CNwt
QwBU=@a
Qmi
C
O}yW
=
=Q u H
xm
jQ@ w ?
= x@
u s v
CavY
QO xS} w EL=@t
100 Q=DWwv
CU=Q} w
|vWwOv|wUwt O}aUO}U
s_mousavi@pwut.ac.ir
1391
R}}=B
x=oWv=O
?r=]t CUQyi
O
1
1
1
1
3
5
6
6
7
7
8
8
9
10
10
Q=DioV}B
v tR |=y|QU pw= pYi
| =
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. . . . . . . . . . . . . . . . . . . |= Q
. |rY Q}}e O | = Q s} Q
. . . . = Q s} Q
. . . | = Q Ov
. |rY Q}}e O
. . . . . . . . . . . . . . . | = = Q x RH
............... = O
. . . . . . . . . . R = xi w x RH
. . . . . . . . . . . . . . . . . . . . |oDU@t
. . . . . . . T = u}o =} h Qa
. . . . . |oDU@t T = w h Qa
. |oDU@t w T = w w ` w h Qa
. |oDU@t w x@ =L
. . . . =o |oDU@t
v tR | U
i C=
D w
vwQ '
y| U
v tR | U
i C=
D w
v tR | U
U D
U D
1111
J
2111
vwQ
3111
111
v tR | y| U
yp t
QO
y
r -t
}
D
}
D
y w
yO N w
221
v t
}
D
131
v } Q=w m
}
D
231
}
D
331
yO N
Q
v
hr=
v } Q= m D=
U
@= D
t
1331
y
2331
21
121
y
v } Q=w w
11
31
16
16
16
17
17
17
19
19
19
20
21
21
21
23
23
24
25
26
26
27
27
27
28
30
30
30
32
34
34
35
35
36
37
37
} B |iO=YD |=ypOt swO pYi
x =
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O
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.
=ov|oDU@ty
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. . . . . x Ok
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R
= x}@
112
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212
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312
X= N
412
}
QO |R U
w swO
x@DQt
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t
t
w
. . . . . . . . . . . . . . .|
. . . . . . . . . . . . . x Ok
. . . . . . . . . . . . h Qa
. . . . . . . QU =kD Qort
. | =Y = x@ Q w
. . . . . . . . . p =i Qort
..........
= x}@
. . . . . . . . . . . . . . w} Q
. . . . . . . . . . . . h Qa
. w} Q w | =DU = | =DU
. . AR(1) O x@ Q w
. . . AR(1) Ov Q =o |oDU@t
. . . . | R |oDU@t w ` =
..........
= x}@
. . . . . . xD = Q = O
=YD
iO
122
D
222
a
322
D t X= N
422
}
w
iO
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D uORs o swO
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v=
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a
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=
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}=
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332
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y
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x W |R U
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| tO | U
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}= |=ypOt swU pYi
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. . . . . . . . . . . . . . QLD u}o =}
. . . . w h Qa MA(q) Ov Q
= x}@
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. O = x}@ Q x xD = Q O
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. . . . . . . . . . . . . . . h Qa
. . . . . . . . . O x@ Q w
. . . . . . . . . . | QH p}rL ARMA
. . . . . . . . . . Q = x}@
l
X= N w
|R U
W w Q
}
t
y %
113
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x W |R U
W | U
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233
D %
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13
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VR= @ w |R U
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v
=
v t | yp t
=
O
| yp t
143
33
43
37
39
40
40
41
42
43
44
44
45
47
47
50
50
51
52
53
54
. . . . . . . . . . . . . . . . . . . . . . . . . . . p O@
}
Q
Q
D Mv | U
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. . . . . . . . . . . . |rY Q} ARIMA
. . . . . . . Q O} w Q p =i
. . . . . . . . . . . . |k}ir O
. . . . . . . . . . = =F h Qa
. . . . . . . . . Q = x}@
. . . . . . . . . . . . |rY ARIMA =
. . . . . . . . . . . . . . h Qa
............
Q x
. . . . . . . . . . . . . . . . ARCH =
. . . . . . . . . SP500 = Q
ARCH O h Qa O = = Q O
. GARCH = O x w Q = \U
xD =
Q GARCH O
= x}@
. . . . . . . . SP500 Q Q Q
. . . . . . . . s}r Q
O==
. . = x}@ |v} V} GARCH
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r D | U w
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534
k= | U QO |Q= } B v
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Q |twta Q=DWwv
Q= i=s v |= @
w x xm CiQo Q=Qk Q} R
CQ Y @
x@
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http://cran.r-project.org/doc/contrib/Mousavi-R-lang_in_Farsi.pdf
j} Q]
R= =Q
Q}kL sy
w |t
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=
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Q |t V}=Q} w
OO o
Q = x}rw= |}=vW 'xS} w EL=@t ?}kaD
Q= i=s v @
=yvW}B
|= @ |O
:::
w
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QyD
u=
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EL@t u}rw=
O@=}|t xt=O=
Q
Qw t
Q |t=Qo xOvv=wN xm CU= u}=
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R= x
R=
|r=N Q=L xR}Hw ,=trUt
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1391
S
QO x } w
|= @
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|vWwOv|wUwt O}aUO}U
u
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pw= pYi
|v=tR |=y|QU
|v=tR
x@ Ov=wD|t
=@
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x@
C= y
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C= y
Q |t O}m =D
=Ut |v=tR pY=wi =@ xDUUo C@F
|w
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w
1,2, ,n
w
CQ Y
w |t
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xm CU= |}=yOv}Qi C@F
QO
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QP
=Wv
x1 x2
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QO "O } B s
x@ |v=tR xawtHt '?U=vt |v=tR
xO=O u
R=
xn =
C=Q}}eD w
T
=}kt
w
w
|v=tR
g
CU= xOW C@F
"
1949-1960
=
xQwO QO Q t
wva Q} R R OwN QO |}=yxO=O p}=i R=
u}= "OyO|t u=Wv
:::
pmW
1
1
1
1111
y|QU s}UQD
u=
=NDv=
?
n
=
u} Qi=Ut O=OaD xm OwW|t xO=iDU= AirPassengers
w
O@t
s}UQD
Q
CQ Y
<=
xt : t = 1 2
'|v=tR |QU
=
| y| U
xDUUo
} w f
OvwQ
11
|QU
w
s}UQD |=Q@
Qiv Q=Ry ?UL Q@ xv=y=t |rrtr=u}@
=Q %p=Ft
> data(AirPassengers)
1
1391 '
|vWwOv|wUwt
2
> AP <{ AirPassengers
> plot(AP, ylab = "Passengers (1000's)")
O w| x
w window() = x R =
t
m OQ=O O Hw
s v
@
u @ R QO |
Qo}O `@=D "CU= q=@
= O
Q
| y )
m |= H=
pY=L
11
pmW
400
300
100
200
Passengers (1000’s)
500
600
v= D
1950
1952
1954
1956
1958
1960
Time
u} Qi=Ut xv=y=t |v=tR
Q
| U
V}=tv
%11
O}vm xHwD Q} R p=Ft x@ "OyO CUOx@
"
pmW
=Q
|v=tR
Q
x wtHt Q} R l}
| U R= |= a
> x <{ AirPassengers
> y <{ window(x, 1950, 1951)
CU= Q} R
"
w
CQ Y
x@
u
xH}Dv xm
Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec
1950 115 126 141 135 125 149 170 170 158 133 114 140
1951 145
"
O}vm xHwD Q} R p=Ft x@ x=t ,qFt 'O}W=@ xDW=O pQDvm p=U
R=
|rYi
|wQ
O}y=wN@ Qo =
> x <{ AirPassengers
> y <{ window(x, 1950, c(1951,0))
CU= Q} R
"
w
CQ Y
x@
u
xH}Dv xm
Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec
1950 115 126 141 135 125 149 170 170 158 133 114 140
O}vm xHwD Q} R p=Ft x@ =}
"
> x <{ AirPassengers
> y <{ window(x, c(1950,2), c(1951,3))
|v=tR
3
=
Q
| y| U
1
pYi
Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec
1950
126 141 135 125 149 170 170 158 133 114 140
1951 145 150 178
O M} Q=D xm O}vm xHwD Q} R
x = = |v=tR
"OQ= v
V1 V2 V3 V4 V5
3.59 2.20 4.53 7.01 10.45
9.60 19.12 21.93 24.20 25.22
3.13 3.97 17.52 11.77 9.65
2.76 4.17 6.71 10.57 9.87
1.22 2.81 6.25 9.07 28.12
...
...
...
...
1
2
3
4
5
x@
wtv sUQ
"O
=Q
=yu TBU
V6
43.86
42.28
18.26
7.58
20.00
...
Q sy QU CWB
w xO=O Q= k
"
CU= Q} R KQW x@
V7
59.97
37.32
72.26
15.45
36.48
...
=
=
Q
| U
x@ uwvm =
V8 V9 V10 V11 V12
92.29 47.00 19.16 12.45 8.84
14.11 6.75 4.40 2.62 2.82
32.45 12.09 5.12 3.03 2.46
29.85 4.25 1.74 1.12 1.04
45.32 12.19 2.98 2.85 2.19
...
...
...
...
...
=
O}=@ =OD@= xv=y=t
=Q |Q t | yp U
w
| v y t
= O
pw= CQ Y | y )
m "O=O s
=Hv=
=
| yxO=O
= u}=
s}UQD
w |t
=Q Q m
Q
|= @
w
u= D
CQ Y wO
> x <{ matrix(scan("F:/R_les/data/ghar.txt"), ncol=12, byrow=T)
> y <{ t(x)
> y <{ matrix(y,ncol=1, byrow=T)
> ts.plot(y)
CU= Q} R KQW x@
"
w
= O
swO CQ Y | y )
m
> x <{ read.table("F:/R_les/data/ghar.txt")
> y <{ as.matrix(y)
> y <{ t(x)
> y <{ matrix(y, ncol=1, byrow=T)
> ts.plot(y)
2111
v tR |QU OvJ
| =
O}rwD p}=i
1990
u}=
w
=
p U
=D
1958
u)D ?UL Q@
"OQ=O O Hw =
=}r=QDU=
qmW
w p U R=
C
qFt
QO ,
=Wv
"O=O u
w
w x
s-= D CQ Y @ =Q
|v=tR
QD}r uw}r}t ?UL Q@ Q}aWr=<=t 'Ca=U
w |t xOv=wN
"O W
RO
)
m
\UwD
Q
| U
OvJ
w |t
u= D
wr}m ?UL Q@
C=w
online
w x
Q
CQ Y @ | U
Q
j @
xU
> "http://www.massey.ac.nz/~pscowper/ts/cbe.dat"
> CBE <{ read.table(www, header = T)
CBE1:4, ]
CU= Q} R
"
w x
=
= O |HwQN
CQ Y @ q @ | y )
m
1391 '
|vWwOv|wUwt
4
choc beer elec
1 1451 96.3 1497
2 2037 84.4 1463
3 2477 91.2 1648
4 2785 81.9 1595
|=Q=O w
Q
OvQ=Ov M} Q=D Qo}O
wtv p}O@D |v=tR
|= @ "O
=@ x@ "OvDUy x=t
CQ a
Q
| U
x@
ts() ` =
=
w p U
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@ D R= xO
Ok=i q=@
=}
=Q < W=
=
w |t x_Lqt xm Qw]u=ty
| yxO=O O W
u}= O}=@ u}=Q@=v@ "OvDU}v |v=tR
O}vm xHwD Q} R
"
Q
=D =
| U Q N U
= O x@ Q=m u}=
| y )
m
> "http://www.massey.ac.nz/~pscowper/ts/cbe.dat"
> CBE <{ read.table(www, header = T)
> Elec.ts <- ts(CBE, 3], start = 1958, freq = 12)
> Beer.ts <- ts(CBE, 2], start = 1958, freq = 12)
> Choc.ts <- ts(CBE, 1], start = 1958, freq = 12)
> plot(cbind(Elec.ts, Beer.ts, Choc.ts))
"
CU= q=@
= O
10000 14000
6000
150
6000
2000
Choc.ts
100
Beer.ts
200 2000
Elec.ts
cbind(Elec.ts, Beer.ts, Choc.ts)
1960
1965
1970
1975
1980
1985
Time
|v=tR
1. trend
2. seasonal variation
Q
| U
OvJ V}=tv
%21
pmW
Q
| y )
m |= H=
1990
pY=L
wtv
21 Q=O
|v=tR
5
=
Q
| y| U
3111
i C=Q}}eD w OvwQ
|rY
3 CQB Q}O=kt '2 |rYi
= =}
,v L= w
|rm Cr=L
'
u}@D
=
Q t
Q
QO w
Q}}eD '1 OvwQ xmr@ OyO|t CUOx@
Q
C=
CU= |rYi
Q}}eD p=U Qy
C=
O u} QDxO=U "Ov} wo OvwQ 'OvwW|tv Qy=_ |@ w=vD
w |t 'OvwQ K=w
u= D
= |t Qy=_ R}v
QO 'OR U
w x xm |v=tR
|= @ p t
`}tHD =@ |rYi C=Q}}eD Q}F-=D
wor= =yvD xv |v=tR
=Q | U |
wor= Q=QmD q=@ p=Ft
QO
CQ Y @
uOQw
CUOx@
Q
=Q
Q
sUQ
| U
=]N Q}O=kt
l}D=tDU}U
| U QO
pYi
1
Q}}eD
C=
Q CU= |]N Vy=m w V}=Ri= 'OvwQ
|= @ "
Qy Q}O=kt xYqN "OwW|t s=Hv= aggregate() `@=D R= xO=iDU= =@ R QO Q=m u}= "OQ@ u}@ R= xvq=U Q=t x@ xv=y=t
