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This paper is published in AGIFORS 32
A RELABILITY ASSESSMENT OF A PRESSURE CONTROL UNIT FOR ALYEMDA DASH 7 AIR CRAFT USING BAYESIAN INFERENCE

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### A relability assessment

1. 1. AGIFORS 32A RELABILITY ASSESSMENT OF A PRESSURE CONTROL UNIT FORALYEMDA DASH 7 AIR CRAFT USING BAYSEIAN INFERENCE October 4-9 1992 By: Dr Jairam Singh ADEN University Engr Mohamed salem AYEMDA Airline ABSTRACT The reliability of the pressure control unit has been assessedby extending the Bayesian concept of statistical inference making use ofless number of field observations. The Maximum Likelihood ( ML )function has been estimated by the parametric analysis of the data. The subject knowledge about the data was expressed as a uniformdistribution function. The range of this distribution was decided by trail anderror using koiomogorov test repeatedly. The point estimation and intervalestimation were done to determine the parameters of distribution and MeanTime To Failure ( MTTF ). SYMBOLS cdf Cumulative Density Function. f(  .y ) Joint pdf of  & y . f( Y ) Marginal pdf of Y. f(Y) Estimate pdf. g ( Y/  ) Conditional pdf of Y given  . h ( ) Prior of  . h(t) Hazard rate function. L ( / Y ) Likelihood function.  Failure rate a random variable ( r.v. ) 1 Lower range of failure rate.  2 Upper range of failure rate. 3 Mean range of failure rate. n Sample size. R(t) Estimate of reliability function. S Number of failure
2. 2. T Failure time for Ith failure.  Sample space. TBPI Two sided Bayesian Probability Interval.  Weibull Distribution.  Scale parameter of Weibull Distribution.  Shape parameter of Weibull Distribution.  Location parameter of Weibull Distribution. INTRODUCTION Reliability is one of the most important parameters for preventivemaintenance planning of service producing equipment such as computers,radars, and aeroplanes. As the system ages its reliability decreases. In order toassess and estimate this stochastic process of reliability deterioration themethod of statistical inference is currently riding a high tide of popularity invirtually areas of statistical application. The work of several authors, notably De finetti (1937), Jeffreys (1961),Lindley (1965), and Savage (1954), has provided a philosophical basis for themethod. The other theories of inference are based on rather restrictiveassumptions which provide solutions to a limited set of problems, where asBayesian method can be used and Bayesian inference is shown in Fig.2. The statistical inference based on sampling theory is usually morerestrictive than Bayes due to exclusive use of sample data. The Bayes use ofrelevant past experience, which is quantified by the prior distribution producesmore informative inferences. The Bayesian method usually requires less sampledata to achieve the same quality of inference than the method based onsampling theory. This is the practical motivation for using the Bayesian method in thoseareas of application where sample data may be difficult to obtain. In this paper an extension of Bayes theorem to form the likelihoodfunction in the process of ascertaining the reliability function on the assumedprior making use of the rare sample data has been discussed. The method hasbeen applied to find out the reliability of the pressure control unit of thepressurization system of dash 7 aircraft. The data have been obtained form thecatalogues of its performance.
3. 3. THE MODEL Let T be random variable ( r.v. ) with probability density function ( pdf )( T ) which is dependent on  which is another random variable. We describeour prior belief or ignorance in the value of  by a pdf h (  ) . This is analystssubjective judgement about the behaviour of  and should not be confusedwith the so-called objective probability assessment derived from long-termfrequency approach. Consider a random sample of observations T ,T2 …,T from f ( T ) and  f T /   where f Ti /   is conditional probability ndefine a statistic Y = i i hdistribution T for a given  . Then there exists a conditional pdf g ( Y/  ) for Ygiven  . The joint pdf of Y and  can be given by f (  ,Y) = h (  ) g (Y/  ) (1) And also form conditional probability concept, we have f ( , Y ) g (  /Y) = f 2 (Y ) Where f(  ,Y) is joint probability distribution and f 2 (Y) is marginalprobability distribution. Substituting f (  ,Y) from ( 1 ) we get h( / Y ) g (Y /  ) g (  /Y) = f 2 (Y)  0 (2) f 2 (Y )Where f 2 (Y) is given byf 2 (Y) =   f ( , Y )d    h( ) g (Y /  )d (3)for  as a continuous r.v. Equation ( 2 ) is simply a form of Bayes theorem. Here h (  ) is the priorpdf which expresses our subjective knowledge about the value of  before thehard data Y become available. The g(  /Y) is the posterior pdf of given thehard data Y.Since f 2 (Y) is simply a constant for a fixed Y, the posterior distribution g(  /Y)is the result of the product of g (Y/  ) and h (  ).
