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Maine.month.ts

<

{ ts(unemploy, start = c(1996, 1), freq = 12)

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Maine.annual.ts

<

{ aggregate(Maine.month.ts)/12

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layout(1:2)

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plot(Maine.month.ts, ylab =

"

unemployed (

%

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plot(Maine.annual.ts, ylab =

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unemployed (

%

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"CU= q=@ |=yO)m |=QH=pY=L 41 Q=Owtv

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wave.dat

<

{ read.table (www, header=T)

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par(mfrow=c(2,1), mar=c(2,4,2,2))

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iO

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xi

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t |

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U

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o

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t

>

data(AirPassengers)

>

AP

<

{ AirPassengers

>

AP.decom

<

{ decompose(AP,

"

multiplicative

"

)

>

plot(ts(AP.decom

$

random7:138]))

>

acf(AP.decom

$

random7:138])

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y

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v

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1391'

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O

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w

U

w

t 14

Time

ts(AP.decom$r

andom[7:138])

0 20 40 60 80 100 120

0.90

0.95

1.00

1.05

1.10

AirPassengers

|

iO

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D

xi

r

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t%10

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"O}vmxHwDQ} R |=yO)mx@=yQ=}at h=QLv=x@U=Lt |=Q@ "CU= 0.03 Q@=Q@|rYi

>

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])

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0.2

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0.6

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Series AP.decom$random[7:138]

AirPassengers

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0.2

0.4

0.6

0.8

1.0

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wn

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t

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t =w

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1391'

|v

Ww

O

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w

U

w

t 22

>

for (t in 2:1000) xt]

<

{ xt - 1] + wt]

>

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l

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o for

xkr

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>

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

>

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0 200 400 600 800 1000

0

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04

06

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x

|

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0.4

0.8

α =0.7

lag k

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0 2 4 6 8 10

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lag k

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Q

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t G=

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U= x.ar$asy.var >

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

pOtx@DQ

1] 1

>

x.ar

$

ar

pOtQDt=Q=B

1391'

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Ww

O

v|

w

U

w

t 28

0 20 40 60 80 100

−

1

0123

Index

x

0 5 10 15 20

−0.2

0.2

0.6

1.0

Lag

AC

F

5 10 15 20

−0.2

0.2

0.4

0.6

Lag

P

ar

tial A

CF

V

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=

y

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yO

w

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2

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W

1]0.6009459

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1]0.4404031 0.7614886

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number of parameters

C

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Global

<

{ scan(www)

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Global.ts

<

{ ts(Global, st = c(1856, 1), end = c(2005, 12), fr = 12)

>

Global.ar

<

{ ar(aggregate(Global.ts, FUN = mean), method =

"

mle

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>

mean(aggregate(Global.ts, FUN = mean))

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=

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0

2

4

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plot(x, type =

"

l

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"

time t

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>

acf(x, main=

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0

200

400

600

800

1000

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2

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time t

x

0

5

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15

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25

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arima()

>

set.seed(1)

>

b

<

{ c(0.8, 0.6, 0.4)

>

x

<

{ w

<

{ rnorm(1000)

>

for (t in 4:1000)

f

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<

{ xt] + bj]

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g

>

x.ma

<

{ arima(x, order = c(0, 0, 3))

>

x.ma

Call:

arima(x = x, order = c(0, 0, 3))

Coefficients:

ma1

ma2

ma3 intercept

0.7898 0.5665 0.3959

-0.0322

s.e. 0.0307 0.0351 0.0320

0.0898

sigma^2 estimated as 1.068: log likelihood = -1452.41, aic = 2914.83

k

=

k;1

C

i

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w

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>

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

intercept

-0.596966371 0.502703368 -0.006571345

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3

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www

<

{

"

http://www.massey.ac.nz/

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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

1391'

|v

Ww

O

v|

w

U

w

t 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

>

acf(resid(x.arma))

0

1

2

3

−0.2

0.0

0.2

0.4

0.6

0.8

1.0

Lag

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p

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1391'

|v

Ww

O

v|

w

U

w

t 42

Time

Elec.ts

1960 1965 1970 1975 1980 1985 1990

2000

8000

14000

Time

diff(Elec.ts)

1960 1965 1970 1975 1980 1985 1990

−1500

0

1000

Time

diff(log(Elec.ts))

1960 1965 1970 1975 1980 1985 1990

wQUB p=kDv= Qorta CQwYx@ uwvm = "CU= pOtQDt=Q=B u QOxmx

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= =Q q=@O)m u=ty Q=m xm OQ=O OwHw |=xv=N@=Dm`@=D

R

QOxD@r=

"CU=Q} R CQwYx@ xOW |R=Ux}@WsDU}U uwvm =

+ ma = 0.3), n = 1000)

sigma

^

2 estimated as 1.079: log likelihood = -1457.45, aic = 2920.91

=Q |

QDy

@ VR=

Q

@ARI p

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t h

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v >

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

Q

t=

u

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r

wy

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u

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yO

C

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x

@ =Q

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t p

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x]

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=

T

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U=

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C

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w

v u=

w

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|m

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w

m

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m

x

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x

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wv

m = "

C

U= |

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sD

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w

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O

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@CSS >

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)

+ 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]]