ML20062K151
| ML20062K151 | |
| Person / Time | |
|---|---|
| Issue date: | 07/29/1982 |
| From: | Silling S NRC OFFICE OF NUCLEAR MATERIAL SAFETY & SAFEGUARDS (NMSS) |
| To: | Knapp M NRC OFFICE OF NUCLEAR MATERIAL SAFETY & SAFEGUARDS (NMSS) |
| References | |
| REF-WM-1 NUDOCS 8208170065 | |
| Download: ML20062K151 (6) | |
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JUL 2 91932 g7 4 n %
3105.3.2/SAS/82/07/28/0 Distribution:
gv,,WtfHL file WMHL r/f I W,,~
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3105.3.7 JBMartin REBrowning tidBell MEIORANDUM FOR:
Malcolm R. Knapp SASilling & r/f High-Level Waste Licensing HJMiller Management Branch J0 Bunting Division of Waste Management PDR FR74-Stewart A. Silling' High-Level Waste Licensing Management'6 ranch Division of Waste Management 3UBJECT:
ERROR ANALYSIS FOR TRADE-OFF PLOTS I
The statistical techniqup/1 sed in constructing the trade-off plots presented in the 10<CFR 6b rationale document predicts the probability of fa'ilure of the assumed standard based on the number of failures in a finite samp?e.
The e,uestion of what confidence to place in these estimates arises.
The attached dccument describes how a confidence interval can be j
determined for the contours in the trade-off plots.
The analysis indicates that the probabilities shown on the trade-off plots are within 12% of the actual probabilities, with a confidence of 90%.
ORIGI::n sic:a rz Stewart A. Silling High-Level Waste Licensing Managment Branch Division of Waste Management
Enclosure:
As stated WMHL {;g :
0FC :
NAME :SASilling:1mc DATE : 7/E7 /82 8208170065 820729
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ERROR ANALYSIS FOR TRADE-0FF PLOTS A trade-off plot is constructed by connecting points on a plane which have the same frequency of failure within a finite sample.
If this frequency of failure is to be interpreted as a measure of the probability of failure, it is of interest to examine the uncertainties introduced by the finiteness of the sample.
The failure or non-failure of a vector is a Bernoulli trial (a random event with two possible outcomes) with parameter p, which is the proba-bility of failure.
This probability is the quantity which the contours on the trade-off plot attempt to measure.
Let the sample size be N vectors.
If the parameter is p, the probability of obtaining exactly F failures in the sample may be derived from classical probability theory (ref.1):
P(Flp)=pg p (1-p)N-F (1)
F C
where P(Fjp)=conditionalprobabilityofFfailuresifpisgiven C = binomial coefficent pN N!
~ (N-F)!F!
i Equation 1 may be approximated by a normal distribution (ref. 2):
N(f-p) y
,2p(1-p)
P(flp)=
~
2) e
/2np(1-p)/N where P(flp) = conditional probability of failure fraction f if p is given f = failure fraction in sample = F/N Equation 2 gives the probability of having a certain fraction of failures for a given parameter p.
However, what is needed is the probability of having a certain parameter p given an observed failure fraction f.
This inversion of the problem may be accomplished with the help of Bayes' theorem (ref. 3), which may be applied as follows:
1
..m9
P(plf)=
(3) el P(p')P(flp')dp' 0
where P(plf)=probabilityofhavingBernoulliparameterpgiven failure fraction f in a sample P(p) = assumed probabilit/ density function for p based on a priori knowledge of system (also called the " prior distribution")
The prior distribution for p will be assumed to be a uniform distribution, since past experience with trade-off plots has shown a fairly even range of contours between 0 and 1.
In other words, it is assumed that nothing is known about the behavior of p before the sample is taken.
Substituting Equation 2 into Equation 3 using P(p) = 1 and cancelling the normalization constants from numerator and denominator, N(f-p)2 P(plf)=
p)fDII-DI)
(4) 8
,1 2p'(1-p') ge(1_p))-1/2 dp' i
e
<0 Equation 4 may be evaluated numerically. The resulting curves of P(plf) behave as shown in Figure 1 for the special case f = 0.30 for a few values of N.
Note that the curves become narrower for larger N.
This makes sense because the larger the sample size, the more confidence one has
~
that the observed failure fraction is a good approximation to the true probability of failure p.
The lopsidedness of the curves may be under-.
stood by considering the extreme case p = 0.
In that case, there is no possibility.that even a single vector in the sample would fail, hence f=0.
The same is not true of the point p = 2f, which is the same distance from f as 0 is, but could possibly be the parameter.
A confidence interval is any pair of numbers a and b such that there is a 1-a probability that the random variable is in the interval (a,b) where 1-a is the degree of confidence. One way of finding such a pair of numbers is to take a such that the probability of having the random variable in the interval (-sa) is a/2 and b such that the probability of having the random variable in the interval (b,=) is also a/2. This procedure was used to find confidence intervals for the probability distribution 2
c.
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4 Figure 1.
, Conditional probability of p given f = 0.3.
(Unnormalized i
probability density function.)
t
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i b
a f
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k i
3-a t
i 2
--n-wm
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h for p described by Equation 4 with N = 50. Table 1 shows the resulting confidence intervals for various values of f and various degrees of confidence.
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i Along the 90% confidence column in Table 1, a confidence interval of l
0.12 is sufficient to include any of the predicted intervals. Hence one may conclude that the probabilities plotted on the contour lines of a trade-off plot have at least a 90% chance of being within 12%
t of the actual probabilities.
It should be noted that the above is a conservative estimate for the confidence interval when La" tin Hypercube Sampling is used. However, it is difficult to quantify how much the use of Latin Hypercube i
Sampling improves the confidence interval for a given sample size.
' References l
(1)
H. J. Larson, Introduction to Probability Theory and Statistical l
Inference, Second Edition, John Wiley & Sons, New York (1974),
j page 125.
i (2)
Ibid., page 217.
(3)
Ibid., page 56.
e P
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l i
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Observed Degree of Confidence (1-a)
Fraction (f) 99%
95%
90%
80%
-70%
i a
.037
.048
.054
.063
.070 0.10 b
.266
.221
.199
.176
.161 4
a
.098.
.117
.129
.143
.153 0*20 b
.379
.334
.312
.286
.270 a
.169
.195
.210
.229
.242-0*30 b
.482
.439
.416
.391
.373 a
.247
.280
.297
.319
.334 0*40 b
.578
.538
.516
.491
.474 a
.332
.369
.389
.412
.429 0*50 b
.668
.631
.611
.588
.571 a
.422
.462
.484
.509
.526 0.60 b
.753
.720
.703
.681
.666 i
a
.518
.561
.584
.609
.627 i
0*70 b
.831
.805
.790
.771
.758 a
.621
.666
.688
.714
.730 0*80-b
.902
.883
.871
.857
.847 a
.734
.779
.801-
.824
.839 0.90 t
b
.963
.952
.946
.937
.930 1
1 Table 1.
Confidence intervals for N = 50. There is a 1-a probability that the Bernoulli parameter p lies in the interval (a,b) if a fraction f of the l'
vectors fail.
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