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Some simulation results as to weakly correlated processes
Mathematics and Computers in Simulation 29 (1987) 191-208 North-Holland
SOME SIMULATION J. vom SCHEIDT Ingenieurhochschule
RESULTS
AS TO WEAKLY CORRELATED
191
PROCESSES
and B. FELLENBERG Zwickau, Zwickau, DDR 9.540, German Dem. Rep.
The investigation of linear functionals of weakly correlated processes which arise e.g. from the solutions of random differential equation problems are dealt with. A brief survey of the theory of weakly correlated processes is given, and especially, approximations of moments and distributions of these functionals are deduced which depend on the correlation length. Furthermore, functions of such functionals are considered. In the main part these results are confirmed by simulation. For this, a method of simulation of weakly correlated processes is given and the simulation results are used for the comparison of the theoretically-calculated with the statistically-determined values of some characteristics. Several examples of simulations are added.
1. Introduction The theoretical investigation of linear functionals of weakly correlated processes f,(x, w), x E R, leads to approximations of moments and distributions which depend on the correlation length 6 of the process (cf. [4,5]). The definition of weak correlations is connected with the interpretation that f,(x, w) and f,( y, w) are independent if the distance of x and y exceeds 6 > 0. The exact definition introduced in [4] is given by a separation of all moments (n~_,f,( xi)). There are many problems and technological applications having linear functionals of such processes as solutions, e.g. random vibrations, random wave propagations or random heat equations investigated in [l-3]. Thereby the theoretical results of the theory of weakly correlated processes are used for the determination of some characteristics like moments and probabilities which are of practical interest. In some cases comparisons with estimations from measured results were possible and good agreement could be stated. Here, we will confirm generally the theoretically obtained results by a method of simulation of weakly correlated processes and subsequently by a statistical analysis of the simulation results. The special aim will be the corroboration of the latest results concerning higher order approximations.
2. Weakly correlated processes and a method of simulation At first, we repeat the definition of weakly correlated processes. Let 1. 1 denote the norm on R and let the mean of a stochastic process be denoted by ( . ). A stochastic process f,( X, w), x E D c R with (f,(x)) = 0 is called weakly correlated with correlation length e if the relation
037%4754/87/$3.50
0 1987, Elsevier Science Publishers B.V. (North-Holland)
192
J. vom Scheidt, B. Fellenberg / Some simulation results
is satisfied for all kth moments, k = 2, 3,. . . , where I = (1, 2,. . . , k} and {xi, i E Ii}, p with Uy=,1, = I denote the so-called maximally e-neighbouring subsets of { xl, i E I}. j=l >***> The subset {xi, i E Ij} is maximally r-neighbouring if a permutation ();;;;.:%,) of Ii = { i,, . . . , ir } existswith (x~~-x~~+~]
=0
and
(p,“) =u”(u,)=u,”
exist. Then it follows immediately L(x,
from the independence
u) = b,(+,(o)
+ $(x)P,+,(~)
where u,=(~+(p/N)(j3-a),
p=O ,..., x - up
b,(x) =
)
up+1 is a weakly correlated
for k=2,
ap
3,..., that the special process on D = [CY,p]
for x E [up,
ap+l]y
(1)
N,
z;,(x) = 1 -l&x),
p=O,...,N-1
process having the correlation
length
(2)
E = 2( p - Cy)/N. Its correlation
function
is given, for x E [up, a,,,],
by
&(x3 Y) = u&M(Y)> ~,W,-l~Yb~~
I
=
b,Wp(Y)a,2
[Up-l, a,], b,bPp(Yb;+l~ Y E [up’ up+11 YE
+
Y
[a,+,, a,+,],
~,b)b,,l(YbJ,z,l~
YE
0,
otherwise.
(3)
Further, we will investigate three cases of the distribution of the random variables p,(w): (i) normal, (ii) uniform and (iii) logarithmic normal distribution. Thereby, we use for simulation the following methods. The generation of independent uniformly distributed random variables on [OJ] based on the algorithm ik+i = (53,
+ 645497) mod 999997,
TJ~+~= q,+,/999997 where qk+l is a realization. uniformly distributed random 5 = a,/~
and
YjO-C 999997,
Then, the number p k+l = 2~j~+~s-s gives a realization variable on t-s, s]. Furthermore, the number
of a
cos(2.lTn,),
is a realization of a zero-mean Gaussian random variable with variance ai, where ql, q2 denote two independent realizations of a uniformly distributed random variable on (0, 11. Finally, the formula 4 = exp(5) leads to a realization
- exp( u,2/2) of a centralized
logarithmic
normally
distributed
variable.
J. vom Scheidt, B. Fellenberg / Some simulation results
193
3. Distribution and moments of linear functionals Now, we consider the random variables r,,(w) = /‘F;(x)f,(x. a
j=l,2
a> dx,
(4)
being linear functionals of the weakly correlated process fc(x, w). The functions E;(x) are assumed to be continuously differentiable on D = [a, ,8] and fi(x, w) possesses continuous sample functions almost sure as well as (]f,(x)]~)~c,
forp=l,2,....
Then, it is possible to deduce expansions relative to E of kth moments (H,“=,Y, c) and of density functions of ( rt,, . . . , r,,) (cf. [4,5]). This has been done by a straightforward consideration of the moments and their integration domains as well as by using two approaches to the characteristic function of random vectors and a generalized Gram-Charlier-series. We give here only the results which will be confirmed by simulation. They depend on some characteristics “A, given below. Firstly, we have the moments up to the 4th order, j, k E { 1, 2)) (rjc)
=
O,
(54
(y-&
=*A,(E;F,)c+*A,(E;Fk)~*+o(~*),
(~JY~~)
=3A,(E;FF,Y)~2 + o(E*)
(5:)
cw
for p + q = 3,
= 3(2A,jF,2)~2 E* + [62Al(~2)2A2(~2)
+ [32A1(~F~)2A2(~2)
+473(q4)]~3
+ 32A,(F,2)2A2(F,Fk)
(54
+ o(c3),
+4A=,(F,3F,)]c3
for j#k,
+o(c3) ($&)
(5c)
(54
= [ZA,( F,*)‘A~( Ft) + 2(2A,( k;F,))*]c* +[*A,(q*)*A,(F,2)
+ZA,(F;)‘A,(<*)
+4*A,(qFk)‘A2(F,Fk)
+4A=,(q*~,t)]c3+
o(c’>
for j#
k.
(5f)
Secondly, the approximation of the density function of rjr( w) can be written as
,ff&) c+o(c) , j 24(2A1,j) 72(2AI,j) 1 i (64
2A2, + -H*(Z) j
22Al,
+
4z3j ’
*H,(E)+
(3A2,j !2
194
J. vom Scheidt, B. Fellenberg / Some simulation results
where
and Hk(x)
denote
the Chebyshev-Hermite-polynomials.
I&(x)=x, H,(x)
H2(x)=x2-1,
&(x)=x3-33x,
= x5 - 10x3 + 15x,
The two-dimensional PjkcU1,
density
u2)
function
=Pjk,l(U1,
Especially,
and
it yields H,(X)
= 1,
I&(x)=x4-6x2+3
H,(x)=x6-15x4+45x2-15.
of (rj,(o),
~~,(a)) possesses
the form
u2)
*l+
c k,+k,=3
i
3;q2( ~+Pl~~*-P*) .,,+$=3
(4
3A,(
-PI)!@2
-P2)!
.
qqp)
PI!P2!
k,-p:> 0
6-b) where k,, pr = 0, 1,. . . i.il=zlI/~, E71 =
ii,=
E;/\IB,,,
4
=
(-42%
(-424
+
+
Q42v[~%(w22
B,,E;)/(B,,B,,
-
-
B:2)y2>
g2y2,
Bjk = 2A,( E;F’) and the first approximation Pjk,l(%~
‘2)
=
Pjk,l is given by 1 2m( B,,B22 - B:,)l12 exp
B,,u,~ - 2 B,,u,u, 4w322
-
+ B,,u; %-2)
.
J. vom Scheidt, B. Fellenberg / Some simulation results
195
Thirdly, the characteristics pA, can be deduced in the following form. Defining K,(x) = { Y : 1x - y 1 < 6 } and using the correlation function R,( x, y) of the weakly correlated process we introduce the expansion
y) dy = a(x)r
IK,(x)
R,(x,
+ b(x)e2
+ o(e”).
Then it follows 2~,(~&)
= j’<(x)F,(x)a(x)
dx,
where R,(x,
Y) dY
1
for 24E [0, l]
.X=C-+
and u(u, p> = FrI& ‘J e K,(x)n[a,Pl
R,(x,
Y) dY
I x=p+ru
for UE [-l,O].