4. 4. g (/ )  g ( / ) h( ) (4) Given the sample data T, f ( / ) may be regarded as a function not ofi , but of  . When so regarded Fisher ( 1922 ) refers to this as theLIKELIHOOD FUNCTION of  given i , which usually written as L ( / ) toinsure its distinct interpretation apart form f (/ ) . The likelihood function isimportant in Bayes the rem and is the function through which the sample dataT modify prior knowledge of  ; we can write the Bayes theorem as: g ( / Y )  h( ) L( / Y ) (5)or Pr iorDistribution  LikelihoodPosterior Distribution = M arg inalDistri bution The statistical decision theory concerns the situation in which a decisionmaker has to make a choice from a given set of available actions ( a 1 ,....., a n )and where the loss of a given action depends upon the state of the nature which is unknown. In Bayesian decision theory,  is assumed to have a priordistribution. The decision maker combines the prior knowledge of  andstochastic information of  and then chooses the action that minim zes theexpected loss over the posterior distribution.Therefore, the decision theory will have the following steps: 1. A sampling experiment is conducted and an observable r.v., T 1 , is obtained, defined on a sample space = (T i ) such that when  is true state of nature, T i is obtained which has probability distribution f(T/  ) 2. The identification and selection of the model which describes the observed set of data i.e. the action. 3. The selection of a suitable prior distribution of  , defined on the sample space. 4. A determination of loss function L(  ,a) representing the loss incurred when action a is taken and the state of nature is  . The loss incurred in estimating  by  (   ( T ) Where T i is the observed value of ( T ) should reflect the discrep amcybetween the value of  and the estimate  . For this reason the loss function Lin and estimation is often assumed to be of the form L( , )h ( )( -λ ) (6)
5. 5. Where  is a non-negative function of the error ( -  ) .When  is one dimensional, the loss function can often be expressed as L (  ,  ) a /  -  / b (7) If b = 2, the loss function is a squared error loss function which lensitself to mathematical manipulation. It represents a second orderapproximation of a more general loss function  ( -  ) .The Bayes risk will be given by R ( ,  )  a E ( -  ) 2 (8) The Bayes estimator for any specified prior distribution h (  ) , will bethat estimator that minimizes the posterior risk given by    E a ( -  ) 2 / Y   a ( -  ) 2 g ( -  ) d (9)By adding and subtracting E ( /Y ) and simplifying; we get   E a (  -  ) 2 Y  a  - E (/Y) 2  a Var ( /Y ) ( 10 )Which is clearly minimized when   E ( /Y )     g ( /Y )d ( 11 )The minimum posterior risk is  ( Y ) a Var (/Y ) ( 12 ) ILLUSTRATIVE EXAMPLE Six failure time were observed for a pressure control unit in apressurization system of DASH 7 aircraft in Alyemda. This mechanism containsand expensive sub-assembly that must be completely replaced after failure andthat management is attempting to forecast maintenance costs over the next fiveyears. In this situation a knowledge of the reliability and the MTTF ( MeanTime To Failure ) would be useful. We shall use the Bayesian estimation forevolution of these parameters. The observed failure time were 1352, 1956,2082, 2109, 2122, 2172, flight operating hours. The data can be expressed inflight operating years as 0.1543, 0.2232, 0.2376, 0.2407, 0.2422, 0.2479.Analysis Using Bayes theorem expressed in equation ( 5 ) we shall have toestimate the likelihood function L (  / Y ) and we have to select a suitableprior h (  ) to assess the posterior distribution of  for a given Y ( i.e T1 , T2, ..., T6 ).
