A Practical Approach to Expert Elicitation for Bayesian Reliability Analysis of Aging
B. Villain & B. Vérité
Electricité de France, Research and Development Division, 6, Quai Watier, 78401 Chatou Cedex, France {Benjamin.Villain & Bruno.Vérité}@edf.fr
C. Biernacki & G. Celeux
INRIA, Rhônes-Alpes, 655, Avenue de l’Europe, 38330 Montbonnot St-Martin, France {Christophe.Biernacki & Gilles.Celeux}@inria.fr
ABSTRACT : In 1998, EDF (Electricité de France) and INRIA worked in partnership to develop a practical approach to elicitation from expert’s opinions, suitable for supplying quantitative information within the framework of a Bayesian Reliabily Analysis (BRA) of component subjected to aging. The purpose of this paper is to present and illustrate the main results of this cooperation and to show its usefulness in a test case.
The approach is intended to form part of a BRA in which the expert provides subjective information to complement a small amount of objective data obtained from feedback experience. The difficulty is to translate expert's opinions into "prior" information on reliability model’s parameters (Weibull distribution in the case of aging). The mix between the two information sources contains all available information about the studied event and can be used to evaluate the predicted reliability at different mission times, for an aging component. The SURVBAYESã software presented at the ESREL’98 conference in Trondheim is used for the analysis at EDF.
1 INTRODUCTION
The capability of quantifying reliabilistic component aging phenomena, in other words the increase in the failure rate with time, is an important requirement at EDF. This is why EDF has been interested in the development and use of powerful Reliabily Analysis methods (Classical, Bayesian or Structural) to study this phenomenon for many years. The use of this type of method (Villain et al. 1996) can help
decision-making concerning the lifetime evaluation and extension for sensitive components (valves, pumps, alternators, pipes, etc.). It also forms part of the OMF-Structures (“Reliability Centered Maintenance”) method (Bryla et al. 1997), (Vérité et al. 1998) of optimizing preventive maintenance programs or in-service inspection campaigns on passive components in French nuclear power plants.
When only a small amount of feedback experience data or life tests results are available, the Bayesian Reliability Analysis (BRA) is used to compensate for this lack of “objective” data derived from the observation of components considered to be identical and independent, by making use of
“subjective” data elicited from experts. In this case the main problem is to quantitatively translate expert opinions into the form of a prior probability density on the parameters of a reliability model.
EDF uses the SURVBAYES software (Souchois et al. 1998) for BRA of non-reparable components starting from a Weibull type reliability model (Clarotti et al. 1996). The software uses expert opinions to determine a prior density in the form of a credibility histogram p(q,p), where q is the unknown shape parameter in the Weibull model (h is the scale parameter), and p is the reliability at the mission time :
Weibull model survival function :
q
q h
h x
x
F( | , )=exp - (1)
Reliability at the mission time :
p tm
exp h
q
(2) Let q0 be the boundary between acceptable aging (q£q0) and unacceptable aging (q>q0), p0 be the boundary between acceptable reliability (p³p0) and unacceptable reliability (p<p0), and let c=(c1,c2,c3,c4) be the credibilities of the four quadrants thus defined (where 0£ci£Csup for i=1,
…,4); the credibility histogram p(q,p) is then expressed as follows :
£
£
£
£
0 0
4
0 0
3
0 0 2
0 0
1
0 0
0 ,
1 ,
1 ,
1
0 , 1 ) , ,
| , (
p p c
p p c
p p c
p p c
c p p
q q
q q
q q
q q q
q
p
(3)As we will see later, the concept of acceptability is related to reliability objectives fixed at the design stage and accepted by the decision-maker. This is the context in which experts have to give their opinions about the values of the parameters q0, p0 and c.
From a practical Bayesian point of view, prior distributions of the credibility histogram type (improper) are used to express a decision-maker's waiting strategy, in other words the role of
“objective” data is then to eliminate steps in the histogram that extend to infinity. We then have a prior density that represents “vague knowledge”
about unknown parameters. Within a BRA, a sensitivity analysis will eventually be carried out on the choice of the q0, p0 and c parameters.
In the following, we will mention the various techniques used for interviewing experts through the results of a bibliographic study. This helped us to build up a specific strategy for interviewing experts in order to determine quadrants credibility. Finally, the strategy was validated by using various relevant artificial situations.