Furthermore, it yields
with
and (11) with
- UXx)L(Y*)>(f,(Y,)f,(y,))]
- <.L(x).L(Y2)>(f,(Y&(Y3)) dy,
dy2
dy3,
where W(x)
= ((x,
Y, ,...,
Y,)c[~,
p]““:(x,
y, ,...,
y,)c-neighbouring),
q=2,
3.
196
J. vom Scheidt, B. Fellenberg / Some simulation results
4. Distribution and moments of random functions Besides the consideration of linear functionals it is also possible to make expansions of functions of functionals. We consider here some selected results for two functions of two functionals. General results can be found in [5]. Let d,, d, be functions of (yr, vz) having the representation, k = 1, 2, 2 d/h,,
~2)
=
4,
+
2
c 4,,~a CZ=l
+
c a,b=l
d,,,oay,
(12) where r > 3/2 and g, is bounded on K,(O) for S > 0. Furthermore, we assume the existence of all moments of dkr( w) A d,(r,,, Yap), i.e. ( 1d,, 1”) < cp < 00 for k = 1, 2 and all p. Then, we consider especially the difference z,,J w) = dk( o) - d,,, and the 1st and 2nd moments are given by (13a)
n,b=l
+
i n,b=l
dj od/c,b2A2(F,4)
+
’
abdkcd
+
Cdj >
3
+
where here and in the following, moment can be written as (Cd, - (dj>)(d,
B,cBbd
+
B,dBbc)
the abbreviation
- (dk)))
(dj
, abdk 7c +
d/c,abdj,c)3Az(E’bF,E)
dj,abcdk,d + dk,a6cdj,d)
a,b,c,d=l
‘(B,bBcd+
i a,b,c=l
1
e2 +
0k2),
Bab = 2A,( F,F,) is used again. The 2nd central
= (ZjZk) - (Zj)(Zk).
(134
Formulas of higher moments can be also formulated but we renounce the presentation. The density function of zk( w), k = 1,2, has the following form (the index k in d, = d,,, is neglected) Pkb)
= /--&
exp(-fi2/2)
+ $4~3+2&(;3~22+2A;1) (
+32A21 +2A;4+
22A;,)H,(ir) (14)
J. vom Scheidt, B. Fellenberg / Some simulation results
197
where
a,b=l
‘A; = 8 i
daBBOb,
‘A2 = h2 $
a,b=l
3A”21= b2
d,db2A2(
FaFb),
a,b=l
;
d,d,dc3A2(
FaFbFc),
3A”22= ii2
o,b,c=l
4:3 = 6”
;
dabdc3A2(FaFbFc),
a,b,c=l
i
d,dbd,d,4F3(
FaFbFcFd),
‘a& = ii3
a,b,c,d=l
2A;2=52
f
dabdcdB&d,
2A;3=d2
a,b,c,d=l
2X14= 6”
;
d,,d,d,B,,B,,,
o,b,c,d=l
dabcddBabBcd,
; a,b,c,d=l
i a,b,c,d,e,f=l
da~c&ded$td&_Bcf,
2
2& = &”
c a,b,c,d,e,f=l
2& = 66
dabdcddedfBodBbeBcf,
d,bdcddedfdghB,.BbfB,gBdh
i
7
a,b,c,d,e,f,g,h=l
32A;, = 6”
5
d,,d,d,d,
3A2 ( FO&Fd) B,, .
a,b,c,d,e=l
5. Calculation of the statistical characteristics Now, we calculate the characteristics pA, for the process f,( X, o) defined by (1). Thereby, we have to regard the fact that no bound effects come into being, i.e. the relation (x - C, x + C) C (a, p) yields always in the limit terms. Firstly, we consider 2A1 and 2A2 using (7)-(9). Taking into account the expansions a; = fJ’(q) ui?,l
=
u’(x)
= u’(x) +
+ a2’(x)(a;
u2’(x)(ui+l
-x)
-x) +
+ o(h),
o(h),
where x E [ai, ai+r], i = 1,. . . , N - 2 and h = c/2 we can derive
= u2(x)h
+ o(h2) = +u2(x)
+ o(c2).
198
J. vom Scheidt, B. Fellenberg / Some simulation results
And by this, it follows u(x) Furthermore.
= $u’(x)
and
b(x)
= 0.
we have
-1 K(x> E / K(x)n[a,Pl
$((l Y) dY
2u)u;
+ 4uo:),
:(2(1-u)a:+(2u-l)0,2),
O
and we obtain
a(u,
$(l + 2U)C+
for 0 6 u < $
:a,2
for + < u < 1
a) =
and
1 a(u,
J0
a) du = &cJ,‘.
Hence, we have
1lu(u,a) du-a(cY)
= -&a;
0
and by a similar calculation
J’ a(u, p) du-a@)=
-&u;.
-1
Then the characteristics
Secondly,
(cf. (8) and (9)) are given by
2A,(qFk)
= :Jp~(x)&(x)u2(x)
2A,(F,C)
= -~~F,(ol)Fx(a)uW
we consider
dx,
(Isa)
+ q(P)I;,(B)uW).
Wb)
3A2 and 4z3 using (10) and (11). Defining
the sets Jp i (u~+~_~, u~+,__~)
i=2,3 ,..., N - 3, and taking into consideration for p = 1,2,. . . , 5 for XE[U,,U~+~], pendence of pi and x E J3 = (ai, u,+r) we have
Then the determination
/ M2(x)/
of the integrals
(L(x)f,(~r)L(~2))
results in dYr dY2 = h2(bi(x)e?
+ bi(x)e:+r)*
the inde-
J. vom Scheidt, B. Fellenberg / Some simulation results
199
Considering an analogous expansion of of (cf. crf above) it follows
and by this a3 = $a3( x). The calculation of the function a4( x) can be carried out in the same manner, of course with more extensive considerations concerning the set M3( x). We renounce the derivation of this function, it follows i,(x)
= +( u”(x) - 3(uz(x))2). Thus, the expressions 3A2 and “x3 are 3A,(I$F,F,)
= ~~~F,(x)~~(~)F,(x)o~(x)
4z3(y~~/~m)
(154
dx, - 3(02(x))‘)
=;,~~(x)~~(x)~(x)F~(+~(x) cx
dx.
(154
6. Simulation and analysis of linear functionals To obtain realizations of the linear functionals rj,( w) we determine realizations of p,(w), 1,. . .) N, and with these realizations of rj,( w) by using the formula
p = 0,
N-l r,c(w>=
where cjP =
(djO-cjCl)PO+
C p=l
(djp-cjp+cjp-l)Pp+CjN-lPN~
Op+l J
and the two p&s 2x + 1. Defining
b,(x)q.(x)
dx and djp=
Qp+l 1
of functions FI(x) = 1, F,(2)
djo - Cl0>
P = 0,
djp-cjp+cjp_l,
p=l,...,
cjN-l,
P = N,
4(x) =x
dx. Thereby, we consider (Y= 0, fi = 1 as well as F3(x) = sin(2Tx),
F4(x) =x2
+
N-l,
the exact moments are given by, j, k, 1, m E { 1, 2) or { 3, 4}, (Tjrr/cr) = E
(164
gjpgkp”,2,
p=o N ( ‘jerks rlc > =
c p=o
g/p gkpglp”i
7
(16b)
J. vom Scheidt, B. Fellenberg / Some simulation results
200
N Crj,rk,r~,rmc)
=
C p=o
~jpgkp~lpgmp”~
N ’
c
(164
(gjpgkpg,~g,,fgjpgk~glpg,,fgjpgk,g,,S,p)~~u~,
Piclz=qO
because of the independence of the random variables pP( w) and ( pp) = 0. To compare the theoretically calculated moments given by (5a)-(5f) and the above exact moments with those obtained by simulation, the usual formulas of estimation of moments are used. Let m be the number of realizations xii) of rlc( w ) and Ej, Ejk, . . . , denote the simulation results (estimations) of ( r,z), (t-jerk<), obtained by .
.
.
,
m jjjj=
C ,p,
+
m
mjk=
~
(Xlj)-,j)(Xl(k)-mk)r....
,~
r=l
i=l
The comparison of density functions is carried choose the intervals with c = cj = : Jr:c A,,j I -,=(--co,
(kc,(k+l)c]
I = k
i ((k-l)c,
The theoretical
‘jk,l
-6~1,
kc]
probabilities
’
$ lkPj,,(u)
I,=(6c,
co)
and
fork=
-1,
-2 ,...,
for k=l,2
are determined d”,
out by calculation
,...,
of x2-values.