= |t u}at =yxO=O R= l} Qy |=Q@
"OR U
pYi cycle() `@=D w OwW|t XNWt boxplot() `@=D \UwD pYi
=Q
> data(AirPassengers)
> AP <{ AirPassengers
> layout(1:2)
> plot(aggregate(AP))
> boxplot(AP ~ cycle(AP))
"
Q |t x_Lqt xm Qw]u=ty
w |t xOv=wN
"O W
RO
)
m
\UwD
online
= O
Q
| y )
m |= H=
w x xv=y=t
CQ Y @
pY=L
=m}@
|Q
wtv
31 Q=O
=
| yxO=O
%p=Ft
2000
5000
aggregate(AP)
OO o
CU= q=@
1950
1952
1954
1956
1958
1960
100
400
Time
1
=yv
ts() ` =
@ D R=
|v=tR
Q
| U
boxplot
x@ =yv p}O@D
`}tHD `@=D \UwD xvq=U \UwDt
2
3
O
w x W
Q
|= @ w
4
6
7
`}tHD xvq=U
u QO
8
9
Q
V}=tv
| U
OvDU}v |v=tR
Q |t x_Lqt
"OO o
5
Q
10
| U
freq
11
12
%31
pmW
=yxO=O u}= "CU=
=v
Q
?w D xQwO w `w W
unemploy Q}eD
QDt=Q=B xm
w |t
'O W
=iDU=
xO
w |t u}at s}UkD Qorta
"O W
> www <{ "http://www.massey.ac.nz/~pscowper/ts/Maine.dat"
> Maine.month <{ read.table(www, header = TRUE)
3. outliers
=
t s v
w
1391 '
|vWwOv|wUwt
6
> Maine.month.ts <{ ts(unemploy, start = c(1996, 1), freq = 12)
> Maine.annual.ts <{ aggregate(Maine.month.ts)/12
> layout(1:2)
> plot(Maine.month.ts, ylab = "unemployed (%)")
> plot(Maine.annual.ts, ylab = "unemployed (%)")
CU= q=@
= O
Q
| y )
m |= H=
pY=L
wtv
41 Q=O
3 4 5 6
unemployed (%)
"
1996
1998
2000
2002
2004
2006
4.5
3.5
unemployed (%)
Time
1996
1998
2000
2002
2004
Time
=yv \UwDt
=m}@
w |Q
Q
| U
V}=tv
%41
pmW
|v=tR
|=y|QU
21
x} RHD
1
2
1
=ypOt
=ypOt u}=Q@=v@
w |t xOy=Wt |rYi
"O W
O = |t Q} R
" W @
Q}}eD =}
C=
w
OvwQ =y|QU
w x xm 'CU= 4 |atH pOt '|v=tR
CQ Y @
Q
R=
|r}N
=
| U xO U
O
QO ' W
x_Lqt ,q@k xm Qw]u=ty
x} RHD l} "OvDUy |}=yxirw-t
|=Q=O
xt = mt + st + zt
s
xirw-t "CU= |rYi C=Q}}eD xOv}=tv t
w
OvwQ xv=Wv
mt xi w
r -t "
CU= xOW xOy=Wt
Q
| U
O
=
w u tR
|t u=Wv
" yO
u
xt
=Q
t
u QO
xm
z
=]N R}v t
=Wv Q} R CQwYx@ xm 'CU= 5 |@ Q pOt l} ?U=vt pOt 'OwW Qy=_ OvwQ l} CQwYx@ |rYi C=Q}}eD Qo =
w |t
"O W
xt = mt :st:zt
0
4. additive
5. multiplicative
0
0
xO=O
|v=tR
7
Q |t p}O@D |atH pOt x@ |@ Q pOt 'Q}N= x]@=Q
"OO o
R=
=
Q
| y| U
pYi
1
sD} Q=or uDiQo =@ OwW|t x_Lqt xm Qw]u=ty
yt = ln(xt ) = ln(mt ) + ln(st ) + ln(zt )
= mt + st + zt
0
0
0
R QO
Qo = "Ovm|t
x@
Q
OQw @ l
w Q
=F
QLDt u}ov=}t
Q
=iDU= =@
VwQ R= xO
Q |t O=H}=
\ @ t xO=O p t |= @ "OO o
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51
Q=O
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C=
wtv xU =@ pmW l} x=ov
pmW |@ Q
w
|atH pOt
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w
w x Q} R
CQ Y @
decompose() ` = R
plot() ` = p
w `=
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Q}o Q=Qk
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1
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m QO 'q @ p t j @
pmW OOQo|t x_Lqt =Hm} |rYi C=Q}}eD w OvwQ Q=Owtv l} |wQ xQNq=@ w OwW|t p=@vO |@ Q pOt =yO)m
"61
trend
seasonal
14000
8000
12000
2000
1.10 2000 6000
1.00
0.94
random
1.06
0.90
1.00
observed
Decomposition of multiplicative time series
1960
1965
1970
1975
1980
1985
1990
Time
Q
= xirw-t V}=tv
| U | y
%51
pmW
31
|oDU@ty
|oDU@ty u}vJ O}=@ =yv p}rLD
=
| yxO=O
j} Q]
w
x} RHD
Q
w Ovy=wN xDU@ty |r=wDt
|= @ "O @
w |t h} QaD |oDU@ty `@=D \UwD |v=tR
R= w O W
=
Q
| y| U QO
= Q}eDt
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Q |t
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QO
Q u}at
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Q O
O =Wt
OQw @ x W x y
|vWwOv|wUwt
8
2000
4000
6000
8000 10000
14000
1391 '
1960
1965
1970
1975
1980
1985
1990
Time
|rYi
Q}}eD
C=
w
OvwQ V}=tv
pmW
%61
Tv=}
Q}eDt u}ov=}t
x@ xm CU=
E (x) u Q =v
CU= xat=H u}ov=}t `k=w
w |t
}= @ @ "
w
QO 'O W
E (x )2 ]
= QLv= `@ Qt u}ov=}t
;
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Q=w w
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w |t
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2 =
@
CU= h} QaD p@=k Q} R
"
E
Q
=@ xm |=} Q O}t=
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V}=tv R}v
xm OwW|t xDN=vW
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w
1
3
1
u}ov=}t h} QaD
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CQ Y @
= x
x Q}eD
@
CU=
m
t
x
Tv=} Q=w
2
3
1
h} QaD
(x y) Q}eD
t wO
Qo =
(x y) = E (x x)(y y )]
;
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n
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xR= v=
x wtv l} Qo =
|= v
CU= Q} R
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w x
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w
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(x y)
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x]@=Q xm Owtv
Cov(x y) =
;
t wO
Q
OQw @ =Q
u}@ |]N
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R}t `k=w
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QO
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u= D
(xi yi)
X
(xi ; x)(yi ; y)=n
w x
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t wO
u}@
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| y
w |t h} QaD Q} R
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(x y)
= Q}eDt u}@
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x y
x y
;
;
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Q
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+1
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9
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Q
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pYi
1
w x xvwtv |oDU@ty ?} Q
CQ Y @
Cov(x y)
sd(x) sd(y)
Cor(x y) =
|oDU@tyOwN
w
Tv=}
Q=wmwD=
3
3
1
`@=wD h} QaD
w x |v=tR |QU l} u}ov=}t `@=D "OwW|t xDN=OQB u}ov=}t x@ =OD@= |v=tR |=y|QU |oDU@ty h} QaD |=Q@
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O = |t
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(t) = E (xt )
Q
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QO
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Q
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x =
O = |t Q} R
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t
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CU= Q} R
w
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CU= Q} R
w x
"
n
X
R=
x wtv
CQ Y @ u |= v
xt =n
i=1
w x CU= =DU}= u}ov=}t
CQ Y @ '
QO
xm |v=tR
Q
| U
l} Tv=} Q=w `@=D
2 (t) = E (xt )2]
;
w x
x wtv OQwQ@ "OOQo|t
CQ Y @ u |= v
2 C =
x@ p}O@D
@ F
Var(x) =
=
=
x
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P
CU= Q} R
"
(xt ; x)
n;1
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2(t)
w
2
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| ys o Cw
"
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xm O}Owtv x_Lqt
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w x
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R=
wtv u=}@ Q} R
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k = E (xt )(xt+k )]
;
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w |t h} QaD Q} R
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k = k2
O = |t
" W @
6. stationary
7. lag
0 = 2
Q
= }R '
8. Autocorrelation
CU=
0 = 1 x
m
CiQo xH}Dv
9. Serial correlation
w |t q=@ h} QaD
u= D
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R= xO
1391 '
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10
1331
yOwN x@U=Lt
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x y
n 1
;
w |t "OW=@ xDUUo |v=tR
Q
u= D
x1 x2 l}
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xn
n O}v
O =Wt
x y
Q
wvm =
m Z i u
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(x1 x2 ) (x2 x3 )
Q
Z i swO
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u=
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w pw=
(xn 1 xn)
;
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CU= Q} R
QO =Q x y
w
"
nP
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"
;
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C= y
;
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n
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w |t j} Q] u}ty x@ "CU=
u= D
Q
n
P
x = n1
t=1
xt
u QO
xm
k = 0 1 2
2
u}=
nP
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xt x1 = n 1 1 xt
x
t=2
t=1
x Q}eD u} |oDU@t `
x
r1 O = R | = O x n Q
(xt ; x) (xt+1 ; x)
(xt ; x)
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2
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n
P
n
P
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nP
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;
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t=1
wtv x@U=Lt
xt+1 x(2)
x2 = n 1 1
CU= hrDNt
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t=1
CU=
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"
r1 =
;
;1 ;
2 nP
x@ |oDU@ty ?} Q 'O}vm
CQ Y
xt x(1) xt+1 x(2)
;
=Q
Q x@U=Lt
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Q
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r
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r
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w
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10. correlogram
|v=tR
11
Q}oxvwtv |v=tR xrY=i x@ |oDU@ OyO|t u=Wv
"OQ=O |
=F
p t u=
wvax@
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xO
R
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lag.plot() ` =
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sy x@ C@Uv hrDNt
=
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W OO o
N-D Q
U D
t
J
D
| yxO=O '
pYi
1
=yQ}N-=D xm =ypw]
= Q}N-=D s}UQD
Q
| y
> lag.plot(LakeHuron, lag=4, labels=F, do.lines=F, diag.col = "red")
C lag=4
Q
pm
Q | s} Q Q} = =y = LakeHuron =
|= @ "71
Q
| y| U
U=
w
=
|= @
=F
u J 'q @ p t QO
LakeHuron
576 577 578 579 580 581 582
LakeHuron
576 577 578 579 580 581 582
LakeHuron
lag 3
lag 4
576 577 578 579 580 581 582
Q}N-=D
CU= Q} R
"
w x
CQ Y @ u
576 577 578 579 580 581 582
lag 2
LakeHuron
lag 1
|rm pmW
w |t
"O W
=
| yQ=O
=iDU=
xO
wtv
%71
pmW
acf() ` = R
@ D R=
QO Q
=ov|oDU@ty s}UQD
w
rk x@
=Lt
U
acf(x, lag.max=NULL, type=c("correlation","covariance","partial"), plot=TRUE)
u QO
x
w Q} = Oa lag.max
Q | x@ =L 10log(N)
= QD =
xD
type
CU= =yxO=O
"
x]@=Q
|
R=
xm CU=
Q V}B