2 EXPERT ELICITATION METHODS 2.1 Elicitation: a scientific approach
Elicitation of subjective probabilities must be based on a rigorous approach if it is to be credible and efficient. As for any scientific research, it must respect five basic principles (Cooke 1991), namely reproducibility, responsibility, empirical check, neutrality and equity.
In the case of a medical system, Robert (1991) shows that the performance of a diagnostic system increases with the rigor of the elicitation approach.
The choice of the expert and the combination of several expert opinions are also important elements.
2.2 The two main biases
Meyer & Booker (1991) identified two main types of biases (the motivation bias and the knowledge bias) that must be taken into account in the elicitation approach.
The motivation bias indicates that the elicitation process alters the expert's answers, for example due to group pressure or poor understanding.
The knowledge bias indicates that the expert does not give his answers in the expected standard, like probabilities.
2.3 The six steps in an elicitation process
Six steps have been identified in the elicitation process (Meyer & Booker 1991).
2.3.1 Step 1
This step defines the purpose of the project, and therefore it is not specific to the elicitation approach.
2.3.2 Step 2
The purpose of step 2 is to reduce the motivation bias by giving recommendations about the choice of questions. Questions must be simple, precise and possibly illustrated by the context, definitions or even figures. It is preferable to break excessively complex questions down into several simple questions.
2.3.3 Step 3
Experts must be chosen by their peers. They must be motivated, available and preferably know the type of answer expected in order to reduce knowledge bias. The use of several experts from different backgrounds helps to reduce both types of bias.
2.3.4 Step 4
The components of the elicitation (elicitation types, communication types and answer types) have to be selected.
There are three sorts of elicitation types: group meeting (organized or not), DELPHI method, individual interview. Although it takes time, the individual interview reduces the motivation bias better than the other two methods, and thus makes it easier to combine expert opinions using a rigorous mathematical method.
Communication types are face to face, telephone, mail, etc. Logistic constraints such as cost or time always have to be considered.
Finally, the types of expected answers must be fairly familiar to the expert in order to reduce the motivation bias. Experts can be calibrated by specific practice questions. Nevertheless, types of answers with an inevitable knowledge bias must be avoided, for example, preference should be given to quantiles over extreme values.
2.3.5 Step 5
This step deals with combinations of opinions, and if it is done objectively, it can reduce both types of bias (Moslesh et al, 1988). When opinions are in the form of probability distributions, Lind and Nowak
(1988) presented four desirable characteristics for a combination operator but which cannot all be done together. These characteristics are the preservation of symmetry, continuity property, preservation of shape and Bayesian consistency (i.e. the result of the combination is unchanged if it is combined before or after the Bayesian update of opinions by data).
Genest & Zidek (1986) recommend weighted arithmetic and geometry averages. The advantage of a geometric combination is that it satisfies three of the previous properties (including Bayesian consistency) and avoids multi-modality which is always a problem when making decisions. It is recommended that equal weights should be chosen (Meyer & Booker 1991).
2.3.6 Step 6
The final step concerns documentation and satisfies the five scientific requirements set down by Cooke.
The objective is to make the process and results credible to the outside world and to experts, to critically appraise, etc. The entire process including the identity of experts, etc. must be outlined.
3 CONSTRUCTION OF A STRATEGY 3.1 Resetting parameters
In order to reduce biases, the q0 (shape) and p0 (reliability at mission time) parameters must be reset to make them more natural for the experts. The s0=Np0 parameter transforms a reliability into a number of items of equipment that survive the mission time among N. The r0 2q01 parameter transforms aging (abstract concept) into a magnification factor which represents the increase in the number of failures between half mission time and complete mission time. However, credibilities are unchanged.
3.2 Profile of experts
There are two types of experts, namely “design”
experts and “maintenance” experts. “Design” experts are competent to define what aging and reliability are acceptable, and will therefore be queried on r0
and p0. “Maintenance” experts know where equipment is located with respect to “design” expert classes. Therefore, they make a decision about c1,
…,c4 after obtaining the opinion of “design” experts.
