Teknik Peramalan: Materi minggu kedelapan
 Model ARIMA Box-Jenkins
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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
after lag q
Dies down
AR(p): autoregressive of order p
Dies down
Cuts off
after lag p
ARMA(p,q): mixed autoregressivemoving average of order (p,q)
Dies down
Dies down
AR(p) or MA(q)
Cuts off
after lag q
Cuts off
after lag p
No order AR or MA
(White Noise or Random process)
No spike
No spike
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 Firstorder 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
ACF
PACF
PACF
Theoretically of ACF and PACF of The Second-order
Moving Average Model or MA(2) … [Graphics illustration] … (2)
ACF
ACF
PACF
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
ACF
PACF
PACF
Simulation example of ACF and PACF of The Firstorder 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  = (112)
 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
ACF
PACF
PACF
Theoretically of ACF and PACF of The Second-order
Autoregressive Model or AR(2) … [Graphics illustration] … (2)
ACF
ACF
PACF
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 AutoregressiveMoving Average Model or ARMA(1,1)
The model
Zt =  + 1 Zt-1 + at  1 at-1 , where  =  (11)
 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 AutoregressiveMoving Average Model or ARMA(1,1) … [Graphics illustration] … (1)
ACF
ACF
PACF
PACF
Theoretically of ACF and PACF of The Mixed AutoregressiveMoving Average Model or ARMA(1,1) … [Graphics illustration] … (2)
ACF
ACF
PACF
PACF
Theoretically of ACF and PACF of The Mixed AutoregressiveMoving Average Model or ARMA(1,1) …
[Graphics illustration] … (3)
ACF
ACF
PACF
PACF
Simulation example of ACF and PACF of The Mixed AutoregressiveMoving Average Model or ARMA(1,1) …
[Graphics illustration]