Z i
O = |t
" W @
"correlation"
Q
R
|=Q=O
10
P
=v@t
|
= w
'OO o h L u t oQ
Q
= w
P
'OO o h L u t oQ
QO u
sD} Q=or
u}= Qo = "CU= Q_v
= x wtv
w y v
u}= Qo = "Ovm|t u}at
=Q
O
xR= v=
acf ` =
N-D O=
OQ t
Nx
m 'OO o
w
CU=
@ D ` v w
xm
t
y
"
U
Q
Q=O @ %
D %
t
m =Q m R= |=
WQ %
CU= `@=D u}=
Q V}B
Z i
Q |t sUQ Q=ov|oDU@ty Q=Owtv CQwY u}= QO w CU= TRUE ZQiV}B |=Q=O |k]vt u=twoQ u}= %plot
"OO o
w |t
"O W
R=
=
A J
=yQ}N-=D
=yv x@ |UQDUO
k=0 Q
o= "
Q
|=R=
x@
acf Q
}O
=kt
Q |t xQ}NP
|= @ w OO o
w |tv sUQ xOW xDio
wtv 'OW=@
w O W
acf(x)$acf
Q=O
w x
Q
CQ Y @ |Q=O @ QO
FALSE
= w
u}= Qo =
u t oQ
Q}O=kt u}= xm O}W=@ xDW=O xHwD
CU= acf$acfk+1] CQwYx@ q=@ Q=OQ@ u}=Q@=v@ "OwW|t xO=iDU= Q=OQ@ T}Ov= u=wva x@ Q}N-=D u=tR
"
O}}=tv x_Lqt
"
=Q
acf$acf1] Ok
acf ` = =F wv x
CU= l} Q@=Q@
Q} R
= O
| y )
m
Q=
@ D p t u=
O =
t ' W @
wvm =
a @ u
1391 '
|vWwOv|wUwt
12
> www <{ "http://www.massey.ac.nz/~pscowper/ts/wave.dat"
> wave.dat <{ read.table (www, header=T)
> par(mfrow=c(2,1), mar=c(2,4,2,2))
> plot(ts(waveht), ylab = 'wave height (mm)')
> acf(waveht, main="")
CU= q=@
= O
Q
| y )
m |= H=
pY=L
wtv
81 Q=O
−500
0
wave height (mm)
500
"
100
200
300
400
−0.5
0.0
ACF
0.5
1.0
0
0
5
=ov|oDU@ty
u Q
10
w
15
|v=tR
Q
| U
w |t x@U=Lt Q} R
"O W
> acf(waveht)$acf2]
1] 0.4702564
w |t x@U=Lt Q} R
"O W
w x O =
CQ Y @ ' W @
20
V}=tv
25
%81
pmW
CQ Y @
k=1
k=1
Tv=} Q=wwmwD= x@U=Lt Qw_vt Qo = ,=OOHt
w x
QO
acf
QO
Q=
Okt p@k p=Ft xt=O=
QO
> acf(waveht, type = c("covariance"))$acf2]
1] 33328.39
r
O N
xrY=i xv=Wv xm OvQ=O OwHw =y k u}}=B
O = |t =yxvwtv
" W @
r
xR= v=
u}@ R= k Q}O=kt Qo =
"O
x@ O}=@ =Hv}=
xD@r=
"O m OQ =Q
uwQO
Q}O=kt xm
\N
wO
QO
xrY=i
Q}o|t Q=Qk
Q
xO
=
u QO
xm 'CU=
=iDU= OQwt
%5 K]U
u}J]N
w q @ QO
QO
Q
| U
Q@=Q@ %5
2
p
w x \N
CQ Y @
N
w
}
QiY
CU= |vFDUt xOa=k u}=
u swO "
O}vm|t x_Lqt xm Qw]u=ty
|vat K]U =@
uO @ Q=O
|oDU@ty u}}aD |=Q@
k = 0 |va
wO
Q
u R=
xm CU=
R=
=D =
|Q N U
w |t x=ov 'Ot
Z i u= D
OvDUy u=v}t]=
w
Q}@ Q}N=
uw
r0 = 1 x
u}=
=D
wO ' yxO=O O=
m
pw= "
rk Q
=kt
}O
w]N
\
CW=O xHwD xDmv xU
Q}N-=D 40 |=Q@ ,qFt %5 K]U QO xm u swU "OvQ=Ov |Q=O|vat Cw=iD QiY =@ =t= OvW=@v QiY ,=k}kO CU= umtt
"
Q
O}vmv
=
|Qw=O u T U=
=ov|oDU@ty
O
QO " yO
Q@
|t
O}v=Ov =]N pwtWt
w
=Q =Wv xDUy Vy=m l}
u
r0
<
RH@
w x
Q}@ =D
uO i= uw
CQ Y @ Q
=ov|oDU@ty
=
OaD =@ ?U=vDt
w
QO =Q O N
=yxO=O OvwQ ,qwtat
|v=tR
13
w
CU=
Q
x@=Wt Q}O=kt
| U QO
|W=v xm C@Ft
R=
R Q}O=kt
O}vm|t x_Lqt
Q
OvDUy syx@ l}ORv
"
1
acf(AirPassengers)
pmW
91
w nQ @
=
| y| U
=
| yxO=O
=
=Q "
pYi
=
| yu tR QO
−0.2
0.0
0.2
0.4
ACF
0.6
0.8
1.0
Series AirPassengers
0.0
0.5
1.0
1.5
Lag
AirPassengers
=
Q
xvq=U
QO ' yxO=O | U
=v
=
| yxO=O Q
=ov|oDU@ty
w |t Qy=_ Q=ov|oDU@ty
?w D "O W
%91
pmW
O = xDW=O OwHw R}v |rYi
|wQ ' W @
Q}}eD Qo =
C=
x]@=Q xm CU= p=Um} Q}N-=D QO |oDU@ty Q=Okt QFm =OL "OwW|t x_Lqt ?w=vD u}ty =@ R}v u Q=ov|oDU@ty
=D
w
Tmar=@ "OyO|t u=Wv
Q iQ u J
=@ |ivt x]@=Q xm Ov=xOW =OH sy
"
xm |rYi
P
h L
Ov=xOW =OH sy
=Q
=
R= p U
0.5
=
x t
=
R= x t
6
12 \
Q}}eD
w
OvwQ
P
h L R=
TB |v=tR
Q
Q |t s}UQD
111
pmW
| U QO
=ov|oDU@ty
QO Q
pkDUt
July
D
D w
w x
U=
=
W
U
m y
QO
R=
|rY=
=
w 101
}= t
=iDU= =@ Q} R O)m
xO
pmW
m
QO
Q
TB Q=}at
=}at
QLv= xm O}vm|t x_Lqt
h=
QLv=
h=
w
CU=
41 O
Q O
|= @ " vQ=O
111
P
vwQ h L R=
w
=Hv=
|
TB 'Q=}at
h=
O
h y
CU= "Ov=xOW OQwQ@
Q
| U
pmW
TBU
QLv= "CU=
Q |t
"OO o
=ov|oDU@ty
QO Q
w@v QFw-t ,qt=m
xO
QDW}@ |UQQ@ Qo = "Ovm|t
'O W s
sy
QO "
|iO=YD xirw-t
OyO|t u=Wv
Q}}eD
C=
|oDU@tyOwN X}NWD Q=ov|oDU@ty |rY=
= xirw-t x@ x} RHD =Q} R OUQ|tv CUQO Q_v x@ Q}@aD u}= "CU=
| yxO=O | U Q
R=
u}@ |]N C@Ft
xm q=@ Q}O=kt p=Ft
CQ Y @ =Q x t xOR=wO ' y
Q}}eD K}LYD
C=
=DU@=D
MQ u
> data(AirPassengers)
> AP <{ AirPassengers
> AP.decom <{ decompose(AP, "multiplicative")
> plot(ts(AP.decom$random7:138]))
> acf(AP.decom$random7:138])
|rY
Q}}e K}LY C pm | wv}U = rk | Q} x
i C=
GwR
w x Q}O=kt 'CU= |Uwv}U ,=@} QkD |rYi
OvwQ xirw-t w K}LYD |rYi C=Q}}eD decompose() `@=D R=
"OO o
(xt xt+12 )
CQ Y @
CU= u=DUtR lJwm Q}O=kt p=@vO x@ OyO|t
C=
w xm
U D
Q
OQw @
109 Q Q June =
@= @
D
|vWwOv|wUwt
14
0.90
0.95
1.00
ts(AP.decom$random[7:138])
1.05
1.10
1391 '
0
20
40
60
80
100
120
Time
AirPassengers |
=YD xirw-t
iO
O}vm xHwD Q} R
"
= O x@ =yQ=}at
| y )
m
> AP <{ AirPassengers
> AP.decom <{ decompose(AP, "multiplicative")
> sd(AP7:138])
1] 109.4187
> sd(AP7:138] - AP.decom$trend7:138])
1] 41.11491
> sd(AP.decom$random7:138])
1] 0.0333884
%101
pmW
QLv= x@U=Lt
h=
Q CU=
|= @ "
0.03 Q Q
@= @
|rYi
|v=tR
15
0.4
−0.2
0.0
0.2
ACF
0.6
0.8
1.0
Series AP.decom$random[7:138]
0
5
10
15
20
Lag
AirPassengers |
=YD xirw-t Q=ov|oDU@ty
iO
%111
pmW
=
Q
| y| U
1
pYi
swO pYi
x}=B |iO=YD |=ypOt
|}=yQorta ,=vt "OwW|t EL@ 'OQ=O |Oa@ pwYi QO |O=} R OQ@ Q=m xm |iO=YD |=ypOt R= |=xQ=B pYi u}= QO
O QD?U=vt X}NWD
p t
Q |t QDK=w
"OO o
w
Q
OO
R
|= @ | } H Q= @= w
O@=}|t
Ovvm|t
=iDU= |r}N =yv
xO
QDW}@ \U@ R}v xDWPo
|
= EL@
| y
R=
|v=tR
=
Q
| y| U
= u}vJty
R= |=xQ B
xm
Q |t |iQat
OO o
Q}o|t Q=Qk |UQQ@
1 O}iU xiwv
Ovm|t u=}@
xt |va
t
=
= O =t}k=@
} yx v
O
O =Wt
=
u tR QO =Q p t w x y
Q
w
| U CQ Y
u}=
Q
| yxO=O | U
CU=
QO "
Zw
u}@
Cw
=iD
R=
y
|W=v
Qit pOt \UwD ^t |va}
|
=]N
O
O l}
'p t
1
1
2
O =t}k=@
QO x v
Q Q}O=kt
u x W OQw @
w
CU= Q} R
"
xt = yt y^t
;
1. White noise
16
OQ t
12
xtOkt
"
w
"O
yt
Q
| U
w x
CQ Y @
x}=B |iO=YD
17
Tmavt Q=ov|oDU@ty
QO
|rYi
OvW=@ xDU@ty O}=@v =yxOv=t}k=@
Q}}eD =}
C=
Q
w
OvwQ Ovv=t |v=tR
Q
Q |t Q} R h} QaD ?Hwt xO}= u}= "OyO u=Wv
"OO o
2
=
Q
= w xm OW xO}O u}vJty
wn
xH}Dv
p
w |t
u= D
w |t
wor= `wv I}y O}=@v =yv Q=ov|oDU@ty =Pr
|
w |t x=ov 'Ovm
u= D
w : t = 1 2
n
= Q}eDt Qo = 'Ov} wo 2 (DWN) xDUUo O}iU xiwv =Q f t
| y
qkDU= C}Y=N
|w
2 T =
OvW=@
R= "
v } Q=w w
Q}B p=tQv p=tDL= |r=oJ `@=D
QiY u}ov=}t =@ u=Um} `} RwD
g |v=tR |QU
pkDUt sy
|=Q=O w
CU=
"
m
m
W=O
Uw o
U
|a@
O \UwD xOW
=Q p t
= x}@W
Q
|R U
w |t
| U "O W
=iDU= =yxO=O 3 |R=Ux}@W
xO
Q R}=tDt xOW xOy=Wt
"OO o
Q
w |t
|= @ u R= u= D
rnorm(100)
w
O}=tv O=H}= xOv}
=F
Q O = |t
p t |= @ " @ }
Ovm O=H}=
"
=Q
Q
|= @ =Q
= =
|}=yw} Q=vU Ov=wD|t
R
|@ wN@ Q=m u}=
u t U
OvDUy
=
| yxO=O R=
i v | U
3
1
2
Q xDi=}
VR= @ p t
O =D
= Q
=F
wtv
QO "O
Q CU=
=iDU=
O
xO
=Um} `} RwD
OQ= v U= p t v u
|= @
|=Q=O
"
O}it Ov}Qi
= QDt=Q=B
p t | y
= Qy
QO
xm |vat u}O@ 'OW=@ u=Um}
= |}=D100 hrDNt
| y
= Q
| y= H=
= x}@W
|R U
=@ OwW
P
h L
"
Q
= Qy
|= H= Q @
`@=D u}= Qo = =t=
CU=
12
Q
`w W
pmW q=@
2 T =
v } Q=w w
=v}t]=
=
xR @
= O x@ uwvm =
= O
xH}Dv
Q
w |t
v= D
| y )
m
set.seed() ` =
100 x = Q Q
u |= @
=Um} OOa
Q=ov|oDU@ty w swO
Tv=} Q=wwmwD= `@=D |=Q=O 'OW=@
u
O}vm xHwD Q} R
u
Q
|R U
100 O
Q |t O=H}=
| y )
m |= H=
=
| yxO=O
xm pkDUt Q}eDt
x]kv xm CU=
"OO o
O
= x}@W
|
> set.seed(1)
> w <{ rnorm(100)
> plot.ts(w, type = "l")
Q @
Q
=D Ov} wo R}v 4 |oDN=U
'p t |= @ "
i o
x}@W
=
=
R=
Cor(wi wj ) = 0 i 6= j x C Q
wt N (0 2) x C | = O}i x w Q
R QO |R U
C kw=
w
2
1
2
= Q}eDt Qo = "CU=
R= y
QO
"O W
h} QaD
w1 w2
pYi
= xirw-t xm OW x_Lqt p@k pYi
| y| U ,a v
|YNWt
=Q
O
| U | y
=t= 'OvDUy xDU@ty |v=tR
| U
=
| yp t
@ D
t v @ |= H=
Q |t pY=L
"OO o
x@DQt
=iDt
Cw
4
1
2
X=wN
w
QiY u}ov=}t |=Q=O w xDU@ty=v w f t g CQwY x@ |iO=YD Q}eDt Qo =
8
>
< 2
(k) = >
:0
CU= Q} R
"
k=0
k=0
6
Q
= }R
2. Discrete white noise
Cov(wt wt+k ) = E (wt ; )(wt+k ; )]
3. Simulation
4. Synthetic
|vWwOv|wUwt
18
−2
−1
0
w
1
2
1391 '
0
20
40
60
80
100
Time
O}iU xiwv |v=tR
Q
V}=tv
| U
%12
pmW
= E (wt) = E (wt+k ) = 0
w |t xH}Dv x=ov 'CU=
8
>
< ( 2)
"O W
E wt = a2
E (wt wt+k ) = E (wt)E (wt+k ) = >
:0 0
k=0
k=0
%R=
8
>
<1
x@ pta
QO
=t= OvW=@ QiY Q@=Q@ O}=@
|vat u}O@ u}=
w
6
OvwW|t `k=w u=v}t]=
Q}@ 'Q}N-=D 20 w %5
O
k=0
u t uw
w
uO @ Q=O
:0
|=R=
x@ xOW
6
CUDQ=@a |oDU@tyOwN `@=D
6
= x}@W O}iU xiwv
|R U
=
| yxO=O
r
= pN=O =t= 'CU}v QiY ,=k}kO =y k Q}O=kt
xR @
|oDU@tyOwN `@=D
Q}oxvwtv
|
r
O}vm xHwD Q} R p=Ft x@ uwvm = "CU= p=mW= q@ R}v =y k
OQ=O O Hw
"
U=
@ D
QO
r= "OQ=O
@ D
|oORuwQ}@ l}
Ovm|t x@U=Lt
}= "
|
7 Q}
Q_v
O}vm xHwD Q} R p=Ft x@
"
Q}}eD Cra
C=
|vat K]U x@ xHwD =@ |DL "OvQ=Ov QiY =@ |Q=O|vat Cw=iD =yv xm CU=
"
> set.seed(2)
> acf(rnorm(100))
w |a = R
xD@
C ARMAacf() ` = u
w
k=0
k=0
k = cck = >
0
w
u J
=
N-D QO
xm 'CU=
w x
CQ Y @ =Q
"O
> wn <{ ARMAacf(ar=0, ma=0, lag.max=20)
wtv s}UQD
22
pmW q=@
= O
w l}
R= OQ t
Q
| y )
m |= H=
xH}Dv
r
O}iU xiwv l} k |oDU@tyOwN `@=D xm
plot() ` =
=
w |t
@ D @ u= D
=Q
`@=D u}= pY=L
x}=B |iO=YD
19
=
O
2
pYi
| y )
m |= H=
xH}Dv
| yp t
0.4
−0.2
0.0
0.2
ACF
0.6
0.8
1.0
Series rnorm(100)
0
5
10
15
20
Lag
O}iU xiwv |oDU@tyOwN `@=D V}=tv
> plot(wn, type='h')
QiY Q@=Q@ Q}O=kt =yQ}N-=D Q}=U
Q
|= @ w
CU= l} Q@=Q@
k=0
QO
%22
pmW
xm 'CU=
pmW q=@
32
= O
Q
CU=
"
5 |iO=YD uORs=o
1
2
2
xtOkt
Q}o =Qi pOt
|
Q |DL ,qwtat "CU= |iO=YD
R= u VR= @
= O
Q |@U=vt
| y vwQ |= @
Q =@ = |iO=YD
VR= @ , r e
h} QaD
Q
CU= |iO=YD
xt
=
uORs o f
g x=ov "CU=
|v=tR
Q
| U
l}
xt
f
=
uORs o
ARIMA
CU= QD?U=vt
"
% o=
22
g
w
u J
2
2
2
xm O}vm
Q
Z i
xt = xt 1 + wt
;
=@ u}vJty
= xrO=at
w q @
QO
xt 1 = x t 2 + w t
;
x
;
1
;
|v} Ro}=H =@ "CU= O}iU xiwv
w |t xH}Dv 6 wQUB |v} Ro}=H
% m O W
5. Random walk
VwQ
xt = wt + wt 1 + wt 2 +
6. Back substitution
;
;
u}= xt=O=
w
Q
wt
x
xt 3 |v Ro =
| U f
|r@k
QO
g u QO
;
}
m
} H
|vWwOv|wUwt
20
0.0
0.2
0.4
wn
0.6
0.8
1.0
1391 '
5
10
15
20
Index
O}iU xiwv
|
Q_v |oDU@tyOwN `@=D V}=tv
P
w |t
%= r "O W
Q
`w W
t=1
%32
=
u tR R= w
xt = w1 + w2 +
pmW
CU}v C}=yv|@
w
Q
j i | U
pta
QO
+ wt
w x Qorta l} u=wD|t u}=Q@=v@ "OOQo|t p=ta= R}v xO}J}B |v=tR |=y|QU |=ypOt |=Q@ wQUB |v} Ro}=H
CQ Y @
wtv h} QaD Q} R
"O
7 wQUB p=kDv=
CU= Q} R
"
B (xt ) = xt
3
2
2
Qorta
w x xm
CQ Y @
B Qort
a
1
;
=@ Q
O xDio Qorta Qo = "Ov} wo R}v Q}N-=D Qorta u x@ OQ=wt R= |=xQ=B QO xm 'OwW|t xO}t=v wQUB p=kDv= Qorta
, D t x W
=ov
%x
B n(xt ) = xt
Q
QmD
=
wvm =
'OO o Q=
n
;
w |t xDWwv Qorta u}= =@ |iO=YD
"O W
xt = B (xt ) + wt
(1 B )xt = wt
xt = (1 B ) 1 wt
= (1 + B + B 2 + )wt
xt = wt + wt 1 + wt 2 +
)
;
)
)
7. Backward shift operator
;
;
;
;
uORs o u
x}=B |iO=YD
21
|iO=YD
"
CU= Q} R
w x
CQ Y @
x = 0 x
@
k (t) = Cov(xt xt+k ) = Cov
CU= Q} R
"
l}ORv
R=
t
X
i=1
uORs=o swO
xHwD =@ |iO=YD
wi
t+k
X
j =1
!
