681.5.015
1
. .
,
2
. .
1
«
2
»,
,
[email protected], [email protected]
,
-
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:
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.
New approach for complex systems analysis based on discovering of direction impact between
processes using self-organizing algorithms is proposed. Input factors filtering, exogenous and endogenous factors sets determining and system processes relationships structure synthesis procedures
are offered. Technique proposed makes it possible to reveal possibility of causality sign reversing
between processes in the look-ahead period and significantly decrease time of the models system
synthesis.
Key words: cause-effect relation, influence direction, GMDH, system of processes, structure synthesis.
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[1])
-
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x1, x2,
m
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, xm,
.
, 2009
-
149
. .
,
,
. .
X (dimX = n?m)
-
:
X
xj, j = 1,
x11
x 21
x12
x22
x1m
x2 m
xn1
xn 2
xnm
,m
i = 1,
,
, n.
F
-
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.
Cm1
M
Cm2
... Cm1
[2]
-
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2m 1
.
«
»
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[3],
[2],
:
n
*
min
F
.
,
i(N)
,
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,
13
3
xi(+1) = f(a0, x1(0), x1( 1), x1( 2), x1(
3)
,
:
x2(0), x2( 1), x2( 2), x2(
13? 252
1- , 2- , , 13-
3)
x13(0), x13( 1), x13( 2), x13( 3)).
(i = 1, ,13),
-
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[4]
[2-3],
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, . .
[2-3]
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-
.
.
150
, 2009
[5-7]
(
).
[5]
.
:
x1 = f1(x2, x3, , xm);
x2 = f2(x1, x3, , xm);
.
xm = fm(x1, x2, , xm-1).
:
1 S
n , (B)
F
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S
i, i=1, ,m.
( ) 1.
m.
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[6]
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[5].
[5],
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[6]
-
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I(3).
[6]
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[7]
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[5-6]
[2].
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n , I(3).
[2-7],
-
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(
«
-
»
-
)
.
.
(3), (4)
xi(t + s ) = fi (xj(t), xj(t
), , xj(t
xj(t + s ) = fj (xi(t), xi(t
), , xi(t
xk+s,i = fi (xk,j, xk 1,j, , xk l,j),
, 2009
xi, xj,
(1), (2):
l )),
l )),
(1)
(2)
(3)
151
. .
xk+s,j = fj (xk,i, xk 1,i,
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. .
, xk l,i),
(4)
;s
); l
(
(
-
).
(
)
, s, l.
(3)
(4)
:
xi
i
xj
xj,
xi.
I
i
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j
,
i
,
min{ i, j} =
min{ i, j} =
,
(5), (6)
(3), (4)
:
I
=
j=
i
i
-
xj
xj
I
j
xi,
xi.
(5)
(6)
-
-
,
.
[7].
:
xk+1,i = fi (xk,i, xk 1,i,
xk+1,j = fj (xk,j, xk 1,j,
, xk l,i, xk,j, xk 1,j,
, xk l,j, xk,i, xk 1,i,
, xk l,j),
, xk l,i),
(7)
.
[7]
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»)
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(5), (6),
xj, xi
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(7)
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152
xj.
xj.
, 2009
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ji
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ij
ji
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ij
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[0,100],
:
ij
ji
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i
0,01
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0,01
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xi
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(3)
[8]
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1995
.)
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(4),
-
1
2
.).
1999
.
-
, 2009
153
. .
,
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.
(3)
1
: s = 1, l = 10, NA = 27, NB= 22, N = 49,
1
2.
.1
(4),
.3.
1
=1
.
.2,
2
2
7000
6000
.
5000
4000
3000
2000
0
10
20
30
40
1.
50
.
1
( 1),
(
,
( 2)
),
,
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1
(3), (4),
2
.2
,
.3.
-
1
2:
x1, k 1 = 6339,09 8349360 x2,k 5 x2,k1/3 x2,k 8 1 x2,k 2 1 x2,k 9 1/3 +87,3463 x2,k 7
x2,k 1 x2,k 31/3 x2,k 10 1 x2,k 4 1 14,5702 x2,k 6 x2,k 1 x2,k 10 2/3 x2,k 2 1/3 5262,91 x2,k2/3
x2,k 21/3 x2,k 8 2/3 x2,k 9 4/3 5130,33 x2,k 9 x2,k 71/3 x2,k 91/3 x2,k 2 1 x2,k 3 1.
