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STK352

Analisis Deret Waktu

MODEL ARIMA MUSIMANPertemuan 12

Farid Mochamad AfendiDepartemen Statistika IPB

27 Mei 2008

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MATERI PEMBAHASAN

MATERIPEMBAHASAN

PENGANTAR

MODEL ARMAMUSIMAN

ILUSTRASI


PENGANTAR

MODEL ARMA MUSIMAN

ILUSTRASI

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PENGANTAR

MATERIPEMBAHASAN

PENGANTAR

Pengantar

MODEL ARMAMUSIMAN

ILUSTRASI


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Pengantar

MATERIPEMBAHASAN

PENGANTAR

Pengantar

MODEL ARMAMUSIMAN

ILUSTRASI


Model ARIMA juga dapat digunakan untuk fitting data yangberpola musiman.

Langkah awal adalah penentuan s atau panjang periodemusiman.

Proses identifikasi model ARIMA musiman analog denganmodel ARIMA non musiman.

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MODEL ARMA MUSIMAN

MATERIPEMBAHASAN

PENGANTAR

MODEL ARMAMUSIMAN

Model MA(1) Musiman

Model MA(Q)Musiman

Model AR(1) Musiman

Model AR(P )

Musiman

ILUSTRASI


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Model MA(1) Musiman

MATERIPEMBAHASAN

PENGANTAR

MODEL ARMAMUSIMAN

Model MA(1) Musiman

Model MA(Q)Musiman

Model AR(1) Musiman

Model AR(P )

Musiman

ILUSTRASI


Bila periode musiman s = 12 maka model musiman MA(1)

Z t = at −ΘZ t−12

Mudah ditunjukkan bahwa series tersebut memiliki autokorelasitidak nol hanya untuk lag 12 saja.

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Model MA( Q ) Musiman

MATERIPEMBAHASAN

PENGANTAR

MODEL ARMAMUSIMAN

Model MA(1) Musiman

Model MA(Q)Musiman

Model AR(1) Musiman

Model AR(P )

Musiman

ILUSTRASI


Secara umum, model musiman MA(Q) adalah

Z t = at −Θ1Z t−s −Θ2Z t−2s − . . .−ΘQZ t−Qs

dengan persamaan polinomial ciri MA musimannya

Θ(x) = 1−Θ1xs−Θ2x

2s− . . .−ΘQx

Qs

Model ini invertible bila nilai mutlak dari akar Θ(x) = 0semuanya lebih dari 1.

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Model MA( Q ) Musiman (lanjutan)

MATERIPEMBAHASAN

PENGANTAR

MODEL ARMAMUSIMAN

Model MA(1) Musiman

Model MA(Q)Musiman

Model AR(1) Musiman

Model AR(P )

Musiman

ILUSTRASI


Seperti model MA(q ) non musiman, MA(Q) musiman memilikiautokorelasi yang tidak nol untuk lag s, 2s , . . . ,Qs.

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Model AR(1) Musiman

MATERIPEMBAHASAN

PENGANTAR

MODEL ARMAMUSIMAN

Model MA(1) Musiman

Model MA(Q)Musiman

Model AR(1) Musiman

Model AR(P )

Musiman

ILUSTRASI


Untuk model AR(1) musiman masih dengan s = 12

Z t = ΦZ t−12 + at

Dapat ditunjukkan bahwa ρ12k = Φk untuk k = 1, 2, . . . denganautokorelasi lag lain bernilai 0.

Dengan kata lain, autokorelasi kelipatan periode musimanadalah tail off sementara autokorelasi lag lain bernilai nol.

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Model AR( P ) Musiman

MATERIPEMBAHASAN

PENGANTAR

MODEL ARMAMUSIMAN

Model MA(1) Musiman

Model MA(Q)

Musiman

Model AR(1) Musiman

Model AR(P )

Musiman

ILUSTRASI


Secara umum, model AR(P ) musiman adalah

Z t = Φ1Z t−s + Φ2Z t−2s + . . . + ΦP Z t−Ps + at

dengan persamaan polinomial ciri AR musimannya

Φ(x) = 1− Φ1xs− Φ2x

2s− . . .− ΦP x

Ps

Model ini stasioner bila nilai mutlak dari akar Φ(x) = 0semuanya lebih dari 1.

