesel failure rates from NUREG/CR-1362 Reactor Manufacturer!
Failure Rates 2 Babcock & Wilcox
.028*,
.130,
.094*,
.042,
.110 Combustion Engineering
.097,
.080,
.080*,
.083*,
.028*,
.078*
.017*,
.049,
.028,
.042,
.140*,
.220*,
.056*,
.074,
.110,
.035*,
.014*,
.042*,
.049,
.056*,
.028 l
.200*,
.014,
.100,
.014*,
.013,
.014*,
.042*,
.014,
.027,
.028,
.014*,
.014*,
4
.028*,
.009*,
.046 Notes:
1The diesel generators for these plants are manufactured by Alco (2),
Cooper-Bessemer (2), De Laval (1), Fairbanks Morse (11), General Motors (23), Nordberg Manufacturing (1) and Worthington (2).
j 2The failure rates with asterisk are for plants with diesel generators manufactured by General Motors.
I 5
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Table 2 Diesel generator failure frequency distribution (a)
Class interval = 0.025 Interval no.
failure rate class Frequency 1
0
.025 10 2
.026
.050 16 3
.051
.075 3
4
.076
.100 7
5
.101
.125 2
6
.126
.150 2
7
.151
.175 0
8
.176
.200 1
9
.201
.225 1
(b)
Class interval = 0.033 Interval no.
failure rate class Frequency 1
0
.033 17 2
.034
.067 11 3
.068
.100 8
4
.101
.133 3
5
.134
.167 1
6
.168
.200 1
7
.201
.233 1
6
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3.
EXAMINATION OF PRIOR AND POSTERIOR DISTRIBUTIONS i
3.1 Discretized Failure Rates For the discrete Bayesian approach used in this study, it was necessary to define discrete diesel failure rates for which probabilities were to be associ-ated.
The selected failure rates are given in Table 3 and consisted of all values, to one significant figure, between 1x10 3 and 9x10 2; failure rates outside this range were assumed to have zero probability for all distributions studied.
Table 3 Selected discrete diesel failure rates 1x10 3 1x10 2 1x10 1 2x10 3 2x10 2 2x10 1 3x10 3 3x10 2 3x10 1 4x10 3 4x10 2 4x10 1 5x10 3 5x10 2 5x10 1 6x10 3 6x10 2 6x10 1 7x10 3 7x10 2 7x10 1 8x10 3 8x10 2 8x10 1 gx10 3 9x10 2 9x10 1 3.2 Candidate Prior Probability Distributions Based on the evaluations of the estimated diesel failure rates given in NUREG/CR-1362, the lognormal and loguniform distributions were treated as being reasonable candidate priors.
To obtain probabilities at the discrete failure rate points, the lognormal and loguniform density functions were integrated over intervals centered at each failure rate point.
The end points of the intervals were selected to be midway on the natural log scale.
As representative of the lognormal family which reasonably describes the LER experience, a lognormal having a median of 4 x 10 2 and a 90% error factor of 4.5 was used (the 5th and 95th percentiles are 4-02/4.5 and 4-02 x 4.5, respect-ively).
As representative of the loguniform family, a specific loguniform was selected such that there was a 90% probability for failure rate lying between 1x10 2 and 1x10 1 inclusive.
As tails to the loguniform, a 1% prob-ability was assigned to each of the discrete failure rate values above 1x10 1 and to each of the 5 values immediately below 1x10 2 The density function for the particular lognormal [4-02, 4.5] is shown in Figure 4.
A loguniform density with equal area between 9x10 3 and 1.8x10 1 is shown for comparison.*
The discretized lognormal and loguniform priors obtained from the continuous distributions are given in Table 4; the loguniform contains the added tail The loguniform shown in the figure is slightly different from the previously defined loguniform, extending from 9-03 to 1.8-01 instead of 1-02 to 1-01.
The difference was found to have negligible impact in the evaluations performed in this study.
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In addition to these priors, three other priors are considered as candidate priors:
a median-smoothed loguniform, a i
uniform, and an average of the loguniform and uniform.
The smoothed loguni-form was obtained by replacing each discrete loguniform probability value by the median of the 3 values centered around that value.
The uniform assigned equal probability (9%) to each diesel failure rate point between 1x10 2 and 1x10 1 The average consisted of a simple arithmetic averaging of the uniform and loguniform discrete probabilities.
4 Table 4 lists the prior probabilities versus diesel failure rates for the different priors defined.
Figures 5 through 7 illustrate the discrete priors j
l for the lognormal, loguniform, and uniform; Figure 8 presents all the priors 1
for comparison.
3.3 Posterior Probabilities For Hypothetical Test Data Table 5 gives posterior complementary cumulative probabilities (CC0Fs) for hypothetical diesel test data.
The diesel test data consists of numbers of tests and numbers of failures.
The posterior CCDFs are shown for the log-normal, loguniform, and aveceae of the uniform and loguniform.
The results for the smoothed loguniform were sery similar to those for the loguniform and are not shown.
Failure rate value; of.09 and greater are only shown in the table.
The probabilities in Table 5 are for the diesel failure rate being greater i
that the value shown.
As an example, for 5 tests and 1 failure and for the
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lognormal prior, the probability is 0.17 that the failure rate is greater than 0.1.
For discrete probability distributions as used in this report, the i
probability of being greater than a value is not in general the same as the l
probability of being greater than or equal to a value.
In Table 5 again for i
the lognormal distribution, the probability that the failure rate is greater i
than or equal to 0.1 is 0.32, which is the same as the probability of the failure rate being greater than 0.09.
The CCDF for failure rate values of
.09 and higher is of particular importance in risk analysis since diesel failure can become an important contributor to risk when its failure rate is this high (i.e.,.1 to.2).
Based on Table 5 and the other such tables obtained from the computer code, the posterior CCDFs for failure rate values of.09,.1 and.2 were jud ed not to be significantly different for the different candidate priors.
This is not surprising since all the prior CC0Fs were constrained to have approximately the same values in this region.
With regard to posterior details that do depend on the prior shapes, it is observed that the lognormal concentratos probabilities on failure rates of 0.1 and 0.2, while for the other priors the probabilities are more dispersed.
Table 6 compares classical confidence and Bayesir.a posterior probability intervals (" Bayesian confidence intervals") for zero failures in given numbers of diesel tests.
The Bayesian prior illustrated is the loguniform; the other priors gave similar posterior behaviors.
The classical confidence and Bayesian posterior probabilities shown are for the failure rates being less than or equal to the corresponding diesel failure rate value.
To be comparative with the Bayesian results, the classical confidences were evaluated at the diesel failure rate plus one half the next interval on a log scale.
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.001 002
.005
.01
.02
.05
.10
.20
.50 10 Diesel Failure Rate
(
i t
(
Fioure 8.