For this, we
-6,
6.
by
1=1,2,3andk=-I
,...,
7,
where pj,, denotes the density function of cr( w) in dependence on the order of approximation 1 used, i.e. 1= 1 is the share of p,(u) without the terms multiplied by & and e etc. Then the corresponding x2-values are given by the comparison with the relative frequencies Hjk in Ik: 7 x;.r=m
(Hjk-Pjk,1)2
c k=
-7
(17)
q’k,l
Analogously a x2-value for the two-dimensional case pjk( ur, u2) can be defined using the above interval section in both directions. Thereby, we consider (interval-) areas IP4 with p, q = - 3, j = 1, 3 and corresponding frequencies Hjp4 as well as -2,... ,3, densities P~,+~( ul, u2), ak( x) of the probabilities P,pq,l. For concrete simulations we have to specify the functions random variables pP( o) in f,( X, w), i.e. (pi) = ak( aP), p = 0, 1,. . . , N. We consider with respect to the distribution of pP the cases (x E [OJ]): (i) Normal distributron (il) a2(x) = 1, (i2) 02(x) = 1 -4(x - 0.5)2. Then, we have u3(x) = 0 and u4(x) = ~(u~(x))~.
J. uom Scheidt, B. Fellenberg / Some simulation results
201
(ii) Uniform distribution on [ - d( a ;), d( ai)] where d(x) = d, - 4( d, - d,)( x - 0.5)2 and (iil) d, = d, = J”;, (ii2) d, = idi = +fi. Then, it follows a2(x) = :d2(x), a3(x) = 0 and a4(x) = ld4(x). (iii) Logarithmic normal distribution with a2(x) = S(x)( S(x) - l), S(x) = exp( d(x)) and (iiil) d, = 0.5, d, = 0.1, i.e. d, > d,, (iii2) d, = 0.1, d, = 0.5. Here, it yields a3(x) = /w’ (S3(x) - 3S(x) + 2) and u”(x) - ~(D~(x))~ = S2(x)(S6(x)
- 3S2(x) + 12S(x) - 6). Considering t&se three cases we have obviously: 3A1=4z3 = 0 for (i); 3A2 = 0, 4z3 # 0 for (ii) and 3A2 # 0, 4x3 # 0 for (iii). Now, we can compare our theoretically obtained results with simulation results as well as possible exact results. Thereby in general m = 1000 realizations were generated. First, we consider the moments up to the 4th order and we put (r$-s4+1c) = m:,,,, t = 1, 2, ex, si where t = ex denotes the exact moments given by (16a)-(16c), t = si means the estimated moments m and t = 1, 2 denotes the order of approximation in (5a)-(5f), i.e. for example: m:,,, = 2A,( FlF2)c but mz,,, = 2A,( F:)c +2A2( Ft)c2 etc. For the 2nd moments the values summarized in Tables 1 and 2 are computed. We can see that mf,,, are better than ml,,,, with respect to m:r(,, in all cases. These values are obtained for N = 10. It can be remarked that this agreement still improves with increasing N (see also Fig. 1). Furthermore, we can see the usefulness of the simulation procedure, i.e. the agreement of m$q and mf,pq. This is important for the analysis of density functions and of random non-linear functions where the exact values are unknown (see below). An investigation of the dependence on the number of realizations m shows the better coincidence of ynzPq and rnzrpq with increasing m. This is plotted in Fig. 1 for the case (iil) and N = 20, i.e. E = 0.1 where the same behaviour can be stated for the other cases and moments. Now, we consider briefly the other moments for N = 10. Some estimations of the 1st moments are contained in Table 3 where we note that rni p4 = 0 for t = 1, 2, ex. Simulation results of the 3rd moments are given in Table 4 for s = 1. Here, we have mi,p4 = 0, t = 1, 2 ex- for the cases (i) and (ii). Further, it yields mt,pq = 0 for the case (iii). Therefore, the values of m3,p, and rn:Tpy are also contained in Table 4. Some selected results as to the 4th moment are written in Table 5 where we restrict us to the case (i). For the other cases similar tendencies can be observed. We can find that the statements made for the 2nd moments (see above) can be generalized to the other moments. Secondly, we consider the one-dimensional density functions. In case (i) the exact density function p,(u) is known as the normal distribution with the exact moments given above. Because of 3A2 = 0 we have for the first two approximations p,,i( u) =P,,~( u). Then, a comparison of P,(U) with PJu), I= I, 3, is possible. This results in the fact that pj,3( u) always agrees better with pj( u) than p,,i(u). For the case (ii) we have again p,,, = pI,3 and a comparison of the X2-values leads to the fact x:,3
G
x3,1
- 4S3(x)
for all cases (il)-(iii2).
Remarkable deviations between the approximations can be established, especially in the case (iii), because only P~,~( U) is a symmetric function (see Fig. 2). The negative values are a
1
0
2
1
0
1
1
3
3
3
2
1
0
2
1
0
P
2
1
0
2
1
0
s
1
1
1
3
3
3
m&l
2
1
0
2
1
0
4
0.074
- 0.046
0.047
0.031
0.048
0.095
t = ex
(il)
0.068
- 0.042
0.045
0.030
0.046
0.100
t = si
2nd moments
t = si
0.035
- 0.028
0.034
0.020
0.033
0.034
- 0.027
0.032
0.021
0.034
0.066
0.074
- 0.046
0.047
0.031
0.048
0.075
- 0.046
0.045
0.032
0.047
0.093
t = si
0.030
- 0.024
0.030
0.018
0.030
0.029
- 0.023
0.029
0.018
0.031
0.062
t = si
0.030
- 0.025
0.032
0.018
0.031
0.061
t=2
0.061
t = ex
(ii2)
0.031
0.074
0.095
t = ex
0.066
t = ex
0.087
(iil)
0.036
0.074
- 0.048
- 0.025
0.032
0.050
0.050 - 0.048
0.018
0.031
0.033
0.031
0.047
0.061
0.095
t=l
0.050
t=2
(ii2)
0.100
t=l
(iil)
(i2)
- 0.029
- 0.048
- 0.048
0.087
0.036
0.050
0.050
0.033
0.067 0.020
0.048
0.095
-t =1,2
(i2)
0.031
0.033
0.050
Exact and estimated
Table 2
2
1
0.100
t=l
P
s
t=2
(il)
m:.Pq
4
of the 2nd moments
1
Approximations
Table
0.035
- 0.028
0.034
0.020
0.034
0.068
t = ex
(iiil)
0.038
- 0.029
0.036
0.021
0.034
0.069
t=l
(iiil)
0.032
- 0.025
0.032
0.019
0.033
0.066
t = si
0.036
- 0.029
0.036
0.020
0.034
0.068
t=2
0.040
- 0.018
0.015
0.012
0.016
0.033
t = ex
(iii2)
0.051
0.036
- 0.016
0.013
0.012
0.017
0.034
t = si
0.038
0.015 - 0.019
- 0.019
0.012
0.016
0.032
t=2
0.015
0.014
0.018
0.037
t=l
(iii2)
203
J. vom Scheidt, B. Fellenberg / Some simulation results t
m31 PQ
Gpq t P’O, q=2
I
404
-
403
---
t= 1
-
t=2,ex
-
t= 1,2,ex
p =2, q=o
t’ 0,024 -o,O? L f 18
- 0,03
2
3
G
5
c.ld c m t=7,2ex
P=q”l c
Fig. 1. Comparison of the moments in dependence on m for the case (iil), p + q = 2 and N = 20.
