数学建模时间序列分析讲解

Holt
x^t xt1 rt
r~xtt
xt (~xt
(1 )(~xt1
~xt1 ) (1
rt1 )
)rt1
~x0 x1
r0
xn1 x1 n
Holt
l
x^T l ~xT l rT
2.7
1978----2000 Holt
~x0 x1 51259
xtn1 )
5
xt 4
xt 3
xt 2
xt 1
xt
~xt
xt 4
xt 3
xt 2 5
xt 1
xt
2.3:
5
5
5
1.1
4
1.2
7
1.3
8
1.4
11
1.5
18
2.1
3
2.2
8
2.3
10
2.4
13
2.5
21
3.1
6
3.2
9
9.6
3.3
13
9.4
3.4
17
9.6
3.5
2.2.2
Holt
~xt xt (1 )xt1 (1 )2 xt2
~xt xt (1 )~xt1
~x0 x1
0.050.3
x^T 1 ~xT
xT (1 )xT1 (1 )2 xT2
p 0q 0
E
(
t
)
0Var(
t
)
2
,
E(
t
s
)
0,
s
t
Exs t 0,s t
0 0 ARMA( p, q)
ARMA( p, q)
xt 0 1xt1 p xt p t 1t1 qtq
ARMA
^k ^kk
AR(P)
P
MA(q)
q
ARMA(p,q)
: k
N
n
(xt x )2
^(0) t1
n 1
k
^k
^ (k ) ^(0)
nk
(xt x )(xtk x )
^k t1 n
(xt x )2
,0 k n
t 1
^kk
D^ k D^
,0
k
n
1 ^1
^1
1 ^1
^ k 1
D^k ^1
1
^2
D^ ^1 1
^ k 2
^k1 ^k2
^k
^k1 ^k2
1.4
S-plusMatlabGaussTSPEviews SAS
----SAS SAS SAS/ETSSAS/ETS SAS
.
2.1
2.1.1
xt a bt It
E
(
I
t
)
0,Var
(
I
t
)
2
2.1 1981----1990
n
1
n
( (
x
t
1 2
n1 2
x
t
n
2
xt n11 xt xt n11
2
2
xt n 1 xt xt n 1
2
2
1 2
x
t
n1
)n
2
x
t
n
)n
2
5
xt 2
xt 1
xt
xt 1
xt 2
~xt
xt 2
xt 1
xt 5
xt 1
xt 2
n
~xt
1 n (xt
xt1
Q(~)
n
2 t
n
[xt
t
i xt1]2
i 1
i 1
i 1
57 OVERSHORTS
MA(1)
xt 4.40351 (1 0.82303 B) t
Var(^
2
)
2178
.929
1880-1985
ARMA(1,1)
r0
x23 x1 23
4325
0.15
0.1
.ARMA
ARMA ARMAARMA
t
t
1Ext (t )
2 rt,tk rt,tk rk
xt
t -- 0
2
ARMA
K
nk
(xt x )(xtk x )
^(k) t1
nk
n
xi
x i1 n
Tt ln Tt
a ln a b ln b
Tt a bt ct2
Tt a bt
2.2
Tt a bt ct 2 t2 t2
Tt 502 .2517 0.0952 t 2
2.2
2.2.1
n n
n
~xt
1
Y
1.
nk
(xt x)( xtk x)
^k t1 n
(xt x)2
t 1
^kk
D^ k D^
2.
^ k
q
^kk
P
AR(P) MA(q)
ARMA(p,q)
^k ^kk
k ^k ^kk
^k ^kk
()d 95 2 () d
ARMA1950--1998
1.1 1.2 1.3 1.4 2.1 2.2 2.3 2.4 3.1 3.2
y
166 52 140 733 224 114 181 753 269 214
4
280 295 307.875 315.500 323.625 341.750 357.875 374.875
3.3 3.4 4.1 4.2 4.3 4.4 5.1 5.2 5.3 5.4
6.5
18
30 25 20 15 10
5 0
1 4 7 10 13 16 19 22 25 28
2.4
.
1.1
1
1.2
2
1.3
3
1.4
4
2.1
5
2.2
6
2.3
7
2.4
8
3.1
9
3.2
10
.
166
3.3
52
3.4
140
4.1
733
4.2
224
4.3
114
4.4
181
5.1
753
5.2
269
5.3
x^T 2 x^T 1 (1 )xT (1 )2 xT 1 x^T 1 (1 )x^T 1 x^T 1
l
x^T l x^T 1 , l 2
2.5 xt
xT 10 ~xT1 10.5 0.25
(1) x^T 2
(2) x^T 2 xT 1 x^T1 ~xT 0.25xT 0.75~xT1 10.3
5.6
x^T 2
1 4
x^T 1
xT
xT 1
xT 2
5.6
5 5.4 5.8 4
5.45
2 x^T2
1 4
x^T 1 xT
xT 1 xT 2
1 4
1 4
xT xT 1 xT 2 xT 2
xT
xT 1
xT
2
5
16
xT xT 1 xT 2
1 16
xT
3
xT
5 16
^ 2 ^12
^1
^2
^2 ^12 1 ^12
3.11:MA(1)
MA(1) xt t 1 t1
0
1
(1
12
)
2
1
2
1
1 0
1 1 12
^1
1
1
2^1
4^12
3.12:ARMA(1,1)
ARMA(1,1) xt 1xt1 t 1 t1
1
1 0
(1 1)(111) 112 211
2.4
1
2
3
4
5
6
7
8
9 10 11 12
130 380 330 410 440 390 380 400 450 420 390
N=412
N4
y^12
y11
y10 4
y9
y8
415
n412
312.5 390 392.5 405 402.5 405 412.5 415 418.75
2008-8-2
y
210 860 345 203 233 922 324 224 284 822
4 397.750 405.875 407.375 418.000 423.125 423.125 432.125 426.000 417.000 427.000
6.1 6.2 6.3 6.4 7.1 7.2 7.3 7.4 8.1 8.2
. . . ARMA
.
