Teknik Peramalan: Materi minggu kedelapan

Model ARIMA Box-Jenkins Identification of STATIONER TIME

SERIES Estimation of ARIMA model Diagnostic Check of ARIMA model Forecasting

Studi Kasus : Model ARIMAX (Analisis Intervensi, Fungsi Transfer dan Neural Networks)

General Theoretical ACF and PACF of ARIMA Models

Model ACF PACF

MA(q): moving average of order q Cuts off Dies down after lag q

AR(p): autoregressive of order p Dies down Cuts off after lag

p

ARMA(p,q): mixed autoregressive- Dies down Dies down moving average of order (p,q)

AR(p) or MA(q) Cuts off Cuts off after lag q after lag p

No order AR or MA No spike No spike (White Noise or Random process)

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) … [Graphics illustration]

ACF

ACF PACF

PACF

Simulation example of ACF and PACF of The First-order Moving Average Model or MA(1) … [Graphics 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) … [Graphics illustration] … (1)

ACF PACF

ACF PACF

Theoretically of ACF and PACF of The Second-order Moving Average Model or MA(2) … [Graphics illustration] … (2)

ACF PACF

ACF PACF

Simulation example of ACF and PACF of The Second-order Moving Average Model or MA(2) … [Graphics illustration]

Theoretically of ACF and PACF of The First-order Autoregressive Model or AR(1)

The model Zt = + 1 Zt-1 + at , where = (1-1)

Stationarity condition : –1 < 1 < 1

Theoretically of ACF

Theoretically of PACF

Theoretically of ACF and PACF of The First-order Autoregressive Model or AR(1) … [Graphics illustration]

ACF PACF

ACF PACF

Simulation example of ACF and PACF of The First-order Autoregressive Model or AR(1) … [Graphics illustration]

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) … [Graphics illustration] … (1)

ACF PACF

ACF PACF

Theoretically of ACF and PACF of The Second-order Autoregressive Model or AR(2) … [Graphics illustration] … (2)

ACF PACF

ACF PACF

Simulation example of ACF and PACF of The Second-order Autoregressive Model or AR(2) … [Graphics illustration]

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

ACFTheoretically 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) … [Graphics illustration] … (1)

ACF PACF

ACF PACF

Theoretically of ACF and PACF of The Mixed Autoregressive-Moving Average Model or ARMA(1,1) … [Graphics illustration] … (2)

ACF PACF

ACF PACF

Theoretically of ACF and PACF of The Mixed Autoregressive-Moving Average Model or ARMA(1,1)

… [Graphics illustration] … (3)

ACF PACF

ACF PACF

Simulation example of ACF and PACF of The Mixed

Autoregressive-Moving Average Model or ARMA(1,1) … [Graphics illustration]