Table 3 Estimated 1st moments
msis.nq S
P
4
1 1 3 3
1 0 1 0
0 1 0 1
(ill
62)
(iil)
(ii2)
(iiil)
(iii2)
- 0.0088 - 0.0038 - 0.0017 0.0069
- 0.0053 - 0.0031 - 0.0035 0.0006
- 0.0085 - 0.0010 - 0.0095 0.0151
- 0.0064 - 0.0041 0.0034 - 0.0055
0.0052 0.0016 0.0003 0.0013
- 0.0005 0.0002 - 0.0010 0.0028
Table 4 Exact, approximated and estimated 3rd moments
f ms.Pq
(ill
62)
(iil)
(ii2)
(iiil)
S
P
4
t = si
t = si
t = si
t = si
t = si
t = ex
t=2
t = si
.t = ex
t=2
1 1 1 1
3 2 1 0
0 1 2 3
-
0.0005 0.0004 0.0003 0.0002
- 0.0004 0.0002 0.0004 0.0003
-0.0020 0.0010 0.0005 0.0003
0.0109 0.0055 0.0032 0.0021
0.0160 0.0080 0.0045 O.QO28
0.0161 0.0080 0.0046 0.0028
0.0038 0.0019 0.0014 0.0011
0.0041 0.0020 0.0016 0.0014
0.0059 0.0030 0.0025 0.0023
0.0005 0.0002 0.0001 0.0001
(iii2)
Table 5 Exact, approximated and estimated 4th moments m:.pq
61)
S
P
4
t = ex
t = si
t=l
t=2
t = ex
t = si
t=1,2
1 1 1 3 3 3
4 3 2 2 1 0
0 1 2 2 3 4
0.0271 0.0136 0.0074 0.0078 - 0.0103 0.0166
0.0237 0.0119 0.0066 0.0068 - 0.0098 0.0164
0.0309 0.0150 0.0083 0.0089 - 0.0124 0.0225
0.0270 0.0135 0.0074 0.0083 - 0.0106 0.0160
0.0131 0.0065 0.0035 0.0027 - 0.0029 0.0036
0.0116 0.0060 0.0034 0.0023 - 0.0025 0.0032
0.0133 0.0067 0.0036 0.0030 - 0.0032 0.0039
W
204
J. vom Scheidt, B. Fellenberg / Some simulation results
-- .-O,?
u
w
Fig. 2. Approximations p4,,( u), I = 1,2, 3 of p4( U) in the case of (iiil).
consequence of the expansion in c They vanish with increasing values of N, i.e. decreasing values of C. The plotted densities show the deviation from the normal distribution P~,~. Thereby, it can be noted that gene_rally the characteristic 3A2 represents the deviation from the symmetrical distribution and “x3 describes the deviation from the normal distribution. It is possible to
-f=
-7-6-5
m =lOOO m =SOUff
-4-3 -2-l
Fig. 3. Theoretical probabilities pdk,, and relative frequencies Hdk in the case of (iiil).
J. uom Scheidt, B. Fellenberg / Some simulation results
205
Table6 Theoretical probabilities P3p4,, and relative frequencies H3pq
q=-3
p=-3
p=-2
p=-1
p=o
p=l
p=2
p=3
0
0
0
0 0
0
0.0011 0.0004 0.0004
0.0049 0.0038 0.0046
0.0094 0.0088 0.0087
0.0071 0.0070 0.0069
0.0122
0.0362
0.0329
0.0095
0.0749
0.0931
0.0364
0.0050
0
0
0
0
0.0014 0.0011 0.0014
0 0
0.0014
0.0190
q=-2
q=-1
0 0
0
c
0.2298
q=o
q=l
q=2
q=3
1
confirm these by simulation. is drawn Fig. 3 the theoretical P4k,lfor I = 1, 2, 3 represented as polygon lines are compared with the relative frequencies Hdk_ Thirdly, we make some comments concerning the analysis of the two-dimensional density function. In principle the same conclusions as in the one-dimensional case can be made. Table 6 contains the relative frequencies HJIpqin comparison with the probabilities Pjp4,/,I= 1, 3 in the case (iii), N = 20, m = 5000 and j = 3. The best agreement can be stated for P3pq,3. 7. Simulation and analysis of random functions Now, we describe an application of the results as to functions of linear functionals rj,( w) of weakly correlated processes. Thereby, we consider the eigenvalue problem (A,+B(w))U=AU,
(18)
where A,,=
fors=1,3.
206
J. vom Scheidt, B. Fellenberg / Some simulation results
The averaged problem AJ = A,U has the simple eigenvalues eigenvalues of the problem (18) can be calculated from A 112,~= ~l~&s(~)rrs+l(~)) and the expansions
of d,(y,,
A,, = - 2 and
= :( r, + c+~ + 1 f [(3 + r, - rS+1>2 + 16]1’2),
y2) can be written
(1%
as
$Yl+ ZY2- $(Yl -Y2) + $CYr
d,(Y,>Y2) = -2+
A,, = 3. The
-Y2J3+
~2(Yl~Y2~=3+~Y~+fY2+$(Yl-Y2~2-~(y,-Y2)3+
***3
***9
and it follows (cf. (12))
and d 1,111
d 1,112
d 2,111
d 2,112
1 - 1
-1
1 - 1
1
Using (13a)-(13c) we can determine the approximations eigenvalues A,,, denoted by MS’,, , t = 1, 2 (A,,,>
=AH+
d k,ob’Al(~+s-,F,+s-,)r
i
-1 1
of the first
+ ok)
two moments
= M& + o(e),
of the
k = 1,2
a,b=l
and
(Ml,,
-
(4,s>)“(A2,,
-
@2,s>)“>
=
:;:I
+
44
+
occ21
7
i
where SE {l,...,
4) denotes
again the considered
functions
F’(X) in the functionals
Table 7 Approximated and estimated 1st moments of A,,, M’s.k
(ix)
(il)
s
k
t=2
t = si
t=2
t = si
1 1 3 3
1 2 1 2
- 2.0011 3.0011 - 2.0074 3.0074
- 2.0058 2.9932 - 2.0103 3.0051
- 2.0007 3.0007 - 2.0042 3.0042
- 1.9982 3.0052 - 2.0026 3.0041
rs( w). For
J. vom Scheidt, B. Fellenberg / Some simulation results
201
Table 8 Approximated and estimated 2nd moments of A,,,
x,,
61)
s
P
4
t=l
t=2
t = si
t=l
t=2
t = si
1 1 1 3 3 3
2 1 0 2 1 0
0 1 2 0 1 2
0.0413 0.0553 0.0813 0.0422 - 0.0106 0.0202
0.0388 0.0524 0.0770 0.0320 - 0.0108 0.0184
0.0376 0.0505 0.0743 0.0343 - 0.0116 0.0176
0.0269 0.0376 0.0557 0.0163 - 0.0079 0.0152
0.0265 0.0372 0.0554 0.0142 - 0.0071 0.0144
0.0254 0.0359 0.0536 0.0130 - 0.0064 0.0132
(iiil)
the simulation we take N= 10 (6 = 0.2), [(Y, p] = [0, l] an d m = 1000 realizations. Furthermore, simulation results of (A, ,( w), A,,,(w)) are obtained from simulation results of ( rs’,(w), r,, I( w)) by replacing these values’ in (19). The resulting estimates of the above moments are denoted by Msi s,, . Then, we obtain the values in the Tables 7 and 8 where we have only given the results for the cases (il) and (ml). Similar tendencies can be observed for the other cases. Concerning the first moments we can note that in the majority of the considered cases the it can be relations (A,,,) G A,, and (A,,) > A,, are confirmed by simulation. Furthermore, remarked that with decreasing values of E (N = 20, 50) these relations will be fulfilled in all cases. If we compare the 2nd moments we find a better coincidence between the simulation it4sfPq than in comparison with M&q. results MsS,Pqand the 2nd approximation Secondly, we consider the density function pk( u) of A,,,(w) - A,, given by (14) where we have again approximations pk,,( u), I= 1, 2, 3 according to the order of 6 used. Then, the same conclusions can be made as in the previous section. That means, if we also consider the theoretical probabilities with respect to 1 and relative frequencies, that the best coincidence can be observed for the highest order I = 3. Some X2-values are given for this in Table 9, where the and two values of c with m = 6000 realizations are considered. The case (il), &,&) -A,, influence of the correlation length is also obvious. Finally, we can conclude that the simulation results confirm the theoretically obtained moments as well as distributions and also the better efficiency of the higher order approximations with respect to e could be proved.
Table 9 x2-values for the case (il) and LI~,~(w)- AkO in dependence on the order 1 of approximation I=1
I=2
I=3
E= 0.2
69.0
39.1
23.7
E= 0.1
35.1
36.7
22.8
e = 0.2
118.5
113.7
28.6
E = 0.1
56.0
34.9
28.1
k=l
k=2
208
J. vom Scheidt, B. Fellenberg / Some simulation results
References fields, FMC-Series 19 (1986) [l] B. Fellenberg and J. vom Scheidt, Probabilistic analysis of random temperature 15-24. [2] Proc. 2nd Conference on Stochastic Analysis, 27-30 October 1986; Special Issue of “Wiss. Beitr.” (IH Zwickau, 1986). [3] J. vom Scheidt, ed., Problems of stochastic analysis in applications, Special Issue of “Wiss. Beitr.” (IH Zwickau, 1983). [4] J. vom Scheidt and W. Purkert, Random Eigenvalue Problems (Akademie-Verlag, Berlin and North-Holland, New York, 1983). [5] J. vom Scheidt, Random Equations of the Mathematical Physics (Akademie-Verlag, Berlin) in preparation.
