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TRANSCRIPT
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Pemodelan Data Deret Waktu(AR dan MA)
Dr. Kusman Sadik, M.SiDepartemen Statistika IPB, 2016
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The forecasting methods based on the smoothing may be inefficient and sometimes inappropriatebecause they do not take advantage of the serial dependence in the observations in the most effective way. way.
To formally incorporate this dependent structure, in this course we will explore a general class of models called autoregressive integrated moving average models or ARIMA models (also known as Box-Jenkins models).
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AR(p) : Autoregressive ber-ordo pI(d) : Integrated ber-ordo d
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I(d) : Integrated ber-ordo dMA(q) : Moving Average ber-ordo q
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Model Stasioner:AR(p), MA(q), dan ARMA(p, q)
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Model Tidak-Stasioner:ARI(p, d), IMA(d, q), dan ARIMA(p, d, q)
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E(Yt) = E(Yt-1 ) = … = E(Yt-k ) Konstan V(Yt) = V(Yt-1 ) = … = V(Yt-k ) Konstan
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Misalkan diketahui data deret waktu sebagai berikut: 2, 3, 2, 5.Hitung secara manual penduga fungsi autokorelasi (ACF) untuk k = 1 dan 2, kemudian bandingkan hasilnya dengan keluaran Minitab.
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r1 = {(3-3)(2-3)+(2-3)(3-3)+(5-3)(2-3)}/{(-1)2+(0)2+(-1)2+(2)2} = - 0.333 r2 = {(2-3)(2-3)+(5-3)(3-3)}/{(-1)2+(0)2+(-1)2+(2)2} = 0.167
= 3
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(4.1.4)
14et ~ Normal(0, σe)
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Karena et ~ Normal(0, σe2) maka: E(et) = E(et-1 ) = … = E(et-k ) = 0 V(et) = V(et-1 ) = … = V(et-k ) = (σe)2
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t t-1 t-k e E(et.ek) = 0 untuk semua t ≠ k
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V(Yt) = V(et - θet-1) = V(et) + θ2V(et-1) = σe2 + θ2σe2
Cov(-θet-1 , et-1) = -θCov(et-1 , et-1) = -θVar(et-1)= -θσe2
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ACF / Autocorrelation Function : ρk = (γk/γ0)
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t = k Cov(αet , θek) = (αθ)σe2
t ≠ k Cov(αet , θek) = 0
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Var(Yt) = Var(φYt-1 + et) = φ2Var(Yt-1) + Var(et)
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1. Misalkan diketahui data deret waktu sebagai berikut: 4, 3, 6, 7, 5, 9. Hitung secara manual penduga fungsi autokorelasi (ACF) untuk k = 1, 2, dan 3, kemudian bandingkan hasilnya dengan keluaran Minitab.
2. Melalui Minitab, bangkitkan data yt, (n = 125), berupa MA(2) dengan θ1 = 0.75 dan θ2 = - 0.85 serta et ~ Normal(0,1). Gunakan 100 data terakhir, kemudian buat correlogramnya. Apa yang dapat disimpulkan dari correlogram tersebut?dari correlogram tersebut?
3. Melalui Minitab, bangkitkan data yt, (n = 125), berupa AR(1) dengan Φ = 0.9 dan et ~ Normal(0,1). Gunakan 100 data terakhir:a. Buatlah correlogramnya. Apa yang dapat disimpulkan dari
correlogram tersebut?b. Buatlah plot antara yt dengan yt-1. Apa kesimpulan Anda?c. Buatlah plot antara yt dengan yt-2. Apa kesimpulan Anda?
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Cryer, J.D. and Chan, K.S. 2008. Time Series Analysis withApplication in R. Springer.
Montgomery, D.C., et.al. 2008. Forecasting Time Series Analysis2nd. John Wiley.
Wei, William, W.S. 1990. Time Series Analysis, Univariate andMultivariate Methods. Adison-Wesley Publishing Company Inc,Canada.
Abraham, B. and Ledolter, J. 2005. Statistical Methods forForecasting. John Wiley.
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Bisa di-download di
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http://www.stat.ipb.ac.id/en/index.php?page=dr-kusman-sadik
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