Comparison of the Three Prior Probability Mass functions j
l l
l i
i Table 5 Posterior CCDF's Prior Distributions Diesel Tests /
failure Log-Log-failures rate uniform normal Uniform Average 5/1 0.09 0.32 0.32 0.21 0.27 0.1 0.073 0.17 0.063 0.067 0.2 0.049 0.038 0.043 0.045 0.3 0.028 0.009 0.025 0.026 0.4 0.013 0.002 0.012 0.012 0.5 0.004 0.001 0.004 0.004 5/2 0.09 0.66 0.66 0.50 0.57 0.1 0.39 0.50 0.34 0.36 0.2 0.33 0.18 0.29 0.30 0.3 0.25 0.068 0.22 0.23 0.4 0.15 0.024 0.13 0.14 0.5 0.064 0.008 0.056 0.059 10/1 0.09 0.23 0.21 0.14 0.19 0.1 0.018 0.072 0.016 0.017 0.2 0.007 0.008 0.006 0.007 0.3 0.002 0.001 0.002 0.002 0.4 0.5 10/2 0.09 0.45 0.47 0.27 0.35 0.1 0.097 0.27 0.080 0.085 0.2 0.056 0.050 0.046 0.049 0.3 0.024 0.008 0.020 0.021 0.4 0.007 0.001 0.006 0.007 0.5 0.001 0.001 0.001 25/3 0.09 0.40 0.35 0.22 0.30 0.1 0.02 0.10 0.016 0.02 0.2 0.003 0.005 0.003 0.003 0.3 0.4 0.5 l
l l
18
s I
Table 6 Bayesian Versus 1Cassical Bounds for Zero Failures Failure Classical Bayesian CDF Tests / failures rate confidence (loguniform) 10/0
.01
.12
.26
.02
.21
.50
.03 29
.63
.04
.36
.72
.05
.42
.79
.06
.48
.84
.07
.53
.87
.08
.58
.90
.09
.62
.92
.1
.75
>.995
.2
.93
.3
.98
.4
.996
.5 25/0
.01
.28
.37
.02
.45
.64
.03
.58
.78
.04
.68
.86
.05
.76
.90
.06
.81
.93
.07
.86
.95
.08
.89
.97
.09
.92
.98
.1
.97
>.995
.2
.999 l
100/0
.01
.70
.70 l
.02
.89
.93
.03
.96
.98
.04
.99
.99
.05
.995
>.995 i
l 19
As observed in Table 6 for a given confidence level, the Bayesian upper bound failure rate is less than the classical upper bound failure rate.
The two results merge as the number of tests increase; at 100 tests the two approaches give very similar results.
However, even at 10 tests the Bayesian and clas-sical results are not much different.
For example, at 90% confidence level, the classical upper bound failure rate is between.1 and.2 and the corresponding Bayesian upper bound is.08.
To investigate the effects of tests results on the entire posterior distribution and not just its tail, Table 7 gives the lognormal posterior probabilities for various diesel tests and failure results.
As the number of tests increase, the spread of the posterior distribution decreases.
After 100. tests, the failure rate is known within approximately a factor of 2; the classical confidence interval is essentially the same as the Bayesian interval.
After 25 tests, the failure rate is known to about a factor of 5 from lower 5% value to upper 95% value.
Even af ter as small as 5 tests, the posterior changes significantly from the prior and clearly indicates high failure rate values when large numbers of failures occur.
The results for the other priors were in consonant with those from the lognormal prior.
Based on these and similar results from more complete computer runs, the candidate priors selected were judged to have large enough dispersion so that the posterior probabilities were responsive to test results, particularly those involving large numbers of failures.
To better understand how the posterior distribution incorporates test data results, plots were made of the posterior distribution for the lognormal prior.
Figures 9 and 10 show the posterior probabilities for the lognormal for 25 tests with 5 failures and for 100 tests with 23 failures, respectively.
For these large number of tests, the posterior is observed to center about the diesel failure rate value equal to the number of diesel failures divided by the number of diesel tests, which is the usual estimate as obtained from classical statistical theory.
For actual data, the preceding evaluations indicate that if diesel prior distributions are to be based on LER data, then any of the candidate priors are adequate as long as gross interval probabilities are calculated.
Because of uncertainties in LER data and grossness in the LER data analysis, even these results should only be believed to be no better than order of magnitude pre-cision.
Of course, if priors are not to be based on LERs or if a wider range of sensitivity studies are desired, then the candidate set of priors presented may not be applicable, or at least should be expanded.
1 20
Table 7 Lognormal Posteriors Versus Prior (for test / failures)
Diesel Failure Posterior Posterior Posterior Posterior Posterior Posterior Rate Prior 5/1 5/5 10/1 25/1 25/5 100/23
.001
.002
.001
.003
.003
.001
.004
.005
.001
.002
.005
.007
.001
.003
.006
.008
.001
.002
.004
.007
.01
.002
.003
.006
.008
.01
.002
.004
.007
.009
.01
.003
.004
.008
.01
.07
.02
.03
.05 m
.02
.17
.08
.11
.19
.001
.03
.14
.10
.13
.19
.004
.04
.11
.10
.12
.15
.01
.05
.09
.09
.11 11
.02
.06
.07
.08
.09
.08
.03
.07
.05
.07
.08
.06
.04
.08
.04
.06
.06
.04
.05
.09
.03
.05
.05
.03
.06
.1
.09
.16
.002
.14
.06
.23
.001
.2
.06
.13
.04
.06
.01
.47
.91
.3
.06
.03
.07
.01
.06
.09
.4
.005
.01
.10
.001
.004
.5
.002
.002
.13
.6
.001
.15
.7
.16
.8
.17
.9
.18
l l
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Pr.ior
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seseessesseesses Posterior i
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.40 Diesel f ailure Rate Figure 9 Lognormal Posterior Versus Prior for 25 Tests with 5 Failures 22
1 00 Prior essessessessesses Posterior 50 40 m30
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.40 Diesel Failure Rate figure 10 Lognormal Posterior Versus Prior for 100 Tests with 23 Failures 23 n mmiiur mii
4.
CALCULATION OF EXCEEDANCE PROBABILITIES -
4.1 Definition of Unacceptable Numbers of Failures One of the purposes of this study is to investigate a Bayesian approach for defining unacceptable numbers of failures in given numbers of tests.
For a given number of diesel tests, as the number of failures increases, the posterior probability that the diesel failure rate is higher than some given value also increases.
The probability that the diesel failure rate is greater than or equal to some value will be termed the exceedance probability for that diesel failure rate value.* Thus, as the number of failures increases, the exceedance probability for a given failure rate increasec.
For a given number of tests, the number of failures can be considered to be unacceptable when the exceedance probability for some criterion failure rate value is higher than some minimum probability level.
For the given number of tests, the minimally unacceptable number of failures can be defined to be the minimum number of failures for which the exceedance probability is larger than the minimum probability level.
For ease of writing, the symbol f(n,A,o) will denote the minimal unacceptable number of failures for n tests for a defined failure rate value A and probability level a.
When the number of failures is greater than or equal to f, the exceedance probability for A is greater than or equal to a.
(Equivalently, whenever the number of failures is greater than f(n,A,a), the probability that the failure rate is less than A, is less than 1-a).
Figure 11 is a plot of f(n,A,a) versus number of tests n for various failure rate criterion A and probabilities a.
