A relability assessment

AGIFORS 32

A RELABILITY ASSESSMENT OF A PRESSURE CONTROL UNIT FOR
ALYEMDA 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 assessed
by extending the Bayesian concept of statistical inference making use of
less 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 uniform
distribution function. The range of this distribution was decided by trail and
error using koiomogorov test repeatedly. The point estimation and interval
estimation were done to determine the parameters of distribution and Mean
Time 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

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 preventive
maintenance planning of service producing equipment such as computers,
radars, and aeroplanes. As the system ages its reliability decreases. In order to
assess and estimate this stochastic process of reliability deterioration the
method of statistical inference is currently riding a high tide of popularity in
virtually 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 the
method. The other theories of inference are based on rather restrictive
assumptions which provide solutions to a limited set of problems, where as
Bayesian method can be used and Bayesian inference is shown in Fig.2.

The statistical inference based on sampling theory is usually more
restrictive than Bayes due to exclusive use of sample data. The Bayes use of
relevant past experience, which is quantified by the prior distribution produces
more informative inferences. The Bayesian method usually requires less sample
data to achieve the same quality of inference than the method based on
sampling theory.
This is the practical motivation for using the Bayesian method in those
areas of application where sample data may be difficult to obtain.

In this paper an extension of Bayes theorem to form the likelihood
function in the process of ascertaining the reliability function on the assumed
prior making use of the rare sample data has been discussed. The method has
been applied to find out the reliability of the pressure control unit of the
pressurization system of dash 7 aircraft. The data have been obtained form the
catalogues of its performance.

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 describe
our prior belief or ignorance in the value of  by a pdf h (  ) . This is analysts
subjective judgement about the behaviour of  and should not be confused
with the so-called objective probability assessment derived from long-term
frequency approach.

Consider a random sample of observations T ,T2 …,T from f ( T ) and

 f T /   where f Ti /   is conditional probability
n
define a statistic Y = i
i h

distribution T for a given  . Then there exists a conditional pdf g ( Y/  ) for Y

given  . 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 marginal
probability 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 by

f 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 prior
pdf which expresses our subjective knowledge about the value of  before the
hard data Y become available. The g(  /Y) is the posterior pdf of given the
hard 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 (  ).

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 the
LIKELIHOOD FUNCTION of  given i , which usually written as L ( / ) to
insure its distinct interpretation apart form f (/ ) . The likelihood function is
important in Bayes the rem and is the function through which the sample data
T modify prior knowledge of  ; we can write the Bayes theorem as:

g ( / Y )  h( ) L( / Y ) (5)

or
Pr iorDistribution  Likelihood
Posterior Distribution =
M arg inalDistri bution

The statistical decision theory concerns the situation in which a decision
maker 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 prior
distribution. The decision maker combines the prior knowledge of  and
stochastic information of  and then chooses the action that minim zes the
expected 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 amcy
between the value of  and the estimate  . For this reason the loss function L
in and estimation is often assumed to be of the form

L( , )h ( )( -λ ) (6)

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 lens
itself to mathematical manipulation. It represents a second order
approximation 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 be
that 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 a
pressurization system of DASH 7 aircraft in Alyemda. This mechanism contains
and expensive sub-assembly that must be completely replaced after failure and
that management is attempting to forecast maintenance costs over the next five
years. In this situation a knowledge of the reliability and the MTTF ( Mean
Time To Failure ) would be useful. We shall use the Bayesian estimation for
evolution of these parameters. The observed failure time were 1352, 1956,
2082, 2109, 2122, 2172, flight operating hours. The data can be expressed in
flight 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 to
estimate the likelihood function L (  / Y ) and we have to select a suitable
prior h (  ) to assess the posterior distribution of  for a given
Y ( i.e T1 , T2, ..., T6 ).

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

1
and 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 )
    

- 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 parameters
which 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 above
sampling scheme without withdrawls prior to test termination, can be written
as
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 n
items are simultaneously tested until s failure occurs. Here  is rescaled total
time on test.

Now making use of equation ( 5 ), the posterior distribution of  can be
given 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-informative
prior can be obtained as follows:

s
Step 1. Let L (  / Z ) = In TT f ( i /  ) denote the log-likelihood of the
i 1

sample.

 1 L 
Step 2. Let J (  ) = ( - ) where  is the ML estimator of  .
n  z

Step 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

2 2 YS e -y
  
- 
 s
e d  dy
1 1  s 1

1
  ( s  1,  2  ) -  ( s  1, 1  ) 
 s 1

The posterior distribution ( 22 ) assumes the form

 s 1 s e -
h (  ; s, 1 ,  2 )  ( 23 )
( s  1,  2 ) - ( s  1, 1 )

S 6
Where    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
 0.01681

The 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 =450
approximately using kolmogorov test repeatedly.

Point and Interval Estimation of  .

As it has been shown earlier that the Bayesian estimator which
minimizes 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 dy
E ( / ; S, 1 ,  2 )  1

  ( S  1,  2 ) -  ( S  1, 1 ) 

 ( 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/
 0.2348

Interval 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 ) 2

And

( S 1,  2 ) - ( S  1,  ) 
Pr (    |  ; S, 1 ,  2 ) 

 ( 26 )
( S  1,  2 ) - ( S  1, 1 ) 2

Taking   0.05 the equation ( 25 ) and ( 26 ) were solved  ( lower limit )
and  ( upper limit ) were calculated as

  191.5   442.5

Then the mean time to failure ( MTTF ) for Weibull distribution for the
expected failure rate can be found out as

  1
MTTF    , refer Marts ( 1982 )

 0.02348  ( 5/4 )
= 0.2128 flight operating year

Similarly 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.

Finally the three graphs showing posterior distribution, and 8. using
equation ( 23 ) and following formula.

R ( t ) = e (t /  )

 t  -1
H(t)= ( )
 

CONCLUSIONS

This paper shows that the distribution of the failure rate can be easily
determined using insufficient test data. It also shows that the subjective prior
distribution of failure rate can be decided by Fisher information. Mean Time
To Failure ( MTTF ) and reliability variation with time can be predicted be the
use of Bayesian approach.

These data can be used for a better preventive maintenance planning of
such equipment.

REFERENCES

1. 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.

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.

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A relability assessment

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