Box-Jenkins Forecasting - PDF document

Department of Logistics Management Box-Jenkins Forecasting 傳統預測方式 • 觀察數據變化的型態，選擇影響數據變化的因素 – Trend, seasonal factors, causal forecasting • 如果預測模式合理，預測誤差 (residuals) 接近隨機變化 Box-Jenkins 預測 • 將難以解釋的數據變化輸入 black box • 觀察 black box 產生的預測誤差是否接近 white noise – White noise: 隨機變化的數值，沒有關聯，看不出任何變化型態 • 如果預測誤差不像是 white noise ，再換另一種 black box – Three types of black boxes: MA, AR, ARMA 1 Department of Logistics Management Correlation vs. White Noise • 雖然有各種因素影響與隨機變化，時間相近的產品銷售量 通常具有相關性，而不是獨立無關。 • 如果可從過去紀錄估計各期銷售量之間的影響幅度，則可 以目前資料預測未來銷售量。 8 6 4 white noise 2 theta 0.8 0 ‐ 2 ‐ 4 上方的數列有時間相關性，下方數列為隨機變化 2 1

Department of Logistics Management Correlation Between Random Variables 衡量兩個隨機現象或 covariance cov(X, Y)=E[X-E(X)][Y-E(Y)] 兩組數據間共同變化 的程度 1. cov(X, X)=Var(X) 2. If X and Y are independent, then cov(X,Y)=E(XY)-E(X)E(Y)=0        x x y y i i s cov(X, Y) ≈ XY  n 1 兩組數據間 標準化，不受 X, Y 大小的影響 的相關係數 cov(x,y) corr(x,y)  correlation -1 ≤ corr(x, y) ≤ 1   x y n       x x y y i i    ( X , Y ) i 1 n n     ( x x ) 2 ( y y ) 2 i i 3 i  1 i  1 Department of Logistics Management I. Autocorrelation 同一組銷售數據，時間相近的銷售量可能具有相關性         (t, ) cov(y ,y ) E(y )(y ) autocovariance   t t t t      (t, ) ( ) covariance stationary ( 只與時間差距  有關，與時間 t 無關 )      (0) cov(y ,y ) var(y ) t t t Autocorrelation 標準化，不受單位轉換的影響     cov(y , y ) ( ) ( )       t t  ( )   (0) var(y ) var(y ) (0) (0)  t t 4 2

Department of Logistics Management Estimating Autocorrelation       E (y ) (y )    t t  ( )  =1, 2, 3, …   2 E[(y ) ] t T 1      ( y y )( y y )   t t   T T 1         t 1 y y ˆ ( ) t T T 1   2  ( y y ) t 1 t T  t 1 ˆ ˆ ˆ   ˆ      Partial Autocorrelation y c y y       t 1 t 1 t 考慮其他各期也在同時影響 5 Department of Logistics Management A Simple Example of Autocorrelation 6 3

Department of Logistics Management Autocorrelation Function 觀察各個 autocorrelation 的大小，以挑選適合的 black box: MA, AR, ARMA models one-sided gradual damping gradual damped oscillation 7 Department of Logistics Management 隨機擲銅板，正面贏 $1 ，反面輸 $1 ，觀察累積金額 8 4

Department of Logistics Management Computing Autocorrelations on MINITAB significance level Autocorrelation Function for winnings (with 5% significance limits for the autocorrelations) 1.0 0.8 0.6 0.4 Autocorrelation 0.2 0.0 -0.2 -0.4 -0.6 -0.8 -1.0 1 5 10 15 20 25 30 35 40 45 50 55 60 65 Lag 9 Department of Logistics Management No Autocorrelations for White Noise White noise~N(0,  2 ): 隨機變化，相鄰數值沒有關聯 Time Series Plot of white noise 3 Autocorrelation of white noise 2 1 Autocorrelation Function for white noise white noise (with 5% significance limits for the autocorrelations) 0 1.0 0.8 -1 0.6 0.4 Autocorrelation -2 0.2 0.0 -3 1 20 40 60 80 100 120 140 160 180 200 -0.2 I ndex -0.4 -0.6 white noise process -0.8 -1.0 1 5 10 15 20 25 30 35 40 45 50 Lag 10 5

Department of Logistics Management Example: Canadian Employment Index 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 1 2 3 4 5 6 7 8 9 10 11 12 Autocorrelations Quarterly index form 1962.1~1993.4 11 Department of Logistics Management Computing Autocorrelations of CANEMP Autocorrelation Function for caemp (with 5% significance limits for the autocorrelations) 1.0 0.8 0.6 0.4 Autocorrelation 0.2 0.0 -0.2 -0.4 -0.6 -0.8 -1.0 1 5 10 15 20 25 30 Lag 12 6