wj =
x@DQt
=
X=wN
4
2
2
= Ov}Qi
i=j
swO
w x |oDU@tyOwN ?} Q uwvm = "CU= =DU}==v Ov}Qi u}=Q@=v@ 'CU= u=tR
R=
Ov)m |r}N
}= @ @ "
w x
U=
J m |=
L
t
w Oy=wN C@Ft Q}O=kt
CQ Y @ w O @
@ k CQ Y @
|=Q=O
@
|iO=YD
v
Q=
=
Q
w
X= N
|a@=D Tv=} Q=wwm
2
1
p
k (t) = p Cov(xt xt+k ) = p 2 t
=
Var(xt ) Var(xt+k )
t (t + k)2
1 + k=t
k Ok u Q =v C QDm w x_ q p = w x t x C@U k Ok = Qy
t
x@DQt
Cov(wi wj )
CQ Y @
Q=
pYi
2
uORs o
X
O
| yp t
t p L
R
=
t
@ nQ @ | y
=ov|oDU@ty
uORs o |= @ Q
w
Q
|= @
CU= l} x@
O = |t Vy=m l} Q=Okt
" @ }
8 p=iD
x
5
2
2
Qorta
xm OW=@ |iO=YD uORs=o f t g |QU Qo = p=Ft |=Q@ "Ovm p}O@D =DU}= |QU x@ =Q =DU}==v |QU Ov=wD|t Qorta u}=
xt xt 1 = wt
;
;
w x
CQ Y @ u
x
x]@=Q xm Ovm|t O=H}= =Q O}iU xiwv |QU f t g pw= x@DQt p=iD 'CU= =DU}==v
CU=
"
w |t h} QaD Q} R
"O W
xt = xt xt
r
Q
;
=} QUB p=kDv= Qorta ?UL Q@ Ov=wD|t
"OO o u @ w
r
1
u}=Q@=v@ "CU=
r
xt = (1 B )xt x
CU= Q} R
n
|va} p=iD Qorta
;
"
r
w x
CQ Y @ r
;
m
O}W=@ xDW=O xHwD
w x p=iD Qorta QDq=@
CQ Y @
= x@DQt
| y
= (1 ; B )n
|R=Ux}@W
6
2
2
w |t Qy=_ =yQ=Owtv QO pOt |rY= C=}YwYN w CU= O}it |v=tR |QU l} xar=]t |=Q@ |R=Ux}@W ,=@r=e
"O W
wkr=@ u=mt= l}
x
wvax@ Ov=wD|t pOt
u=
=Wv OwN
'O=O u
R= =Q
x@=Wt
=} wYN |N} Q=D
C Y
xO=O
xm |Dkw u}=Q@=v@
Q |iQat
"OO o
"
> set.seed(2)
> x <{ w <{ rnorm(1000)
8. Dierence operator
Ovm
= x}@W
|R U
=Q
|iO=YD
= l} Ov=wD|t Q} R
uORs o
Q
u |= @
= O
| y )
m
1391 '
|vWwOv|wUwt
22
> for (t in 2:1000) xt] <{ xt - 1] + wt]
> plot(x, type = "l")
= for xkr
Q| |
Ok x x =vt
uORs o
xkrL
L "OO o
t
yOQ=
xm Owtv O=H}= |}=yO)m =@
R= u QO
@ ,
t
=Q
w
|iO=YD
OyO|t C@Uv
=
= x}@W
uORs o |R U
wx
O}iU xiwv
@ =Q
Q
wDUO u}rw=
| U Q
w |t xD@r= "Ovm|t O=H}=
u= D
=Q
|iO=YD
O = OWv
" W @ x
=iDU=
xO
> set.seed(2) # so you can reproduce the results
> v <{ rnorm(1000) # v contains 1000 iid N(0,1) variates
> x <{ cumsum(v) # x is a random walk
> plot(x, type = "l")
Q
=ov|oDU@ty
CUOx@
uOQw
Q CU= xOW
|= @ "
=Wv
xO=O u
42
pmW
O
=H}=
QO x W O
Q
| U |wQ
Qy@
0
20
x
40
60
| U Q
0
200
400
600
800
1000
Index
|iO=YD
= V}=tv
uORs o
%42
pmW
w |t xOy=Wt 52 pmW QO xH}Dv TBU "OOQo|t =QH= Q} R QwDUO 'p@k |=yxt=vQ@ xt=O= QO 'xOW |R=Ux}@W
"O W
> acf(x)
Q |t O}iU xiwv
"OO o
p=iD Qorta p=ta=
'
62
pmW
QO
=
Q
| yxO=O | U
x@ p}O@D p=iD l}
Q CU= jO=Y R}v xOW
|= @ "
xH}Dv TBU
= x}@W
|R U
w |t =QH= Q} R QwDUO 'Q}N=
"O W
QO
|iO=YD
Q
= O
uORs o ' W
w
x_Lqt ,q@k xm Qw]u=ty
O xDio ?r]t xm O}O O}=@ uwvm =
| U OQ t QO x W
= x = Q xt=O=
| y t v @
CU= OwHwt
QO "
R di() ` =
QO
@ D
w |t xOy=Wt
"O W
> acf(di(x))
u
=v}t]=
=
xR @ R=
|oORuwQ}@
w
w |tv x_Lqt Q=ov|oDU@ty
OQ t wO "O W
Q |YNWt
|= @
wor= 'Q}N= pmW
|
QO
= x}@W |QU CiQ|t Q=_Dv= xm Qw]uty u}=Q@=v@ "CU}v =vDa= p@=k %5 uOw@ Q=O|vat K]U QO xm OQ=O OwHw
|R U
x}=B |iO=YD
23
=
O
| yp t
pYi
2
0.0
0.2
0.4
ACF
0.6
0.8
1.0
Series x
0
5
10
15
20
25
30
Lag
|iO=YD
=
=ov|oDU@tyOwN V}=tv
%52
uORs o Q
Ovm|t
"
pmW
Q}B |iO=YD
|w
= Ov}Qi l}
uORs o
w}UQoQwD=
AR(p)
=YDN= x@
Q
xt = 1 xt 1 + 2xt 2 +
;
1
3
2
=kDv= Qorta ?UL Q@ Q}N= xrO=at Qo = "OvDUy
p = 0 =
6
D t
w}UQoQwD= Ov}Qi
O
= QDt=Q=B
@ p t | y
i
w
O}iU xiwv
w Oy=wN Q} R
;
;
;
;
w}UQoQwD=
u
x@ ?@U u}ty x@
w
1 = 1
CU= xDWPo
u QO
=
xm CU=
=
| yu tR QO
wt
f
w x
g u QO
Q
=} QUB
AR(1)
x
=
X N
Cr=L |iO=YD
x
=y x@ C@Uv t uw}UQoQ `k=w
=mv x@ uwvm =
C
=
uORs o O u}=
QO p t
Ov} wo
"
= CU= Q@=Q@
% @
x^t = 1xt 1 + 2xt 2 +
;
;
xm
pB p)xt = wt
O}vm xHwD Q} R
O = |t
Q
| U
CQ Y @ 'OO o u @ w
"
" W @
|v=tR
g
;
"O @
p(B )xt = (1 1B 2B 2
xt
=Q f
+ pxt p + wt
;
p
p x@ Q
O
x W
32
|=ypOt
h} QaD
Qo = 'Ov} wo
x
R=
+ t pxt
;
t
=
Q
u tR QO OQw @ 1391 '
|vWwOv|wUwt
24
0.0
0.2
0.4
ACF
0.6
0.8
1.0
Series diff(x)
0
5
10
15
20
25
30
Lag
p=iD p=ta= =@ |iO=YD
=
=ov|oDU@tyOwN V}=tv
uORs o Q
Q |t
"OO o
Q =]N
OQw @
=a@ Qt pk=OL
C
w}UQoQwD=
O Qo = "OW=@ \rDNt =}
Q k
= Ov}Qi
uORs o
u}}aD
w
|k}kL
= xW} Q
| y
CU= =DU}= Ov}Qi x=ov
QO "
Q OL=w
'OO o
R=
Q Q} R p=Ft Q=yJ x@ uwvm = "CU= =DU}==v Ov}Qi
;
1
2
B=0x
QUB p=kDv= Qorta
w
w O W
t
xrtHOvJ u}= xW} Q "OwW|t xH}Dv
QDoQR@
R=
w
w
= QDt=Q=B
;
m
m
2
3
2
|}=DU}=
p(B ) = 0 x
=Q
J | y
}Q s
;
} i
w x
CQ Y @
}
}= v w
AR(1)
CU= QDoQR@ OL=w
R=
}
t
;
"
R=
|m} =Q} R 'CU= =DU}==v t
;
} ;
;
U=
"
|va}
U=
| y
L=w R=
1 + 41 B 2 = 0
} Q |=Q=O
oQ @
;
rO
t
;
w
;
|=
t Q k = } R "OO o
xt =
B p
;
t
x
}
1
4 t;2
}= v
+ wt
v=
H
@
p t 3
a R= xO
J "O W
U
Q=
w x
CQ Y @
}=
t
L
CQ Y @
;
;
= xW} Q =Q} R 'CU= =DU}=
| y
;
}
O
x = 21 xt 1 + 12 xt 2 + wt w x AR(2) O
1
1)(B + 2)xt = wt |va 12 (B 2 + B 2)xt = wt QU =kD Qort
=iD =
2 (B
C B = 1 2 = xW
(B ) = 21 (B 1)(B + 2) xrt Ov w | xH}D
C O
QD R B = 2 jr] O Q
Q | | =DU = ?@ B = 1 Ok \k
CU= OL=w =yxW} Q
"
D
p t 1
rO
;
iO
p t 2
;
UO @
t
xm CU=
CQ Y @
;
}
m
=at
rO
D
;
U=
H D
;
;
CU=
| y
x = xt 1 41 xt 2 + wt w x AR(2) O
1
2)2 xt = wt |va 41 (B 2 4B + 4)xt = wt x =a
4 (B
B = 2 x O | C x (B ) = 14 (B 2)2 x =a p =
=iDU= =@ =Q} R "CU= =DU}= t
|=
O = |t OL=w
H
x = 12 xt 1 + wt
=at xW} Q =Q} R 'CU= =DU}= t
rO
R= xO
" W @
|}=DU}==v
|=
"
w
x@ pOt
B ) xrt Ov = xW =t jr]
| B = 1 x C = 1 B | =Y
O}v x w AR Ov Q | =DU = | =DU
QDW}@ p (
"
@= @
VwQ
Ov=wD|t xm Ovt=v xYNWt xrO=at
|=Q=O
|= @
B=2QQ 1
QO
pmW
%62
U= @
t
t
AR(2)
v
i
O
p t 4
x}=B |iO=YD
25
2QQ
= xW} Q
@= @ y
R=
l} Qy jr]t QOk "OW=@|t
i=
p
;
1
w
CU= \rDNt
O
O= a=
xm OvDUy
CU= OL=w
"
xrO=at
O
x W
= xW} Q
| y
xDio `@=D =@
w |t xm
u= D
=yJ
sQ
w
OQ=O O Hw
w
w
polyroot()
=F
w s U OQ= t 'p t u=
wvax@
= x@ |a@=D
s v
Q |UQQ@
"O m
=Q
R
QO |=
=
2
pYi
B = 2i
QDoQR@ xm OW=@|t
R=
xrtHOvJ
=y|QU |}=DU}=
O
| yp t
= xW} Q x@U=Lt
| y
wtv x@U=Lt
w O
Q
|= @
xYNWt
=Q
w |t x@U=Lt
"O W
w
=F
%s U p t
> polyroot(c(-2,1,1))
1] 1-0i -2+0i
%R=
OvDQ=@a =yxW} Q jr]t QOk
> Mod(polyroot(c(-2,1,1)))
1] 1 2
=yJ p=Ft
%sQ
> polyroot(c(1,0,1/4))
1] 0+2i 0-2i
%R=
OvDQ=@a =yxW} Q jr]t QOk
> Mod(polyroot(c(1,0,1/4)))
1] 2 2
AR(1) p t swO
O
"
w x
CU= Q} R
CQ Y @
AR(1)
O
x@DQt
O
3
3
2
X=wN
O =Wt ,q@k xm Qw]u=ty
p t ' W x y
xt = xt 1 + wt
;
xm
=Wv
O=O u
w |t "OW=@|t
u= D
2 T =
v } Q=w w
QiY u}ov=}t =@ O}iU xiwv
wt
f
g u QO
xm
x = 0
k = k 2=(1 2)
;
xm CWwv
w |t
u= D
< 1 Ov Q
j
j
} i
|}=DU}=
Q
\ W w
B Qort
=iDU= =@
a R= xO
(1 ; B )xt = wt
wtv x@U=Lt
"O
xt = (1 B ) 1wt
;
;
= wt + wt 1 + wt 2 +
;
2
;
=
1
X
i wt
i
;
i=0
=Q
xt
w |t uwvm =
u= D
1391 '
|vWwOv|wUwt
26
"
1
X
E (xt ) = E
!
i wt
=
i
;
i=0
1
X
O}vm xHwD u}ov=}t x@U=Lt x@ uwvm =
iE (wt i) = 0
;
i=0
CU= Q} R KQW x@ Tv=} Q=wwmwD=
!
"
k = Cov(xt xt+k ) = Cov
=
1
X
i=0
X
j =k +i
= k 2
wt i
i
1
X
;
j =0
O}vm s}UkD Tv=} Q=w Q@
=Q
;
X1
i=0
QDW}@ CaQU =@ QDmJwm
|
O |t
72
" }
pmW
QO
2i = k 2=(1 2)
;
Ov}Qi
4
3
2
Q=ov|oDU@ty
=
t w
Q
|= @
(k 0)
| y
|iv
;
Tv=} Q=wwmwD= xm CU= |i=m |oDU@tyOwN `@=D x@U=Lt
k = k
p}t QiY CtU x@
j
;
ij Cov(wt i wt+k j )
AR(1)
"
j wt+k
|=R=
x@ Q=ov|oDU@ty u}=Q@=v@ "CU=
C@Ft Q}O=kt
Q =ov|oDU@ty
|= @ Q
j
<1
j
=F xt=O=
wO p t
u QO
xm
Ovm|t
QO "
> plot(0:10, rho(0:10, -0.7), type = "b", xlab="lag k")
> rho <{ function(k, alpha) alpha^k
> par(mfrow=c(2,1), mar=c(4,4,2,4))
> plot(0:10, rho(0:10, 0.7), type = "b", xlab="lag k",
+ ylab=expression(rk]), main=expression(alpha==0.7))
> plot(0:10, rho(0:10, -0.7), type = "b", xlab="lag k",
+ ylab=expression(rk]), main=expression(alpha==-0.7))
> abline(h=0, lty=2)
9
x
x@ \ki t Qo = |DL 'OvDUy QiY hr=Nt =yQ}N-=D s=tD
|} RH |oDU@tyOwN `@=D
Q =y|oDU@tyOwN Q}O=kt
|= @
k
k Q}
'
N=
5
3
2
xrO=at j@=]t
x
=y|oDU@ty QF= xm CU= |oDU@ty R= |awv ' Q}N-=D QO |} RH |oDU@tyOwN `@=D "OW=@ xDW=O |oDU@ t;1
w Oy=wN QiY
"O @
CU=
'
AR(k)
k>1
s
O xDi=}
p t
=tD
AR(1) | R
? Q u} k =
Q
|= @
Q
VR= @
} H
} t=
@
|oDU@tyOwN `@=D p=Ft
|oDU@tyOwN `@=D Q=ov|oDU@ty u}=Q@=v@ "CU= QiY
w
x@U=Lt
Q
|= @
pacf()
= x@ |a@=D
s v
R
Q}o Q=Qk
QO "O
k Q} =
QQ k>p
Q@=Q@
N-D QO
@= @
=iDU=
xO
w
OQ t
Q
P
QDmJwm Q}N-=D =@
|} RH |oDU@tyOwN `@=D '|rm Cr=L
O w | AR Ov Q
|=R=
v= D
QO
x@ k ?}=Q AR(p) Ov}Qi QO |va}
t
w
"OQ=O O Hw
9. Partial autocorrelation function
Q
|= @ "OO o h L
} i
x@DQt u}}aD
Q |} RH
|= @
|} RH |oDU@tyOwN `@=D s}UQD
x}=B |iO=YD
27
=
O
| yp t
2
pYi
0.0
0.4
rk
0.8
α = 0.7
0
2
4
6
8
10
8
10
lag k
0.0
−0.5
rk
0.5
1.0
α = − 0.7
0
2
4
6
lag k
= 0:7 ;0:7
Q
|= @
AR(1) Ov Q
=ov|oDU@ty V}=tv
} i Q
%72
pmW
|R=Ux}@W
QO
|} RH |oDU@tyOwN
w
|oDU@tyOwN `@=wD
|oDU@tyOwN Q=ov|oDU@ty QO
k>1Q
=kt
}O
w
Ov}Qi
O
= x}@W Q} R
w x W |R U
w x
CQ Y @
R
QO
6
3
2
AR(1)
O
p t
Q O}vm|t x_Lqt xm Qw]u=ty "OOQo|t s}UQD 82 pmW
|= @
O
w
|vat |oDU@ty '|} RH
"OQ= v O Hw |Q=O
xOW |R=Ux}@W |=yxO=O Q@
x.ar
Q
Q
R= xO
=iDU= =@
| U |= @
u
w}UQoQwD= pOt 'Q} R |=yO)m
u
xDi=}
VR=Q@ |=ypOt
7
3
2
xDi=}
VR=Q@ |=ypOt
8
3
2
O = |t VR=Q@ =yxO=O Q@ ar() `@=D \UwD R QO
QO " @ }
10 |@v=Ht Tv=} Q=w xm 'O@=}|t
Q O
VR= @ x W xO=O
QDt=Q=B
w
%95 u=v}t]=
w |t
"O W
> set.seed(1)
> x <{ w <{ rnorm(100)
> for (t in 2:100) xt] <- 0.7 xt - 1] + wt]
> x.ar <{ ar(x, method = "mle")
> x.ar$order
pOt
x@DQ
pOt
QDt=Q=B
1] 1
> x.ar$ar
10. asymtotic
AR(p)
= = O
O
p t
= x}@W
xR @ @ x W |R U
QNDU=
G=
x.ar$asy.var
|vWwOv|wUwt
28
−1
0
x
1
2
3
1391 '
0
20
40
60
80
100
0.6
0.2
−0.2
ACF
1.0
Index
0
5
10
15
20
0.2 0.4 0.6
−0.2
Partial ACF
Lag
5
10
15
20
Lag
V}=y|oDU@tyOwN
w
AR(1) Ov Q
V}=tv
Q@ 'OW
=iDU= =yO)m
} i
%82
pmW
1] 0.6009459
> x.ar$ar + c(-2, 2) sqrt(x.ar$asy.var)
pOt
QDt=Q=B
u=v}t]= xR=@
1] 0.4404031 0.7614886
?
