1 Chap 9 Box-Jenkins Models • Box-Jenkins 模模模模模模 stationary 模模 • Stationary series ( 模模模模 ) 模模: The statistical properties of the time series are constant through times. E(Y t ) =μ , var(Y t ) =σ 2 , cor(Y t ,Y t+k ) =ρ k for all t • 模模模模模模模模模模模 stationary, 模模模模模模 stationary • 模模模模模
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1 Chap 9 Box-Jenkins Models Box-Jenkins 模式用於描述 stationary 序列 Stationary series ( 平穩序列 ) 定義: The statistical properties of the time series are constant.
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1
Chap 9 Box-Jenkins Models
• Box-Jenkins 模式用於描述 stationary 序列• Stationary series (平穩序列 )
定義: The statistical properties of the time series are constant
through times.
E(Yt) =μ , var(Yt) =σ2 , cor(Yt ,Yt+k) =ρk for all t
• 如果手中的時序資料不是 stationary, 必須將它轉為stationary
• 如何轉換?
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Stationary series
Nonstationary series
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Exp 9.1 The company would like to develop a prediction model that can be used to give prediction interval forecasts of weekly sales of Asorbent Paper Towels. For the past 120 weeks the company has recorded weekly sales of Absorbent Paper Towels.
t y 1stDiff
1 15
214.406
4
-0.593
6
314.938
30.531
9
416.037
41.099
1
5 15.632
-0.405
4
614.397
5
-1.234
5
713.895
9
-0.501
6
814.076
50.180
6
9 16.3752.298
5
1016.534
20.159
2
First Differences Zt = Yt – Yt-1
The original series is not a stationary series
Y(t)
-5
0
5
10
15
20
0 20 40 60 80 100 120
4
Zt = Y(t) -Y(t-1)
-4
-3
-2
-1
0
1
2
3
4
0 120
First Differences series becomes a stationary series
2ndDiff
-4
-3
-2
-1
0
1
2
3
4
5
0 120
Second Differences series is still a stationary series
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圖形觀察:原資料圖、差方資料圖檢定法:
如何檢測 stationarity?( 平穩性 )
Dickey-Fuller test
Phillips-Perron test
Random-walk with drift test
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1. Backward 運算: B(Yt) = Yt-1, B2(Yt) = Yt-1
2. First difference 一階差分 :
3. Second differences 二階差分 :
1 ttt YYY
差分運算
2112 2)( tttttt YYYYYY
ttt YBBYBY )21()1( 222
tttt YBYYY )1( .4 1
5. Difference with lag k : tk
ktt YBYY )1(
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差分功能
一階差分消去直線 trend
二階差方消去二次 trend
4 ttt YYY
12 ttt YYY
消除季節因素
四季節差分
月季節差分
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Fig 9.1 nonstationary series
First difference
Second difference
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9.2 The autocorrelation and partial autocorrelation function
• autocorrelation at lag k : cor(Yt ,Yt+k) =ρk • Sample autocorrelation at lag k, rk