(NMSE)
N
1
NMSE
( x1i
x1i ) 2
( x1i
2
i 1
N
:
0,73
x1 )
i 1
154
, 2009
,
-
2
1:
x 2, k 1 = 8313,01 18056100 x1,k 11/3 x1,k 21/3 x1,k 51/3 x1,k 71/3 x1,k 81/3 0,18474
x1,k 4 x1,k 7 x1,k 10 x1,k 1 x1,k 3 1 21384,29 x1,k 21/3 x1,k 91/3 x1,k 6 1/3 x1,k 1 1/3 x1,k
0,0275 x1,k 2 x1,k 3 x1,k 101/3 x1,k 1 x1,k 4 1/3 0,01916 x1,k3 x1,k 1 1 x1,k 10 1.
2
= NMSE
1/3
= 0,53298.
(x1)
(x2)
7000
6000
.
5000
4000
3000
2000
0
10
20
30
40
50
.
2.
.
(x2)
7000
(x1)
6000
.
5000
4000
3000
2000
0
10
20
30
40
50
.
3.
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,
2
1,
, 2009
2
= 0,53
2,
: 1
2
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1,
-
2.
155
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*
10%,
*
= 0,9,
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. .
= 0,1,
-
,
*
0,9.
(3)
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(4)
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I
= 0,53
2
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12
I
= 0,19,
2
<
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>
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:
1) xj
3) xj
xi,
xi,
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ij
2) xj
4) x j
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ij
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(3), (4), xj X \{xi},
xi
j
j,
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(xi, xj)
xi
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xi, xi
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-
xi
X ,
5-
xi.
(xi, xj), xi, xi
xj X \{xi}
:
{xi}+
156
,
xj
xi;
, 2009
{xi}
,
xj
{xi}0
xi
{xi}±
,
{xi}?
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xj
xi;
xi;
xj
xi;
-
.
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(3)
*
xj
)
xj xi, xj {xi}+,
xj
xi, xj {xi}0,
xi, xj {xi}+.
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(4).
xj ? xi, xj {xi}?,
xi, xj {xi}±,
,
xj
*
*
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0
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(
±
{xi}+
?
{xi} , {xi} , {xi} .
{xi}+, {xi} .
{xi} ,
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{xi}0, {xi}±, {xi}?
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x i.
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1) «
»
+
±
X = {xi} {xi} ,
:
xi
xi
{xi}0
{xi}
2) «
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+
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0
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X = {xi} {xi} {xi} {xi} ,
{xi}?.
xi
xi
{xi} .
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Q(
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4.
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157
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ji
1,
xj
{xi }
{xi } , i
ji
0,
xj
{xi }
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1,
ji
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j
{xi }? , i
j
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{xi }
xj
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{xi } , i
{xi }? , i
j
j
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,
{x1, x2, x3, x4, x5,}
(
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x1
(
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.), x3
-
(%), x4
(
.
.), x5
(
1995 .
xi, xj,
.
.).
1999 .
(3), (4) ,
,
.
(3), (4)
s = 1, l = 10
[8]
,
,
=1
,
*
= 0,9;
*
= 0,1.
-
NMSE.
158
x1
x2, NMSE = 0,73; x2
x2
x1
x3, NMSE = 0,7; x3
x3, NMSE = 0,76; x3
x5
x5
x5
x1, NMSE = 0,7; x1
x2, NMSE = 0,71; x2
x3, NMSE = 0,43; x3
x1
x2
x3
x5
x4, NMSE = 0,47; x4
x4, NMSE = 0,47; x4
x4, NMSE = 0,6; x4
x4, NMSE = 0,84; x4
x2:
x1, NMSE = 0,53,
x3:
x2, NMSE = 0,89,
x1, NMSE = 0,88,
x5:
x5, NMSE = 0,93,
x5, NMSE = 0,93,
x5, NMSE = 0,64,
x4:
x1, NMSE = 0,65,
x2, NMSE = 0,68,
x3, NMSE = 0,72,
x5, NMSE = 0,74,
: x2
x1.
: x2 x3;
: x1 x3.
: x5 x1;
: x5 x2;
: x5 x3.
: x1 x4;
: x2 x4;
: x3 x4;
: x4 x5.
, 2009
Q
(
.5)
(
5.
.6):
-
6.
.
.
*
,
{xi}
{xi}
0
?
{xi} , i = 1,
,m
= 0,9;
,
,
*
= 0,1
,
,
.
