Probabilistic modeling and stochastic process investigations to provide measures of quality of performance and reliability for systems engineering - Chebyshev approximatio

Cryer, J. D.

Leadbetter, M. R.

English

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System K e l i a b i l i t y  rechnology , 
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Techn ica l  l i epor t  No. 1 
ON CERTAIN FUNCTIONALS OF NORMAL PROCESSES 
M. R. L e a d b e t t e r  and J. D. Cryer  
-I 
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Prepa red  flor 
O f f i c e  o f  R e l i a b i l i t y  and  Q u a l i t y  Assurance  
N a t i o n a l  Aeronau t i c s  and Space A d m i n i s t r a t i o n  
Washington, D. C .  20546 
Under C o n t r a c t  NASw-905 
https://ntrs.nasa.gov/search.jsp?R=19660004216 2020-03-16T21:32:39+00:00Z
System R e l i a b i l i t y  Technology 
Technica l  Repor t  No. 1 
F _ ,  . I .  * - = -  
*- 
! ON CERTAIN FUNCTIONALS OF NORMAL PROCESSES 
~ _C" ,< -- 
by  
M. R. L e a d b e t t e r  and J. D. Cryer  
. -  
September 1964 
Prepared  f o r  
O f f i c e  of R e l i a b i l i t y  and Q u a l i t y  Assurance  
N a t i o n a l  Aeronaut ics  and Space A d m i n i s t r a t i o n  
Washington, D. C. 20546 
Under Con t rac t  NASw-905 
i i  
FOREWORD 
L -  
~ . -  
This report is one of  a series of technical reports issued by RTI (Research 
Triangle Institute), Durham, North Carolina, under NASA (National Aeronautics and 
Space Administration) Contract NASw-905, "Development of Reliability Methodology 
for Systems Engineering". 
J .  E. Condon, Director, ORQA (Office of Reliability and Quality Assurance) NASA 
Headquarters, Washington, D. C. 
The contract is administered under the direction of 
The major objective of this contract is to apply probabilistic modeling 
techniques developed at RTI under a previous NASA contract to a NASA in-house 
R and D system and to conduct research necessary to further develop techniques 
appropriate to probabilistic modeling. A complex 250 volt-ampere static inverter 
under design and development by the Astrionics Laboratory of MSFC (Marshall Space 
Flight Center) was selected as a representative NASA system for application and 
demonstration of the techniques, and the majority of the technical reports in this 
series are devoted to documenting results of this effort. The additional research, 
both basic and applied in nature, on general probabilistic modeling is documented 
in several specific reports also included in this series. 
The effort under this contract began in April, 1964 and will continue for a 
period of approximately two years. The studies are being performed jointly in 
the Institute's Solid State Laboratory and Statistics Research Division under the 
general direction of Dr. R. M. Burger with W. S. Thompson serving as project 
leader. 
iii 
PREFACE 
T h i s  r e p o r t  i s  t h e  f i r s t  t e c h n i c a l  r e p o r t  of t h e  s e r i e s  t o  be i s s u e d  under  
C o n t r a c t  NASw-905 and i s  devo ted  s p e c i f i c a l l y  t o  documenting some t h e o r e t i c a l  
r e s u l t s  on p r o b a b i l i s t i c  modeling o b t a i n e d  i n  t h e  b a s i c  r e s e a r c h  e f f o r t  unde r  t h e  
c o n t r a c t .  The b a s i c  r e s e a r c h  s t u d i e s  c o n s i s t  ma in ly  of  i n v e s t i g a t i o n s  i n  c u r v e ,  
l e v e l  and z e r o  c r o s s i n g s  by  c e r t a i n  normal s t o c h a s t i c  p r o c e s s e s  ( b o t h  s t a t i o n a r y  
and n o n - s t a t i o n a r y ) .  
performance and t h e  r e l i a b i l i t y  o f  c e r t a i n  complex sys t ems .  In t h i s  r e p o r t ,  a 
number of such measures  are  d i s c u s s e d .  These i n c l u d e  t h e  t i m e  which t h e  p r o c e s s  
of  i n t e r e s t  spends o u t s i d e  g i v e n  ( u n d e s i r a b l e )  levels  and r e l a t e d  random v a r i a b l e s .  
Such i n v e s t i g a t i o n s  p r o v i d e  measures  of t h e  q u a l i t y  of  
iv 
CONTENTS 
S e c t i o n  
1 System Performance I n d i c e s  
2 Mean and Var iance  of t h e  Z - s t a t i s t i c s  
3 S p e c i a l  Cases 
4 
n 
Computat ional  Aspec ts  and Asymptotic Formulae 
5 Conclus ion  
REFERENCES 
Page 
1 
2 
6 
6 
7 
9 