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Model AR( P ) Musiman (lanjutan)

MATERIPEMBAHASAN

PENGANTAR

MODEL ARMAMUSIMAN

Model MA(1) Musiman

Model MA(Q)

Musiman

Model AR(1) Musiman

Model AR(P )

Musiman

ILUSTRASI


Model AR(P ) musiman memiliki

autokorelasi kelipatan periode musiman tail off sementaraautokorelasi lag lain bernilai nol.

autokorelasi parsial cut off setelah lag P kelipatan periode

musiman, sementara lag lain bernilai nol.

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ILUSTRASI

MATERIPEMBAHASAN

PENGANTAR

MODEL ARMAMUSIMAN

ILUSTRASI

U.S. Air PassengerData

Plot Data Asal

Plot Transformasi Ln

PemeriksanKehomogenan Ragam

ACF DataTransformasi Ln

ACF Musiman DataTransformasi Ln

diff 1

Diff 12

diff 1-Diff 12

Fitting

Diagnosa

Validasi


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U.S. Air Passenger Data

MATERIPEMBAHASAN

PENGANTAR

MODEL ARMAMUSIMAN

ILUSTRASI


Plot Data Asal





diff 1

Diff 12

diff 1-Diff 12

Fitting

Diagnosa

Validasi


Sebagai ilustrasi, disajikan analisis data ’U.S. Air PassengerData’ yang berupa data bulanan dari Januari 1960 hinggaDesember 1977 (Cryer, 1986 p.270).

Data tahun terakhir digunakan untuk validasi

2.42 2.14 2.28 2.50 2.44 2.72 2.71 2.74 2.55 2.49 2.13 2.28 # 1962.35 1.82 2.40 2.46 2.38 2.83 2.68 2.81 2.54 2.54 2.37 2.54 # 1962.62 2.34 2.68 2.75 2.66 2.96 2.66 2.93 2.70 2.65 2.46 2.59 # 196

2.75 2.45 2.85 2.99 2.89 3.43 3.25 3.59 3.12 3.16 2.86 3.22 # 1963.24 2.95 3.32 3.29 3.32 3.91 3.80 4.02 3.53 3.61 3.22 3.67 # 1963.75 3.25 3.70 3.98 3.88 4.47 4.60 4.90 4.20 4.20 3.80 4.50 # 1964.40 4.00 4.70 5.10 4.90 5.70 3.90 4.20 5.10 5.00 4.70 5.50 # 1965.30 4.60 5.90 5.50 5.40 6.70 6.80 7.40 6.00 5.80 5.50 6.40 # 1966.20 5.70 6.40 6.70 6.30 7.80 7.60 8.60 6.60 6.50 6.00 7.60 # 196

7.00 6.00 7.10 7.40 7.20 8.40 8.50 9.40 7.10 7.00 6.60 8.00 # 19610.45 8.81 10.61 9.97 10.69 12.40 13.38 14.31 10.90 9.98 9.20 10.94 # 19710.53 9.06 10.17 11.17 10.84 12.09 13.66 14.06 11.14 11.10 10.00 11.98 # 19711.74 10.27 12.05 12.27 12.03 13.95 15.10 15.65 12.47 12.29 11.52 13.08 # 19712.50 11.05 12.94 13.24 13.16 14.95 16.00 16.98 13.15 12.88 11.99 13.13 # 19712.99 11.69 13.78 13.70 13.57 15.12 15.55 16.73 12.68 12.65 11.18 13.27 # 19712.64 11.01 13.30 12.19 12.91 14.90 16.10 17.30 12.90 13.36 12.26 13.93 # 19713.94 12.75 14.19 14.67 14.66 16.21 17.72 18.15 14.19 14.33 12.99 15.19 # 197

15.09 12.94 15.46 15.39 15.34 17.02 18.85 19.49 15.61 16.16 14.84 17.04 # 197

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Plot Data Asal

MATERIPEMBAHASAN

PENGANTAR

MODEL ARMAMUSIMAN

ILUSTRASI


Plot Data Asal





diff 1

Diff 12

diff 1-Diff 12

Fitting

Diagnosa

Validasi


Plot data asal memperlihatkan pola musiman dengan s = 12 sertaadanya perilaku nonstasioner baik dalam rataan maupun ragam.