Figure 11 utilizes the lognormal prior
[4-02,4.5].
For comparison, Figure 12 is a plot of f(n,A,u) when classical statistical calculations are used.
For the classical calculations a refers to the significance level.
Figure 12 was obtained using binomial tables.
Figures 13 and 14 directly compare the Bayesian and classical statistic results for the same A and a values.
As observed in Figure 12, for the classical statistics approach, zero failures can be unacceptable for small numbers of tests.
This occurs because the classi-cal approach incorporates no a priori information about the diesel failure rate value and, thereby, derives little information from small numbers of tests, especially when zero failures occur.
As the number of tests increases, the Bayesian and classical estimates of f(n,A,a) approach the same value.
This convergence occurs because as the number of tests increase, the test data dominate in the Bayesian approach and the effect of the prior decreases.
To assist in determining f(n,A,a), tables can be constructed for exceedance probabilities versus numbers of failures and tests for various rates A.
Tables 8, 9 and 10 give the exceedance probabilities for A=.05,.1 and.4 as a function of the number of failures and tests.
These tables were obtained by AThe exceedance probability is related closely to the CCDF; as defined here, the exceedance probability is the probability that the failure rate is i
greater than or equal to a value and the CCDF is the probability that it is greater than a value.
24
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Table 8 Exceedance Probabilities for A=0.05 Lognormal Prior [4-02,4.5]
Tests / Failures 0
1 2
3 4
5 6
1 0.78 2
0.76 0.96 3
0.73 0.94
.995 4
0.71 0.93 0.99
>.999 5
0.69 0.91 0.99
.999 6
0.67 0.90 0.98
.998 7
0.65 0.89 0.98
.998 8
0.63 0.87 0.97
.997 9
0.61 0.86 0.97
.995 10 0.60 0.84 0.96 0.99
.999 11 0.58 0.83 0.96 0.99
.999 12 0.56 0.82 0.95 0.99
.999 13 0.55 0.80 0.94 0.99
.998 14 0.54 0.79 0.93 0.99
.998 15 0.52 0.77 0.92 0.98
.997 16 0.51 0.76 0.92 0.98
.997 17 0.49 0.75 0.91 0.98
.996 18 0.48 0.73 0.90 0.97
.995
.999 19 0.47 0.72 0.89 0.97 0.99
.999 20 0.45 0.71 0.88 0.97 0.99
.999 29
l Table 9 Exceedance Probabilities for A=0.1 Lognormal Prior [4-02,4.5]
Testr/ failures 0
,1 2
3 4
5 6
1 0.49 2
0.44 0.81 3
0.39 0.75 0.96 4
0.36 0.71 0.93 0.99 5
0.32 0.66 0.90 0.98
.999 6
0.30 0.62 0.88 0.98 7
0.27 0.58 0.85 0.96 8
0.25 0.54 0.82 0.95 0.99 9
0.23 0.50 0.78 0.94 0.99 10 0.21 0.47 0.75 0.92 0.98
.997 11 0.19 0.44 0.72 0.91 0.98
.996 12 0.18 0.41 0.69 0.89 0.97 0.99 13 0.16 0.38 0.66 0.87 0.96 0.99 14 0.15 0.36 0.63 0.84 0.95 0.99 15 0.14 0.34 0.60 0.82 0.94 0.98 16 0.13 0.32 0.57 0.80 0.93 0.98 17 0.12 0.30 0.54 0.77 0.92 0.98 18 0.11 0.28 0.51 0.75 0.90 0.97 19 0.11 0.26 0.48 0.72 0.89 0.96 20 0.10 0.24 0.46 0.70 0.87 0.96 30
Table 10 Exceedance Probabilities for A=0.4 Lognormal Prior [4-02,4.5]
Tests / Failures 0
1 2
3 4
5 6
1 0.07 2
0.04 0.26 3
0.02 0.16 0.53 4
0.01 0.10 0.37 0.76 5
.009 0.07 0.26 0.60 0.89 6
.006 0.04 0.19 0.46 7
.004 0.03 0.13 0.36 8
.002 0.02 0.09 0.27 0.54 9
.001 0.01 0.07 0.20 0.44 10 (9-04)
.008 0.05 0.15 0.35 0.60 11 (6-04)
.006 0.03 0.11 0.27 0.51 12 (4-04)
.004 0.02 0.08 0.21 0.42 13 (2-04)
.002 0.01 0.06 0.16 0.34 14 (1-04)
.002 0.01 0.04 0.13 0.28 15 (9-05)
.001
.007 0.03 0.10 0.22 16 (6-05)
(7-04)
.005 0.02 0.07 0.17 17 (3-05)
(4-04)
.003 0.02 0.05 0.14 18 (2-05)
(3-04)
.002 0.01 0.04 0.11 19 (1-05)
(2-04)
.001
.008 0.03 0.08 20 (8-06)
(1-04)
.001
.006 0.02 0.06 31
running the computer code described in the next section with the lognormal prior (4-02,4.5).
Using the respective table for A=.05,
.1, or
.4, f(n,A,a) is obtained for a given n and for any given probability level by reading the minimum number of failures for which the exceedance probability is greater than or equal to a.
For example, for n=20, A=.4 and a=.05, from Table 10, f(n,A,a)=5.
The other candidate priors gave results which were very similar to those in the previous tables.
As sensitivity studies for priors other than the candidate priors and not necessarily LER-based, Table 11 gives the exeedance probabilities for A m 0.1 for a lognormal prior [3-02, 3.0].
Table 12 gives the exceedance probabilities for A =.1 for a lognormal [1.6-02, 3.2]; this lognormal is such that there is a 90% probability for the failure rate lying in a range of 5-03 to 5-02.
Table 11 indicates that nominal deviations from the reference lognormal will produce little or no change in the unacceptable number of failures defined.
Table 12 indicates that if the prior is really centered at low failure rates such as used here, the unacceptable number of failure for a given criteria will be greater than for the reference lognormal [4-02, 4.5].
Thus, the reference lognormal is conservative in this sense.