,
-
:
xk+1,1 = f1(xk,1,
xk+1,2 = f2(xk,1,
xk+1,3 = f3(xk,1,
xk+1,4 = f4(xk,1,
xk+1,5 = f1(xk,4,
, xk
, xk
, xk
, xk
, xk
l,1,
xk,5,
l,1, xk,2,
l,1, xk,2,
l,1, xk,2,
l,4, xk,5,
, xk
, xk
, xk
, xk
, xk
l,5);
l,2,
xk,5, , xk l,5);
, xk l,3, xk,5,
l,2, xk,3,
, xk l,3, xk,4,
l,2, xk,3,
l,5).
, xk l,5);
, xk l,4);
.
.
,
x2
x4
, x1
: x1
.
20NMSE1 = 0,335
x4, x2
x4
,
x1, x4
x4
NMSE4 =0,53
x1,
(
10-
)
.
,
,
:
.
NMSE2 = 0,31
x2, x4
NMSE4 = 0,31.
,
x2, x4
20-
, 2009
.
159
. .
,
. .
-
.
1,
2,
,
1995 .
22
(
(
.
.
(%); x4
(
.
.); x9
(
.
("
.); x12
(
.,
.(%
.
.); x17
(
.
.
.(
0,9;
.(
.); x20
.); x21
*
(%); x6
(
.
.); x8
(
.
.); x10
(
.,
-
.); x11
" %); x13
.); x14
); x15
(%
(
x1
.); x2
.); x3
(%); x5
(%); x7
(
1999 .
.
.); x16
$100
. .); x19
(
.); x18
,
,
.);
(
.
.); x22
: l = 10, NA = 27, NB = 19, NC = 3.
-
=1
,
*
=
= 0,1.
.
x3
-
:
x2
x13
x18
x20
= 0,88;
3 = 0,89;
3 = 0,78;
3 = 0,83;
3
= 0,96
13 = 1,06
18 = 1,04
20 = 0,99
2
x8
x14
x19
x21
= 0,87;
3 = 0,9;
3 = 0,82;
3 = 0,8;
3
7.
1
160
= 1,03
14 = 0,1
19 = 1,03
21 = 0,96
8
.
,2
,3
, 2009
x3
:
x3, k 3 = 99,175 0,000023 x22,k x21,k 52 x8,k + 1003909769 x14,k x2,k 1 x21,k 4 2 +
+ 13991031 x13,k 1 x21,k 5 2 x22,k 1 1 + 0,034 x14,k 5 x20,k 5 x14,k 1 x22,k 2 1 87212948
x8,k 2 x19,k 4 1 x20,k 5 1 x21,k 1 + 27754585087 x13,k 3 1 x20,k 5 1 x20,k 4 1 x13,k 1 1.
RE
(
)
NC
| xi
xi |
i 1
RE
N C | xmax
xmin |
x3
:
3
= NMSE
22
= 0,24902, RE = 0,1175.
,
-
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-
63%,
3
.
,
,
,
[2-7].
-
,
,
[2-7].
s.
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(3), (4)
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l(
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(
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s.
(
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,
),
(
Q)
xi, xj
s.
s = 1,
s ( . 8).
o
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*
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, 2009
161
. .
,
. .
, . .
-
,
(
. 9).
-
xi, xj,
,
s.
8.
9.
-
-
xj {xi}
{xi}±.
±
10,
{xi}±
(s = 2, s = 6).
xj
.
,
s
10.
ij
(
<
*
,
{xi}±
*
i <
. 11).
11.
-
162
, 2009
s, s = 1,
Qs,
.
12.
Qs.
s, s = 1,
(
. 12),
-
Q,
s,
.
:
*
1.
*
*
.
*
,
.
{xi}±
( * ,
,
*
).
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*
,
*
*
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,
-
.
2.
,
,
,
: xi = f(xj,xk), xi = f(xj,xk,xp),
,
,
,
.
.
,
.
-
,
, 2009
163
. .
,
. .
,
-
.
.
[1]
:«
. .,
», 1985, - 216
.
[2]
.,
. .
.
. .,
. .,
. .
,
, 1977,
-
.
6, . 27 40.
[3]
.,
. .,
,
.
[4]
. .,
. .,
. .
, 1979,
1, . 25 37.
. .,
. .2.
. .
.
,
,
III.
1979, 4, . 15 23.
[5]
.
. .,
. .,
.
[6]
. .,
.
.
3, . 11 19.
, 1982,
. .,
. .
II.
-
.
, 1983,
[7]
I.
. .,
3, . 3 11.
. .,
. .
III.
.
, 1984,
.
4,
.10 17.
[8]
. .,
. .
,
/
.4,
164
24,
.
, 2004.
. 51 56.
, 2009