=NDv= "CU= Q=wDU= |}=tvDUQO QFm =OL `@=D
=
T U=
Q Ov}Qi
xO
QO VR= @
x]@= x@U=Lt "CU= pOt QDt=Q=B pk=OL Qov=}@ xm OwW|t s=Hv= 12 AIC x]@=
"
QO
xm 11 mle
Ov}Qi x@ \w@ Qt
R= '
CU= Q} R
p
VwQ
Okt
Q=
w x O
CQ Y @ x W
xDio
AIC = ;2 log-likelihood + 2 number of parameters
CUQO x@DQt
Q=
Okt
R=
=
= O xm O}vm xHwD "Ovm|t Q=}DN=
'q @ | y )
m
^ = 0:6 Q Q AR(1)
QDmJwm xm 'Ot CUOx@
O
w
"OQ= v O Hw
E}L u}=
@= @
Q
Q
| U
w |t O}OQD =@ "CU=
u= D
=@ =aO= u}=
= =
O QDt=Q=B
p t
AIC u QDm
}
Q
wtv |@=} R=@
OQw @ "O
Q |t pOt QDt=Q=B Q=Okt pt=W %95 u=v}t]=
x =Nro QF=
|= v
R=
11. Maximum likelihood estimation
AR p t %
|W=v xm OyO|t u=Wv
Q CU= xOW
|Q oR U |= @ "
VR=Q@
P
h L
w =
O
=
=Q tO
|v=yH
12. Akaike Information Criterion
R=
=Q
= =t=
xR @
|W=v
1970
|=x
@ D QO
} p t
"
U=
|=tO |QU
OvwQ V}=Ri=
|iO=YD xO}OB l} p=kDv=
ar() ` =
p = 1 |va O
C = 0:7
O
J m @ p t '
R= |O= }= w OO o
xDi=}
=aO=
Q u} QDy@
=Q VR= @
=
9
3
2
Q
p U R= | U
u}=
Ov}=Ri OvwQ xm Owtv
x}=B |iO=YD
29
"O
wtv
=iDU= u}at `@=wD
xO
R=
xm
O
wtv
u uw @ xO
O}vm xHwD xvq=U
"
=
=
O
|R Up t =Q
Q
| tO | U
=
O
| yp t
2
pYi
OvwQ xirw-t CU= umtt '=yxO=O
u}ov=}t x@ xDi=}
Q
VR= @
AR
O x@
p t
> www <{ "http://www.massey.ac.nz/~pscowper/ts/global.dat"
> Global <{ scan(www)
> Global.ts <{ ts(Global, st = c(1856, 1), end = c(2005, 12), fr = 12)
> Global.ar <{ ar(aggregate(Global.ts, FUN = mean), method = "mle")
> mean(aggregate(Global.ts, FUN = mean))
−0.2
0.0
0.2
0.4
ACF
0.6
0.8
1.0
1] -0.1382628
> Global.ar$order
1] 4
> Global.ar$ar
1] 0.58762026 0.01260253 0.11116731 0.26763656
> acf(Global.ar$res-(1:Global.ar$order)], lag = 50, main="")
0
10
20
30
40
50
Lag
=yxOv=t}k=@ Q=ov|oDU@ty V}=tv
Q}N-=D
QO
rk
O
Okt RH x@ =Q} R "OyO|t u=Wv
Q=
|t Q_v x@ ?U=vt '=yxO=O Q@
" UQ
xO=O
CUOx@
O
= QDt=Q=B
=Q p t | y
w
=Q
O}iU xiwv
AR(4)
xvq=U \UwDt
=
O
Q
| U
%92
pmW
l} '=yxOv=t}k=@
Q P O
Q
92
p t VR= @ = r " vQ=O Q= k u
| tO
u}ov=}t Q=Okt xm Q}N=
pmW Q=ov|oDU@ty
=v}t]=
=
xR @ QO
=yv x}k@
27
= O G}=Dv x@ xHwD =@ uwvm =
| y )
m
CWwv
"
w |t 'CU=
u= D
x^t = 0:14+0:59(xt 1 +0:14)+0:013(xt 2+0:14)+0:11(xt 3 +0:15)+0:27(xt 4 +0:15)
;
;
;
;
;
swU pYi
=DU}= |=ypOt
O}m =
,=
xt
=Q f
g
|v=tR
Q
w |t
| U "O W
|
Q}o}B EL@
xt=O=
u
|va} 'Ovmv Q}}eD u=tR
f (xt1 xt2 Cov(xt xs) T =
w
u
wvm = "OW KQ]t |@r=]t |}=DU}= x@ `H=Q ,q@k
=kDv= QF=
QO p
xt ) = f (xt1 +h xt2 +h
xt +h)
k
w |t pt=W R}v =Q u=tR
v } Q=w m w O W
QO
w `} RwD `@=D Qo = Ov} wo 1 =DU}=
Q
QO | U s= D
k
C@=F Tv=} Q=w w u}ov=}t 'Omw-t |}=DU}= xm O}vm xHwD
O = xDW=O |oDU@
" W @
lQLDt
X=wN w
xrtH
quQ
} N
u}vJty
w
O}iU xiwv
=
|Q H
xrtH
R=
q x@ Q
CU= Q} R
w x
"
1. Strictly stationary
xt = wt + 1 wt 1 +
;
30
u}ov=}t
h} QaD
|]N ?}mQD
D t R=
CQ Y @ u
+ q wt
k= t s
q
;
;
j
Q}N-=D x@ \ki
|=ypOt
MA(q)
%
(MA)
j
Ov}Qi
13
1
1
3
QLDt u}ov=}t Ov}Qi
l
x]@=Q "OW=@|t O}iU xiwv |r@k
=DU}=
31
=kDv= Qorta ?UL Q@
p
w |t
u= D
2
= xrO=at "CU= w
Tv=} Q=w
=Q q @
=
QiY u}ov=}t =@ O}iU xiwv
w
O
| yp t
wt
f
g u QO
wtv |U} wvR=@
"O
xt = (1 + 1B + 2B 2 +
=DU}= CqtH R= |y=vDt `tH pt=W MA |=yOv}Qi xm u}= x@ Q_v "CU=
w
=
x@ C@Uv Tv=} Q=wwmwD=
w
B
'
w
xm
QUB
+ q B q )wt = q (B )wt
|
"OQ=O O Hw u tR
pYi
3
u}ov=}t
=@
QO C F w
qx
xrtHOvJ q
HQO R= |=
u QO
xm
CU= Q=QkQ@ |}=DU}= =Pr 'CU= O}iU xiwv
CU= x@U=Lt p@=k |oO=U x@
"
xt
f
g
Tv=} Q=w
w
u}ov=}t
E (xt ) = E (wt + 1 wt 1 + + q wt q )
= E (wt) + 1 E (wt 1) + + q E (wt q )
=0
;
;
;
;
OvDUy QiY Q@=Q@ =yu}ov=}t
"
R=
l} Qy =Q} R
Var(xt ) = Var(wt + 1wt 1 + + q wt q )
= Var(wt) + 12 Var(wt 1) + + q2 Var(wt q )
= w2 (1 + 12 + + q2)
;
;
;
"
OvDUy pkDUt Qo}Om}
"
8
>
>
>
1
>
>
>
q ;k
>
< i i+k
>
>
>
>
>
:
C
x
qtH pt=W t;1
w
xt
Q
=Um} Tv=} Q=w
CU= Q} R QwYx@
k 0
qtH s=tD =Q} R
|=Q=O C
Q |oDU@tyOwN `@=D
|= @
i
i=0
= }R '
u
k = 1 : : : q
q
k>q
0
O}iU xiwv pkDUt
w2
R= w
k=0
P
(k) = > P 2
>
i=0
;
k>q
CU= QiY Q@=Q@ `@=D
Q CU=
|= @ "
0 = 1
CU= QiY =yv Tv=} Q=wwm u}vJty
"
xrtH
O |y=vDt=v x@DQt =@ =DU}= w}UQoQwD= Ov}Qi ?UL Q@
uw @
w
=} Q} R
"O W u @
wD@ Qo = Ov} wo 2 Q}PBuwQ=w
=Q u u=
x = (1 B )wt x]
w x Ov=wD|t t
CQ Y @
;
@=Q
=@
MA(1) Ov Q
=F
=Q
u QO
w
xm
OvDUy
MA Ov Q
} i
Q CWwv =]N
} i 'p t |= @ "
wt = (1 B ) 1xt = xt + xt 1 + 2xt 2 +
;
;
;
Ovm
"
OL=w
R=
QDoQR@ |oty
2. invertible
(B )
O
j Y
;
|}=Qoty
Q
\ W
= xW} Q jr]t QOk Qo = CU= Q}PBuwQ=w Ov}Qi l}
| y
=D CU=
<1
j
j
MA(q) Ov Q
u QO
xm
|rm Qw]x@
} i '
OvW=@
"
1391 '
|vWwOv|wUwt
32
R | yp
|R=Ux}@W w Q=ov|oDU@ty %
w
xkrL
=iDU= =@ |a@=D Q=m u}=
R= xO
Q
wtv
|= @ "O
R
=}
xO B
w |t
QO u= D
=Q
MA(q) Ov Q
=
2
1
3
=Ft
Q |oDU@tyOwN `@=D
} i |= @
w |t h} QaD
"O W
Q
\ W
> rho <{ function(k, beta) f
+ q <{ length(beta) - 1
+ if (k > q) ACF <{ 0 else f
+ s1 <{ 0 s2 <{ 0
+ for (i in 1:(q-k+1)) s1 <- s1 + betai] betai+k]
+ for (i in 1:(q+1)) s2 <{ s2 + betai]^2
+ ACF <{ s1 / s2g
+ ACFg
Ov Q
Q Q
= O C x@ =L p = MA(q) Ov Q
Q |oDU@t w ` = = = O
=iD =
C
pm
w x 3 = 0:2 = 0:5 1 = 0:7 = QD = = MA(3)
> beta <{ c(1, 0.7, 0.5, 0.2)
> rho.k <{ rep(1, 10)
> for (k in 1:10) rho.kk] <{ rho(k, beta)
> plot(0:10, c(1, rho.k), pch = 4, ylab = expression(rhok]), xlab="lag k")
> abline(0, 0)
}R | y )
m "
U
U= 13
t
@ k
} i |= @
W CQ Y
@
yO N
w
@ D 'q @ | y )
m R= xO
'
| y
0.6
0.8
1.0
"
U=
0.0
0.2
0.4
ρk
} i |= @
0
2
4
6
8
lag k
MA(3)
O
=ov|oDU@ty V}=tv
p t Q
%13
pmW
10
t=Q B
@
U=
@
=DU}=
33
w
|v=tR
Q
| U "O
Q}o|t Q=Qk
=iDU=
xO
w
=ov|oDU@ty s}UQD
OQ t Q
w
MA(3) Ov Q
=
O
| yp t
= x}@W
} i |R U
3
Q Q} R
|= @
pYi
= O
| y )
m
|oDU@tyOwN `@=D Q=Okt xU u}rw= 'CiQ|t Q=_Dv= xm Qw]u=ty "CU= xOW s}UQD 23 pmW QO Q=ov|oDU@ty
O
Q
" vQ=O Q= k u
=v}t]=
=
xR @ QO
CU=
"
|
r
3
=y k Q}O=kt
Q}oxvwtv
'
Q}}eD
C=
R=
R=
QDW}@
= Q}N-=D
| y
Q
|= @ w
OvDUy QiY
R= |Q=O
4
2
0
−2
x
−4
200
400
600
800
1000
0.4
0.0
ACF
0.8
time t
0
5
10
15
20
25
30
Lag
u Q
=ov|oDU@ty
w
Cw
=iD
|=Q=O
|W=v xm OW=@ xDW=O OwHw |oORuwQ}@ CU= umtt sy %5 xD@r=
> set.seed(1)
> b <{ c(0.8, 0.6, 0.4)
> x <{ w <{ rnorm(1000)
> for (t in 4:1000) f
+ for (j in 1:3) xt] <{ xt] + bj] wt - j]
+g
> par(mfrow=c(2,1), mar=c(4,4,2,4))
> plot(x, type = "l", xlab="time t")
> acf(x, main="")
0
|vat
MA(3)
= x}@W Ov}Qi V}=tv
|R U
%23
pmW
1391 '
|vWwOv|wUwt
34
xDi=}
VR=Q@
xOW |R=Ux}@W |QU
= w
=@ pOt x@DQt xm
u t oQ
QUm
u}ov=}t
=Q
w
OQ=O O Hw
Q V}B
Z i
arima()
w x
O
CQ Y @ x W
R
= x@ |a@=D
s v
xDio `@=D
'
O=
x@ xDi=}
Q =
ar() ` =
qN Q@
Q |t
"O W
Q 3 <=O@t
MA(q) O
order=c(0,0,q)
p t
Q xrtH
OQw @
1
2
3
VR=Q@ pOt
w |t s}_vD
@ D h
23
|=ypOt
Q@ Ov=wD|t
QO " @ } VR= @ yxO=O
"OO o
C
MA
R= Z a
Ovm|tv
QO w
=a@ Qt `wtHt '=yQDt=Q=B OQwQ@ |=Q@ arima() `@=D "OvwW|t OQwQ@ |OOa sD} Qwor= l} \UwD pOt |=yQDt=Q=B
O =
" W @
= QDt=Q=B
' y
Q} R
|=R=
w x
CQ Y @
x@ xm CU=
method=c("CSS") = w x | w
w u x MA(q) Ov Q
Q Q | Q
Q | x@ =L Qm
Q wt = O =t} =
u t oQ
CQ Y
w^t |va
}=
@
DQ Y QO
} i VR= @ |= @
yx v
} VOQw @ w
m
k @ "OO o
t
U
xD@r=
= |t pk=OL
OR U
=Q
|]QW
=a@ Qt `wtHt sD} Qwor= K}wD
] W C
t Q
w x =yxOv=t}k=@
=a@ Qt `wtHt
t Q ] @
C
CU=
"
S (^1 : : : ^q ) =
Q =kt
R= | }O
O
'|O a |
n
X
t=1
w^t =
2
n n
X
wHDUH "OvW=@ QiY =@ Q@=Q@ '=yv Q=QmD
pt=W
Q
Q Q}N= xOW
VR= @
= x}@W
|R U
coeff: 2 s.e. of coeff: =@ Qk
, }
D O
=
| yxO=O
Q}o|t Q=Qk
;
w^0 : : : w^t
Q
u
Q@
\w
=a@ Qt `wtHt xm Ovm|t u}at
=Q
=yQDt=Q=B
`w W QO
Q pk=OL q=@
= QDt=Q=B "O@=}|t
+ ^q w^t q )
;
t=1
"OO o
| y
o2
xt ; (^1w^t 1 +
q Q}O=kt xm
;
C
x.ma
%95 =v}t
x@
u
QLDt u}ov=}t
'l
=
]= xR @ QO
O
Q
QWt
= O
p t ' }R | y )
m QO
xt=vQ@ |HwQN
O
Q
QO x W OQw @
=_Dv= xm u=vJty ,=vt "CU= xOW xO=iDU= =yv R= |R=Ux}@W QO xm OwW|t (0.8,0.6,0.4) |=yQDt=Q=B Q}O=kt
O QiY =@
"OQ= v
include.mean=FALSE = w
= u
Q
C include.mean=TRUE
> set.seed(1)
> b <{ c(0.8, 0.6, 0.4)
> x <{ w <{ rnorm(1000)
> for (t in 4:1000) f
+ for (j in 1:3) xt] <{ xt] + bj] wt - j]
+g
> x.ma <{ arima(x, order = c(0, 0, 3))
> x.ma
`@=D
QO
u t oQ R= Q m
"
U=
Call:
arima(x = x, order = c(0, 0, 3))
3. Intercept
|Q=O
|vat
Cw
=iD
wtv s}_vD QiY Q@=Q@
}= |= @ "O
Q V}B u}=Q@=v@
Z i
<=
=Q
O@t
u}ov=}t Q=Okt
w |t
"O W
Q CiQ|t
R= Z a
=iDU=
xO
w |t
u= D
arima()
=DU}=
35
=
O
| yp t
pYi
3
Coefficients:
ma1
ma2
ma3
0.7898
0.5665
0.3959
s.e.