32
Table 11 Exceedance Probabilities for A=0.01 Lognormal Prior [3-02,3.0]
Tests / Failures 0
1 2
3 4
5 6
1 0.14 2
0.13 0.35 i
3 0.12 0.32 0.63 4
0.11 0.30 0.59 0.85 5
0.10 0.27 0.56 6
0.09 0.25 0.52 0.79 7
0.08 0.23 0.48 0.76 8
0.08 0.21 0.45 0.73 0.91 9
0.07 0.20 0.42 0.70 0.89 10 0.07 0.18 0.39 0.66 0.87 0.96 11 0.06 0.17 0.37 0.63 0.85 0.95 12 0.06 0.16 0.34 0.60 0.83 0.94 13 0.05 0.14 0.32 0.57 0.80 0.93 14 0.05 0.14 0.30 0.54 0.78 0.92 15 0.04 0.12 0.28 0.51 0.75 0.90 16 0.04 0.12 0.26 0.48 0.72 0.89 17 0.04 0.11 0.24 0.46 0.70 0.87 18 0.04 0.10 0.23 0.43 0.67 0.86 19 0.03 0.10 0.22 0.41 0.64 0.84 20 0.03 0.09 0.20 0.38 0.62 0.82 33
Table 12 Exceedance Probabilities for A=0.01 Lognormal Prior [1.6-02,3.2]
Tests / Failures 0
1 2
3 4
5 6
1 0.03 2
0.03 0.12 3
0.02 0.11 0.34 4
0.02 0.10 0.30 0.64 5
0.02 0.09 0.27 6
0.02 0.09 0.25 0.55 7
0.02 0.07 0.23 0.51 8
0.02 0.06 0.20 0.48 0.77 9
0.01 0.06 0.19 0.44 0.74 10 0.01 0.05 0.17 0.41 0.70 0.90 11 0.01 0.05 0.16 0.38 0.67 0.88 12 0.01 0.05 0.14 0.35 0.64 0.86 13 0.01 0.04 0.13 0.32 0.60 0.83 14 0.09 0.04 0.12 0.30 0.57 0.81 15
.008 0.04 0.11 0.28 0.54 0.78 16
.007 0.03 0.10 0.26 0.51 0.76 17
.007 0.03 0.10 0.24 0.48 0.73 18
.006 0.03 0.09 0.22 0.45 0.70 19
.006 0.03 0.08 0.20 0.42 0.68 20
.005 0.02 0.08 0.19 0.39 0.65 34
J 5.
A COMPUTER CODE FOR BAYSIAN ANALYSIS OF BINOMIAL SAMPLES (BABS) 5.1 Introduction The computer code BABS was developed to compute a component-specific posterior distribution of diesel failure probability per demand, using Bayesian analysis of binominal sample data.
In terms of Bayesian statistics,-the BABS code com-1 putes the posterior distribution of A, where A is the parameter of the bionomial distribution giving the probability of x diesel failures in n demands.
The j
posterior distribution is computed based on an input prior distribution of A, and on data giving the number of observed failures in n demands experienced by the particular diesel to be analyzed.
5.2 BABS Input The input to the BABS code consists of two groups of data:
observations of t
diesel failures in a given number of demands and input describing the prior distribution of A, the diesel failure probability per demand.
t 5.2.1 Observed Diesel Performance Data This data set provides observations of diesel performance over a given number of demands.
The input consists of pairs of numbers giving the number of demands I
(tests) and the corresponding number of failures observed.
Table 13 shows the format of the input cards for this data.
t Table 13 Observed Diesel Failure Data Input Card Program type Columns variable Format Description 1
1-5 NCASE 15 Number of test / failure pairs to be input on subsequent cards.
NCASE must be less than or equal to 250.
2 1-5 N(I) 15 Number of tests performed on a particular diesel.
i 2
6-10 NX(I) 15 Corresponding number of failures I
observed.
Note:
The number of cards of typa 2 is determined by the value of NCASE on card type 1.
5.2.2 Prior Disttibution Data l
l This data set specifies the prior distribution of diesel failure probability l
per demand.
The g ior distribution is in the form of a probability mass func-tion defined at oiesel probability values provided by the user.
The mass func-tion itself may either be input or computed by the program.
Severi.1 sets of prior distribution data may be input in a single run.
Execution terminates 35 y---,
e
.y
,---r.--,
w.en no further prior distribution data is detected in the input stream.
Table 14 shows the format of the input cards for the prior distribution data.
5.2.3 Sample Input Description Figure 15 is a listing of sample input for BABS, including Job Control Cards for running the code at the NIH computer center.
Card type 1 has NCASE=2, indicating the two cards of type 2 are to follow.
The cards of type 2 request the posterior distributions be computed for the cases 10 tests, 2 failures and 10 tests, 3 failures. The next card is a title for the first prior distribu-tion, a discretized loguniform distribution specified as a probability mass function with 20 points.
The second prior distribution function is a dis-cretized lognormal distribution with a median of 4-02 and a 90% error factor of 4.5.
5.3 BABS Output The output from the BABS code consists of a table showing the prior and posterior distributions of diesel failure probability per demand for etch set of diesel test and failure data.
The distributions are printed in three forms:
(1) the probability mass function, (2) the cumulative distribution function (CDF), and (3) the complementary cumulative distribution function (CCDF).
The CDF value Pcdf(x) is defined by:
Pcdf(x) = P(diesel failure probability <_ x)
The CC0F value Pccdf(x) is defined by:
Pccdf(x) = 1 - Pcdf(x) = P(diesel failure probability > x)
Figures 16(a) - 16(d) show the four tables which are produced when the BABS sample input (Table 15) is executed.
There is one output table for each combination of observed diesel failure data and prior distribution.
5.4 Description of Major Subroutines The BABS code consists of a main program and 2 subroutines written in FORTRAN IV.
The code also calls 2 additional subroutines from the International Mathematical and Statistical Library (IMSL), a statistical sof tware package available through the NIH computer center.
5.4.1 The MAIN Program BABS's MAIN program reads the input cards of types 1-4 and calls subroutine PRIOR to compute (or read) the prior probability mass function.
It then com-putes the posterior distribution according to the following formula:
(") P(I)*(1-P(I))"~*PPRI(I)
PPOST(I) =
I ( )P(I)*(1-P(I))""*PPRI(I)
I where n = input number of tests 36
Table 14 Prior Distribution Data Card Program type Columns variable Format Description 3
1-80 Title 20A4 Title describing prior distribution for output headings.
4 1-5 NP 15 Number of points in probability mass function.
NP must be less than or equal to 100.
If IDTYPE=2, NP must be greater than 2.
4 6-10 IDTYPE 15 Type of prior distribution where:
IDTYPE=1 means that the probability mass function will be input by the user; IDTYPE=2 means a discretized lognormal prior will be generated by the program.
4 11-20 PARAM1 E10.2 If IDTYPE=2, this field must contain the median, m, of the desired log-normal distribution.
Leave PARAM1 blank for IDTYPE=1.
4 21-30 PARAM2 E10.2 If IDTYPE=2, this field must contain the 90% error factor, f, for the desired lognormal distribution.
The 90% error factor, f, is defined by:
P(m/f i diesel failure probability 1 m f) = 0.9.
Leave PARAM2 blank for IDTYPE=1.
th 5
1-10 P(I)
E10.5 The I diesel failure probability for the prior probability mass function.
5 11-20 PPRI(I)
E10.5 If IDTYPE=1, PPRI(I) contains the prior probability mass function value associated with the point P(I).
Leave blank if IDTYPE=2.
Notes:
The number of cards of type 5 is determined by the value of NP on card type 4.
The sum of the values PPRI(I), I=1 to NP must equal 1.
In generating the discretized lognormal prior when IDTYPE=2, the program assigns to point P(I) the area under the lognormal density function over the interval from (InP(I-1) + InP(I))/2 to (InP(I) + InP(I+1))/2.
For this case, the prior probability mass function is generated for points P(l), I=2 to NP-1.
P(1) and P(NP) are used only for determining the intervals about 9(2) and P(NP-1) respectively.