3.3 Taking account of uncertainty
We suggest that experts should be asked to estimate specific values for r0, p0, and c, and also to estimate the uncertainty on these values. The value- uncertainty pair will be used to estimate probability distributions hr, hs and hc1,…,hc4 for s0, p0 and c1,
…,c4 respectively and consequently the hq and hp distributions on q0 and p0 respectively.
Therefore, we define a prior distribution on the prior distribution parameters, namely the credibility history. This type of approach is qualified as hierarchical (Robert 1994).
3.4 Choice of prior distributions
We propose to use Beta distributions for hq, hp and hc1,…,hc4 since, for a Bayesian application, these are classical prior distributions due to their flexibility and the simplicity of the resulting calculations. The Beta distribution has two parameters a and b and is written as follows:
) ( )
1 ) (
( ) (
) ) (
, ,
|
( x 1 x 1I[0,1] x
b a
b b a
a x
B a b
. (4)
This equation is bounded, so that we have to define a maximum value qsup for q0. For example, the value qsup=200 leads to rsup=2199, which is a boundary far greater than orders of magnitude given by experts.
3.5 Questions to be put to experts
Values of the two parameters of the Beta distribution have to be defined for each expert,. We do this by asking the expert to make two choices:
1 a median on r0, p0 or c1,…,c4;
2 an uncertainty among several predefined uncertainties, visually making use of the shape of the density of r0, p0 or c1,…,c4.
These two items of information are sufficient to completely define the Beta distribution for r0, p0 or c1,…,c4. Furthermore, uncertainties must be defined before starting elicitation. We propose four values, namely low, medium, high and very high.
Finally, an order has to be respected in questions about credibilities. The expert chooses the most credible quadrant, and this quadrant is necessarily fixed with a low uncertainty and a maximum credibility Csup; this is the reference quadrant. The other three quadrants are then considered in decreasing order of median credibility, leaving the choice of the uncertainty. This means that fairly natural questions can be asked such as: "By how many times is this quadrant less credible than the reference quadrant?".
3.6 Combination of opinions
When nq experts are questioned on r0 and therefore on q0, nq distributions hq1,...,hqnq are obtained. We select the equal weight geometric combination recommended in the bibliographic study and therefore:
q q q
q q
q
n i
i n
h h
1
/
)]1
( [ )
( . (5)
We proceed in the same way for p0 and c1,…,c4. The combined distribution is then simply a Beta distribution with parameters equal to the arithmetic mean of the parameters of the combined distributions.
It can be verified that this combination is consistent: the uncertainty of the combination of two experts is between the uncertainties of each and an expert who has more confidence in his opinion will have more influence.
3.7 Selection of a (q0 ,p0 ,c) triplet
In fact, the SURVBAYES software needs a (q0,p0,c) triplet in imput. We can use the richness of the distributions obtained to define several triplets from distributions on q0, p0 and c:
an average value: the expectancy of each Beta distribution is used;
a median value: the median value of each Beta distribution is used;
a "risky" value: this leads to more risky maintenance than the median value, using the 25 or 75% quantiles (depending on the case) of each Beta distribution;
a “conservative” value: this leads to a safer value than the median value based on the same principle as the "risky" value, by inverting the 25 or 75% quantiles depending on the case.
Note that median and average values are complementary, and are used to detect excessive values.
3.8 Additional questions about life
In addition to the strategy presented above, experts (and particularly “maintenance” experts) are asked additional questions about a censored interval of the life of equipment. Expert data collected directly in the form of an interval (rather than a median) about failure times will be considered as being equivalent to feedback experience. They will then simply be added to real feedback experience data and processed in the same way.
The advantage of these additional questions is to make the prior distribution more informative about parameters for the Weibull distribution, and thus ensure convergence of the integral expressing the predicted reliability, which is evaluated in the SURVBAYES software.
3.9 Use in the ELICITAEã software
The ELICITAE software written in Matlabã, implements this strategy and facilitates additional questions about life duration.
Its graphic features are useful for displaying density curves associated with the four available uncertainties to the expert. Note that density shapes may vary as a function of the median selected by the expert, and will therefore have to be recalculated.
The software can also be used for practice runs with experts. A documentation module is also included in order to ensure traceability of all minor details in the process.