SOME SIMULATION J. vom SCHEIDT Ingenieurhochschule
RESULTS
AS TO WEAKLY CORRELATED
191
PROCESSES
and B. FELLENBERG Zwickau, Zwickau, DDR 9.540, German Dem. Rep.
The investigation of linear functionals of weakly correlated processes which arise e.g. from the solutions of random differential equation problems are dealt with. A brief survey of the theory of weakly correlated processes is given, and especially, approximations of moments and distributions of these functionals are deduced which depend on the correlation length. Furthermore, functions of such functionals are considered. In the main part these results are confirmed by simulation. For this, a method of simulation of weakly correlated processes is given and the simulation results are used for the comparison of the theoretically-calculated with the statistically-determined values of some characteristics. Several examples of simulations are added.
1. Introduction The theoretical investigation of linear functionals of weakly correlated processes f,(x, w), x E R, leads to approximations of moments and distributions which depend on the correlation length 6 of the process (cf. [4,5]). The definition of weak correlations is connected with the interpretation that f,(x, w) and f,( y, w) are independent if the distance of x and y exceeds 6 > 0. The exact definition introduced in [4] is given by a separation of all moments (n~_,f,( xi)). There are many problems and technological applications having linear functionals of such processes as solutions, e.g. random vibrations, random wave propagations or random heat equations investigated in [l-3]. Thereby the theoretical results of the theory of weakly correlated processes are used for the determination of some characteristics like moments and probabilities which are of practical interest. In some cases comparisons with estimations from measured results were possible and good agreement could be stated. Here, we will confirm generally the theoretically obtained results by a method of simulation of weakly correlated processes and subsequently by a statistical analysis of the simulation results. The special aim will be the corroboration of the latest results concerning higher order approximations.
2. Weakly correlated processes and a method of simulation At first, we repeat the definition of weakly correlated processes. Let 1. 1 denote the norm on R and let the mean of a stochastic process be denoted by ( . ). A stochastic process f,( X, w), x E D c R with (f,(x)) = 0 is called weakly correlated with correlation length e if the relation
037%4754/87/$3.50
0 1987, Elsevier Science Publishers B.V. (North-Holland)
192
J. vom Scheidt, B. Fellenberg / Some simulation results
is satisfied for all kth moments, k = 2, 3,. . . , where I = (1, 2,. . . , k} and {xi, i E Ii}, p with Uy=,1, = I denote the so-called maximally e-neighbouring subsets of { xl, i E I}. j=l >***> The subset {xi, i E Ij} is maximally r-neighbouring if a permutation ();;;;.:%,) of Ii = { i,, . . . , ir } existswith (x~~-x~~+~]
=0
and
(p,“) =u”(u,)=u,”
exist. Then it follows immediately L(x,
from the independence
u) = b,(+,(o)
+ $(x)P,+,(~)
where u,=(~+(p/N)(j3-a),
p=O ,..., x - up
b,(x) =
)
up+1 is a weakly correlated
for k=2,
ap
3,..., that the special process on D = [CY,p]
for x E [up,
ap+l]y
(1)
N,
z;,(x) = 1 -l&x),
p=O,...,N-1
process having the correlation
length
(2)
E = 2( p - Cy)/N. Its correlation
function
is given, for x E [up, a,,,],
by
&(x3 Y) = u&M(Y)> ~,W,-l~Yb~~
I
=
b,Wp(Y)a,2
[Up-l, a,], b,bPp(Yb;+l~ Y E [up’ up+11 YE
+
Y
[a,+,, a,+,],
~,b)b,,l(YbJ,z,l~
YE
0,
otherwise.
(3)
Further, we will investigate three cases of the distribution of the random variables p,(w): (i) normal, (ii) uniform and (iii) logarithmic normal distribution. Thereby, we use for simulation the following methods. The generation of independent uniformly distributed random variables on [OJ] based on the algorithm ik+i = (53,
+ 645497) mod 999997,
TJ~+~= q,+,/999997 where qk+l is a realization. uniformly distributed random 5 = a,/~
and
YjO-C 999997,
Then, the number p k+l = 2~j~+~s-s gives a realization variable on t-s, s]. Furthermore, the number
of a
cos(2.lTn,),
is a realization of a zero-mean Gaussian random variable with variance ai, where ql, q2 denote two independent realizations of a uniformly distributed random variable on (0, 11. Finally, the formula 4 = exp(5) leads to a realization
- exp( u,2/2) of a centralized
logarithmic
normally
distributed
variable.
J. vom Scheidt, B. Fellenberg / Some simulation results
193
3. Distribution and moments of linear functionals Now, we consider the random variables r,,(w) = /‘F;(x)f,(x. a
j=l,2
a> dx,
(4)
being linear functionals of the weakly correlated process fc(x, w). The functions E;(x) are assumed to be continuously differentiable on D = [a, ,8] and fi(x, w) possesses continuous sample functions almost sure as well as (]f,(x)]~)~c,
forp=l,2,....
Then, it is possible to deduce expansions relative to E of kth moments (H,“=,Y, c) and of density functions of ( rt,, . . . , r,,) (cf. [4,5]). This has been done by a straightforward consideration of the moments and their integration domains as well as by using two approaches to the characteristic function of random vectors and a generalized Gram-Charlier-series. We give here only the results which will be confirmed by simulation. They depend on some characteristics “A, given below. Firstly, we have the moments up to the 4th order, j, k E { 1, 2)) (rjc)
=
O,
(54
(y-&
=*A,(E;F,)c+*A,(E;Fk)~*+o(~*),
(~JY~~)
=3A,(E;FF,Y)~2 + o(E*)
(5:)
cw
for p + q = 3,
= 3(2A,jF,2)~2 E* + [62Al(~2)2A2(~2)
+ [32A1(~F~)2A2(~2)
+473(q4)]~3
+ 32A,(F,2)2A2(F,Fk)
(54
+ o(c3),
+4A=,(F,3F,)]c3
for j#k,
+o(c3) ($&)
(5c)
(54
= [ZA,( F,*)‘A~( Ft) + 2(2A,( k;F,))*]c* +[*A,(q*)*A,(F,2)
+ZA,(F;)‘A,(<*)
+4*A,(qFk)‘A2(F,Fk)
+4A=,(q*~,t)]c3+
o(c’>
for j#
k.
(5f)
Secondly, the approximation of the density function of rjr( w) can be written as
,ff&) c+o(c) , j 24(2A1,j) 72(2AI,j) 1 i (64
2A2, + -H*(Z) j
22Al,
+
4z3j ’
*H,(E)+
(3A2,j !2
194
J. vom Scheidt, B. Fellenberg / Some simulation results
where
and Hk(x)
denote
the Chebyshev-Hermite-polynomials.
I&(x)=x, H,(x)
H2(x)=x2-1,
&(x)=x3-33x,
= x5 - 10x3 + 15x,
The two-dimensional PjkcU1,
density
u2)
function
=Pjk,l(U1,
Especially,
and
it yields H,(X)
= 1,
I&(x)=x4-6x2+3
H,(x)=x6-15x4+45x2-15.
of (rj,(o),
~~,(a)) possesses
the form
u2)
*l+
c k,+k,=3
i
3;q2( ~+Pl~~*-P*) .,,+$=3
(4
3A,(
-PI)!@2
-P2)!
.
qqp)
PI!P2!
k,-p:> 0
6-b) where k,, pr = 0, 1,. . . i.il=zlI/~, E71 =
ii,=
E;/\IB,,,
4
=
(-42%
(-424
+
+
Q42v[~%(w22
B,,E;)/(B,,B,,
-
-
B:2)y2>
g2y2,
Bjk = 2A,( E;F’) and the first approximation Pjk,l(%~
‘2)
=
Pjk,l is given by 1 2m( B,,B22 - B:,)l12 exp
B,,u,~ - 2 B,,u,u, 4w322
-
+ B,,u; %-2)
.
J. vom Scheidt, B. Fellenberg / Some simulation results
195
Thirdly, the characteristics pA, can be deduced in the following form. Defining K,(x) = { Y : 1x - y 1 < 6 } and using the correlation function R,( x, y) of the weakly correlated process we introduce the expansion
y) dy = a(x)r
IK,(x)
R,(x,
+ b(x)e2
+ o(e”).
Then it follows 2~,(~&)
= j’<(x)F,(x)a(x)
dx,
where R,(x,
Y) dY
1
for 24E [0, l]
.X=C-+
and u(u, p> = FrI& ‘J e K,(x)n[a,Pl
R,(x,
Y) dY
I x=p+ru
for UE [-l,O].
