Metode Peramalan Deret Waktu – STK 352

Pendugaan parameter dilakukan setelah menentukan model tentatif, berdasarkan data pengamatan 𝑌1, 𝑌2, … , 𝑌𝑛.

Metode yang bisa digunakan:

Metode momen

Metode kuadrat terkecil

Metode kemungkinan maksimum

Method of Moments (MM)

Methods of Moment estimation is a general method where equations for

estimating parameters are found by equating population moments with the

corresponding sample moments:

etc.

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Trivial MM estimates are estimates

of the population mean ( ) and the

population variance ( 2).

The benefit of the method is that the

equations render possibilities to

estimate other parameters.

Model: 𝑌𝑡 = 𝜙𝑌𝑡−1 + 𝑒𝑡

𝜙 = 𝑟1

𝜌𝑘 = 𝜙𝑘 untuk 𝑘 = 1,2, … 𝜌1 = 𝜙 𝜌1 = 𝜙

𝑌𝑡 = 𝜙1𝑌𝑡−1 + 𝜙2𝑌𝑡−2 + ⋯ + 𝜙𝑝𝑌𝑡−𝑝 + 𝑒𝑡

𝐶𝑜𝑣(𝑌𝑡, 𝑌𝑡−𝑘) = 𝜙1𝐶𝑜𝑣 𝑌𝑡−1, 𝑌𝑡−𝑘 + 𝜙2𝐶𝑜𝑣 𝑌𝑡−2, 𝑌𝑡−𝑘 + ⋯

+𝜙𝑝𝐶𝑜𝑣(𝑌𝑡−𝑝, 𝑌𝑡−𝑘) + 𝐶𝑜𝑣(𝑒𝑡 , 𝑌𝑡−𝑘)

𝛾𝑘 = 𝜙1𝛾𝑘−1 + 𝜙2𝛾𝑘−2 + ⋯ + 𝜙𝑝𝛾𝑘−𝑝

𝜌𝑘 = 𝜙1𝜌𝑘−1 + 𝜙2𝜌𝑘−2 + ⋯ + 𝜙𝑝𝜌𝑘−𝑝

dibagi 𝛾0

𝜌𝑘 = 𝜙1𝜌𝑘−1 + 𝜙2𝜌𝑘−2 + ⋯ + 𝜙𝑝𝜌𝑘−𝑝 untuk 𝑘 ≥ 1

Jika 𝑘 = 1,2, … dengan 𝜌0 = 1 dan 𝜌𝑘 = 𝜌−𝑘 , diperoleh

persamaan umum Yule-Walker:

Persamaan yule-walker:

𝜌𝑘 = 𝜙1𝜌𝑘−1 + 𝜙2𝜌𝑘−2 + ⋯ + 𝜙𝑝𝜌𝑘−𝑝

Sehingga:

𝜌1 = 𝜙1+𝜌1𝜙2 𝑟1 = 𝜙1+𝑟1 𝜙2

𝜌2 = 𝜙1𝜌1+𝜙2 𝑟2 = 𝜙1𝑟1+ 𝜙2

dan

Model: 𝑌𝑡 = 𝑒𝑡 − 𝜃𝑒𝑡−1

𝜌1 = −𝜃

1 + 𝜃2 𝑟1 = − 𝜃

1 + 𝜃2

Jika 𝑟1 < 0.5 maka:

𝜃 = −1

2𝑟1±

1

4𝑟12 − 1 =

−1 ± 1 − 4𝑟12

2𝑟1

Menduga 𝛾0 = 𝑉𝑎𝑟 𝑌𝑡 menggunakan ragam contoh:

𝑠2 =1

𝑛 − 1

𝑡=1

𝑛

𝑌𝑡 − 𝑌 2

Untuk model AR(p):

𝜎𝑒2 = 1 − 𝜙1𝑟1 − 𝜙2𝑟2 − ⋯ − 𝜙𝑝𝑟𝑝 𝑠2

Untuk kasus khusus AR(1)

𝜎𝑒2 = 1 − 𝜙𝑟1 𝑠2 = 1 − 𝑟1

2 𝑠2

dengan 𝜙 = 𝑟1

Untuk kasus MA(q):

𝜎𝑒2 =

𝑠2

1 + 𝜃12 + 𝜃2

2 + ⋯ + 𝜃𝑞2

Untuk kasus ARMA(1,1):

Misalkan terdapat data deret waktu sbb:

Jika untuk data tersebut menggunakan model AR(1): 𝑌𝑡 = 𝛼 + 𝜙𝑌𝑡−1 + 𝑒𝑡

Tentukan penduga parameternya, yaitu 𝛼 dan 𝜙 denganmetode momen!

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If the process mean is different than zero

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we expect the plot to suggest a rectangular scatter around a zero horizontal level with no trends whatsoever

Increased variation

Very large magnitudes

quantile-quantile plots are an effective tool for assessing normality

Outliers

To check on the independence of the noise terms in the model, we consider the sample autocorrelation function of the residuals, denoted 𝑟𝑘.

H0: sisaan saling bebas

H1: sisaan tidak saling bebas

Lakukan prosedur uji

Ljung-Box berdasarkan

informasi di atas !

AIC

BIC

MAPE

MSE

1. Cryer JD, Chan KS. 2008. Time Series Analysis with Application with R. New York: Springer.

2. Pustaka lain yang relevan.