3 model arima

MODEL TIME SERIES

ARIMA

(REGULER, MUSIMAN,

CAMPURAN)

TAHAPAN ARIMA

1

• IDENTIFIKASI

• diperoleh Model-model Sementara

2 • PENDUGAAN

PARAMETER

3 • DIAGNOSTIC

CHECKING

Correlogram:

ACF, PACF

ESACF, MINIC

FORECASTING

meme

nuhi

Tidak

memenuhi

FLOWCHART

PEMODELAN

ARIMA

Model ACF PACF

AR(p)

turun cepat secara

eksponensial /

sinusoidal

terputus setelah lag p

MA(q) Cuts off setelah lag q

turun cepat secara

eksponensial /

sinusoidal

AR(p) atau

MA(q) Cuts off setelah lag q Cuts off setelah lag p

ARMA(p,q) Cuts off setelah lag (q-

p)

Cuts off setelah lag (p-

q)

No order AR or

MA(White Noise or

Random process)

No spike No spike

- ACF dan PACF sampel dibandingkan dengan ACF dan PACF teoritis

- Paling umum digunakan

IDENTIFIKASI MODEL ARIMA dengan CORRELOGRAM

KELEBIHAN - relatif mudah

- tingkat kesesuaian yang tinggi bila

- perilaku data Time Series tidak terlalu kompleks

- asumsi-asumsi terpenuhi dengan baik

KELEMAHAN

- tidak mampu memberikan identifikasi yang jelas tentang orde model jika modelnya kompleks

- pertimbangan subyektif yang mengakibatkan hasil dengan kesimpulan yang berbeda

- Model yang dihasilkan kadang tidak cukup memuaskan

PENDUGAAN DAN PENGUJIAN PARAMETER MODEL ARIMA

Estimasi Parameter

Diagnostic Checking

ACF residual

ACF

1

-1

0 Lag k 8

1

-1

0 Lag k 8

1

-1

0 Lag k 8

1

-1

0 Lag k 8

cuts off

dies down (exponential)

dies down (exponential)

dies down (sinusoidal)

no oscillation

oscillation

ACF untuk deret berkala stasioner

Dying down fairly quickly versus extremely slowly

Dying down fairly quickly

Lag k 8

1

-1

0

Lag k 8

1

-1

0

Dying down extremely slowly

stationary time series (usually)

nonstationary time series (usually)

Sample Partial Autocorrelation Function (PACF)

For the working series Z1, Z2, …, Zn : Corr(Zt,Zt-k|Zt-1,…,Zt-k+1)

Perhitungan PACF pada lag 1, 2 and 3

SPACF pada lag 1, 2 dan 3 adalah:

ACF PACF

Dying down fairly quickly Cuts off after lag 2

Contoh deret stasioner

ACF PACF

Dying down extremely slowly Cuts off after lag 2

Contoh deret yang tidak stasioner

t/2 . se(rk) t/2 . se(rk)

+ +

Sample ACF

MODEL AUTOREGRESSIVE (p)

AR (1)

ACF dari AR(1)

PACF dari AR(1)

SIMULASI AR (1) UNTUK

SIMULASI AR (1) UNTUK

THEORETICALLY OF ACF AND PACF OF THE SECOND-

ORDER AUTOREGRESSIVE MODEL OR AR(2)

The model Zt =

+ 1 Zt-1 + 2 Zt-2 + at, where = (112)

Stationarity condition : 1 + 2 < 1 ; 2 1 < 1 ; |2| < 1

Theoretically of ACF Theoretically of PACF

THEORETICALLY OF ACF AND PACF OF THE SECOND-ORDER

AUTOREGRESSIVE MODEL OR AR(2) … [GRAPHICAL ILLUSTRATION] … (1)

ACF PACF

ACF PACF

THEORETICALLY OF ACF AND PACF OF THE SECOND-ORDER

AUTOREGRESSIVE MODEL OR AR(2) … [GRAPHICAL ILLUSTRATION] … (2)

ACF PACF

ACF PACF

SIMULATION EXAMPLE OF ACF AND PACF OF THE SECOND-ORDER

AUTOREGRESSIVE MODEL OR AR(2) … [GRAPHICAL ILLUSTRATION]

THEORETICALLY OF ACF AND PACF OF THE FIRST-ORDER

MOVING AVERAGE MODEL OR MA(1)

The model

Zt = + at – 1 at-1 , where =

Invertibility condition : –1 < 1 < 1

Theoretically of ACF Theoretically of PACF

THEORETICALLY OF ACF AND PACF OF THE FIRST-ORDER

MOVING AVERAGE MODEL OR MA(1) … [GRAPHICAL ILLUSTRATION]