Gambar 1: Time Series Plot Data Asal

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MATERIPEMBAHASAN

PENGANTAR

MODEL ARMAMUSIMAN

ILUSTRASI


Plot Data Asal





diff 1

Diff 12

diff 1-Diff 12

Fitting

Diagnosa

Validasi


Transformasi logaritma berhasil mengatasi ketidakstasionerandalam ragam meskipun ketidakstasioneran dalam rataan masihnampak.

Gambar 2: Time Series Plot Data Transformasi Logaritma

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Pemeriksan Kehomogenan Ragam

MATERIPEMBAHASAN

PENGANTAR

MODEL ARMAMUSIMAN

ILUSTRASI


Plot Data Asal





diff 1

Diff 12

diff 1-Diff 12

Fitting

Diagnosa

Validasi


Gambar 3: Plot Range-Mean data asal

Gambar 4: Plot Range-Mean data transformasi ln

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Pemeriksan Kehomogenan Ragam (lanjutan)

MATERIPEMBAHASAN

PENGANTAR

MODEL ARMAMUSIMAN

ILUSTRASI


Plot Data Asal





diff 1

Diff 12

diff 1-Diff 12

Fitting

Diagnosa

Validasi


Gambar 5: Uji kehomogenan ragam data asal

Gambar 6: Uji kehomogenan ragam data transformasi ln

ACF D T f i L23:49:27


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ACF Data Transformasi Ln

MATERIPEMBAHASAN

PENGANTAR

MODEL ARMAMUSIMAN

ILUSTRASI


Plot Data Asal





diff 1

Diff 12

diff 1-Diff 12

Fitting

Diagnosa

Validasi


Plot ACF data setelah transformasi logaritma menunjukkan polanonstasioner. Perhatikan juga pola ACF untuk lag s, 2s , . . ..

Gambar 7: Plot ACF Data Transformasi Logaritma

ACF M i D t T f i L23:49:27


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ACF Musiman Data Transformasi Ln

MATERIPEMBAHASAN

PENGANTAR

MODEL ARMAMUSIMAN

ILUSTRASI


Plot Data Asal





diff 1

Diff 12

diff 1-Diff 12

Fitting

Diagnosa

Validasi


Plot ACF data setelah transformasi logaritma untuk lag 12, 24, 36,48 menunjukkan pola nonstasioner.

Gambar 8: Plot ACF Musiman Data Transformasi Logaritma

Pl t D t N l Diff i d 123:49:27


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Plot Data Nonseasonal Differencing d = 1

MATERIPEMBAHASAN

PENGANTAR

MODEL ARMAMUSIMAN

ILUSTRASI


Plot Data Asal





diff 1

Diff 12

diff 1-Diff 12

Fitting

Diagnosa

Validasi


Nonseasonal differencing d = 1 berhasil mengatasiketidakstasioneran dalam rataan untuk komponen nonseasonalnya.

Gambar 9: Plot data setelah nonseasonal differencing d = 1

Plot ACF Data Nonseasonal Differencing23:49:27


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Plot ACF Data Nonseasonal Differencing

d = 1

MATERIPEMBAHASAN

PENGANTAR

MODEL ARMAMUSIMAN

ILUSTRASI


Plot Data Asal





diff 1

Diff 12

diff 1-Diff 12

Fitting

Diagnosa

Validasi


Plot ACF data nonseasonal differencing d = 1 mengkonfirmasikestasioneran komponen non musiman (namun perhatikan lag 12,24, dst).

(a)

Plot ACF

(b)

Plot ACF Musiman

Gambar 10: Plot ACF Data Nonseasonal Differencing d = 1

Plot Data Seasonal Differencing D 1223:49:27


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Plot Data Seasonal Differencing D = 12

MATERIPEMBAHASAN

PENGANTAR

MODEL ARMAMUSIMAN

ILUSTRASI


Plot Data Asal





diff 1

Diff 12

diff 1-Diff 12

Fitting

Diagnosa

Validasi


Gambar 11: Plot data setelah seasonal differencing D = 12

Plot ACF Data Seasonal Differencing D 1223:49:27


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Plot ACF Data Seasonal Differencing D = 12

MATERIPEMBAHASAN

PENGANTAR

MODEL ARMAMUSIMAN

ILUSTRASI


Plot Data Asal





diff 1

Diff 12

diff 1-Diff 12

Fitting

Diagnosa

Validasi


Nonseasonal differencing D = 12 berhasil mengatasiketidakstasioneran dalam rataan untuk komponen seasonalnya(namun tidak untuk komponen non musimannya).