P(1) should satisfy 0 < P(1) < P(2',.
P(NP) should satisfy P(NP-1) < P(NP) < 1.
37
1.
//VMBBABS JOB (WDCC,280,A),'FRAN GOLDBERG' 2.
// STEP 2 EXEC FORGCALL,NAME:'WDCCVMB.BABS. LOAD',DISX:PD5006 3.
//GO.FT05F001 DD
- 4.
/WUNN 5.
2 6.
10 2
7.
to 3
8.
LOG-UNIFORM, 0.9, EQUAL-TAIL 9.
20 1
10.
.005 J1 11
.006
.01 12.
.007
.01 13.
.008
.01 14.
.009
.01 15.
.01 1329 16.
.02 1830 17.
.03 1155 18.
.04
.0851 19.
.05
.0675 20.
.06
.0561 21.
.07
.0479 22.
.08
.0419 23.
.09
.0372 24
.1 1329 25.
.2
.31 26.
.3
.01 27.
.4
.01 28.
.5
.01 29.
.6
.01 ca 30.
LOG-NORMAL, MEDIAN: 4.0E-02, ERROR FACTOR:4.5 5 31.
29 2
4.0E-02 4.5 32.
.0007 33.
.001 34.
.002 35.
.003 36.
.004 37.
.005 38.
.006 39.
.007 40.
.008 41.
.009 42.
.01 43.
.02 44.
.03 45.
.04 46.
.05 47.
.06 48.
.07 49.
.08 50.
.09 51.
1 52.
.2 53.
.3 54.
.4 55.
.5 56.
.6 57.
.7 Sh:
h Figure 15. BABS Sample Input 60.
1.0
LOG-UNIFORM, 0.9, EQUAL-TAIL PRIOR DISTRIBUTION: DISCRETE INPUT, NUMBER OF POINTS :
20 DIESEL FAILURE
- PRIOR DISTRIBUTION u*amm*******
- ww*******
POSTERIOR DISTRIBUTION **********
TESTS
2 0.0050 0.0100 1.00E-02 9.90E-01 0.0001 1.47E-04 1.00E+00 0.0060 0.0100 2.00E-02 9.80E-01 0.0002 3.57E-04 1.00E+00 0.0070 0.0100 3.00E-02 9.70E-01 0.0003 6.41E-04 9.99E-01 0.0080 0.0100 4.00E-02 9.60E-01 0.0004 1.01E-03
'9.99E-01 0.0090 0.0100 5.00E-02 9.50E-01 0.0005 1.47E-03 9.99E-01 0.0100 0.1329 1.8 3 E- 01 8.17E-01 0.0075 8.98E-03 9.91E-01 0.0200 0.1830 3.66E-01 6.34E-01 0.0381 4.71E-02 9.53E-01 0.0300 0.1155 4.81E-01 5.19E-01 0.0499 9.70E-02 9.03E-01 0.0400 0.0851 5.66E-01 4.34E-01 0.0602 1.57E-01 8.43E-01 0.0500 0.0675 6.34E-01 3.66E-01 0.0686 2.26E-01 7.74E-01 0.0600 0.0561 6.90E-01 3.10E-01 0.0754 3.01E-01 6.99E-01 0.0700 0.0479 7.38E-01 2.62E-01 0.0805 3.82E-01 6.18E-01 8$ 0.0800 0.0419 7.80E-01 2.20E-01 0.0843 4.66E-01 5.34E-01 0.0900 0.0372 8.17E-01 1.83E-01 0.0868 5.53E-01 4.47E-01 0.1000 0.1329 9.50E-01 5.00E-02 0.3505 9.03E-01
- 9. 6 8 E- 0 2 0.2000 0.0100 9.60E-01 4.00E-02 0.0411 9.44E-01 5.57E-02 0.3000 0.0100 9.70E-01 3.00E-02 0.0318 9.76E-01 2.39E-02 0.4000 0.0100 9.80E-01 2.00E-02 0.0165 9.93E-01 7.43E-03 0.5000 0.0100 9.90E-01 1.00E-02 0.0060 9.99E-01 1.44E-03 0.6000 0.0100 1.00E+00 0.0 0.0014 1.00E+00 0.0 Figure 16(a). BABS Sample Output
LOG-UNIFORM, 0.9, EQUAL-TAIL PRf0R DISTRIBUTIDH: DISCRETE INPUT, NUMBER OF POINTS :
20 DIESEL FAILURE
- PRIOR DISTRIBUTION Wauhue***www
- mumWem POSTER 1AR DISTRIBUTION **=*******
TESTS
3 0.0050 0.0100 1.00E-02 9.90E-01 0.0000 6.30E-06 1.00E+00 0.0060 0.0100 2.00E-02 9.80E-01 0.0000 1.71E-05 1.00E+00 J.0070 0.0100 3.00E-02 9.70E-01 0.0000 3.42E-05 1.00E+00 0.0080 0.0100 4.00E-02 9.60E-01 0.0000 5.95E-05 1.00E+00 0.0090 0.0100 5.00E-02 9.50E-01 0.0000 9.52E-05 1.00E+00 0.0100 0.1329 1.83E-01 8.17E-01 0.0006 7.42E-04 9.99E-01 0.0200 0.1830 3.66E-01 6.34E-01 0.ba66 7.38E-03 9.93E-01 0.0300 0.1155 4.81E-01 5.19E-01 0.0132 2.05E-02 9.79E-01 0.0400 0.0851 5.66E-01 4.34E-01 0.0214 4.19E-02 9.58E-01 0.0500 0.0675 6.34E-01 3.66E-01 0.0308 7.27E-02 9.27E-01 0.0600 0.0561 6.90E-01 3.10E-01 0.0410 1.14E-01 8.86E-01
= 0.0700 0.0479 7.38E-01 2.62E-01 0.0516 1.65E-01 8.35E-01 l
i 0.0800 0.0419 7.80E-01 2.20E-01 0.0625 2.28E-01 7.72E-01 i
c) 0.0900 0.0372 8.17E-01 1.83E-01 0.0732 3.01E-01 6.99E-01 0.1000 0.1329 9.50E-01 5,00E-02 0.3320 6.33E-01 3.67E-01 0.2000 0.0100 9.60E-01 4.00F-02 0.0576 7.21E-01 2.79E-01 0.3000 0.0100 9.70E-01 3.00E-02 0.1161 8.37E-01 1.63E-01 0.4000 0.0100 9.80E-01 2.00E-02 0.0936 9.31E-01 6.95E-02 0.5000 0.0100 9.90E-01 1.00E-02 0.0510 9.82E-01 1.85E-02 0.6000 0.0100 1.00E+00 0.0 0.0185 1.00E+00 0.0 Figure 16(b). BABS Sample Cutput
- -. - _ _ =. -. -
1 i
LOG-NORMAL, MEDIAN: 4.0E-02, ERROR FACTOR:4.5 PRIO3 DISTRIBUTION: DISCRETIZED LOG-NORMAL, NUMBER OF POINTS :
27, NEDIAN : 4.0E-02. ERROR FACTOR = 4.5 DIESEL FAILURE
- PRIOR DISTRIBUTION
- d**********
- POSTERIOR DISTRIBUTION **********
TESTS
2 0.0010 0.3001 1.12E-04 1.00E+00 0.0000 5.93E-08 1.00E+00 i
0.0020 0.0010 1.14E-03 9.99E-01 0.0000 2.23E-06 1.00E+00 0.0030 0.0027 3.80E-03 9.96E-01 0.0000 1.48E-05 1.00E+00 i
I 0.0040 0.0046 8.43E-03 9.92E-01 0.0000 5.32E-05 1.00E+00 0.0050 0.0066 1.51E-02 9.85E-01 0.0001 1.39E-04 1.00E+00
?.0000 0.0085 2.36E-02 9.76E-01 0.0002 2.95E-04 1.00E+00 4
0.0070 0.0102 3.38E-02 9.66E-01 0.0003 5.49E-04 9.99E-01 l
0.0080 0.0117 4.54E-02 9.55E-01 0.0004 9.24E-04 9.99E-01
'0.0090 0.0129 5.33E-02 9.42E-01 0.0005 1.44E-03 9.99E-01
]
0.0100 0.0702 1.28E-01 8.72E-01 0.0035 4.92E-03 9.95E-01 0.0200 0.1680 2.96E-01 7.04E-01 0.0307 3.56E-02 9.64E-01 l
l 0.0300 0.1413 4.38E-01 5.62E-01 0.0534 8.90E-02 9.11E-01 4,
0.0400 0.1108 5.49E-01 4.51E-01 0.0685 1.58E-01 8.42E-01 f.