ACF

ACF PACF

PACF

SIMULATION EXAMPLE OF ACF AND PACF OF THE FIRST-ORDER

MOVING AVERAGE MODEL OR MA(1) … [GRAPHICAL ILLUSTRATION]

THEORETICALLY OF ACF AND PACF OF THE SECOND-

ORDER MOVING AVERAGE MODEL OR MA(2)

The model Zt =

+ at – 1 at-1 – 2 at-2 , where =

Invertibility condition : 1 + 2 < 1 ; 2 1 < 1 ; |2| < 1

Theoretically of ACF Theoretically of PACF

Dies Down (according to a mixture

of damped exponentials and/or

damped sine waves)

THEORETICALLY OF ACF AND PACF OF THE SECOND-ORDER

MOVING AVERAGE MODEL OR MA(2) … [GRAPHICAL ILLUSTRATION] … (1)

ACF PACF

ACF PACF

THEORETICALLY OF ACF AND PACF OF THE SECOND-ORDER

MOVING AVERAGE MODEL OR MA(2) … [GRAPHICAL ILLUSTRATION] … (2)

ACF PACF

ACF PACF

SIMULATION EXAMPLE OF ACF AND PACF OF THE SECOND-ORDER

MOVING AVERAGE MODEL OR MA(2) … [GRAPHICAL ILLUSTRATION]

MODEL ARMA (p, q)

Theoretically of ACF and PACF of The Mixed Autoregressive-

Moving Average Model or ARMA(1,1)

The model Zt =

+ 1 Zt-1 + at 1 at-1 , where = (11)

Stationarity and Invertibility condition : |1| < 1 and |1| < 1

Theoretically of ACF Theoretically of PACF

Dies Down (in fashion

dominated by damped

exponentials decay)

Theoretically of ACF and PACF of The Mixed Autoregressive-

Moving Average Model or ARMA(1,1) … [Graphical illustration] … (1)

ACF PACF

ACF PACF

Theoretically of ACF and PACF of The Mixed Autoregressive-

Moving Average Model or ARMA(1,1) … [Graphical illustration] … (2)

ACF PACF

ACF PACF

Theoretically of ACF and PACF of The Mixed Autoregressive-

Moving Average Model or ARMA(1,1) … [Graphical illustration] … (3)

ACF PACF

ACF PACF

Simulation example of ACF and PACF of The Mixed Autoregressive-

Moving Average Model or ARMA(1,1) … [Graphical illustration]

Representasi AR

Representasi MA

The General [Nonseasonal] ARIMA(p,d,q) models

The Model is

where

is an appropriate pre-

differencing transformation

Do not need

pre-differencing

transformation

Example: ARIMA(1,0,1) model

The Model is

where

Zt = Yt,

Yt data asli

and

Therefore,

Example: ARIMA(1,0,1) model … [other calculation]

The Model is

Therefore,

p=1 d=0 q=1

Example: ARIMA(1,1,1) model … [nonstationary model]

The Model is

where

Zt = Yt – Yt-1

and

Therefore,

Mean (Zt)

Example: MINITAB output … [nonstationary ARIMA model]

Estimation and

Testing

parameter

Diagnostic

Check (white

noise residual)

Yt = 3.0232 + 1.6591 Yt-1 – 0.6591 Yt-2 + at

Forecasting of ARIMA(p,d,q) model

Forecasting of AR(1) model

or

Forecasting of MA(1) model

Example: Daily readings of viscosity of Chemical Product XB-77-5 [Bowerman and O’Connell, pg. 471]

Example: IDENTIFICATION step [stationary, ACF and PACF]

ACF PACF

Dies down [sinusoidal] Cuts off after lag 2

Stationer time series

Example: ESTIMATION and DIAGNOSTIC CHECK step

Estimation and Testing parameter

Diagnostic Check (white

noise residual)

Example: DIAGNOSTIC CHECK step … [Normality test of residuals]

Example: FORECASTING step [MINITAB output]

Calculation: FORECASTING (FITS and FORECAST) [continued]

CONTOH ANALISIS 2

Index

Da

ta A

sli

126112988470564228141

200000

150000

100000

50000

0

Time Series Plot of Data Asli

Plot time series data permintaan Arc Tube daya listrik rendah

Dari TS plot terlihat bahwa data tersebut tidak stasioner dalam mean. Hal yang sama juga diperoleh dari ACF plot.

Jadi dilakukan DIFFERENCING

Transformasi ?

Dari Box-Cox diperoleh nilai lambda terbaik adalah 1 dan selang kepercayaan untuk lambda melewati 1.