Furthermore, it yields
with
and (11) with
- UXx)L(Y*)>
- <.L(x).L(Y2)>(f,(Y&(Y3)) dy,
dy2
dy3,
where W(x)
= ((x,
Y, ,...,
Y,)c[~,
p]““:(x,
y, ,...,
y,)c-neighbouring),
q=2,
3.
196
J. vom Scheidt, B. Fellenberg / Some simulation results
4. Distribution and moments of random functions Besides the consideration of linear functionals it is also possible to make expansions of functions of functionals. We consider here some selected results for two functions of two functionals. General results can be found in [5]. Let d,, d, be functions of (yr, vz) having the representation, k = 1, 2, 2 d/h,,
~2)
=
4,
+
2
c 4,,~a CZ=l
+
c a,b=l
d,,,oay,
(12) where r > 3/2 and g, is bounded on K,(O) for S > 0. Furthermore, we assume the existence of all moments of dkr( w) A d,(r,,, Yap), i.e. ( 1d,, 1”) < cp < 00 for k = 1, 2 and all p. Then, we consider especially the difference z,,J w) = dk( o) - d,,, and the 1st and 2nd moments are given by (13a)
n,b=l
+
i n,b=l
dj od/c,b2A2(F,4)
+
’
abdkcd
+
Cdj >
3
+
where here and in the following, moment can be written as (Cd, - (dj>)(d,
B,cBbd
+
B,dBbc)
the abbreviation
- (dk)))
(dj
, abdk 7c +
d/c,abdj,c)3Az(E’bF,E)
dj,abcdk,d + dk,a6cdj,d)
a,b,c,d=l
‘(B,bBcd+
i a,b,c=l
1
e2 +
0k2),
Bab = 2A,( F,F,) is used again. The 2nd central
= (ZjZk) - (Zj)(Zk).
(134
Formulas of higher moments can be also formulated but we renounce the presentation. The density function of zk( w), k = 1,2, has the following form (the index k in d, = d,,, is neglected) Pkb)
= /--&
exp(-fi2/2)
+ $4~3+2&(;3~22+2A;1) (
+32A21 +2A;4+
22A;,)H,(ir) (14)
J. vom Scheidt, B. Fellenberg / Some simulation results
197
where
a,b=l
‘A; = 8 i
daBBOb,
‘A2 = h2 $
a,b=l
3A”21= b2
d,db2A2(
FaFb),
a,b=l
;
d,d,dc3A2(
FaFbFc),
3A”22= ii2
o,b,c=l
4:3 = 6”
;
dabdc3A2(FaFbFc),
a,b,c=l
i
d,dbd,d,4F3(
FaFbFcFd),
‘a& = ii3
a,b,c,d=l
2A;2=52
f
dabdcdB&d,
2A;3=d2
a,b,c,d=l
2X14= 6”
;
d,,d,d,B,,B,,,
o,b,c,d=l
dabcddBabBcd,
; a,b,c,d=l
i a,b,c,d,e,f=l
da~c&ded$td&_Bcf,
2
2& = &”
c a,b,c,d,e,f=l
2& = 66
dabdcddedfBodBbeBcf,
d,bdcddedfdghB,.BbfB,gBdh
i
7
a,b,c,d,e,f,g,h=l
32A;, = 6”
5
d,,d,d,d,
3A2 ( FO&Fd) B,, .
a,b,c,d,e=l
5. Calculation of the statistical characteristics Now, we calculate the characteristics pA, for the process f,( X, o) defined by (1). Thereby, we have to regard the fact that no bound effects come into being, i.e. the relation (x - C, x + C) C (a, p) yields always in the limit terms. Firstly, we consider 2A1 and 2A2 using (7)-(9). Taking into account the expansions a; = fJ’(q) ui?,l
=
u’(x)
= u’(x) +
+ a2’(x)(a;
u2’(x)(ui+l
-x)
-x) +
+ o(h),
o(h),
where x E [ai, ai+r], i = 1,. . . , N - 2 and h = c/2 we can derive
= u2(x)h
+ o(h2) = +u2(x)
+ o(c2).
198
J. vom Scheidt, B. Fellenberg / Some simulation results
And by this, it follows u(x) Furthermore.
= $u’(x)
and
b(x)
= 0.
we have
-1 K(x> E / K(x)n[a,Pl
$((l Y) dY
2u)u;
+ 4uo:),
:(2(1-u)a:+(2u-l)0,2),
O
and we obtain
a(u,
$(l + 2U)C+
for 0 6 u < $
:a,2
for + < u < 1
a) =
and
1 a(u,
J0
a) du = &cJ,‘.
Hence, we have
1lu(u,a) du-a(cY)
= -&a;
0
and by a similar calculation
J’ a(u, p) du-a@)=
-&u;.
-1
Then the characteristics
Secondly,
(cf. (8) and (9)) are given by
2A,(qFk)
= :Jp~(x)&(x)u2(x)
2A,(F,C)
= -~~F,(ol)Fx(a)uW
we consider
dx,
(Isa)
+ q(P)I;,(B)uW).
Wb)
3A2 and 4z3 using (10) and (11). Defining
the sets Jp i (u~+~_~, u~+,__~)
i=2,3 ,..., N - 3, and taking into consideration for p = 1,2,. . . , 5 for XE[U,,U~+~], pendence of pi and x E J3 = (ai, u,+r) we have
Then the determination
/ M2(x)/
of the integrals
(L(x)f,(~r)L(~2))
results in dYr dY2 = h2(bi(x)e?
+ bi(x)e:+r)*
the inde-
J. vom Scheidt, B. Fellenberg / Some simulation results
199
Considering an analogous expansion of of (cf. crf above) it follows
and by this a3 = $a3( x). The calculation of the function a4( x) can be carried out in the same manner, of course with more extensive considerations concerning the set M3( x). We renounce the derivation of this function, it follows i,(x)
= +( u”(x) - 3(uz(x))2). Thus, the expressions 3A2 and “x3 are 3A,(I$F,F,)
= ~~~F,(x)~~(~)F,(x)o~(x)
4z3(y~~/~m)
(154
dx, - 3(02(x))‘)
=;,~~(x)~~(x)~(x)F~(+~(x) cx
dx.
(154
6. Simulation and analysis of linear functionals To obtain realizations of the linear functionals rj,( w) we determine realizations of p,(w), 1,. . .) N, and with these realizations of rj,( w) by using the formula
p = 0,
N-l r,c(w>=
where cjP =
(djO-cjCl)PO+
C p=l
(djp-cjp+cjp-l)Pp+CjN-lPN~
Op+l J
and the two p&s 2x + 1. Defining
b,(x)q.(x)
dx and djp=
Qp+l 1
of functions FI(x) = 1, F,(2)
djo - Cl0>
P = 0,
djp-cjp+cjp_l,
p=l,...,
cjN-l,
P = N,
4(x) =x
dx. Thereby, we consider (Y= 0, fi = 1 as well as F3(x) = sin(2Tx),
F4(x) =x2
+
N-l,
the exact moments are given by, j, k, 1, m E { 1, 2) or { 3, 4}, (Tjrr/cr) = E
(164
gjpgkp”,2,
p=o N ( ‘jerks rlc > =
c p=o
g/p gkpglp”i
7
(16b)
J. vom Scheidt, B. Fellenberg / Some simulation results
200
N Crj,rk,r~,rmc)
=
C p=o
~jpgkp~lpgmp”~
N ’
c
(164
(gjpgkpg,~g,,fgjpgk~glpg,,fgjpgk,g,,S,p)~~u~,
Piclz=qO
because of the independence of the random variables pP( w) and ( pp) = 0. To compare the theoretically calculated moments given by (5a)-(5f) and the above exact moments with those obtained by simulation, the usual formulas of estimation of moments are used. Let m be the number of realizations xii) of rlc( w ) and Ej, Ejk, . . . , denote the simulation results (estimations) of ( r,z), (t-jerk<), obtained by .
.
.
,
m jjjj=
C ,p,
+
m
mjk=
~
(Xlj)-,j)(Xl(k)-mk)r....
,~
r=l
i=l
The comparison of density functions is carried choose the intervals with c = cj = : Jr:c A,,j I -,=(--co,
(kc,(k+l)c]
I = k
i ((k-l)c,
The theoretical
‘jk,l
-6~1,
kc]
probabilities
’
$ lkPj,,(u)
I,=(6c,
co)
and
fork=
-1,
-2 ,...,
for k=l,2
are determined d”,
out by calculation
,...,
of x2-values.