0.0500 0.0857 6.34E-01
- 3. 6 6 E- 01 0.0762 2.34E-01 7.66E-01 0.0500 0.0665 7.01E-01 2.99E-01 0.0783 3.12E-01 6.88E-01 l
0.0700 0.0521 7.53E-01 2.47E-01 0.0766 3.89E-01 6.11E-01 j
0.0800 0.0412 7.94E-01 2.06E-01 0.0726 4.61E-01 5.39E-01 0.0900 0.0329 8.27E-01 1.73E-01 0.0673 5.28E-01 4.72E-01 0.1000 0.0890 9.16E-01 8.40E-02 0.2053 7.34E-01 2.66E-01 i
0.2000 0.0602 9.76E-01 2.38E-02 0.2166 9.50E-01 4.97E-02 i
i 0.3000 0.0148 9.91E-01 9.02E-03 0.0412 9.92E-01 8.50E-03 0.4000 0.0051 9.96E-01 3.96E-03 0.0073 9.99E-01 1.22E-03 4
0.5000 0.0021
.9.98E-01 1.89E-03 0.0011 1.00E+00 1.29E-04
}
.0.6000 0.0010 9.99E-01 9.19E-04 0.0001 1.00E+00 6.68E-06 0.7000 0.0005 1.00E+00 4.26E-04 0.0000 1.00E+00 0.0 3
- r 0.8000 0.0003 1 00E+00 1.56E-04 0.0 1.00E+00 0.0 O.9000
-0.0002 1.00E+00 0.0 0.0 1.00E+00 0.0 l
Figure 16(c). BABS Sample Output i
LOG-NORMAL, MEDIAN: 4.0E-02. ERROR FACTOR:4.5 P3IOR DISTRIBUTION: DISCRETIZED LOG-NORMAL, NUMBER OF POINTS :
27, MEDIAN : 4.0E-02, ERROR FACTOR = 4.5 DIESEL FAILURE
- aa******
PRIOR LISTRIBUTION
- WFlu
- u*****am POSTERIOR DISTRIBUTION WummuWuMhh TESTS
3 0.0010 0.0001 1.12E-04 1.00E+00 0.0000 4.44E-10 1.00E+00 0.0020 0.0010 1.14 E- 0 3 9.99E-01 0.0000 3.30E-08 1.00E+00 0.0030 0.0027 3.80E-03 9.96E-01 0.0000 3.15E-07 1.00E+00 0.0040 0.0046 F '3E-03 9.92E-01 0.0000 1.47E-06 1.00E+00 0.0050 0.0066 1.51E-02 9.85E-01 0.0000 4.68E-06 1.00E+00 0.0060 0.0085 2.36E-02 9.76E-01 0.0000 1.18E-05 1.00E+00 0.0070 0.0102 3.38E-02 9.66E-01 0.0000 2.51E-05 1.00E+0D 0.0080 0.0117 4.54E-02 9.55E-01 0.0000 4.78E-05 1.00E+00 0.0090 0.0129 5.83E-02 9.42E-01 0.0000 8.31E-05 1.00E+00 0.0100 0.0702 1.28E-01 8.72E-01 0.0003 3.45E-04 1.00E+00 0.0200 0.1680 2.96E-01 7.04E-01 0.0047 5.03E-03 9.95E-01 0.0300 0.1413 4.38E-01 5.62E-01 0.0124 1.74E-02 9.83E-01 0.0400 0.1108 5.49E-01 4.51E-01 0.0214 3.88E-02 9.61E-01 0.0500 0.0857 6.34E-01 3.66E-01 0.0300 6.88E-02 9.31E-01 0.0600 0.0665 7.01E-01 2.99E-01 0.0574 1.06E-01 8.94E-01 0.0700 0.0521 7.53E-01 2.47E-01 0.0431 1.49E-01 8.51E-01 0.0800 0.0412 7.94E-01 2.06E-01 0.0472 1.97E-01 8.03E-01 0.0900 0.0329 8.27E-01 1.73E-01 0.0498 2.46E-01 7.54E-01 0.1000 0.0890 9.16E-01 8.40E-02 0.1707 4.17E-01 5.83E-01 0.2000 0.0602 9.76E-01 2.38E-02 0.4051 8.22E-01 1.78E-01 0.3000 0.0148 9.91E-01 9.02E-03 0.1320 9.54E-01 4.60E-02 0.4000 0.0051 9.96E-01 3.96E-05 0.0363 9.90E-01 9.67E-03 0.5000 0.0021 9.98E-01 1.89E-03 0.0081 9.98E-01 1.53E-03 0.6000 0.0010 9.99E-01 9.19E-04 0.0014 1.00E+00 1.55E-04 0.7000 0.0005 1.00E+00 4.26E-04 0.0001 1.00E+00 6.97E-06 0.8000 0.0003 1.00E+00 1.56E-04 0.0000 1.00E+00 0.0 0.9000 0.0002 1.00E+00 0.0 0.0 1.00E+00 0.0 Figure 16(d). BABS Sample Output
1
}
l x = input number of observed failures th P(I) = The I input value of diesel failure probability per demand PPRI(I) = the input or computed prior probability mass function value at P(I)
("x)P(I)*(1-P(I))"'* = the binomial probability of observing x failures i
in n trials when the failure probability for any given trial is P(I)
(") = the number of combinations of n things taken x at a time.
The IMSL subroutine MDBIN is called to compute the binomial probabilities.
After computing the posterior distribution, the MAIN program prints both the prior and posterior distributions in the form of a table as described in Section 5.3.