Jadi TIDAK dilakukan TRANSFORMASI

Hasil differencing

Data relatif sudah stasioner

Hasil Pengujian Parameter Model

Model Parameter Koefisien P_Val

ARIMA(1,1,0) -0.5501 0

ARIMA(2,1,0) -0.5948 0

-0.0915 0.262

ARIMA(0,1,1) 0.594 0

ARIMA(0,1,2) 0.6242 0

-0.1198 0.164

S I G N I F I K A N

Pengujian Asumsi white noise

Model Ljung - Box

ARIMA

(1,1,0)

lag 12 24 36

Chi-sq 11 25 33.1

DF 11 23 35

P_Val 0.441 0.348 0.558

ARIMA

(0,1,1)

lag 12 24 36

Chi-sq 21.8 51 65.7

DF 11 23 35

P_Val 0.026 0.001 0.001

White noise

Tidak White noise

Pengujian kenormalan: p_value (0.081) > α (0.05) NORMAL

Model ARIMA terbaik hasil Correlogram

0.5501t t-1 t-1y y y

Permintaan Arc Tube daya listrik rendah pada waktu ke-t dipengaruhi oleh

permintaan ArcTube pada waktu ke-(t-1) dikurangi 0.5501 kali permintaan

ArcTube pada waktu ke-(t-1), ditambah kesalahan pada saat ke-t.

Dengan KRITERIA PEMILIHAN MODEL TERBAIK out-sample :

MSE : 34286,1

MAPE : 12,47%.

General Theoretical ACF and PACF of ARIMA Seasonal Models with

L (length of seasonal period).

Model ACF PACF

MA(Q) Has spike at lag L, 2L, …, QL and cuts off after lag QL

Dies down at the seasonal level

AR(P) Dies down at the seasonal level

Has spike at lag L, 2L, …, PL and cuts off after lag PL

AR(P) or MA(Q) Has spike at lag L, 2L, …, QL and cuts off after lag QL

Has spike at lag L, 2L, …, PL and cuts off after lag PL

ARMA(P,Q) Dies down fairly quickly at the seasonal level

Dies down fairly quickly at the seasonal level

No seasonal operator

Has no spikes (contain small ACF)

Has no spikes (contain small PACF)

Theoretically of ACF and PACF of The First-order Seasonal

L=12 Moving Average Model or MA(1)12

The model

Zt = + at – 1 at-12 , where =

Invertibility condition : –1 < 1 < 1

Theoretically of ACF Theoretically of PACF

Dies Down at the seasonal level

(according to a damped

exponentials waves)

Simulation example of ACF and PACF of The First-order Seasonal L=12

Moving Average Model or MA(1)12 … [Graphical illustration]

12

Has spike only at lag 12 (cuts off) Dies down at seasonal lags

Theoretically of ACF and PACF of The First-order Auto-regressive

Seasonal L=12 Model or AR(1)12

The model

Zt = + 1 Zt-12 + at , where = (1-1)

Stationarity condition : –1 < 1 < 1

Theoretically of ACF Theoretically of PACF

Simulation example of ACF and PACF of The First-order Autore-gressive

Seasonal L=12 Model or AR(1)12 …[Graphics illustration]

12

Has spike only at lag 12

(cuts off)

Dies down at seasonal lags

Theoretically of ACF and PACF of The Multiplicative Moving Average

Model or ARIMA(0,0,1)(0,0,1)12 or MA(1)(1)12

The model

Zt = + at – 1 at-1 1 at-12 + 1.1 at-13 , where =

Stationarity condition : |1| < 1 and |1| < 1

Theoretically of PACF Theoretically of ACF

Dies Down at the

nonseasonal and

seasonal level

Simulation example of ACF and PACF of The Multiplicative Moving

Average Model or MA(1)(1)12 … [Graphical illustration]

Dies down at seasonal lags

Dies down at non seasonal lags

Has spike only at lag 1 (cuts off)

Has spike only at lag 12 (cuts off)

Theoretically of ACF and PACF of The Multiplicative Autore-gressive

Model or ARIMA(1,0,0)(1,0,0)12 or AR(1)(1)12

The model

Zt = + 1 Zt-1 + 1 Zt-12 1.1 Zt-13 + at

Stationarity condition : |1| < 1 and |1| < 1

Theoretically of PACF Theoretically of ACF

Dies Down at the nonseasonal and

seasonal level

Cuts off at the lag 1

[nonseasonal] and lag 12

[seasonal] level

Simulation example of ACF and PACF of The Multiplicative Moving

Average Model or AR(1)(1)12 … [Graphical illustration]

Dies down at seasonal lags

Dies down at non seasonal lags

Has spike only at lag 1 (cuts off)

Has spike only at lag 12 (cuts off)