6. 6. Determination of Likelihood Function Since it is a typical variable life test data from field operation, a good estimation of the range of failure rate can be made for the pneumatic pressure control unit as 5 10 -6 to1700 10 -6 failures / hours, according to Green (1978). Therefore, we take the range as Lower range 1  5  10 -6 f/hr = 0.0438 f/Year Upper range 2  17  10 -4 f/h = 14.892 f/year 1   2 5  1700 Mean range 3    10 -6 f/hr 2 2 = 7.46 f/year. By applying kolmogrov Test, it reflected that data do not conform to exponential distribution. In order to determine the likelihood function, we shall have to identify a model which is represented by the observed data. For this a non-parametric estimation of hazard rate h(t ), reliability R(t )and probability density density function  f (t ) versus time, were plotted using the following expressions (Blom/1958). 1 h(t i )  i  1, 2, .. (n - 1 ) ) 13 ( (n - i  0.625 ) (t i 1- ti ) n - n1  0625 R(t i )  i  1, 2,... (n) ( 14 ) n  0.25 1and f(t i )  i - 1, 2 .. ( n - 1 ) ( 15 ) ( n  0.25 )(t i 1 - t i ) These graphs are shown in fig. 3,4,5, comparing these graphs with standard theoretical graphs, the most likely similar distribution turned out to be a Weibull distribution with shape parameter B=4.A three parameter  (  ,  ,  ) will become a two parameter Weibull model  (  ,  ) under guaranteed life test i.e the location parameter    . Here  is scale parameter. Therefore the life test data are independent random variables with density function  t  t  f (t; ,  )  ( )  -1 exp  - ( )  ( 16 )     
7. 7. - 1/The equation ( 16 ) can be reparameterized by letting    we get f ( t;  ,  )    t  -1 exp (-  t  ) ( 17 ) This version of the weibull distribution separates the two parameterswhich simplifies the further manipulation and is referred to  , ( ,  ) distribution. If the failure time T, has a weibull  , ( ,  ) distribution, then T  follows, an  (  ) distribution. Soland(1969) gives the likelihood function in terms of  , ( ,  ) with pdf f ( t ),as follows if z contains the information obtained from the life test.   TT f (t i )   TT 1 - f (t i )  s n L (Z )  n 1  i s 1  ( 18 )     Where F ( t ) is the cdf of the failure time T. Using equation ( 17 ) the likelihood function corresponding to abovesampling scheme without withdrawls prior to test termination, can be writtenas s s s  s n  L ( / Z )    ( TT n 1 t i ) exp  -  (   t i  i 1 t  i  s 1 i )  ( 19 )  s  s   -1 exp ( -  ) Where s  i 1 Ti   ( n - s ) TS Which represents the usual Type II/item censored situation in which nitems are simultaneously tested until s failure occurs. Here  is rescaled totaltime on test. Now making use of equation ( 5 ), the posterior distribution of  can begiven by S e - h (  ) g ( /Y )     e h (  ) d S -  Selection of prior Distribution h (  ) : According to Box and T i a  ( 1973 )and approximate non-informativeprior can be obtained as follows: sStep 1. Let L (  / Z ) = In TT f ( i /  ) denote the log-likelihood of the i 1 sample.
8. 8.  1 L Step 2. Let J (  ) = ( - ) where  is the ML estimator of  . n  zStep 3. The approximate non-informative prior for  is given by h (  ) J 1/2 (  ) . This is known as Jeffrys rule ( 1961 ). Using equation ( 19 ), the joint probability distribution becomes f ( t;  ,   s s   -1 e -Where is the number of failure. L (  / T )  S In   S In   (  - 1 ) In  -  2  L S L S   -  ,  -    2 2 L - S  0 gines   ( M L estimater )   1 S - 1 J( )  |- ( - 2 |    S  - 2 Therefore, non-informative prior for 1 h ( )  - , hence it is locally uniform.  Therefore taking uniform prior according to Harris and Singpurwalla (1968 )  1  1     2 h (  ; 1 ,  2 )    2  1 ( 21 ) 0  else where Substituting equation ( 21 ) into the posterior distribution becomes  s e - h ( / ; s, , )  ( 22 )  1 -e dLetting Y   The denominator of ( 22 ) becomes