4 i
5.4.2 Subroutine PRIOR 1
Subroutine PRIOR reads or generates the prior probability mass function.
If 1
IDTYPE=1, the prior is simply read in.
If IDTYPE=2, the diesel failure proba-bilities, P(I), are read, and then a discretized lognormal probability mass
]
function is generated.
The first step in this process is to take natural logs of the points P(I) and assign them to the array variables PLN(I).
Then an interval (B1(I), B2(I)) is constructed about each point PLN(I) such that B1(I) is midway between PLN(I) and PLN(I-1) and B2(I) is midway between PLN(I) and PLN(I+1).
The median m and 90% error factor f of the lognormal distribution (which were read in on card type 4) are then converted to the mean and standard deviation of the corresponding normal distribution using the following formulas:
p = In(m) o = In(f)/1.64 Finally, subroutine PNORM is called to integrate the normal distribution with parameters p and a over the intervals (B1(I), B2(I)).
The areas so obtained
]
constitute a discretized lognormal prior probability mass function.
]
5.4.3 Subroutine PNORM l
Subroutine PNORM converts the values B1(I) and B2(I) to the equivalent values of the standardized normal distribution by subtracting p and dividing by a.
It then computes the area under the standard normal distribution over the transformed intervals (TB1(I), TB2(I)) and assigns the area to the variable PPRI(I).
The IMSL routine NDNOR is used for computing the standard normal 4
orobabilities.
Appendix A contains comp _lete listing of the BABS program.
j i
I l
i 43
APPENDIX A BABS SOURCE CODE LISTING 1
A-1
FORTRAN IV G1 RELEASE 2.0 MAIN DATE : 81245 09/25/48 PAGE 0001 005*
DIMENSION TITLE (20),P(100),PPRIC100),PPOST(100) 5.
OPSI DIMENSION N(300),NX(300) 6.
C 7.
C VARIABLE DESCRIPTION 8.
C TITLE (20)
TITLE DESCRIBING PRIOR DISTRIBUTION FOR OUTPUT 9.
C P(100)
INPUT POINTS AT WHICH DISCRETE PRIOR IS DEFINED 10.
C PPRI(100)
'rIOR DISTRIBUTION (MASS FUNCTION) 11 C PPOST(100)
P03TERIOR DISTRIBUTION (MASS FUNCTION) 12.
C N(300)
NUMBER OF TESTS 13.
C NX(300' NUMBER OF FAILURES 14.
C 15.
C READ NUMBER OF CASES (PAIRS OF TESTS AND FAILURES) TO BE INPUT 16.
0003 READ (5,80) NCASE 17.
0004 80 FORMAT (215) 18.
C READ NUMBER OF TESTS AND FAILURES 19.
0005 READ (5,80) (N(I),NX(I),I:1,NCASE) 20.
C 21.
C READ TITLE FOR OUTFUT 22.
0006 1 READ (5,81.END:999) TITLE 23.
0007 81 FOR'1A T ( 2 0 A4 )
24.
C 25.
C READ Y;MBER OF POINTS IN PRIOR, PRIOR TYPE, AND PARAMETERS OF PRIOR 26.
0008 REAJC5,82) NP.IDTYPE,PARAM1,PARAM2 27.
0009 82 FUFMAT(2IS,2E10.2) 28.
C 29.
C CALL ROUTINE TO READ OR COMPUTE PRIOR 30.
0010 CALL PRIOR (P,PPRI.NP.IDTYPE.PARAM1,PARAM2) 31.
yo C
32.
' 0011 DD 100 K:1,NCASE 33.
0012 WRITE (6,90) TITLE 34.
0013 90 FORMAT ('1',25X,20A4) 35.
0014 IF(IDTYPE.EQ.1) WRITE (6,91) NP 36.
0015 91 FORMAT ('O',' PRIOR DISTRIBUTION: DISCRETE INPUT, NUMBER OF',
37.
POINTS :',IS) 38.
1 '
0016 IF(IDTYPE.EQ.2) WRITE (6,92) NP,PARAM1.PARAM2 39.
0017 92 FORMAT ('0',' PRIOR DISTRIBUTION: DISCRETIZED LOG-NORMAL, 40.
1 ' NUMBER OF POINTS :',I5,',
MEDIAN
- ',1 PES.1,',
ERROR FACTOR :',
41.
2 OPF4.1) 42.
C 43.
C COMPUTE DENOMINATOR FOR CALCULATING POSTERIOR 44.
0018 DSUM:0.0 46.
0019 NMX:N(K)-NX(K) 47.
0020 DO 200 I:1,NP 48.
0021 CALL MDBIN(NX(K),N(K),PCI),PS,PK,IER) 49.
0022 DSUM = DSUM + PK W PPRI(I) 50.
0023 200 CONTINUE 51.
C 52.
C WRITF HEADINGS 53.
0024 kn:TE(6,93) N(K),NX(K) 54.
0025 93 F0kMAT('0',' DIESEL FAILURE',5X,12('*'),' PRIOR DISTRIBUTION '
55.
I 12('w'),6X,10('*'),' POSTERIOR DISTC;BUTION
',10('*'),
56.
2 3X,' TESTS
=',I4,/,1X, 57.
3 PROBABILITY',8X,' MASS FUNCTION',7X,'CDF',11X,'CCDF',12X, 58.
4 ' MASS FUNCTION',7X,'CDF',11X,'CCDF',8X,' FAILURES :',I4) 59.
C 60.
C INITIALIZE VARIABLES FOR STORING PRIOR AND POSTERIOR CDF AND CCDF 61.
0026 POCDF = 0.0 62.
0027 PRCDF : 0.0 63.
~
FORTRAN IV G1. RELEASE 2.0 MAIH DATE : 81245 09/25/48 PAGE 0002 0028 POCCDF:1.0 64.
0029 PRCCDF: 1.0 65.
C 66.
C START LOOP TO COMPUTE POSTERIOR AT EACH POINT P(I) 67.
0030 DO 210 I: 1.HP 68.
0031 IF(PRCCDF.EQ.0.0) GO TO 230 69.
1,
_0032 PRCDF:PRCDF+PPRICI) 70.
0033 IF(ABS (PRCDF-1.0).GT.I.0E-6) GO TO 220 71.
i.
0034 PRCCDF:0.0 72.
0035
-GO TO 230 73.
0036 220 PRCCDF:1.0-PRCDF 74.
C 75.
^
C COMPUTE POSTERIOR 76.
003' 230 IF(POCCDF.EQ.0.0) GO TO 249 77.
1 0038 CALL MDBIN(NX(K),N(K),P(I),PS,PK,IER) 78.
0039 PPOST(I) : (PK W PPRI(I)) / DSUM 79.
0040 POCDF:POCDF+PPOST(I) 80.
0041 IF(ABS (POCDF-1.0).GT.1.0E-6) GO TO 240 81.
0042 POCCDF:0.0 82.
0043 GO TO 250 83.
]
0044 240 POCCDF: 1.0-POCDF 84.
i 0045 GO TO 250 85.
i 0046 249 PPOST(I):0.0 86.
j 0047 250 WRITE (6,94) PCI),PPRICI),PRCDF,PRCCDF,PPOST(I),POCDF,POCCDF 87.
