```EE-240/2009
EE-240/2009
EE-240/2009
1. Forma Preditora de Modelos ARMA
wk
C( q  1 )
yk
A( q  1 )
A(q-1) = 1 + a1q-1 + ... + anq-n
C(q-1) = 1 + c1q-1 + ... + cnq-n
Problema: Dados w j , j  k , w j ~ N(0,1), estimar yk+r , r  1
yk = wk + g1 wk-1 + g2 wk-2 + ...
= G(q-1) wk
ou,
yk+r = wk+r + g1 wk+r-1 + g2 wk+r-2 + ... + gr-1wk+1 + grwk + gr+1wk-1 + ...
“futuro”
~y
k
“conhecido”
ŷk
EE-240/2009
~
yk  yk  r  yˆ k
var~
yk   varyk  r  yˆ k 
 1  g12  gr2 1
Seja z uma combinação linear de w j , j  k
Então,
yk  r  z  ~
yk  yˆ k  z
varyk  r  z   var~
yk  yˆ k  z 
 var~
yk   varyˆ k  z   var~
yk 
ŷk é o melhor preditor linear de yk+r
EE-240/2009
Para representar ŷk em termos de yk e não wk:
yk+r = wk+r + g1 wk+r-1 + g2 wk+r-2 + ... + gr-1wk+1 + grwk + gr+1wk-1 + ...
D(q  1 )
F(q-1)wk+r
1
A(q )
wk
x A(q-1)
A(q-1) yk+r = A(q-1) F(q-1) wk+r + D(q-1) wk
= A(q-1) F(q-1) wk+r + D(q-1) q-r wk+r
A(q-1) yk+r = C(q-1) wk+r
C(q-1) = A(q-1) F(q-1) + D(q-1) q-r
EE-240/2009
c1 = a1 + f1
c2 = a2 + f1a1 + f2
C(q-1)
=
A(q-1)
F(q-1)
+
D(q-1)
q-r
...
cr = ar + f1ar-1 + ... + fr-1 + d0
...
0 = fr-1an + dn-1
yˆ k 
wk 
D(q 1 )
1
A( q )
wk
A( q  1 )
1
C(q )
yˆ k 
yˆ k
D(q 1 )
1
C(q )
yk
EE-240/2009
2. Forma Preditora de Modelos ARMAX
uk
q  r B(q  1 )
A(q  1 )
wk
C( q  1 )
A( q  1 )
+
yk
+
A(q-1) = 1 + a1q-1 + ... + anq-n
B(q-1) = 1 + b1q-1 + ... + bnq-n
C(q-1) = 1 + c1q-1 + ... + cnq-n
A(q-1) yk = q-r B(q-1) uk + C(q-1) wk
Analogamente ao caso anterior (ARMA), seja:
C(q-1) = A(q-1) F(q-1) + q-r D(q-1)
EE-240/2009
yk  r 
B(q  1 )
A(q
1
C(q  1 )
uk 
)
A(q
1
)
w k r
C(q-1) = A(q-1) F(q-1) + q-r D(q-1)
yk  r 
yk  r 
B(q  1 )
A(q
1
)
uk 
B(q  1 )
A (q
1
)
A(q  1 )F(q  1 )  q  r D(q  1 )
A(q
uk  F(q
1
1
)w k  r 
)
D(q  1 )
A (q
1
)
w k r
wk
A(q-1) yk = q-r B(q-1) uk + C(q-1) wk
A(q  1 )
B(q  1 )
wk 
yk 
uk  r
1
1
C(q )
C(q )
yk  r 
B(q  1 )
A(q
1
)
uk  F(q
1
)w k  r

D(q  1 )  A(q  1 )
B(q  1 )


y

u
k
k r 
1 
1
1
A(q )  C(q )
C(q )

EE-240/2009
yk  r 
B(q  1 )
1
A(q )
y k r  F(q
1
1
uk  F(q )w k  r
) w k r

D(q  1 )  A(q  1 )
B(q  1 )


y

u
k
k r 
1 
1
1
A(q )  C(q )
C(q )

D(q 1 )
 B(q 1 ) q r D(q 1 )B(q 1 ) 
uk

yk  

1

1

1

1
 A(q )
C(q )
A(q )C(q ) 

C(q-1) = A(q-1) F(q-1) + q-r D(q-1)
q-r D(q-1) = C(q-1) – A(q-1) F(q-1)
y k r  F(q
1
) w k r
D(q 1 )


 B(q 1 )
C(q 1 )  A(q 1 )F(q 1 ) B(q 1 ) 


yk 

uk
1
1
1
1


C(q )
A(q )C(q )
 A(q )

B(q  1 )F(q  1 )
1
C(q )
uk
EE-240/2009
3. Controle de Variança Mínima:
y k r  F(q
1
) w k r 
D(q 1 )
C(q
1
)
yk 
B(q 1 )F(q 1 )
C(q
“futuro”
~y
k
1
)
uk
Escolher uk para
anular estes termos
uk  
D(q  1 )
B(q
1
)F(q
1
)
yk
EE-240/2009
yN  1yN1    nyNn  1uN1    muNm  eN
T
aN
1   yN1   yNn uN1  uNm 
  1  n 1  m T

T ˆ
ˆ N1  ˆ N  KN1 yN1  aN
1N
KN1 

1
PNaN1
T
1  aN1PNaN1


T
PN1  I  KN1aN
1 PN
EE-240/2009
uN
yN 

T ˆ
ˆ N1  ˆ N  KN1 yN1  aN
 1N
KN1 
PN1 
yN
T
aN
1  eN

1
PNaN1
T
1  aN1PNaN1
T
I  KN1aN
 1 PN


ˆ N1
ˆ (q  1 )
D
uk  
yk
1 ˆ 1
ˆ
B(q )F(q )
Separação
Equivalência
à Certeza
EE-240/2009
Controle de Minima Varianca
10
8
6
4
2
0
-2
-4
-6
0
20
40
60
80
100
120
140
160
180
200
EE-240/2009
a1
a2
2
2
1
1
0
0
-1
-1
-2
-2
-3
0
10
20
30
40
-3
0
10
b1
20
30
40
30
40
b2
3
2
1
2
0
1
-1
0
-1
-2
0
10
20
30
40
-3
0
10
20
EE-240/2009
EE-240/2009
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