4 APPLICATION TO A TEST CASE
The application presented in this chapter is carried out on a test case in order to illustrate the way in which a BRA of aging of an non-reparable component is performed, starting from:
a Weibull reliability model;
multi-censored feedback experience data;
and expert opinions translated into the form of a credibility histogram (prior distribution) using the ELICITAE process.
The application concerns real “objective” data derived from feedback experience on the failure of a component of industrial systems (nuclear, aeronautics, etc.) and “subjective” data derived from experts.
In this case, we are concerned with the aging of subcomponents for which strongly multi- censored feedback data are available originating from the results of in-service inspections carried out periodically. No maintenance operation is performed on these sub-components to extend their life.
4.1 Description of test cases and available data Available feedback experience data are related to failures observed during periodic inspections carried out on the subcomponent. There are two groups of identically independent sub- components originating from two components for which very few feedback experience data are available. This situation does not provide much information, and in practice causes statistical analysis problems and requires expert opinions.
The BRA was carried out on these two components starting from the feedback experience data summarized in tables 1 and 2 in which the fourth column contains the number of failures ni of sub-components that occurred during the time interval separating the two inspections carried out at the times indicated on the same line in columns 2 and 3; the last line refers to sub-components that are still working
correctly (observable) at the time of the end of the observation.
Table 1: Component No. 1data I Time intervals at which
failures are observed (in hours)
ni
1 0 10220 0
2 10220 35377 24
3 35377 3337
Table 2: Component No. 2 data i Time intervals in which
failures are observed (in hours)
ni
1 0 27043 7
2 27043 35377 27
3 35377 3327
Note that only one inspection for component No.
1 is useable (no failures at the time of the first inspection) contrary to component No. 2, which gives very little objective information about aging, so that expert opinions will have to be taken into account as described below.
4.2 Bayesian Reliability Analysis of aging
The general principles of the BRA are shown in figure A-1 in the appendix. Its main advantage is that it enables mixing of a small number of
“objective” data with “subjective” data derived from experts elicitation. In our test case, the procedure is carried out starting from the Weibull model in order to analyze aging.
4.2.1 Choice of “objective” data
The first important step in the procedure is to select the data sample. For the case of maintenance of the considered components, a first assumption is that we want to optimize maintenance at component level and thus the sample composed of all sub- components of the same component is analyzed, resulting in 3361 exchangeable entities. Therefore, it is decided to use feedback experience data about component No.1 only, which is the least informative case.
4.2.2 Construction of priors by ELICITAE
A prior distribution has to be chosen or built to provide further information about the parameters for the Weibull model before the Bayesian inference can be carried out using this sample.
Several imaginary situations were created in order to study the behavior and elicitation strategy developed in ELICITAE; choice of uncertainties, the combination method, etc. Some situations are
particularly intended to study cases of disagreement between experts, the impact on the final result used by the decision-maker.
Remember that in order to build up a credibility histogram type, it was decided to use mission time that corresponds to a maintenance objective. We considered the predicted probability of failure on a sub-component after 35377 hours + about two operating cycles, or about 50000 hours, knowing that the final observation time is 35377 hours of operation for components No. 1 and No. 2.
4.2.2.1 “ Design ” experts
Several cases are considered for “design” experts who make statements about p0 and r0 (see §3.2):
2 “design” experts disagree on median values of p0 or r0 and agree on the uncertainty (tables 3 and 4);
2 “design” experts disagree on the median and on uncertainties (tables 5 and 6);
1 “design” expert disagrees with the others about the median and the uncertainty (table 7).
Table 3: same low uncertainty, different medians
Case 1 p0
expert C 1 0.9
expert C 2 0.99
uncertainties C1 & C2 low
Table 4: same very high uncertainty, different medians
Case 2 p0
expert C 1 0.9
expert C 2 0.99
uncertainties C1 & C2 very high
Figure 1: result of combination in case 1
Figure 2: result of combination in case 2
In the two cases in figures 1 and 2, note that the geometric combination is intermediate between the two opinions with an equivalent uncertainty. This avoids multi-modality that causes problems when making decisions by balancing expert opinions with identical weights. In case 2, uncertainty is high which results in a large variation between the risky value p0 (0.908) and the conservative value (0.972).
This difference requires a sensitivity study.