0048 94 FORMAT ('O',0PF10.4,2(5X,0PF15.4,1P2E15.2))
88.
0049-210 CONTINUE 69.
C 90.
1 2 0050-100 CONTINUE 91 a
C 92.
"0051 GO TO 1 93.
9 C
94.
1 0052 999 STOP 95.
1 C
96.
.0053 END 97.
1 i
f t
f i
i
I a
FORTRAN IV G1 RELEASE 2.0 PRIOR DATE : 81245 09/25/48 PAGE 0001 0001 SUBROUTIHE PRIOR (P,PPRI,HP,IDTYPE,PARAM1,PARAM2) 98.
C 99.
0002 DIMEk%IDH P(1),PPRI(1)
- 100, 0003 DIMENSIDH PLN(100) 101.
0004 COMMON / BOUNDS / B1(100),B2(100) 102.
C 103.
C 104.
0005 GO TO (100,200). IDTYPE 105.
C 106.
C READ DISCRETE PRIOR 107.
0005 100 READ (5,82) (P(I),PPRI(I),I:1,HP) 108.
0007 82 FORMAT (2E10.5) 109.
0008 RETURN 110.
C 111.
C CREATE LOG-HORMAL PRI3R 112.
0009 200 HPT:NP 113.
0010 HP:NP-2 114.
C READ POINTS AT WHICH PRIOR IS TO BE COMPUTED 115.
0011 READ (5,83) BL,(P(I),I: 1,NP),BU 116.
0012 83 FORMAT (E10.2) 117.
0013 DO 210 I: 1,NP 118.
0014 PLN(I):ALOG(PCI))
119.
0015 210 CONTINUE 120.
C 121.
C FIND MIDPOINTS (OH LOG SCALE) BETWEEN THE POINTS P(I) 122.
0016 B1(1) = (PLH(1) + ALOG(BL))/2.0 123.
0017 B2(1) = (PLHC2) + PLN(1))/2.0 124.
2 0018 B1(HP) : (PLH(NP) + PLHCHP-1))/2.0 125.
L 0019 B2(NP) : (ALOG(BU) + PLHCHP))/2.0 126.
0020 HP1 : HP-1 127.
0021 DO 220 I:2,HP1 128.
0022 B1(1) : (PLH(I) + PLHCI-1))/2.0 129.
0023 B2(I)
(PLN(I+1) + PLH(I))/2.0 130.
0024 220 CONTINUE 131.
C 132.
C COMPUTE PARAMETERS FOR HORMAL DISTRIBUTION 133.
0025 AMU:ALOG(PARAM1) 134.
0026 SIG:ALOG(PARAM2)/1.64 135.
0027 CALL PHORM(PPRI,HP, AMU,SIG) 136.
0028 RETURN 137.
C 138.
0029 END 139.
FORTRAN IV G1 RELEASE 2.0 PHORM DATE : 81245 09/25/48 PAGE 0001 0001 SUBROUTINE PHORM(PPRI,HP, AMU,SIG) 140.
C 14 1 0002 COMM0H / BOUNDS / B 1( 100 ),82(10 0 )
142.
0003 DIMEHSIDH TB1(100),TB2(100) 143.
0004 DIMENSIDH PPRI(1) 144.
C 145.
C TRANSFORM POINTS INTO STANDARDIZED NORMAL DISTRIBUTION WITH 146.
C MEAN:0, STANDARD DEVIATION:1 147.
0005 DD 100 I: 1.NP 148.
0006 TB1(I) : (B1(I)-AMU)/SIG 149.
0007 TB2(I) : (B2(I)-AMU)/SIG 150.
0008
- C0 CONTINUE 151 C
152.
C COMPUTE AREA UNDER TRUNCATED HORMAL DISTRIBUTION BETWEEN LOWER 153.
C BOUND Al AND UPPER BOUND A2 154.
0009 CALL MDNOR(TB2(HP),A2) 155.
0010 CALL MDNOR(TB1(1),A1) 156.
0011 TOTAR : A2-Al 157.
C 158.
0012 DO 200 1: 1,NP 159.
0013 CALL MDHOR(TB1(I),A1) 160.
0014 CALL MDNOR(TB2(I) A2) 161.
C NORMALIZE THE PROBABILITIES SO THAT THEY SUM TO 1 162.
0015 PPRI(I) : (A2-A1)/TOTAR 163.
0016 200 CONTINUE 164.
0017 RETURN 165.
0018 END 166.
Y, a
i
. REPORV NUVBE R (Ass erera ry DOC)
U S. NUCLE AR REGUL ATORY COMMISSION 4
BIBLIOGRAPHIC DATA SHEET flVREG-0873 O.1lTLE AND SUBTITLE LAdd Volume No., of mortproste)
- 2. (Leave bimkJ
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- 3. RECIPIENT'S ACCESSION NO.
j A Bayesian Analysis of Diesel Generator Failure Data
- 7. AUTHOR (S)
- 5. DATE REPORT COMPLL TED W.E. Vesely, P.K. fliyogi, F.F. Goldberg, P.A. Weldon Dc M er IlW1 1
- 9. PERF ORMING ORGANIZATION N AME AND MAILING ADDRESS (lactue les Codel DATE REPORT ISSUED U. S. Nuclear Regulatory Commission uoNTH lvera Office of Nuclear Regulatory Research January 19R2 l
Division of Risk Analysis s (tenve bienal Washington, DC 20555 1
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- 12. SPONSORING ORGANIZATION N AME AND MAILING ADDRESS (Inclue lip Codel Same as 9, above.
- 11. CONTRACT NO.
4 13 TYPE OF REPORT PE RIOD COVE RE D (Inclusive dates)
Technical
- 15. SUPPLEMENTARY NOTES
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- 16. ABSTR ACT Q00 words or less)
A simple Bayesian approach has been developed to evaluate failure rate implications from the number of failures and number of successes in a given number of diesel tests.
For the Bayesian approach, the diesel is modeled as having a constant probability of failure per trial which is unknown and whose possible values are describable by a probability distribution. The approach utilizes discrete probability distributions (probability mass functions) for ease of implementation, i
As a potential tool for the analyst, a computer code has been written to efficiently calculate the diesel posterior failure rate distributions for any input diesel test data and assumed prior distribution.
The code can be used to monitor diesel tests for up-to-date failure rate implications.
In addition, a wide variety of sensitivity analysis can be perfonned using the code.
i 17 KEY WORDS AND DOCUMENT ANALYSIS 17a DESCRIPTCRS Bayesian Analysis Computer Code Diesel Generator i
Probability distribution Failure Data 17b. IDENTIFIERS!OPEN-ENDED TERMS 4
- 18. AVAILABILITY STATEMENT 19 R4 (This reportl 21.NO. OF PAGES Unlimited 2
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- 22. P RICE NCC FORM 335 (7 77)
UNITED STATES NUCLEAA RE GULATORY COMMIISION f
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