PSI - Issue 40

V.N. Syzrantsev et al. / Procedia Structural Integrity 40 (2022) 411–417 Author name / Structural Integrity Procedia 00 (2022) 000 – 000

413 3

Fig. 1. Gas pipeline stress variations during a year of operation

To recover the unknown stress distribution  density function f  (  ) , we use the Parzen-Rosenblatt estimate developed in the framework of the nonparametric statistics theory by Parzen (1962) and Rozenblatt (1956):

n

    h

   ,

1

 

  i 1

i

( ) 

(2)

f

K





 n h

n

n

where: K (…) is the kernel function; h n is the smoothing parameter (Parzen-Rosenblatt bandwidth). The efficiency of using expression (2) to recover function f  (  ) on the basis of the available sample i 

1, i n  is

n h h  at which the information functional [Syzrantsev (2019), Simakhin (2006),

determined by the parameter value * n

Syzrantseva (2017)]:

  

    

1 n

1

1

n

n



(3)





i

j





( ) ln ( ) ( ) J h K f d     

ln

K



  

 

 





n



1 n h

h

 

1

i

j i n 



n

reaches the maximum.

( ) * 

max

.

(4)

n J h

The implementation of this condition means [13], that ( ) ( ) K f    . To date, more than a dozen of different kernel functions have been proposed [Syzrantseva (2017)]. The analysis of their use in processing samples of various random values, including those the density function of which is polymodal, shows [Syzrantseva (2017)] that at the values of n  300 the functional values of * ( ) n J h become almost the same. At the same time, the experience of solving the problem (4) by numerical methods has showed that for a number of kernel functions functional J ( h n ) is not smooth and has discontinuity points. We use a kernel function with a normal kernel [Syzrantsev (2019), Simakhin (2006)]:



   

2

  

  

  

  

1 exp 0,5 2     

  

  

K



,

(5)

i

i

h

h



n

n