Table 5: different uncertainties, different medians
Case 3 p0
expert C 1 0.9
expert C 2 0.99
uncertainty C1 very high
uncertainty C2 low
Table 6: different uncertainties, different medians
Case 4 p0
expert C 1 0.9
expert C 2 0.99
uncertainty C1 high
uncertainty C2 moderate
In cases 3 and 4 (see figures 3 and 4), note that the geometric combination of the two experts assigns the greatest influence to the most confident expert (C2). This could lead to a bias when the expert underestimates his uncertainty about the given median value.
The observed behavior on p0 is the same on the boundary value q0 related to r0.
Figure 3: result of combination in case 3
Figure 4: result of combination in case 4
Table 7: different uncertainties, different medians
Case 5 r0
expert C 1 2
expert C 2 to 15 6
uncertainty C1 moderate
uncertainty C2 to 15 high
In case 5 in which an expert with “moderate”
uncertainty disagrees with 14 experts who agree with each other about the median value of r0 with
“high” uncertainty, figure 5 shows that the isolated expert still has some influence, and this causes a problem. In this case, it appears important to know the basis for the relative certainty of the expert.
Furthermore, this result of 1 against 14 shown in figure 5 reflects the way in which the ratio between
"moderate" and “high” uncertainty is reflected quantitatively. This reflection can be adjusted in the ELICITAE software.
Figure 5: result of disagreement of one expert against 14
4.2.2.2 “ Maintenance ” experts
“Maintenance” experts make a decision about (c1; c2; c3; c4) starting from median values for the (q0,p0) pair obtained by combining the opinions of “design”
experts (see §3.7). The results of the previous analysis on the impact of disagreement or agreement between “maintenance” experts are the same. Note that the strategy used in ELICITAE fixes a low uncertainty to the highest credibility. If there is a disagreement on the most credible quadrant, a median value equal to the arithmetic mean is assigned to each. In this case, less information is provided by experts.
4.2.3 Sensitivity study on the predicted reliability Observed data (table No. 1) and the various credibility histograms are mixed in SURVBAYES in order to study the sensitivity to the choice of (q0,p0,c) triplets obtained with ELICITAE.
4.2.3.1 The Bayesian inference
The posterior density obtained using BAYES theorem summarizes all available knowledge about parameters for the Weibull model. It is written as follows:
p q q p q
q p q q
( , / ) / , . ( , )
. / , ( , ) .
p y L y p p
L y p p d dp
0
0
(6)
The next step is to use this posterior density to deduce the first moments (mean, standard deviation) on the two parameters (q, p) and the predicted reliability at the mission time:
R t( m)
p t( m, ). ( ,q p q p y d dp) q (7)
Posterior moments are used to give an idea about the density shape (variation coefficient equal to the ratio of the standard deviation to the mean) and thus to determine if the confidence zone around the averages is narrow or wide.
4.2.3.2 Tested prior distributions
We considered the case of a “design” expert and a
“maintenance” expert with a very high uncertainty on median values for triplets. This case was chosen because it amplifies the sensitivity on the final reliability analysis. Table 7 presents the various (q0,p0,c) triplets taken into account in prior for the sensitivity study.
Table 7: different sets of (q0,p0,c) triplets
p0 q0 c1 c2 c3 c4
median 1 0.9 3 10 999 100 1
risky 1 0.852 3.14 29 999 63 0.1
conservative 1. 0.937 2.87 2 999 148 8
median 2 0.9 3 1 100 999 10
The two “median 1” and “median 2” triplets are used to make a separate test on the reliability at the mission time when two experts disagree about the most credible quadrant (c2 for expert 1 and c3 for expert 2).
4.2.3.3 Summary of results
The summary table presented in appendix A-1 summarizes the results obtained with SURVBAYES. The first posterior moments and the predictive reliability at the mission time are shown.
These elements are used as a basis for discussions intended to help maintenance or design decision- maker.
In this table, we added firstly the results for the case in which data for components No. 1 and No. 2 are mixed, using a uniform type prior (case in which c1=c2=c3=c4), and secondly the results of the case in which data for component 2 only are taken (which is more informative) using the same uniform prior.