1.1
7000
1.2
:
, X1, X 2 ,, X t ,
:n n
x1 , x2 ,, xt
1.3
1.3.1
11
1.3.2
xt a bt It ,t 1,2,40
E(It ) 0,Var(It ) 2
a^ 8498 .69 , b^ 89.12
2.1.2
Tt a bt ct 2 t2 t 2
Tt abt
Tt a bct
Tt
eabct
1
Tt a bct
y
352 280 295 930 345 320 390 978 483 320
4 435.875 450.750 463.375 467.500 484.375 502.250 525.500 542.750
1 5 9 13 17 21 25 29
1200 1000
800 600 400 200
28
10.0
4.1
4
10.4
4.2
8
11.0
4.3
11
11.6
4.4
16
11.8
4.5
24
12.4
5.1
3
12.4
13.2
5.2
9
11.6
14.6
5.3
10
10.8
14.2
5.4
12
10.4
14.0
5.5
20
9.6
13.6
6.1
1
9.0
13.4
6.2
5
8.6
12.6
6.3
7
8.2
12.4
6.4
10
12.6
0
y^T l
1 n
( yT l1
yT l2
yT ln )
yT li
y^T li
yT
l
i
,l i ,l i
l 1
y^T 1
yT
yT 1 n
yT n1
2.3
4 55.55.86.2
14 x^T 2
2 x^T 2 xT
1
x^T 1
1 4
xT
xT
1
xT
2
xT
3
5
5.4
5.8 4
6.2
1.3 2
2.1 22 AR(1)
57 OVERSHORT
1.1 2 2 1
2.
3. MA(1)
1880-1985
1. 2. 3. ARMA(1,1)
3.
p q 2
1, ,p,1, ,q, ,2
1(1, , p ,1, ,q ) ^1
214
5.4
11 12 13 14 15 16 17 18 19 20
.
210
6.1
860
6.2
345
6.3
203
6.4
233
7.1
922
7.2
324
7.3
224
7.4
284
8.1
822
8.2
21 22 23 24 25 26 27 28 29 30
352 280 295 930 345 320 390 978 483 320
MA
q
MA(q)
xt
t
1 t 1
2 t 2
qtq
q 0
E
(
t
)
0Var(t
)
2
,
E(t
s
)
0,
s
t
0 MA(q)
MA(q)
xt t 1t1 2t2 qtq
ARMA
ARMA( p, q)
xt 0 1xt1 p xt p t 1 t1 q tq
38
11.4
1
0.7^yt 17.500 17.500 18.550 18.235 19.485 19.519 18.703 19.182 17.838 17.946 19.282 20.848
^yt+1
25.000 26.500 26.050 27.835 27.885 26.719 27.403 25.482 25.638 27.546 29.782 32.248
1
--
k ^k ^k
1949----1998
AR
p
AR( p)
xt 0 1xt1 2 xt2 p xt p t
p 0
E(
t
)
0Var(
t
)
2
,
E(
t
s
)
0,
s
t
Exs t 0,s t
0 0 AR( p)
AR( p)
xt 0 1xt1 2 xt2 p xt p t
x^T 2 x^T 1 10.3
2x^T 2 x^T 1 xT (1 )xT 1
xT 0.25
2.6: =0.3
yt 0.3yt
1
25
7.5
2
30
9.0
3
25
7.5
4
32
9.6
5
28
8.4
6
24
7.2
7
29
8.7
8
21
6.3
9
26
7.8
10
32
9.6
11
35
10.5
12
Anna
37
450
400
350
300
250
200
150
100
0
2
4
6
8
10
12
y^tw
1 yt1
2 yt2
N
N ytN
1,2 , , N
N
i
i1 1 N
N 31 1.5,2 1,3 0.5
y^tw
1.5 yt1
yt 2 3
0.5 yt3
n n n
2 11
c
^1
^2 ^1
,
^1
c
c2 4 , c 2 2 c2 4
,c 2 2
,
c 112 2^2 1 ^1
p+q
Q(^) min Q(~)
n
2
min (xt 1xt1 p xt p 1t1 qtq )
t 1
xt 0 , t 0
p
q
(1
,
,p ,1,
Fra Baidu bibliotek
,q ) ^ pq
n
xi
^ x i1
n
^
2
1 ^12 1 ^12
^p2 ^q2
^
2 x
3.10:AR(2)
AR(2) xt 1xt1 2 xt2 t
Yule-Walker
12
1 2 1 11 2
Yule-Walker
^1
1 1