For this, we
-6,
6.
by
1=1,2,3andk=-I
,...,
7,
where pj,, denotes the density function of cr( w) in dependence on the order of approximation 1 used, i.e. 1= 1 is the share of p,(u) without the terms multiplied by & and e etc. Then the corresponding x2-values are given by the comparison with the relative frequencies Hjk in Ik: 7 x;.r=m
(Hjk-Pjk,1)2
c k=
-7
(17)
q’k,l
Analogously a x2-value for the two-dimensional case pjk( ur, u2) can be defined using the above interval section in both directions. Thereby, we consider (interval-) areas IP4 with p, q = - 3, j = 1, 3 and corresponding frequencies Hjp4 as well as -2,... ,3, densities P~,+~( ul, u2), ak( x) of the probabilities P,pq,l. For concrete simulations we have to specify the functions random variables pP( o) in f,( X, w), i.e. (pi) = ak( aP), p = 0, 1,. . . , N. We consider with respect to the distribution of pP the cases (x E [OJ]): (i) Normal distributron (il) a2(x) = 1, (i2) 02(x) = 1 -4(x - 0.5)2. Then, we have u3(x) = 0 and u4(x) = ~(u~(x))~.
J. uom Scheidt, B. Fellenberg / Some simulation results
201
(ii) Uniform distribution on [ - d( a ;), d( ai)] where d(x) = d, - 4( d, - d,)( x - 0.5)2 and (iil) d, = d, = J”;, (ii2) d, = idi = +fi. Then, it follows a2(x) = :d2(x), a3(x) = 0 and a4(x) = ld4(x). (iii) Logarithmic normal distribution with a2(x) = S(x)( S(x) - l), S(x) = exp( d(x)) and (iiil) d, = 0.5, d, = 0.1, i.e. d, > d,, (iii2) d, = 0.1, d, = 0.5. Here, it yields a3(x) = /w’ (S3(x) - 3S(x) + 2) and u”(x) - ~(D~(x))~ = S2(x)(S6(x)
- 3S2(x) + 12S(x) - 6). Considering t&se three cases we have obviously: 3A1=4z3 = 0 for (i); 3A2 = 0, 4z3 # 0 for (ii) and 3A2 # 0, 4x3 # 0 for (iii). Now, we can compare our theoretically obtained results with simulation results as well as possible exact results. Thereby in general m = 1000 realizations were generated. First, we consider the moments up to the 4th order and we put (r$-s4+1c) = m:,,,, t = 1, 2, ex, si where t = ex denotes the exact moments given by (16a)-(16c), t = si means the estimated moments m and t = 1, 2 denotes the order of approximation in (5a)-(5f), i.e. for example: m:,,, = 2A,( FlF2)c but mz,,, = 2A,( F:)c +2A2( Ft)c2 etc. For the 2nd moments the values summarized in Tables 1 and 2 are computed. We can see that mf,,, are better than ml,,,, with respect to m:r(,, in all cases. These values are obtained for N = 10. It can be remarked that this agreement still improves with increasing N (see also Fig. 1). Furthermore, we can see the usefulness of the simulation procedure, i.e. the agreement of m$q and mf,pq. This is important for the analysis of density functions and of random non-linear functions where the exact values are unknown (see below). An investigation of the dependence on the number of realizations m shows the better coincidence of ynzPq and rnzrpq with increasing m. This is plotted in Fig. 1 for the case (iil) and N = 20, i.e. E = 0.1 where the same behaviour can be stated for the other cases and moments. Now, we consider briefly the other moments for N = 10. Some estimations of the 1st moments are contained in Table 3 where we note that rni p4 = 0 for t = 1, 2, ex. Simulation results of the 3rd moments are given in Table 4 for s = 1. Here, we have mi,p4 = 0, t = 1, 2 ex- for the cases (i) and (ii). Further, it yields mt,pq = 0 for the case (iii). Therefore, the values of m3,p, and rn:Tpy are also contained in Table 4. Some selected results as to the 4th moment are written in Table 5 where we restrict us to the case (i). For the other cases similar tendencies can be observed. We can find that the statements made for the 2nd moments (see above) can be generalized to the other moments. Secondly, we consider the one-dimensional density functions. In case (i) the exact density function p,(u) is known as the normal distribution with the exact moments given above. Because of 3A2 = 0 we have for the first two approximations p,,i( u) =P,,~( u). Then, a comparison of P,(U) with PJu), I= I, 3, is possible. This results in the fact that pj,3( u) always agrees better with pj( u) than p,,i(u). For the case (ii) we have again p,,, = pI,3 and a comparison of the X2-values leads to the fact x:,3
G
x3,1
- 4S3(x)
for all cases (il)-(iii2).
Remarkable deviations between the approximations can be established, especially in the case (iii), because only P~,~( U) is a symmetric function (see Fig. 2). The negative values are a
1
0
2
1
0
1
1
3
3
3
2
1
0
2
1
0
P
2
1
0
2
1
0
s
1
1
1
3
3
3
m&l
2
1
0
2
1
0
4
0.074
- 0.046
0.047
0.031
0.048
0.095
t = ex
(il)
0.068
- 0.042
0.045
0.030
0.046
0.100
t = si
2nd moments
t = si
0.035
- 0.028
0.034
0.020
0.033
0.034
- 0.027
0.032
0.021
0.034
0.066
0.074
- 0.046
0.047
0.031
0.048
0.075
- 0.046
0.045
0.032
0.047
0.093
t = si
0.030
- 0.024
0.030
0.018
0.030
0.029
- 0.023
0.029
0.018
0.031
0.062
t = si
0.030
- 0.025
0.032
0.018
0.031
0.061
t=2
0.061
t = ex
(ii2)
0.031
0.074
0.095
t = ex
0.066
t = ex
0.087
(iil)
0.036
0.074
- 0.048
- 0.025
0.032
0.050
0.050 - 0.048
0.018
0.031
0.033
0.031
0.047
0.061
0.095
t=l
0.050
t=2
(ii2)
0.100
t=l
(iil)
(i2)
- 0.029
- 0.048
- 0.048
0.087
0.036
0.050
0.050
0.033
0.067 0.020
0.048
0.095
-t =1,2
(i2)
0.031
0.033
0.050
Exact and estimated
Table 2
2
1
0.100
t=l
P
s
t=2
(il)
m:.Pq
4
of the 2nd moments
1
Approximations
Table
0.035
- 0.028
0.034
0.020
0.034
0.068
t = ex
(iiil)
0.038
- 0.029
0.036
0.021
0.034
0.069
t=l
(iiil)
0.032
- 0.025
0.032
0.019
0.033
0.066
t = si
0.036
- 0.029
0.036
0.020
0.034
0.068
t=2
0.040
- 0.018
0.015
0.012
0.016
0.033
t = ex
(iii2)
0.051
0.036
- 0.016
0.013
0.012
0.017
0.034
t = si
0.038
0.015 - 0.019
- 0.019
0.012
0.016
0.032
t=2
0.015
0.014
0.018
0.037
t=l
(iii2)
203
J. vom Scheidt, B. Fellenberg / Some simulation results t
m31 PQ
Gpq t P’O, q=2
I
404
-
403
---
t= 1
-
t=2,ex
-
t= 1,2,ex
p =2, q=o
t’ 0,024 -o,O? L f 18
- 0,03
2
3
G
5
c.ld c m t=7,2ex
P=q”l c
Fig. 1. Comparison of the moments in dependence on m for the case (iil), p + q = 2 and N = 20.