0.0307
intercept
0.0351
-0.0322
0.0320
sigma^2 estimated as 1.068:
0.0898
log likelihood = -1452.41,
ARMA
Ov}Qi
aic = 2914.83
33
%|@}mQD |=ypOt
1
3
3
h} QaD
CU= Q} R
"
w x
CQ Y @
p x@ Q
D t R=
w}UQoQwD= Ov}Qi
xt = 1 xt 1 + 2xt 2 +
;
MA AR Ov Q x |D = O
(ARMA) QLD u}o =} w} Q
w
} i
l
m
v t
CU= Q} R
w Ov}Qi |=Q=O
w x
"
CQ Y @ u
xt = 1xt 1 + 2xt 2 +
;
"
;
x]@=Q
xt
f
w |t
=Wv
"O W
= QDt=Q=B =y i
| y
|v=tR |QU
g
xO=O u
w
w
O}iU xiwv f t g
Q |t pY=L 'OvwW|t xOwRi= sy x@
ARMA(p,q) =
@ w
;
;
w x
w |t
CQ Y @ u= D
xm
u QO
"OO o
CU=
+ pxt p + wt + 1wt 1 + 2 wt 2 +
CWwv wQUB p=kDv= Qorta
xm Qw]u=ty
;
O}it xDUO "OvDUy pOt
U oQ D=
g |QU 'OW xOy=Wt ,q@k
+ pxt p + wt
;
kw yp t R= |
t
xt
f
(p,q) x@ Q
D t R=
+ q wt
;
w x]@=Q "CU= O}iU xiwv
=Q j i
wt
f
;
q
xm
g u QO
p(B )xt = q (B )wt
"
"
OvW=@ OL=w
OvW=@ OL=w
"
R=
QDoQR@
R=
QDoQR@
wYN
X
QDW}@ |}Q=m ,=@r=e
QO |
ARMA(p,q)
w
OQ t QO
Q} R
1
= xW} Q s=tD jr]t QOk Qo = CU= Q}PBuwQ=w Ov}Qi
2
| y
CU=
ARMA(p,0)
p t X N
CU=
ARMA(0,q)
p t X N
ARMA
w
OQ= t
= xW} Q s=tD jr]t QOk Qo = CU= =DU}= Ov}Qi
| y
"
"
=yQDt=Q=B
CU= xHwD QwNQO
O
=
Cr=L
AR(p)
p t 3
O
=
Cr=L
MA(q)
p t 4
w |t s=Hv=
p t 'O W
"OQ=O
O
|}=yvD x@
O
O
Q |Dkw %QDt=Q=B pk=OL
VR= @
AR = MA
} w
=
O x@ C@Uv
| yp t
5
1391 '
|vWwOv|wUwt
36
pOt swO
C
qtH ?UL Q@ =Q
O Q}o@ Q_v
" }
x
OD@= QO xm CU= QDy@ 'f t g |v=tR
u =
QO =Q
ARMA(1,1)
O
=F
ARMA(p,q)
Q
| U
Q OvDUy pkDUt sy
p t p t |= @ "
xt = xt 1 + wt + wt
=Q
Q}N= xrO=at "CU=
Var(wt) = w2 E (wt )
w
O
" } QO
R=
2
3
3
X=wN
swO
x@DQt X=wN u}}aD |=Q@
=yv =Q} R "O}U} wv@ O}iU xiwv
1
;
;
Q@
x@DQt
u
O}iU xiwv
Tv=} Q=w
w
wt
u}ov=}t "CU= O}iU xiwv
= xirw-t ?UL Q@ xm
| y
xO
wtv ?DQt
w
|Q ]
u QO
xt
xm
=
T U=
xt = (1 B ) 1(1 + B )wt
;
;
CU= Q} R
w x
"
xt = (1 + B + 2B 2 +
=
1
X
i=0
= 1+
w
CU=
E (wt i) = 0 = i
;
y
Q \U@
h ]
)(1 + B )wt
!
iB i (1 + B )wt
1
X
i=0
i+1B i+1 +
= wt + ( + )
Tv=} Q=w
= x]@=Q CU=Q
CQ Y @ q @
=tD
s
|=R=
1
X
1
X
i=0
i 1wt
;
;
i=0
!
iB i+1 wt
i
E (xt ) u}o =}
x@ =Q} R 'CU= QiY Q@=Q@
"
Var(xt ) = Var wt + ( + )
1
X
= x]@=Q x@ xHwD =@
v t 'q @
#
i 1wt
;
i
;
i=1
= w2 + w2 ( + )(1 + 2)
;
1
w |t xH}Dv
"O W
k>0
Cov(xt xt+k ) = ( + )k 1w2 + ( + )2w2 k
;
Q
|= @
1
X
i=0
2i
k T =
2
1
;
w |t xH}Dv k |oDU@tyOwN `@=D
"O W
k = k =0 = Cov(xt xt+k )= Var(xt )
k 1
= ( + )(1 +2 )
1 + + ;
Q
;
= ( + )k 1w2 + ( + )2w2 k (1 ; 2)
;
ww
v } Q=w m D= |= @
Q
|= @
=DU}=
37
k = k
|@ QHDp}rLD
;
CiQo xH}Dv
1
ARMA
%
=
O
| yp t
3
w |t Q}N= x]@=Q
u= D
w |t
w |t EL@
u= D
c(p,0,q) x@ Q
=
D t @
Oa@ pYi
=Q O W
arima() ` =
@ D
|
ARIMA
ARMA(p,q)
QO
\UwD Ov=wD|t
xm
= Ov}Qi
| y
O
wtv
p t "O
u R=
R=
43
|=ypOt
1
4
3
VR=Q@ w |R=Ux}@W
`@=D \UwD
pYi
ARMA Ov Q
R arima.sim()
QD|twta
= x}@W
|R U
w
} i
QO
Q
"O=O VR= @
O TBU
p t
w |t
'O W
l}ORv xvwtv
R=
= x}@W
|R U
w
= 0:5 = 0:6 = ARMA(1,1) Ov Q
;
w
= QDt=Q=B
| y
@
Q =
Q} R
} i |= @ yxO=O
Q CiQ|t Q=_Dv= xm Qw]u=ty "O@=}|t
OQw @
Q =yv Q@
VR= @
= O
| y )
m QO
ARMA(1,1)
CU= q=@
"
QO
=yv Q}O=kt
> set.seed(1)
> x <{ arima.sim(n = 10000, list(ar = -0.6, ma = 0.5))
> coef(arima(x, order = c(1, 0, 1)))
ar1
ma1
-0.596966371
intercept
0.502703368 -0.006571345
p}O@D
x]@=
33
w
O@=}|t
pmW "OW=@|t =yxO=O
CU= lJwm
w '
Q p}O@D
VR= @
Q
Q
|=Q=O
ARMA(1,1) AR(1) MA(1)
=v O ARMA(1,1) Ov Q O Q |
x O | =W
ARMA(1,1) O
Q@
Q ?U
|= @ | D
= |oDU@tyOwN
| y
Q
Mv | U
w
t p t
m
yO
} i " vO o
t u
Ovm|t O}}=D
"
'
v =Q
O u}=
=Q p t
t
2
4
3
=
= O
O
Q
m QO
| yp t ' } R | y )
xU}=kt sy =@ =yv
AIC
O =t}k=@ Q=ov|oDU@ty
p t x v
R= xO
=iDU=
w
OvDUy O}iU xiwv =@ Q=oR=U
> www <{ "http://www.massey.ac.nz/~pscowper/ts/pounds_nz.dat"
> x <{ read.table(www, header = T)
> x.ts <{ ts(x, st = 1991, fr = 4)
> x.ma <{ arima(x.ts, order = c(0, 0, 1))
> x.ar <{ arima(x.ts, order = c(1, 0, 0))
> x.arma <{ arima(x.ts, order = c(1, 0, 1))
> AIC(x.ma)
1] -3.526895
> AIC(x.ar)
1] -37.40417
> AIC(x.arma)
1] -42.27357
> x.arma
MQv |QU
1391 '
|vWwOv|wUwt
38
Call:
arima(x = x.ts, order = c(1, 0, 1))
Coefficients:
ar1 ma1 intercept
0.892 0.532 2.960
s.e. 0.076 0.202 0.244
sigma^2 estimated as 0.0151: log likelihood = 25.1, aic = -42.3
0.4
−0.2
0.0
0.2
ACF
0.6
0.8
1.0
> acf(resid(x.arma))
0
1
2
3
Lag
p}O@D MQv
Q
Q
| U |= @
ARMA(1,1)
Q
O =t}k=@ Q=ov|oDU@ty
| U x v
%33
pmW
sQ=yJ pYi
=DU}==v |=ypOt
OvwQ
w
|rYi
Q}eD
C=
QF-=Dt =Q} R 'OvDUy =DU}==v =y|QU
R=
u}= QO "OW p}O@D =DU}= |QU x@ p=iD Q=@ l} =@ xm
CU=
"
QLDt u}ov=}t
l
w
w}UQoQwD=
w =yv
"O @
=}U@ Ot xDWPo
R= |Q
R=
|m} |iO=YD uORs=o p=Ft
qtH pt=W xm O@=}|t \U@ |iO=YD
C
w |t xO}t=v 3 ARIMA Q=YDN= x@ xm 'CU= 2 |oJQ=Bm} =}
"O W
SARIMA
w |t
w x
CQ Y @ =Q u u= D
O = |t |v=tR
" W @
`}=W |r=t |v=tR
=
=
Q
= |t =DU}==v
w OR U
=}U@ p}rLD
| y| U R= |Q
Q
| y| U QO
= pYi
| y
=Q u
QO |
xm CU= |rYi
w
1 `}tHD
wvax@ "OvW=@|t
=
O pYi
uORs o p t
=DLt |r=iD
qtH pt=W
ARIMA
?r]t u}= "Ovm|t Q}eD Kww Qw]x@ Tv=} Q=w =DU}==v
xmQw]u=ty
G
C
Ovtv=wv Q=R@= |rYi
u=
QO
=
Q
| U
ARIMA Ov Q
} i
=
O
| yp t "O=O u
Q
| y| U R=
=Wv
|a@
QO
w}UQoQwD= pOt '=y|QU `wv u}= |R=UpOt |=Q@ =yOQm} wQ R= |m} "OQ=O RwQ@ u=mt= R}v |t}rk= |=yOQwmQ QO "CU=
5
GARCH
=YDN= x@
Q
u
xDi=} s}taD `wv
w |t xO}t=v 4 ARCH pOt Q=YDN= x@
"O W
w
CU= Tv=} Q=w
Q
|= @
w |t xO}t=v
"O W
1. aggregated
2. integrated 3. AutoRegressive Integreted Moving Avrage 4. AutoRegressive
Conditional Heteroskedastic 5. Generalised AutoRegressive Conditional Heteroskedastic
39
1391 '
|vWwOv|wUwt
40
|rYi Q}e
jQ@
w
x= N w
|iO=YD
p=iD u}rw=
= pFt OW=@ |iO=YD x=wN 'Ovm
uORs o
xt = Xt 1 + wt |va
'
}
;
|iO=YD
P
h L =Q
=
ARIMA
O}rwD
OvwQ Ov=wD|t
|QU w
f
xt
Q@ p=iD
=ta=
p
w pFt OW=@ u}at
uORs o QO "
|OQ t
xt = xt xt 1 = wt
xt = a + bt + wt Ov Q p =k
O = |t =DU}= xm CU=
" W @
1
1
4
p=iD
g |QU
CU= |]N OvwQ xm
14
pOt
r
;
;
xt = xt xt 1 =o C O}i x w = =] = |] O l x
Q R
arima() ` =
=W MA(1) =
w | x C =DU
QLD u}o =} Ov Q l
O}
Q R \ w u}a O Q |r =i O O} wN =t Q
=t |t =H C = Ok = = O
=iD = |r =i Q Q ARMA O
w | xD@
w | s}_v Qi u}o =} = |r =i Q =o
Q d=0 include.mean=FALSE arima() ` =
xt = yt = b + wt + wt 1 u Q =v O = xt = a + bt + wt x O}v Q ARIMA O = =@
Pt
Q MA(1) Q O | C x xt = x0 + i=1 yi x] yt = xt
wma h Qa
=iD = C
O
= x}@
xt T = u Q =v C -1 Q Q wt 1 ? Q O = Q yt |r =i | = Q
r
;
VR= @
'
v '
QO
U= @
U=
U
@ D "O=O u
yO VR= @
xO
x
;
U D
D | U
t
vwQ
N @
v
@
@ =Q
i v| y
N
@ =Q u u= D
D p t
p t u= D
y=
t
@
@
x W |R U
W f
t
W
t
UO @
g
v } Q=w
}= l
W
D
v R
Y
t
t
@= @
v t
@ F Q=
v t @
w
t
@
t QO
} i
@ =Q
m Z i
r
T
D | U x
v
@ D R=
p t @ \ DQ= QO
t
}
} ' @ } VR= @ f
;
}
yp t
w
m
U=
} i
U=
t
@=Q
}= @ @ "
m
m
o = "OQ
}= @ @ ' W @
;
o = " }
} QO
r= "O W
"O=O VR= @
r
vwQ
D R= xO
g
U= @ "
D
U=
v tR | U
Ci=} Oy=wN V}=Ri= s}kDUt \N pwL
"
O}vm xOy=Wt Q} R
"
yt = xt
= xt xt 1
= a + bt + wt
= b + wt + wt
QO
O}v=wD|t
=Q
Q}N=
r
;
;
a + b(t 1) + wt
; f
;
1
;
)
x0 +
t
X
i=1
yi = x0 +
t
X
(b + wi ; wt 1)
;
i=1
= x0 + bt + wt = xt
;
1g
Q
=
=} R
h= o =Q B C } H
=DU}==v
41
Q |t
"OO o
x0 = a
w0 = 0 x
xH}D xt 1 Q
CU=
w
w |t
"O W
xt = xt
= xt
= xt
= xt
...