On Certain Functionals of Normal Processes 
. -  
I 
1. System Performance Indices 
When considering the performance quality or reliability of complex systems, 
it is sometimes convenient to define certain performance "indices" or "measures" 
based on the characteristics of a stochastic process associated with the system. 
In some cases, the indices and their statistical properties are very simple, for 
example, when we are concerned with a particular property of the system at one 
instant of time. On the other extreme, we may be interested in properties of the 
system for all values of time during some operational period. For example, in a 
missile system, the angular error in rocket attitude can be considered as a 
stochastic process which, for good performance, should be kept small during the 
entire mission period. 
Suppose then that we have such a stochastic process x(t) of interest and that 
for good performance x(t) should never become "too large". Specifically, suppose 
that there is a known function u(t) such that for good performance x(t) should 
always be kept less than u(t). Let h be a function which is zero for negative 
arguments and strictly positive for positive arguments. Define 
(1.1) 
T 
Z = - 1 J h[x(t) - u(t)]dt . 
T O  
(We assume that the behavior of h is  such that, with probability one, the inte- 
gral will exist.) 
Now the event (Z=O] is equivalent to the event (x(t) 5 u(t), t€[O,T]} (if 
x(t) is continuous), and hence 
P(Z=O} = P(x(t) <_ u(t), tc[O.,T]I 
- 2 -  
. -  
and Chebyshev bounds on P[Z=O} (using the mean and variance of Z) give bounds on 
P{x(t) 5 u(t), tc[O,T]} . This latter quantity represents, of course, the relia- 
bility, or probability of a successful mission from this point of view. 
We wish to investigate forms of the function h which lead to tractable 
Chebyshev bounds. One particular, amenable choice for h is 
( xn , i f x 1 0  
h(x) = hn(x) = 
We then write Z for Z with the corresponding h as integrand. Note that Z is 
the proportion of time which the process spends above the curve u(t) on the inter- 
val [O,T] and TZ1 is the - which the process cuts off above the curve in the 
same interval. 
n n 0 
The Z ' s  describe excursions of x(t) above u(t) in various ways - for example, n 
zO Z2, . . .  do. 1' Z takes no account of the size of such an excursion whereas Z 
and Z have been considered previously in Leadbetter [1963] and Cryer [1963] for 
a normal, stationary process x(t). The generalization of these results to the Z 
statistics is the purpose of this report. 
0 
1 
n 
2. Mean and Variance of the Zn-- statisrics 
Let x(t) be a normal stationary stochastic process with zero mean and co- 
variance function r('c), assumed such that, with probability one, the sample 
functions are everywhere continuous. Sufficient conditions for this are given, 
for example, in Belaev [1961] .  
For  convenience, we use  the following notation: 
- 3 -  
_ -  
We have 
= r(.r)/02 
(2n)1/2 
2 - x 12  
e , the normal den 1 - ity fun tion, 
X 
= @(t)dt, the normal distribution function, 
-co 
By the definition of hn we find 
Y 
W 
- a n [x-vtIn Q(x)dx 
t V 
and hence 
( 2 . 1 )  
n T m  
T 
0 J J [x-vtIn @(x)dx dt , e[zn] = - 
Using the binomial expansion for [x-v(t)] n the integral of the form 
al 
[x-cIn @(x)dx may be evaluated as a finite sum of incomplete garmna functions 
C 
This would give a useful form for computing purposes. 
For the variance of Z we note that n’ 
- 4 -  
T T  . T T  
T T  
Now 
where Q(x,y; p) is the standardized bivariate normal density with correlation 
coefficient p=p(t-s) . From Cramer [1946] p. 290,  we have the expansion 1 
c -  
where 
dx 
Using this in ( 2 . 3 )  we obtain 
If 1 < - j 5 n, repeated integration by parts gives 
a3 
(x-v)n-j@(x)dx (j+l) j n! 
a3 
(n-j)! J (x-v)~ @ (x)dx = (-1) 
V V 
and if j > n , 
a3 ( j + l )  (j -n> 
J (x-v)" @ (x)dx = (-l)n n! [ - @  (v) 3 . 
V 
J - 5 -  
Hence ( 2 . 4 )  may be written 
co co 
j n-j n 
j=O 
2n( y €L [+I2 J (x-v,) @(x)dx ./ (y-vsIn-j@(y)dy ( 2 . 5 )  S[yn(t)yn(s)l = 0 
V t S j! (n-j). V 
Note that the first term (j,O) is 
m j  (j-n) (j-n) 
+ c 5 (n!>2 O(vt> O(vs> . 
j=n+1 3 .  
Substituting this into equation ( 2 . 2 )  gives the final result 
where it is understood that the first summation does not appear if n = 0 . 
- 6 -  
3. Special Cases 