Table 3 Estimated 1st moments
msis.nq S
P
4
1 1 3 3
1 0 1 0
0 1 0 1
(ill
62)
(iil)
(ii2)
(iiil)
(iii2)
- 0.0088 - 0.0038 - 0.0017 0.0069
- 0.0053 - 0.0031 - 0.0035 0.0006
- 0.0085 - 0.0010 - 0.0095 0.0151
- 0.0064 - 0.0041 0.0034 - 0.0055
0.0052 0.0016 0.0003 0.0013
- 0.0005 0.0002 - 0.0010 0.0028
Table 4 Exact, approximated and estimated 3rd moments
f ms.Pq
(ill
62)
(iil)
(ii2)
(iiil)
S
P
4
t = si
t = si
t = si
t = si
t = si
t = ex
t=2
t = si
.t = ex
t=2
1 1 1 1
3 2 1 0
0 1 2 3
-
0.0005 0.0004 0.0003 0.0002
- 0.0004 0.0002 0.0004 0.0003
-0.0020 0.0010 0.0005 0.0003
0.0109 0.0055 0.0032 0.0021
0.0160 0.0080 0.0045 O.QO28
0.0161 0.0080 0.0046 0.0028
0.0038 0.0019 0.0014 0.0011
0.0041 0.0020 0.0016 0.0014
0.0059 0.0030 0.0025 0.0023
0.0005 0.0002 0.0001 0.0001
(iii2)
Table 5 Exact, approximated and estimated 4th moments m:.pq
61)
S
P
4
t = ex
t = si
t=l
t=2
t = ex
t = si
t=1,2
1 1 1 3 3 3
4 3 2 2 1 0
0 1 2 2 3 4
0.0271 0.0136 0.0074 0.0078 - 0.0103 0.0166
0.0237 0.0119 0.0066 0.0068 - 0.0098 0.0164
0.0309 0.0150 0.0083 0.0089 - 0.0124 0.0225
0.0270 0.0135 0.0074 0.0083 - 0.0106 0.0160
0.0131 0.0065 0.0035 0.0027 - 0.0029 0.0036
0.0116 0.0060 0.0034 0.0023 - 0.0025 0.0032
0.0133 0.0067 0.0036 0.0030 - 0.0032 0.0039
W
204
J. vom Scheidt, B. Fellenberg / Some simulation results
-- .-O,?
u
w
Fig. 2. Approximations p4,,( u), I = 1,2, 3 of p4( U) in the case of (iiil).
consequence of the expansion in c They vanish with increasing values of N, i.e. decreasing values of C. The plotted densities show the deviation from the normal distribution P~,~. Thereby, it can be noted that gene_rally the characteristic 3A2 represents the deviation from the symmetrical distribution and “x3 describes the deviation from the normal distribution. It is possible to
-f=
-7-6-5
m =lOOO m =SOUff
-4-3 -2-l
Fig. 3. Theoretical probabilities pdk,, and relative frequencies Hdk in the case of (iiil).
J. uom Scheidt, B. Fellenberg / Some simulation results
205
Table6 Theoretical probabilities P3p4,, and relative frequencies H3pq
q=-3
p=-3
p=-2
p=-1
p=o
p=l
p=2
p=3
0
0
0
0 0
0
0.0011 0.0004 0.0004
0.0049 0.0038 0.0046
0.0094 0.0088 0.0087
0.0071 0.0070 0.0069
0.0122
0.0362
0.0329
0.0095
0.0749
0.0931
0.0364
0.0050
0
0
0
0
0.0014 0.0011 0.0014
0 0
0.0014
0.0190
q=-2
q=-1
0 0
0
c
0.2298
q=o
q=l
q=2
q=3
1
confirm these by simulation. is drawn Fig. 3 the theoretical P4k,lfor I = 1, 2, 3 represented as polygon lines are compared with the relative frequencies Hdk_ Thirdly, we make some comments concerning the analysis of the two-dimensional density function. In principle the same conclusions as in the one-dimensional case can be made. Table 6 contains the relative frequencies HJIpqin comparison with the probabilities Pjp4,/,I= 1, 3 in the case (iii), N = 20, m = 5000 and j = 3. The best agreement can be stated for P3pq,3. 7. Simulation and analysis of random functions Now, we describe an application of the results as to functions of linear functionals rj,( w) of weakly correlated processes. Thereby, we consider the eigenvalue problem (A,+B(w))U=AU,
(18)
where A,,=
fors=1,3.
206
J. vom Scheidt, B. Fellenberg / Some simulation results
The averaged problem AJ = A,U has the simple eigenvalues eigenvalues of the problem (18) can be calculated from A 112,~= ~l~&s(~)rrs+l(~)) and the expansions
of d,(y,,
A,, = - 2 and
= :( r, + c+~ + 1 f [(3 + r, - rS+1>2 + 16]1’2),
y2) can be written
(1%
as
$Yl+ ZY2- $(Yl -Y2) + $CYr
d,(Y,>Y2) = -2+
A,, = 3. The
-Y2J3+
~2(Yl~Y2~=3+~Y~+fY2+$(Yl-Y2~2-~(y,-Y2)3+
***3
***9
and it follows (cf. (12))
and d 1,111
d 1,112
d 2,111
d 2,112
1 - 1
-1
1 - 1
1
Using (13a)-(13c) we can determine the approximations eigenvalues A,,, denoted by MS’,, , t = 1, 2 (A,,,>
=AH+
d k,ob’Al(~+s-,F,+s-,)r
i
-1 1
of the first
+ ok)
two moments
= M& + o(e),
of the
k = 1,2
a,b=l
and
(Ml,,
-
(4,s>)“(A2,,
-
@2,s>)“>
=
:;:I
+
44
+
occ21
7
i
where SE {l,...,
4) denotes
again the considered
functions
F’(X) in the functionals
Table 7 Approximated and estimated 1st moments of A,,, M’s.k
(ix)
(il)
s
k
t=2
t = si
t=2
t = si
1 1 3 3
1 2 1 2
- 2.0011 3.0011 - 2.0074 3.0074
- 2.0058 2.9932 - 2.0103 3.0051
- 2.0007 3.0007 - 2.0042 3.0042
- 1.9982 3.0052 - 2.0026 3.0041
rs( w). For
J. vom Scheidt, B. Fellenberg / Some simulation results
201
Table 8 Approximated and estimated 2nd moments of A,,,
x,,
61)
s
P
4
t=l
t=2
t = si
t=l
t=2
t = si
1 1 1 3 3 3
2 1 0 2 1 0
0 1 2 0 1 2
0.0413 0.0553 0.0813 0.0422 - 0.0106 0.0202
0.0388 0.0524 0.0770 0.0320 - 0.0108 0.0184
0.0376 0.0505 0.0743 0.0343 - 0.0116 0.0176
0.0269 0.0376 0.0557 0.0163 - 0.0079 0.0152
0.0265 0.0372 0.0554 0.0142 - 0.0071 0.0144
0.0254 0.0359 0.0536 0.0130 - 0.0064 0.0132
(iiil)
the simulation we take N= 10 (6 = 0.2), [(Y, p] = [0, l] an d m = 1000 realizations. Furthermore, simulation results of (A, ,( w), A,,,(w)) are obtained from simulation results of ( rs’,(w), r,, I( w)) by replacing these values’ in (19). The resulting estimates of the above moments are denoted by Msi s,, . Then, we obtain the values in the Tables 7 and 8 where we have only given the results for the cases (il) and (ml). Similar tendencies can be observed for the other cases. Concerning the first moments we can note that in the majority of the considered cases the it can be relations (A,,,) G A,, and (A,,) > A,, are confirmed by simulation. Furthermore, remarked that with decreasing values of E (N = 20, 50) these relations will be fulfilled in all cases. If we compare the 2nd moments we find a better coincidence between the simulation it4sfPq than in comparison with M&q. results MsS,Pqand the 2nd approximation Secondly, we consider the density function pk( u) of A,,,(w) - A,, given by (14) where we have again approximations pk,,( u), I= 1, 2, 3 according to the order of 6 used. Then, the same conclusions can be made as in the previous section. That means, if we also consider the theoretical probabilities with respect to 1 and relative frequencies, that the best coincidence can be observed for the highest order I = 3. Some X2-values are given for this in Table 9, where the and two values of c with m = 6000 realizations are considered. The case (il), &,&) -A,, influence of the correlation length is also obvious. Finally, we can conclude that the simulation results confirm the theoretically obtained moments as well as distributions and also the better efficiency of the higher order approximations with respect to e could be proved.
Table 9 x2-values for the case (il) and LI~,~(w)- AkO in dependence on the order 1 of approximation I=1
I=2
I=3
E= 0.2
69.0
39.1
23.7
E= 0.1
35.1
36.7
22.8
e = 0.2
118.5
113.7
28.6
E = 0.1
56.0
34.9
28.1
k=l
k=2
208
J. vom Scheidt, B. Fellenberg / Some simulation results
References fields, FMC-Series 19 (1986) [l] B. Fellenberg and J. vom Scheidt, Probabilistic analysis of random temperature 15-24. [2] Proc. 2nd Conference on Stochastic Analysis, 27-30 October 1986; Special Issue of “Wiss. Beitr.” (IH Zwickau, 1986). [3] J. vom Scheidt, ed., Problems of stochastic analysis in applications, Special Issue of “Wiss. Beitr.” (IH Zwickau, 1983). [4] J. vom Scheidt and W. Purkert, Random Eigenvalue Problems (Akademie-Verlag, Berlin and North-Holland, New York, 1983). [5] J. vom Scheidt, Random Equations of the Mathematical Physics (Akademie-Verlag, Berlin) in preparation.