=
;
;
;
;
;
;
;
;
;
= x0 + bt + wt + (1 + )
t
X
i=1
;
;
3
;
wt
CUDQ=@a Tv=} Q=w
%R=
Q@=Q@ Tv=} Q=w
w
CQ Y
u}=
QO
QUB |v} Ro}=H =@
|= @ w
;
+ b + wt + wt 1
2 + b + wt 1 + wt 2 + b + wt + wt 1
2 + 2b + wt + (1 + )wt 1 + wt 2
3 + 3b + wt + (1 + )wt 1 + (1 + )wt 2 + wt
1
;
pYi
4
O}W=@ xDW=O xHwD
m
v
O
| yp t
xm
Var(xt ) = w2 f1 + (1 + )2(t ; 1)g
O = = ;1 x u Qo O = | V R R}
W @
m
}=
t ' @ }
t
}= i=
v
Vv=} Q=w
t
Q=
Okt V}=Ri= =@ xm
2
CU= w
"
sD} Q=or
Q
| U '
=
=F
Q
Q}o|t
w
| yxO=O p t |= @ "O
|v=tR
=F |r=iD
p t
Q
| U 14
Q
pmW
CQ Y
di \
R
w
U D
wtv xU Q} R
QO
p=iD u}rw= 'OW x_Lqt ,q@k xm Qw]u=ty
= O O Q}o@ Q_v
| y )
m " }
Q=O
xDU} QDmr= O}rwD x@
QO =Q
w |t xOy=Wt xm Qw]u=ty "OyO|t u=Wv
| U wO QO O W
sD} Q=or |r=iD
=Q
w Q |a}@]
\ @ t
Q
| U w
|r=iD
w |tv Qy=_ Qo}O OvwQ xirw-t 'xOW xDio
"O W
> www <{ "http://www.massey.ac.nz/~pscowper/ts/cbe.dat"
> CBE <{ read.table(www, he = T)
> Elec.ts <{ ts(CBE, 3], start = 1958, freq = 12)
> par(mfrow=c(3,1), mar=c(4,4,2,4))
> plot(Elec.ts)
> plot(di(Elec.ts))
> plot(di(log(Elec.ts)))
6 |k}irD pOt
=@
w
Q
Ov} wo
d x@ Q
D t R=
|k}irD pOt
O = |t wQUB p=kDv= Qorta
| U " W @
=ov
=Q u x
B
u QO
wt
Q
'OO o f
xm CU=
g
d
r
O}iU xiwv p=iD u}t=d
(1 ; B )d x
m
s}O}O
R=
TB
w |t
"O W
Qo = 'CU=
xt
f
2
1
4
g |QU
xO=O u
d x@ Q
|k}irD
D t R=
(1 ; B ) = wt
di() ` = w | C I (1) = C = | =Y
C Q p = di(x, d=2) = di(di(x)) p
=Wv
Qo =
I (d)
xt
f
g
d
=F
wvax@ "CW=O R}v q=@ x@DQt =@
p t u=
wva Q} R `@=D u}=
u=
Qo}O u=twoQ
QO |
6. Integrated model
=Q
"
@ D u= D
U=
mP
@ k
t "
U=
X N
} w
r L
iO
= O =
D uORs o " W @
=iD Q=@ wO
Q
|= @
|vWwOv|wUwt
42
8000
2000
Elec.ts
14000
1391 '
1960
1965
1970
1975
1980
1985
1990
1980
1985
1990
1980
1985
1990
1000
0
−1500
diff(Elec.ts)
Time
1960
1965
1970
1975
0.00 0.10 0.20
−0.15
diff(log(Elec.ts))
Time
1960
1965
1970
1975
Time
|tD} Q=or |r=iD
Q
| U w
u}= "CU= OL=w Q@=Q@ ZQiV}B CQwYx@
xirw-t
di(x, lag=12)
=F
Q
|r=iD
Q
|v=tR
Q
| U '
Okt "Ovm XNWt
u Q=
Q |atH pOt
p t |= @ "OO o
QO
| U
=Q
|rYi
V}=tv
%14
pmW
p=iD Q}N-=D Ov=wD|t xm CU= OwHwt lag
Q}}eD xirw-t
C=
P
h L
?@U Ov=wD|t u=twoQ
=iD CU= umtt =Hv}= QO "O}=tv|t hPL xv=y=t |=y|QU QO =Q |atH pOt |rYi C=Q}}eD QF= w |]N OvwQ
Cw
u
O)m TBU 'O}vm xHwD
wt = vt vt
;
;
3
=F x@ s=y@= `iQ
p t
lag d
Q O = xDW=O s=y@=
|= @ " W @
"
w
CU= Q} R
w x
CQ Y @
= w
u t oQ wO
u}@
di() ` =
@ D QO
> w <{ di(v, lag=2, d=1)
=yp=Ft
Ov}Qi x@ p}O@D
xt
f
g |QU
B )yt = q (B )wt
= x Qo = uwvm = "CU= p (
| H @
p=iD u}t=d Qo = 'CU=
x
=ov
w
ARIMA(p,d,q) Ov Q
O = yt = (1 ; B )d xt Q
xt | = Q
Q ARMA(p,q)
Q} Q xt Q}eD yt
} i |=Q=O f
' W @
3
1
4
h} QaD
g
v tR | U
o = "OO o
xm OwW|t xH}Dv
'O
o Q= k
t
p(B )(1 B )dxt = q (B )wt
;
ARIMA
=
O
=F OvJ x@ uwvm = "OvDUy q w p x@DQt R= ?}DQD x@ |}=y|=xrtH OvJ q w p
| yp t R= p t
u QO
xm
O}vm xHwD
"
=DU}==v
43
QUB p=kDv= Qorta
w x
w
CQ Y @ u
wvm = "CU= pOt QDt=Q=B
=
O
4
1
Ov}Qi
| yp t
x = xt 1 + wt + wt
xm t
u QO
;
;
pYi
w |t xDWwv
"O W
xt xt 1 = wt + wt
;
;
QLDt u}ov=}t |k}irD pOt
l
1 )
;
u
x@
CU=
w '
ARIMA(0,d,q)IMA(d,q) |r
m
(1 ; B )xt = (1 + B )wt
ARIMA(0,1,1)
Cr=L
w |t
=Wv
QO "O W
xt Q
IMA(1,1)
w x
g | U 'xU}=kt =@
CQ Y @ f
xO=O u
u}=Q@=v@
w x xm Ov} wo
CQ Y @
CU=
"
Qorta
w x
CQ Y @ u
wvm = "CU= pOt QDt=Q=B
u QO
x = xt 1 + xt
xm t
;
1 ;
;
xt 2 + wt Ov Q
} i ;
w |t xDWwv wQUB p=kDv=
"O W
xt xt
;
|rm Cr=L
1;
;
w |t
QO "O W
xt 1 + xt 2 = wt
;
xO=O u
;
=Wv
ARI(1,1)
)
(1 ; B )(1 ; B )xt = wt
w x xm Ov} wo |k}irD w}UQoQwD=
CQ Y @
CU=
"
O
p t u
x@
ARI(p,d)ARIMA(p,d,0)
4
1
4
VR=Q@ w |R=Ux}@W
u}}aD
=
|R U
c(p,d,q) = O x@ Q x O = |
Q R
arima() ` =
x}@ xt = 0:5xt 1 + xt 1 + wt + 0:3wt 1 x w Q
=
@ p t
W
D t
;
m
@ }
t VR= @
QO
;
;
w |t
"O W
w
@ D
=@ =yxO=O Q@
@ \ @ t | yxO=O
Q =yv Q@
xO=O VR= @
ARIMA(p,d,q) Ov Q
} i
Q} R |=yO)m
=F
Q
Q |t
QO p t |= @ "OO o
ARIMA(1,1,1)
O TBU
p t
O
w x W
> set.seed(1)
> x <{ w <{ rnorm(1000)
> for (i in 3:1000) xi] <{ 0.5 xi - 1] + xi - 1] - 0.5 + xi - 2] + wi] + 0.3 wi - 1]
> arima(x, order = c(1, 1, 1))
Call:
arima(x = x, order = c(1, 1, 1))
Coecients:
ar1 ma1
0.4235 0.3308
s.e. 0.0433 0.0450
sigma^2 estimated as 1.067: log likelihood = -1450.13, aic = 2906.26
=
"OQ=O s v
arima.sim() ` =
@ D
u}= "OyO|t s=Hv=
= O)m u=ty Q=m xm
=Q q @
CU= Q} R
"
w
x =N@=Dm `@=D
OQ=O O Hw |= v
w x O
QO
xD@r=
= x}@W sDU}U uwvm =
CQ Y @ x W |R U
> x <{ arima.sim(model = list(order = c(1, 1, 1), ar = 0.5,
R
1391 '
|vWwOv|wUwt
44
+ ma = 0.3), n = 1000)
> arima(x, order = c(1, 1, 1))
Call:
arima(x = x, order = c(1, 1, 1))
Coecients:
ar1 ma1
0.5567 0.2502
s.e. 0.0372 0.0437
sigma^2 estimated as 1.079: log likelihood = -1457.45, aic = 2920.91
ARIMA
|rYi
24
|=ypOt
1
2
4
h} QaD
|atH pOt |rYi C=Q}}eD QF= =D Ovm|t xO=iDU= (s) pYi xR=Ov= x@ |Q}N-=D =@ p=iD R= ARIMA |rYi pOt
s Q}
u}ov=}t xrtH Qov=}@ p=iD
O
s Q}
O = |t
p t " W @
CWwv Q} R
= =@
N-D
l
=
N-D w
Ovm|t
QLDt u}ov=}t
w
P
OvwQ xirw-t p=iD Qorta OL=w Q}N-=D =@
h L =Q
w}UQoQwD=
w x QUB p=kDv= Qorta ?UL Q@
w |t
CQ Y @ w
u= D
=Q
P
ARIMA |rY O C QLD
ARIMA(p d q)(P D Q)s |rY
qtH pt=W
C
Q
"OO o h L
i p t "
U= l
t
i
P (B s)
p(B )(1 ; B s)D (1 ; B )d xt = Q(B s)q (B )wt
|twta Cr=L
xrtH OvJ
|=
OvW=@|t
QO "
q Qp P
w
'
qtH xYNWt xrO=at
= xrtH OvJ ?}DQD x@
R= | y
= xW} Q jr]t QOk
C
|rYi pOt p=Ft OvJ x@ uwvm =
x@DQt
'
| y
w
q Q p P
D=d=0Q
o=
w Oy=wN =DU}= x=ov 'OvW=@ OL=w
"O @
R=
'
w
'
QDoQR@ |oty Q}N= xrO=at AJ
O}vm xHwD
=
=
p U x t |wQ
w x xm
CQ Y @
ARIMA(0 0 0)(1 0 0)12 12 |rY
'
xDWPo p=U x=t xm
u
Q@
O =
\w
QWt CU= ?U=vt xv=y=t
1=12
" W @ j
;
j
>1Q
o=
=v =@
i ?w D
=
AR
=
O
xO U p t Q |rOt u}vJ "CU=
| yxO=O |= @
CU= =DU}= pOt u}= "OW=@ xDW=Po QF=
|}=H@=H =@
' }
t
"O W
'
U=
\U@ t
;
;
;
;
UO @
t
t
"O=O
} w
Y L
}=
@
}= v p t
" W @
;
t
W v u= D
}= "OQ=O
t
v r
r
W o p U QO x t
} Q |=Q=O
m
U=
i
m
@=Q
CQ Y @ =Q p t
;
} u tR u
;
t =Q q @
;
} i '
H D @ "
CQ Y @ u= D
;
=
|Q H
y QO
H
D
t W
@
@
oQ
@
t
@
m
H D
}=
m
u tR QO
J h ] |=
@ p t @=Q |
}
H
Q
h ]
ARIMA
x = xt 1 + xt 12 xt 13 + wt w x w | = Ov O
O | C x 1 (B 12 )(1 B )xt = wt = (1 B 12 )(1 B )xt = wt x]
Q} wD =
w | p = ARIMA(0 1 0)(1 0 0)12 Ov Q |rY ARIMA |r x] = xU =k = x
?r] u x x w = C w w | R}
xt = xt 12 + wt w x O u x O}v x w
C =DU = O u
xD P =
= |va
=
=t
Q}}e x |oDU t =
Q}}e x
O = | B = 1 xW
x C (1 B ) xrt p = A
Q
xrt Ov Q
w
xm
=t= "CU= =DU}==v pOt u}=
"
xt = x12 + wt
u QO
m i
D
m
m
J = }R
=DU}==v
45
x = (1 B 4)wt = wt wt
u}= "CU= t
;
OvW=@ |iO=YD OvwQ
'
w x
;
x = xt 1 + wt wt
w |t w 'CU= t
=v
;
;
4
;
O
O
pYi
4
QLDt u}ov=}t pOt l}
CQ Y @
=yxO=O Qo = "CU= ?U=vt OvwQ
|=Q=O
CQ Y @ u= D
?w D QO
w x pYi Q=yJ
4
;
=
| yp t
l
=
Q \ki
uw @ | yxO=O |= @
CU= =DU}= pOt
w
xm O}=}|t \U@ pw= x@DQt p=iD CQwYx@ pOt
1 0)(0 0 1)4
p=iD 'OW=@ |iO=YD OvwQ |=Q=O |rYi CqtH Qo = "CWwv R}v ARIMA(0
w x
O
CQ Y @ =Q u w ' yO
|t CUOx@
xt = xt 4 + wt wt
=Q
;
;
"
CW=O l} Q}N-=D =@ p=iD
w |t u}=Q@=v@ 'Ovm|t
u= D
CWwv
P
4
;
x]@=Q
w |t R}v
u= D
|]N OvwQ
h L =Q
Q |t p=ta= R}v |rYi
w OO o
ARIMA(0 0 0)(0 1 1)4
s Q}
= = p=iD xm CW=O xHwD O}=@
N-D @
'
QLDt u}ov=}t CqtH x=ov 'OQ=O OwHw |]N OvwQ xm |r=L QO OOQo p=ta= p=iD QO l} Q}N-=D Qo = #Q}N =}
l
Q |t |iQat O}iU xiwv
CQ Y @
w x
=F
wvax@
w
|atH
O =t}k=@
t] x
"OO o
O l} CQwYx@ O}iU xiwv w |rYi C=Q}}eD w |]N OvwQ
p t
u QO
xm
4
=v = |v=tR |QU x@
'p t u=
?w D @
"
O}vm xHwD
OQ=O O Hw
xt = a + bt + s
t] + wt
O}vm xHwD
"
4 Q}
= = p=iD u}rw= x@ uwvm =
N-D @
s
4] = s
t
4]
;
u}=Q@=v@ "OyO|t u=Wv
(1 ; B 4 )xt = xt ; xt 4
= a + bt ; (at + b(t ; 4)) + s
t] ; s
t
= 4b + wt ; wt 4
=Q
4Q t
@
x v
m
;
4]
;
+ wt ; wt
4
;
;
u}rw= xm O}vm
Q
Z i u
wvm =
wtv u=}@
"O
4b C =
@ F
=ov
x
ARIMA(0,0,0)(0,1,1) w x
=t 4 Q} = = p =i
p@ l Q} =
xrtH =@
Q
'OO o p
w | t xm
CQ Y @ =Q u u= D
N-D @
a=
D R=
k
}
= p=iD x@DQt
N-D @
(1 ; B 4 )(1 ; B )xt = (1 ; B 4)(b + s
t] ; s
t 1] + wt ; wt 1)
= wt ; wt 1 ; wt 4 + wt 5
;
;
;
CU= C@=F Q=Okt Ok=i xm CWwv
"
;
;
ARIMA(0 1 1)(0 1 1)4
w x
w |t xm
CQ Y @ =Q u u= D
VR=Q@
2
2
4
x} wQ
u}=Q@=v@ "OW=@ CqtH R= |@}mQD w OOaDt |=yQDt=Q=B pt=W Ov=wD|t xwkr=@ CQwYx@ '|rYi ARIMA |=ypOt
R=
QD?U=vt pOt
=NDv=
?