a. If u(t) is in fact a constant level u(t) = u ,  for all t, then putting 
v = u / a  we have 
(3.1) 
a3 e[Zn] = 0 n 1 (x-v)n@(x)dx, and 
V 
(3.2) 
Var zn = 20 
T 2 [ J  (x-v) @(x)dx] 1 (T-?)pj(T)dT 
co n- j 1 
j=l j ! [ (n-j) ! ]  v 
n 
( T) [ c 2n n! 2 
0 2 
where use is made of the fact that for an even function f(x) 
T T  T 
J J f(t-s)ds dt = 2 J (T-T)f(T)dT . 
0 0  0 
b .  
c. The results for n = 1, u(t) = u constant (in addition to other versions 
The cases n = 0 and n = 1 (u(t) not constant) were obtained in Cryer [1963] .  
of the problem) were reported in Leadbetter [1963] . 
d. The case n = 0 and u(t) = u constant is well known; see, e.g., Siddiqui [1961] .  
4 .  Computational Aspects and Asymptotic Formulae 
Calculation of the mean, e[Zn], can be carried out, as already noted, using 
tables of the incomplete gamma function. 
needed t o  obtain the final integration over t, although in the special cases where 
u(t) = u ,  a constant, the final integration evsr t is t r i v i a l ;  
Presumably numerical methods would be 
The formula for the variance is always convergent (if u(t) is at least bounded 
on [ O , T ] ) ,  but the number of terms which must be computed for a good approximation 
- 7 -  
varies with the form of the covariance function r('r). 
A typical correlation function is r(z) = exp(-0llt I ) ,  i.e. , the covariance 
for a normal Markov process. If v, i.e. u(t), is constant and in the range 
1 to 5. Leadbetter [1963] has shown that for n = 1 and T not too small, five terms 
of the infinite series for Var Z are a good approximation. Also, for large T 
we have the approximation 
1 
where T = z.(v) is the j - t h  tetrachoric function (see, e.g., Pearson [1931]). 
j J  
Similar limiting results hold for other covariance functions which are 
integrable over (0,a). For non-integrable covariance functions the results can 
be simpler. For example, if 
then 
2 co n-1 
Var Z - 2 0 n[ (x-v) @(x)dx] A log T/T . 2n n 
V 
5. Conclusion 
The above investigation has been concerned with the first two moments of 
the Z -statistics, for n = 0, 1, 2 ... . A s  noted, for each fixed n we may 
obtain an upper bound to the probability that the process will never exceed a 
given curve u(t) in the time period 0 <_ t 5 T. 
n 
Specifically we have 
using a one-sided Chebyshev inequality. 
. 
- 8 -  
If it is very desirable that the process never exceed u(t) in the time 
period T, or more specifically, if exceedance of the curve by the process 
induces system failure, then the left-hand side of Equation (5.1) is just 
the probability of correct performance. That is, Equation (5.1) provides 
an upper bound for the reliability of the system when this particular failure 
mode is considered. Equations (2.1) and (2 .6)  provide the appropriate quan- 
tities to be inserted on the right of (5.1). For each fixed n, we obtain 
such a bound. It is not known at present whether some of these bounds are 
sharper than others - and this is an interesting question. Finally, we 
note that these techniques lead to upper bounds for system reliability. The 
calculation of lower bounds may be accomplished by different methods, as 
described, for example, in Leadbetter and Lewis  [1962]. 
e 
- 9  - 
REFERENCES 
Belaev, Yu K. [1961], "Continuity and HGlder's conditions for sample functions 
of stationary gaussian processes," Proc. 4th Berkeley Symp., 2, pp. 23-33. - 
f 
Cramer, H. [1946], Mathematical Methods of Statistics, Princeton University 
Press. 
Cryer, J. D. [1963] , "On excursions outside arbitrary curves by stationary 
normal stochastic processes," Research Triangle Institute Technical Report 
GU-68 NO. 6. 
Leadbetter, M. R. [1963], "On the normal stationary process - areas outside 
given levels," J. Roy. Stat. S O C . ,  By Z N o .  1, pp. 189-194. 
Leadbetter, M. R. and Lewis, J. B. [1962], ffReliability of a linear system 
with random inputs - an example of the use of the spectral moments," 
Research Triangle Institute Working Paper GU-68 No. 5. 
Pearson, K. [1931], Tables for Statisticians and Biometricians, Vol. 11, 
Cambridge University Press. 
Siddiqui, M. M. [1961], "Some properties of the empirical distribution 
function of a random process,"J. Res. Nat'l. Bur. Stand., 65, pp. 117-127. 


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