Jaźwiński J., Smalko Z. The expert method in estimating the parameters of the triangular and beta distributions.

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THE EXPERT METHOD IN ESTIMATING

THE PARAMETERS OF THE TRIANGULAR

AND BETA DISTRIBUTIONS

Jaźwiński J., Smalko Z.

Air Force Institute of Technology, Księcia Bolesława 6, Warszawa, Poland

Abstract: Numerous significant events in the course of flying its mission should be

included into the only rarely occurring ones. Therefore, the usually insufficient statistical data base should be supplemented using the expert method. In the work the expert method has been applied to determine the parameters of the triangular and beta distribution.

1. Formulating the problem

Specialists in aircraft operation observe various events, just to mention such as the following ones:

 a failure to an aircraft while flying a mission,

 an occurrence of a prerequisite for an air accident in the course of the aircraft flying its mission,

 an occurrence of an air accident,

 failure detection in the course of the condition-checking procedure(s),  the on-time/out -of-time accomplishment of various operational procedures.

The above-mentioned, by way of example, events are random in their nature because of many and various factors that affect their occurrence. Random variables and random processes of different kinds are closely related with them. Therefore, what can be given consideration is, e.g. the number of failures in the course of an aircraft’s flight for some definite time interval, the number of flights between failures, the service life of an aircraft subassembly up to its failure, time after which the operator starts performing a task. Other events, such as: prerequisites for air accidents, air accidents themselves, failing to perform tasks, etc. can be analysed in a similar way. Events are described with the probability of their happening (e.g. the probability that an air accident will happen). A discrete random

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variable (i.e. the number of events within a given time interval) and a continuous one (i.e. time between events) are both described with probability distributions.

On the grounds of statistical information gained from experiments one can estimate parameters of distributions of different random variables and random events which occur in the course of operation.

It is practically impossible to notice and watch a large number of events such as air accidents, prerequisites for air accidents as referred to individual aircraft’s subassemblies, failures to aircraft units and / or subsystems. For that reason statistical studies do not provide enough necessary information. The efforts are therefore made to supplement it using the experts’ method. In the further considerations the expert method has been applied to determine both the parameters of the triangular distribution and beta distribution.

2. The expert method as applied to determine parameters of the

triangular distribution

2.1. Non-symmetrical triangular distribution – its functional and numerical characteristics [7]

Density. Density diagram of the distribution has been shown in fig. 1. For x < x1 and

x > x2 the density is zero. The triangle field x1, fm, x2 equals one. Hence: 1 2 m x x 2 f   (1)

Equations of density in intervals x1, xm and xm, x2 can be derived as two point (xa, ya) and

(xb, yb) form of the equation of a straight line y(x)



a



a b a b a x x x x y y y y      (2)

The density is given with the formula:



















































2 2 m 2 m 1 2 2 p m 1 1 m 1 2 1 l 1

x

for

0 f

x

for

x

2 f

f

x

for

x

2 f

f

x

for

0 f

(3)

The cumulative distribution function. The equations of the particular interval of the cumulative distribution function results from the integration of the density equations (3) in the corresponding intervals:













 











































2 2 m 2 m 1 2 2 2 m 2 2 1 2 1 m p m 1 1 m 1 2 2 1 l 1

x

for

1 F

x

for

x

F

x

for

x

F

x

for

0 F

(4)

x

2

x

m

x

1

0 f

m

f

Fig. 1. Density of the triangular distribution

It is not hard to prove that F(x2) = 1.

Intensity. Intensity is given with the formula:









 

















































2 2 m 2 p m 1 2 1 1 m 1 2 1 l 1

x

for

0 x

x

for

x

2 x

x

for

x

2 x

x

for

0

(5)

(4)

As it can be seen from the formula (5): m 2 x x 2    for xxm  



for x1 x2

Modal value. The modal value of the triangular distribution is xm (fig. 1). It corresponds

with its density value defined with the formula (2) and with its distribution function value, which according to (4) is:

 

1 2 1 m m x x x x x F   

It is essential, if

F

 

x

m



1 /

2

, or

F

 

x

m



1 /

2

, because according to that the mode is

in the interval x1,xm or xm,x2 . For

F



x

m



1 /

2 

the distribution is

symmetric.

Mediana. It should be checked if the distribution function for the modal value is greater or smaller than 1/2, because according to that another type of the formula (4) is applied. After having substituted in these formulae F = 1/2, it is obtained:

 when the mediana is in x ,1 xm interval:



 



2 x

x

Me

2 1 m 1 1







(6) and:



 

_{ }

_

2 2 m 2 m 1 m 1 m 1 2 2

x

2 x

x

Me











(7)

 when the mediana is in the interval xm,x2 .