The sensitivity of the results for sample No.1 alone compared with the various prior distributions (conservative, median 1, median 2 and risky, columns 1 to 4), is high for the predicted reliability at 50000 hours which varies from 0.718 to 0.931.
The variation coefficient on p varies strongly from 5% to 43%, whereas the variation coefficient q varies little, from 39% to 55%.
The results (column 5) show that the information provided by sample N°2 alone using a uniform prior
distribution gives a more informative posterior density than the information provided by sample No.1, as demonstrated by the coefficients of variation for p (7%) and q21%).
Furthermore, the sample for component No. 1 (not very informative) mixed with the sample for component No. 2 makes very little difference to the results (column 6); it makes a small reduction to the variation coefficient on p (from 7% to 6%).
In the end, the decision maker will either make his decision based on the Bayesian evaluation of reliability at the mission time that includes all uncertainties, or he considers that he must have further feedback experience data (to attempt to reduce the variation coefficients), in which case further data must be added to the sample of exchangeable data to be considered, for example by adding a third component.
This choice will obviously depend on the fixed target reliability. For example, if it is required that the predicted reliability should be greater than 80%
at the mission time (50000h), then it would be reasonable (median scenario) to consider that component No.1 is sufficiently reliable for the mission, after seeing the predicted reliability results (column 2 or 3). This conclusion is valid even if experts disagree about which is the most credible quadrant.
Otherwise (conservative scenario), one possibility would be to attempt to reduce uncertainties in order to guarantee an optimum decision, but obviously this carries the attendant risk of grouping components that may not be identical.
5 CONCLUSION
There are several advantages in the proposed strategy for eliciting and using expert opinions. Due to the nature of the way in which parameters are set and the questions asked, the result is a simple and reliable procedure for eliciting expert’s opinions. A hierarchical analysis of credibilities can be used to modulate prior about the acceptable probability at the mission time. The choice of Beta prior distributions and a geometric combination of expert opinions makes the calculations simple and consistently takes account of information supplied by experts. Furthermore, graphics output from ELICITAE make it easy to measure the importance of low uncertainties on posterior distributions and help to check the justification for the Bayesian analysis.
The test case study that was presented shows that the main advantage of carrying out a BRA of aging is that control over uncertainties in a decision-making
process can be maintained. It is used to perform a sensitivity analysis on prior ideas and to create the data sample to be used in order to give confidence to the decision maker.
This type of BRA making use of the most general feedback experience data helps to evaluate the predicted reliability of equipment that age during operation. This evaluation provides an appropriate quantification of uncertainties about the appearance of unusual failures for which very little observation data or expert opinions are available. It is essential before making an optimum maintenance decision to the extent that the analysis provides additional help about whether or not the level of information (data, expert opinions) is sufficient, before the decision is made. The decision maker can then make a choice between obtaining more information and taking action, with full knowledge of the facts, by using a quantitative indicator representing the predicted reliability of the component evaluated making use of all available information.
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APPENDIX A: TABLES AND FIGURES
SURVBAYES
EXPERT Prior
Feedback experience data Likelihoo
d
Posterior
Mission
time t
Reliability indicator at t
Sensitivit y study BAYES theorem
yes Posterior / likelihood
consistency no
no
Figure A-1. SURVBAYES: general principles.
Figure A-2. Choice of uncertainties (low, moderate, high and very high) in case 1
Table A-1. Summary of sensitivity results obtained with the SURVBAYES software
Prior Conservative Median 1 Median 2 Risky 0(q, p) = cte 0(q, h) = cte
distribution Exchangeable
sample Component
No. 1 Component
No. 1 Component No.1 Component No.1 Component
No. 2 Component
No. 1 and 2 Expected value
of q 8.9 6.1 6.9 5.9 6.8 6.8
Standard deviation of q
4.9 3 2.7 2.4 1.4 1.4
Coefficient of
variation 55% 49% 39% 41% 21% 21%
Expected value
of p 0.718 0.907 0.888 0.931 0.885 0.900
Standard
deviation of p 0.312 0.159 0.173 0.046 0.064 0.055
Coefficient of
variation 43% 18% 19% 5% 7% 6%
Predicted reliability at
50000 h 0.718 0.907 0.888 0.931 0.885 0.901
Precision 10-3 10-3 10-3 10-3 10-3 10-3