(a)

Plot ACF

(b)

Plot ACF Musiman

Gambar 12: Plot ACF data seasonal differencing D = 12

Plot Data Differencing d 1 D 1223:49:27


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Plot Data Differencing d = 1, D = 12

MATERIPEMBAHASAN

PENGANTAR

MODEL ARMAMUSIMAN

ILUSTRASI


Plot Data Asal





diff 1

Diff 12

diff 1-Diff 12

Fitting

Diagnosa

Validasi


Gambar 13: Plot data setelah differencing d = 1, D = 12

Plot ACF-PACF Data Differencing 23:49:27


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g

d = 1, D = 12

MATERIPEMBAHASAN

PENGANTAR

MODEL ARMAMUSIMAN

ILUSTRASI


Plot Data Asal





diff 1

Diff 12

diff 1-Diff 12

Fitting

Diagnosa

Validasi


Kedua komponen telah stasioner. Identifikasi komponen nonmusiman adalah ARIMA(0,1,2).

(a)Plot ACF

(b)Plot PACF

Gambar 14: Identifikasi ARIMA komponen non musiman

Plot ACF-PACF Musiman Data Differencing 23:49:27


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d = 1, D = 12

MATERIPEMBAHASAN

PENGANTAR

MODEL ARMAMUSIMAN

ILUSTRASI


Plot Data Asal





diff 1

Diff 12

diff 1-Diff 12

Fitting

Diagnosa

Validasi


Identifikasi komponen musiman adalah ARIMA(0,1,1)12, sehinggamodel tentatif adalah ARIMA(0,1,2)×(0,1,1)12.

(a)Plot ACF Musiman

(b)Plot PACF Musiman

Gambar 15: Identifikasi ARIMA komponen musiman

Fitting ARIMA(0 1 2)×(0 1 1)1223:49:27


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Fitting ARIMA(0,1,2) ×(0,1,1)12

MATERIPEMBAHASAN

PENGANTAR

MODEL ARMAMUSIMAN

ILUSTRASI

U.S. Air Passenger

Data

Plot Data Asal





diff 1

Diff 12

diff 1-Diff 12

Fitting

Diagnosa

Validasi


Pengepasan model (Fitting ) ARIMA(0,1,2)×(0,1,1)12

Gambar 16: Fitting ARIMA(0,1,2)×(0,1,1)12

Plot ACF-PACF Residual23:49:27


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Plot ACF PACF Residual

MATERIPEMBAHASAN

PENGANTAR

MODEL ARMAMUSIMAN

ILUSTRASI

U.S. Air Passenger

Data

Plot Data Asal





diff 1

Diff 12

diff 1-Diff 12

Fitting

Diagnosa

Validasi


Identifikasi komponen musiman adalah ARIMA(0,1,1)12, sehinggamodel tentatif adalah ARIMA(0,1,2)×(0,1,1)12.

(a)Plot ACF Residual

(b)Plot PACF Residual

Gambar 17: Plot ACF-PACF residual

Diagnosa Model23:49:27


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Diagnosa Model

MATERIPEMBAHASAN

PENGANTAR

MODEL ARMAMUSIMAN

ILUSTRASI

U.S. Air Passenger

Data

Plot Data Asal





diff 1

Diff 12

diff 1-Diff 12

Fitting

Diagnosa

Validasi


Diagnosa model dari plot residual

Gambar 18: Diagnosa model dari plot residual

Validasi Model23:49:27


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Validasi Model

MATERIPEMBAHASAN

PENGANTAR

MODEL ARMAMUSIMAN

ILUSTRASI

U.S. Air Passenger

Data

Plot Data Asal





diff 1

Diff 12

diff 1-Diff 12

Fitting

Diagnosa

Validasi


Validasi Model menggunakan data Tahun 1977: (MAD=0.344 danMAPE=2.14%)

Gambar 19: Validasi Model dari data Tahun 1997

23:49:27


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MATERIPEMBAHASAN

PENGANTAR

MODEL ARMAMUSIMAN

ILUSTRASI

U.S. Air Passenger

Data

Plot Data Asal





diff 1

Diff 12

diff 1-Diff 12

Fitting

Diagnosa

Validasi


TERIMA KASIH

arima musiman

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