9. 9. 2 2 YS e -y    -   s e d  dy 1 1  s 1  1   ( s  1,  2  ) -  ( s  1, 1  )   s 1The posterior distribution ( 22 ) assumes the form  s 1 s e - h (  ; s, 1 ,  2 )  ( 23 ) ( s  1,  2 ) - ( s  1, 1 ) S 6Where    i 1 Ti ,   4 as a single has used without replacement T14  T24 T34  T44  T54  T64 ( 0.1583 ) 4  ( 0.2232 ) 4  ( 0.2376 ) 4  (0.2407 ) 4  ( 0.2422 ) 4  ( 0.2479 ) 4 0.01681The ML estimator gives  S 6     356.93  0.01681 = 357 failure / operating year.By trail and error, we ascertain the limiting values of 1 = 175 and  2 =450approximately using kolmogorov test repeatedly.Point and Interval Estimation of  . As it has been shown earlier that the Bayesian estimator whichminimizes the squared error loss is expected value of the posterior distribution. 2    S 1 S e - d  E ( / ; s, 1 ,  2 )  1  ( S  1,  2 ) -  ( S  1, 1 )Letting y =  2   y S1 e -y dyE ( / ; S, 1 ,  2 )  1   ( S  1,  2 ) -  ( S  1, 1 ) 
10. 10.  ( S  2,  2 ) -  ( S  2, 1 )  ( 24 )    ( S  1,  2 ) -  ( S  1, 1 ) This is incomplete gamma function which can be readily evaluated.Hence E ( / ; s  6, 1 ,  2 )  329.0646 failure/op erating year.  m  m  ( -m  1/  0.2348Interval estimation A symmetric 100 ( 1 -  )  two sided Bayesian probability interval (TBPI ) for  (  and  ) can be found out as follows ( S  1,  ) - ( S  1, 1 ) Pr (    |  ; S, 1 ,  2 )   ( 25 ) ( S  1,  2 ) - ( S  1, 1 ) 2And ( S 1,  2 ) - ( S  1,  ) Pr (    |  ; S, 1 ,  2 )    ( 26 ) ( S  1,  2 ) - ( S  1, 1 ) 2Taking   0.05 the equation ( 25 ) and ( 26 ) were solved  ( lower limit )and  ( upper limit ) were calculated as   191.5   442.5Then the mean time to failure ( MTTF ) for Weibull distribution for theexpected failure rate can be found out as   1 MTTF    , refer Marts ( 1982 )   0.02348  ( 5/4 ) = 0.2128 flight operating yearSimilarly for 95% TBPI, of the pressure control unit cab be calculated for   0,2179 and    0,2688 as MTTF = 0,1975 flight operating year. MTTF = 0.2436 flight operating year.
11. 11. Finally the three graphs showing posterior distribution, and 8. usingequation ( 23 ) and following formula.  R ( t ) = e (t /  )  t  -1 H(t)= ( )   CONCLUSIONSThis paper shows that the distribution of the failure rate can be easilydetermined using insufficient test data. It also shows that the subjective priordistribution of failure rate can be decided by Fisher information. Mean TimeTo Failure ( MTTF ) and reliability variation with time can be predicted be theuse of Bayesian approach. These data can be used for a better preventive maintenance planning ofsuch equipment. REFERENCES1. Blom, G. ( 1958 ). STATISCAL ESTIMATES AND TRANSFORMED BETA VARIABLE.2. Box, G.E.P. & Tiao, G.C. ( 1973 ). BAYESIAN INFERNCE IN STATISTICAL ANALYSIS.3. De Finnetti, B. ( 1937 ). STUDIES, IN SUNECTIVE PROBABILITY ( English Translation by H.E. kyburg , Jr & H. E. Smokler ( 1964 ), Wiley New York PP. 93 – 158.4. Fisher, R.A. ( 1922 ). ON THE MATHEMATICAL FOUNATION OF THEORETICAL STATICS. Fhil. Tr Roy. Soc. Series A, vol. 222 pp. 308 .5. Green, A.E. & Hourne, A.I. ( 1978 ). RELIABILITY TRCHNOLOGY John Wiley & Sonc. Pp. 535 – 540.6. Harris, C.M. & Singpurwalla, N.D. ( 1968 ). LIFE DISTRIBUTION DERIVED FROM STOCHASTIC HAZARD FUNCTIONS. IEEE Transactions on Reliability, vol. R – 17 pp. 70 – 79.7. Jeffrey, H. ( 1961 ). THEORY OF PROBABILITY ( Third Edition ) Claveudon Press, Oxford.
12. 12. 8. Lindley D.V. ( 1970 ). MAKING DECISTION. Wiley Interscience New York.9. Martz, H.F. & Waller, R.A. ( 1982). BAYESIAN RELIABILITY ANALYSIS. John Wiley & Sonc Inc. pp. 89 – 90.10. Solaved, R.M. ( 1969 ). BAYESIAN ANALYSIS OF WEIBULL PROCESS WITH UNKNOWN SCALE AND SHAPE PARAMETERS. IEEE Transaction on Reliability, vol. R – 18. pp. 181 – 184.11. Savage, L.T. ( 1954 ).THE FOUNATION OF STATISTICS. Wiley New York. ***********************