Q
w
=LDt= =yxO=O Q@
|= @ w O W u
Q
Q
O
VR= @ |= @ p t R=
|a}Uw
xOw
OLt xm CU= xDU}=W
xiwv CQwYx@ xOv=t}k=@ Q=ov|oDU@ty |=Q=O ,=k]vt 'xOW xDi=} VR=Q@ ?U=vt pOt "OOQo xO=iDU= AIC x]@=
"
xm
O
w
OQ=O O Hw
x W
seasonal
xDiQo Q_v
QO
wva Q} R |v=twoQ `@=D u}=
u=
xDU} QDmr= O}rwD
=
| yxO=O
sD} Q=or
w |t
QO "O W
Q
| U
=iDU=
xO
Q} R p=Ft
Q
Q
VR= @ |= @
Q |t
QO "OO o
CU= O}iU
arima() ` =
GQO u QO
R
@ D R= '
|rYi
QO
= xirw-t
| y
1391 '
|vWwOv|wUwt
QDy@
=Q |
Q
VR= @
ARI
46
O
Ovm|t
p t "
P
h L =Q
|]N OvwQ xm CU=
d=1
w
OQ t wO
Qy
QDmJwm
"OQ=O |
QO
p=iD QDt=Q=B "CU=
AIC
Q
O
= } R ' yO
|t u=Wv
> www <{ "http://www.massey.ac.nz/~pscowper/ts/cbe.dat"
> CBE <{ read.table(www, he = T)
> Elec.ts <{ ts(CBE, 3], start = 1958, freq = 12)
> AIC (arima(log(Elec.ts), order = c(1,1,0),
+ seas = list(order = c(1,0,0), 12)))
1] -1764.741
> AIC (arima(log(Elec.ts), order = c(0,1,1),
+ seas = list(order = c(0,0,1), 12)))
1] -1361.586
`@=D Qt= u}=
VwQ
QO
CrwyU
Q
w |t
|= @ "O W
|Dkw OQm} wQ u}= "OyO CUOx@
O}vm xHwD Q} R
"
xO
=Q
=iDU= =]N
w
|aU
QD?U=vt pOt
= O x@ uwvm = "CU=
| y )
m
R=
QD?U=vt pOt uDi=}
AIC x] =
=
@ T U=
Q wDU= sD} Qwor=
| DQ=
w
> www <{ "http://www.massey.ac.nz/~pscowper/ts/cbe.dat"
> CBE <{ read.table(www, he = T)
> Elec.ts <{ ts(CBE, 3], start = 1958, freq = 12)
> get.best.arima <{ function(x.ts, maxord = c(1,1,1,1,1,1))
+f
+ best.aic <{ 1e8
+ n <{ length(x.ts)
w
=ypOt xvt=O pQDvm
Q@ =D CWwv
OyO|t
+ for (p in 0:maxord1]) for(d in 0:maxord2]) for(q in 0:maxord3])
+ for (P in 0:maxord4]) for(D in 0:maxord5]) for(Q in 0:maxord6])
+f
+ t <{ arima(x.ts, order = c(p,d,q),
+ seas = list(order = c(P,D,Q),
+ frequency(x.ts)), method = "CSS")
+ t.aic <{ -2 t$loglik + (log(n) + 1) length(t$coef)
+ if (t.aic < best.aic)
+f
+ best.aic <{ t.aic
+ best.t <{ t
+ best.model <{ c(p,d,q,P,D,Q)
+g
+g
+ list(best.aic, best.t, best.model)
+g
> best.arima.elec <{ get.best.arima( log(Elec.ts),
+ maxord = c(2,2,2,2,2,2))
> best.t.elec <- best.arima.elec2]]
> acf( resid(best.t.elec) )
> best.arima.elec 3]]
1] 0 1 1 2 0 2
w
w |t
u= D
?= H
=Q
Q
|= @
|mJwm
QDy@ OW=@
CSS
=DU}==v
47
=
O
| yp t
pYi
4
> ts.plot( cbind( window(Elec.ts,start = 1981),
+ exp(predict(best.t.elec,12)$pred) ), lty = 1:2)
QDq=@ ?D=Qt uwtR "CU=
pmW "OvDUy
34
ARIMA(0 1 1)(2 0 2)12
swO
pmW O}iU xiwv ,=@} QkD xOW xDio pOt
x@DQt
R=
QDC@U=vt pOt
=
= O
= O =t}k=@ =Q} R 'OUQ|tv Q_v x@
| yx v
O
|t u=Wv
O
=Q x W
Q
=@
O
|Qw p t
|v}@V}B Q}O=kt
24
8000
10000
12000
14000
" yO
Q
'q @ | y )
m |= H=
1982
1984
1986
1988
1990
1992
Time
Q O}rwD xv=y=t
j @
Q
|v}@V}B V}=tv
| U
%24
pmW
ARCH
SP500 | y| U
=
w
"OQ=O O Hw
R
QO
MASS x =N =D
v
@ m QO
McGraw-Hill | =Bt
v
m
x@
34
|=ypOt
w Q
\ @ t
Q
SP500
1
3
4
=
| yxO=O
> library(MASS)
> data(SP500)
> par(mfrow=c(2,1), mar=c(4,4,2,4)) > plot(SP500, type = 'l')
> acf(SP500, main="")
=@ "OyO|t
=Wv
u
=Q
=DU}= Ov}Qi
Okt u} QDoQR@ 'QN ErF
Q=
=ov
pw= x
QO u Q=
Okt
w
O
|t
QO " yO
u} QDmJwm
Q
=Wv
u
| U
O
=Q x W
\Uw ErF
xDio
QO
= q]= |v=tR
C a
Q
| U 44
pmW
Tv=} Q=w xm OwW|t swrat QDW}@ CkO
1391 '
|vWwOv|wUwt
48
0.4
0.0
0.2
ACF
0.6
0.8
1.0
Series resid(best.fit.elec)
0.0
0.5
1.0
1.5
2.0
Lag
Q O}rwD xv=y=t
j @
w
OW=@v C@=F u=tR x@ C@Uv Tv=} Q=w Qo =
Q
Q}eDt
pFt
Q
=
OO o u tR
Q
| U Q
=iDt u=tR
Cw
x@ xDU@=w
w
w |tv s=Hv= |t_vt Q}Ut
Q O}rwD
j @
OW=@ |W}=Ri= Ow} QB
w
|}=t}B=wy
Q
xm u}= p}rO x@ CU= =DU}==v |QU =t= 'OQ=Ov O}iU
Q
u \ W
x@ OvDUy Tv=} Q=w
=at
pO
@= @
"O W
t
i o
t UO
D
v QO
} v=S
D
"
R=
U=
} v=
=
| yxO=O
i v @ |Q=O
?? pm
W
=a@ Qt u}= "OyO|t u=Wv
R=
}= @ @ "
} R CQ Y @
U=
x W
Ovv=t
=@ x@
CQ a
Q}o s=Hv= s_vt
'O
"OQ=O =Q
Q}}eD
C=
u
| U
C
0.048 Q Q 1999 Q@ = 31 = 1900 x w 2 SP500
w | xD Q Q_
1469 = 360 T O u Q =v C
C Q
w x SP500 O
V}=Ri= Qo}O
QO
Tv=} Q=w Qo = "Ov} wo
`@ Qt Q=ov|oDU@ty x=ov 'OyO|t u=Wv =Q O}iU xiwv Q=ov|oDU@ty
u}ov=}t xm
pmW
heteroskedastic
heteroskedastic Q x =o SP500
xw =
|va
=i SP500 = Ct}
|=Q=O | U
=ov|oDU@ty xm O}W=@ xDW=O xHwD "Ov} wo |]QW
QO
%34
"O W
x@ x=ov O@=}|t V}=Ri= OvwQ =@ Tv=} Q=w xm xO=O
= O =t}k=@ V}=tv
| U | yx v
xO=O
u}ov=}t
t Cw
@ x
v D y
k
j@=]t Qo = "CU= Q}eDt Tv=} Q=w
=
=Q u tR
x@ C@Uv
Q s}_vD QiY
"OO o
Q}}eD Q}O=kt
C=
w x =yv
CQ Y @
|mJwm Q=Okt Tv=} Q=w =@ xU}=kt
QO
xm 'CU=
K}LYD u}ov=}t =@ `@ Qt Q}O=kt Q=ov|oDU@ty
library(MASS)
data(SP500)
acf((SP500 - mean(SP500))^2)
O
" yO
|t u=Wv
O==
=Q |Q= } B v w
|]QW
heteroskedastic
w
K w
x@
54
pmW
=DU}==v
−2
−6
SP500
2 4
49
0
500
1000
1500
2000
2500
0.4
0.0
ACF
0.8
Index
0
5
10
15
20
25
30
35
Lag
SP500
=
Q
| yxO=O | U
V}=tv
%44
pmW
0.0
0.2
0.4
ACF
0.6
0.8
1.0
Series (SP500 − mean(SP500))^2
0
5
10
15
20
25
30
Lag
O
x W
K}LYD u}ov=}t =@ Q}O=kt `@ Qt V}=tv
%54
pmW
35
=
O
| yp t
4
pYi
1391 '
|vWwOv|wUwt
50
ARCH p t
h} QaD
O
x@ uO}UQ
Q Ovm u}@D
|= @ "
=Wv
xO=O u
Tv=} Q=w
=Q
ARCH(1) =
xm CU=
@
QO
|]QW
Q}}eD xm
C=
CUOx@ |rOt O}=@
O=O
heteroskedastic |
=D
'Q iQ
u}=
Q
O
f g |QU '?r]t
Qo =
q
t = wt 0 + 1 2t
Q
uO m p t |= @
w}UQoQwD= x@DQt u}rw=
Q
] W
2
3
4
%|Q=O}=B=v uOQm pOt
u}=
w |t
'O W
1
;
OvDUy pOt
"
= QDt=Q=B
| y
1 0
'
w
w
CU= OL=w Tv=} Q=w w QiY u}ov=}t =@ O}iU xiwv f t g q=@ x]@=Q
O Q}o@ Tv=} Q=w Q}N= x]@=Q
" }
R=
O =
QO " W @
volality
=}
Q
u @ |= @
Var(t ) = E (2t )
= E (2t )E (0 + 1 2t 1 )
= E (0 + 1 2t 1 )
= 0 + 1 Var(t 1 )
;
;
;
AR(1) Ov Q
=
} i @ =Q
Q}N= xrO=at Qo = 'CU= QiY u}ov=}t |=Q=O f
w
OL=w Tv=} Q=w |=Q=O f t g xm u}= x@ Q_v
g w
O}vm xU}=kt
xt = 0 + 1 xt 1 + wt
ARCH(1) O v Q T = x O } v | x _ q
ARCH O
Q x Ov | XNW = =] ` Q
;
O v m| t Q= D iQ
"
AR(1) p F
#
t
C UQO
CU= ?U=vt
} i
p t VR= @
v } Q=w
m
m
m
m
t
t y
GARCH | yp t
=
w x
CQ Y @
ARCH(p) Ov Q
t
O
x@
L
N
w | t p Y= L
t "O W
= |oDU@tyOwN
@ t | y
3
3
4
\w@ Qt |=y\U@
O = \U@ q=@ |=yQ}N-=D =@ p x@DQt Ov}Qi l} x@ Ov=wD|t ARCH pOt x@DQt u}rw=
} i " @ }
CU= Q} R
"
v
u
p
X
u
t
t = wt 0 + p2t;i
i=1
"
ARCH(p) x
CU= GARCH(q,p)
m '
Qo = 'OW=@|t
CU= OL=w Tv=} Q=w
ARCH
w x
CQ Y @
GARCH(q,p)
O
QiY u}ov=}t =@ O}iU xiwv
O
p t R=
t
p t |=Q=O f
p
w
wt
f
g u QO
xm
|r=t |=yxO=O OQ@ Q=m =@ QD`}Uw |=y\U@
g |QU "CU=
GARCH(0,p)
=
X N
Cr=L
t = wt ht
ht = 0 +
OvDUy pOt
"
p
X
i2t i +
i=1
u QO
j ht
xm
j
;
;
j =1
(j = 0 1 : : : q) i (i = 0 1 : : : p)
= QDt=Q=B j
| y
q
X
w
w
CU=
=DU}==v
51
xDi=}
ht = 0 + t 1 + t
;
CU= 64 pmW
"
1
;
a = wt ht
p
xm t
O
4
GARCH p t w |R U
O
GARCH(1,1)
1 + 1 < 1 O = O =
w x
CQ Y @
w x =ov|oDU@ty w OW=@
CQ Y @ Q
VR=Q@
=
| yp t
=
4
3
4
x}@W
O =@ |}=yxO=O Q} R
p t
Q
w |t
} @ |Q= } B |= @ "O W
pYi
= O
| y )
m QO
= x}@W CU=
|R U
> set.seed(1)
> alpha0 <{ 0.1
> alpha1 <{ 0.4
> beta1 <{ 0.2
> w <{ rnorm(10000)
> a <{ rep(0, 10000)
> h <{ rep(0, 10000)
> for (i in 2:10000) f
+ hi] <{ alpha0 + alpha1 (ai - 1]^2) + beta1 hi -1]
+ ai] <{ wi] sqrt(hi])
+g
> par(mfrow=c(2,1), mar=c(4,4,2,4))
> acf(a)
> acf(a^2, main="")
0.4
0.0
ACF
0.8
Series a
0
10
20
30
40
30
40
0.4
0.0
ACF
0.8
Lag
0
10
20
Lag
=yv `@ Qt
Qy=_ |oDU@ty Q}O=kt `@ Qt
QO
xm
w
OQ=O
Q}O=kt Q=ov|oDU@ty V}=tv
%64
xDU@ty=v Q}O=kt xm OyO|t u=Wv
pmW
=Q
GARCH
O
w
p t X= N
a
Q
| U
Q |t
"OO o
1391 '
|vWwOv|wUwt
garch() ` =
52
=iDU= =@ xm O@=}|t
Q O
@ D R= xO
= x}@W
VR= @ x W |R U
=
| yxO=O |wQ
|@=} R=@ xQ=@ wO pOt |=yQDt=Q=B xm OOQo|t x_Lqt "OwW|t Ci=} tseries
= w xm CU= GARCH(1,1) Q@=Q@ Q}N= `@=D
,a v
w |t
"O W
=iDU=
xO
GARCH
Q@
R
= w
=F
Q xDU@ QO xm OwW|t s=Hv=
Q V}B "OQ}o|t Q=Qk %95 u=v}t]=
u t oQ R=
Q
|Q= i=s v
Z i
order=c(p,q)
O
p t ' } R p t QO
QDq=@
=
xR @ xOw
= x@DQ
OLt
w |t
QO w O W
Q =t= 'CU= woNU=B
| y
|= @
> set.seed(1)
> alpha0 <{ 0.1
> alpha1 <{ 0.4
> beta1 <{ 0.2
> w <{ rnorm(10000)
> a <{ rep(0, 10000)
> h <{ rep(0, 10000)
> for (i in 2:10000) f
+ hi] <{ alpha0 + alpha1 (ai - 1]^2) + beta1 hi -1]
+ ai] <{ wi] sqrt(hi])
+g
> library(tseries)
> library(quadprog)
> library(zoo)
> a.garch <{ garch(a, grad = "numerical", trace = FALSE)
> connt(a.garch)
Q |t pY=L Q} R G}=Dv
"OO o
=
= O
Q
'q @ | y )
m |= H=
=@
2.5 %
97.5 %
a0 0.0882393 0.1092903
a1 0.3307897 0.4023932
b1 0.1928344 0.2954660
w x
O
=@ =Lt pL=Qt
CQ Y @ =Q |O a C U
O
" yO
|t CUOx@
w
Q wDU=
=Q | DQ=
OyO|t CUOx@
O
|O a VwQ
=@ =Lt xYqN
=Q C U
grad="numerical"
trace=F
= w
O
Q
wt
R= f
g
|va}
GARCH
O
O =t}k=@
p t x v
Q
O = |t
| U " @ }
Q
VR= @
SP500
Q
| U
VR=Q@
Q@
5
3
4
GARCH
O
p t
w |t x@U=Lt
"O W
w x O}=@ xH}Dv x=ov 'OW=@ ?U=vt =yxOv=t}k=@ |QU
CQ Y @
GARCH(1,1) C =
h^ t = ^0 + ^12t 1 + ^h^ t
;
1
Q
|= @
GARCH
O Qo =
p t
w |t Qy=_ OL=w Tv=} Q=w
r L QO "O W
;
|tv u=Wv |r}iD
Q@
w^t = pt
h^ t
QiY u}ov=}t =@ O}iU xiwv
=F
u t oQ " yO
SP500 | U
Q} R x]@=Q
= w
u t oQ p t QO
w
=DU}==v
53
=@
Q=
Okt
u
u}=Q@=v@ "CU}v OwHwt xOv=t}k=@
O |t CUOx@ =yxOv=t}k=@
" }
=a@ Qt
C
Q
| U Q=
Q
Okt u}rw=
O =t}k=@
| U w x v
Q
"O
| U Q
|=R=
o
t s
v
v=
=k}kLD OL=w x@
O s}UQD
x W
74
w Q
1850-2007
\ @ t pmW
O
=
CU= xOW s=Hv= |D=ar=]t s}rk=
O w
QO w x W x v= N
Q_v
w
Q
CU= xDiQo Q=Qk pta
OQ t | U "
l
"
CU= xOW p}O@D |vwDU x@
Q]U Cr=L
|
R=
t() ` =
=iDU= =@ pOt
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> stemp <{ scan("http://www.massey.ac.nz/~pscowper/ts/stemp.dat")
> stemp.ts <{ ts(stemp, start = 1850, freq = 12)
> plot(stemp.ts)
> stemp.best <{ get.best.arima(stemp.ts, maxord = rep(2,6))
> stemp.best3]]
0.0
−0.5
−1.0
stemp.ts
0.5
1] 1 1 2 2 0 1
1850
1900
1950
2000
Time
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> stemp.arima <{ arima(stemp.ts, order = c(1,1,2),
+ seas = list(order = c(2,0,1), 12))
> t( connt(stemp.arima) )
ar1
ma1
ma2
sar1
sar2
sma1
2.5 % 0.8317389 -1.447400 0.3256699 0.8576802 -0.02501886 -0.9690530
97.5 % 0.9127947 -1.312553 0.4530474 1.0041394 0.07413434 -0.8507036
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> stemp.arima <{ arima(stemp.ts, order = c(1,1,2),
+ seas = list(order = c(1,0,1), 12))
> t( connt(stemp.arima) )
ar1
ma1
ma2
sar1
sma1
2.5 % 0.8304007 -1.445057 0.3242728 0.9243495 -0.9694897
97.5 % 0.9108115 -1.311246 0.4508999 0.9956639 -0.8679223
ARIMA
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> stemp.res <- resid(stemp.arima)
> par(mfrow=c(2,1), mar=c(4,4,2,4))
> acf(stemp.res)
> acf(stemp.res^2, main="")
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ACF
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Series stemp.res
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Lag
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`H=Qt
1] Cowpertwait, Paul S.P., Metcalf Andrew V. (2009) Introductory Time
Series with R, Springer, 254p.
2] Ihaka Ross, (2005) Time Series Analysis, University of Auckland, 105p.
3] Shumway Robert H., Stoer David S., (2011) Time Series Analysis and
Its Applications (With R Examples), Third edition, Spriger, 596p.