Expected value. The expected value is given with the formula:







 





x



 

x



x



dx

x

2 x

dx

x

2 x

E

1 m 1 2 2 2 x m x 1 m 1 2 1 m x 1 x













After calculating it is obtained:

3 x x x

E_ 1 m  2 ₍₈₎

The distribution function F(E) value depends on the relation between the xm value and the

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when: m x E , that is

2 x

x

1 2 m





,

then according to (4) and (8):

 



_



_{ }



_



2 m 1 2 2 2 m 2 2 m 1 1 2 1 m p

x

9 x

x

9 x

2 x

x

E

F









(9)  when: m x E , that is

2 x

x

1 2 m





,

then according to (4) and (8):

 

_



_{ }



_

2 m 1 2 2 2 1 m l

x

9 x

x

2 x

E

F









(10) when:

2 x

x

1 2 m





,

then, as it can be proved with each of the formulae (9), (10) it is obtained: 2

1 F F_p  _l 

Variance. The random variance is given with the formula:







 





x x



 

x



x



dx x x x dx x x x x x x x S m x m m m x 2 1 2 2 2 1 1 2 1 2 2 2 2 2 1        

_

and after having computed:

18 x x x x x x x x x S 1 m 1 2 m 2 2 2 2 m 2 1 2 _      ₍₁₁₎

Standard deviation. The standard deviation is given with the formula:

18 x x x x x x x x x S 1 m 1 2 m 2 2 2 2 m 2 1 2 _      ₍₁₂₎

2.2. Estimation of the triangular distribution parameters with expert methods engaged

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The basic parameters of the triangular distribution are the lower bound x1, the upper

bound x2 and the modal value xn.

To determine parameters of the distribution x1, x2, xn we appoint n experts. Each expert

answers the three following questions:

1. What is the least value of the random variable X realization? 2. What is the greatest value of the random variable X realization?

3. What is the most frequent met value of the realization of the random variable X? We will get three sets of the experts answers: x₁_i,x₂_i,x_mi; i= 1, 2, ... , n.

The gained sets we arrange into non-decreasing sequences:

) n ( 1 ) 1 b ( 1 ) 2 ( 1 ) 1 ( 1

x

...

x

...

x



) n ( 2 ) 2 b ( 2 ) 2 ( 2 ) 1 ( 2

x

...

x

...

x



(13) ) n ( m ) m 2 b ( m ) m 1 b ( m ) 2 ( m ) 1 ( m

x

...

x

...

x

...

x



Estimation of the parameters x1, x2, xn values of the triangular distribution are determined

from the formula:

) i ( 1 1 b 1 i 1 1

x

b

1 x





 ) i ( 2 n 2 b i 2 2

x

b

n

1 x







 (14) ) i ( m m 2 b m 1 b i m 1 m 2 m

x

b

1 x







where the b1 , b2 , b1m , b2m values mean the selected, extreme experts estimations.

Taking into account the collected information the standard deviations can be determined:





2 ) i ( 1 1 i b 1 i i 1 x

x

1 b

1 _















2 ) i ( 2 2 n 2 b i 2 2 x

x

1 b

n

1 _









 (15)





2 ) i ( m m m 2 b m 1 b i m 1 m 2 xm

x

b

1 _











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Using the determined values of the standard deviation, parameters of the triangular distribution can be determined with greater likelihood. Assuming that:

1 x 1 1 1 x k x    (16) 2 x 2 2 2 x k x   

As the modal value can be accepted:











b m b b m n b m



n 1 x_m  ₁_m  ₂_m ₁_m   ₁_m (17) where: m x m

k

x

m







m x m

k

x

m







(18) m x m

Parameters x1, x2, xn of the distribution permit us to determine another functional and

numerical characteristics of the random variable of the triangular distribution.

3. The parameters of the triangular distribution applied to

determine the parameters of the beta distribution

3.1. Beta distribution [2]

In a general case the beta distribution has the following form:

 

_

_{ }



_{ }



_

 



 



1 1 1 x A B x A B ! 1 ! 1 ! 1 x f ____              



A,B



x (19)

The expected value of the random variable of the beta distribution W_B is given with the formula:

 









                 A B 2m A B W E B (20) where:

m - a modal value of the random variable W ;B













2 1 B 1 A W M m B ____        ₍₂₁₎

The normalised form of the beta distribution, i.e. x

 

0,1 has been assumed in further studies. The beta distribution takes then the following form:

 

_



_{ }



_

_x 1



₁ _x



1 ! 1 ! 1 ! 1 x f  _           ₍₂₂₎

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



     B W E (23)

Further on the beta distribution will be labelled B(, ). 3.2. Determining the parameters for the beta distribution

Comparing the expected value and the modal value for the triangular distribution with the expected value and the modal value for the beta distribution, assuming that A = x1 , B = x2 ,

we determine the parameters



and  for the distribution.

For the case when A = 0 and B = 1 the equation, which determine



and  parameters have the following form:

2 1 m         , (24)       x where: m x

m_  - is given with the formula (17);







  2 m 1

x

E

x

- is given with the formula (8).

After having solved the above equation we gain:

        x x m 1 m 2 (25)







   _     1 x x m 1 m 2

4. Conclusions

1. The presented method can be applied to determine another distributions, e.g. the Johnson distribution.

2. Effectiveness of the presented method can be checked with the Monte Carlo method.

References

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2. Jaźwiński J. Salamonowicz T., Smalko Z.: Using the beta distribution for assessment of availability and reliability of transport systems. Archives of Transport, Volume 13, 2001.

3. Jaźwiński J., Smalko Z., Żurek J.: The Monte Carlo Method as Applied to Determine Subjective Probability of Dangerous Events in Air Transport, with Expert Method Engaged. International Conference on Monte Carlo Simulation, MCS 2000, in Monte Carlo, Principality of Monaco, June 2000.

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6. Jaźwiński J., Smalko Z.: Wykorzystanie rozkładu trójkątnego w modelowaniu bezpieczeństwa systemu transportowego. Materiały VI Sympozjum Bezpieczeństwa Systemów, Kiekrz, 1996.

7. Marcinkowski J.: Rozkłady prawdopodobieństwa przydatne w rozwiązywaniu problemów transportu. Oficyna Wydawnicza Politechniki Wrocławskiej, Wrocław 1997.