Department of Logistics Management Computing Partial Autocorrelations Partial Autocorrelation Function for canemp (with 5% significance limits for the partial autocorrelations) 1.0 0.8 0.6 Partial Autocorrelation 0.4 0.2 0.0 -0.2 -0.4 -0.6 -0.8 -1.0 1 5 10 15 20 25 30 Lag Question: 如何挑選符合上一頁與本頁圖形的 black box ？ 13 Department of Logistics Management 觀察 Autocorrelations 與 Partial Autocorrelations ，以選擇適當 的 black box 估計 black box 的參數 如果選擇適當的 black box ，預測誤 差 (residual) 應該接近 white noise 14 7

Department of Logistics Management II. Moving Average Models y t = c +  t +  t -1  t is white noise The MA(1) Process 本期的誤差 上期的誤差 1.8 Whitenoise 1.6 MA1 1.4 1.2 1 0.8 0.6 0.4 0.2 0 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61  =0 → y t 等於 white noise  >0 → 數據呈現正相關 15 Department of Logistics Management MA(q) Process y t = c +  t +    t -1 +    t -2 +…+  q  t -q    WN(0, 2 ) t MA(2)  1>0,  2>0 MA(2)  1<0,  2>0 Autocorrelation Function for C3 Autocorrelation Function for C3 (with 5% significance limits for the autocorrelations) (with 5% significance limits for the autocorrelations) 1.0 1.0 0.8 0.8 0.6 0.6 0.4 0.4 Autocorrelation Autocorrelation 0.2 0.2 0.0 0.0 -0.2 -0.2 -0.4 -0.4 -0.6 -0.6 -0.8 -0.8 -1.0 -1.0 1 2 3 4 5 6 7 8 9 10 Lag 1 2 3 4 5 6 7 8 9 10 Lag 顯著的項目數 ≈ q 16 8

Department of Logistics Management Autocorrelation and Partial Autocorrelation of MA processes MA process: autocorrelation 圖形前幾段較顯著， partial autocorrelation 為漸進式降低 17 Department of Logistics Management Select MA(4) as Black Box for CANEMP 18 9

Department of Logistics Management Residual Plot of MA(4) Model for CANEMP y t = 99.926 +  t -1 +  t -2 +  t -3 +  t -4 +  t Time Series Plot of CANEMP, Residual, Fitted value 120 20 Variable CANEMP 110 Residual 15 Fitted value 100 10 90 Data 80 5 Residual 70 0 60 -5 50 40 -10 1 13 26 39 52 65 78 91 104 117 Index 19 Department of Logistics Management Autocorrelations of Residuals • 如果 MA(4) 模式合宜， residuals 應該接近 white noise ， 而且沒有明顯的 autocorrelations Autocorrelation Function for RESMA(4) (with 5% significance limits for the autocorrelations) 1.0 0.8 0.6 0.4 Autocorrelation 0.2 0.0 -0.2 -0.4 -0.6 -0.8 -1.0 1 5 10 15 20 25 30 Lag 20 10

Department of Logistics Management III. Autoregressive Models y t = ϕ 1 y t -1 +  t    2 The AR(1) Process WN(0, ) t 上期的銷售 本期的誤差 y t = ϕ 1 y t -1 + ϕ 2 y t- 2 +  t The AR(2) Process 7 AR1 6 AR2 5 4 3 2 1 0 1 11 21 31 41 51 61 71 81 91 101 111 121 131 141 151 161 171 181 191 21 Department of Logistics Management Autocorrelations of AR(1) Process 1 0.9  =0.4 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 1 2 3 4 5 6 7 8 9 10 11 12  =0.95 What if  <0? 22 11

Department of Logistics Management AR(p) Process y t = ϕ 1 y t -1 + ϕ 2 y t- 2 + … + ϕ p y t-p +  t    WN(0, 2 ) t Autocorrelation Function for AR(2) AR(2) (with 5% significance limits for the autocorrelations) 1.0 0.8 Partial Autocorrelation Function for AR(2) (with 5% significance limits for the partial autocorrelations) 0.6 0.4 Autocorrelation 1.0 0.2 0.8 0.0 0.6 Partial Autocorrelation -0.2 0.4 -0.4 0.2 -0.6 0.0 -0.8 -0.2 -1.0 -0.4 2 4 6 8 10 12 14 16 -0.6 18 20 22 24 Lag -0.8 -1.0 2 4 6 8 10 12 14 16 18 20 22 24 Lag 23 Department of Logistics Management Autocorrelation and Partial Autocorrelation of AR Processes Autocorrelations:        ( ) 1 , 2 , 3 ,  Partial autocorrelations:     , 1     p( )    0, 1  24 12

Department of Logistics Management Residual Plot of AR(2) Model for CANEMP y t = 2.2382 + 1.5235  y t -1  0.5463  y t- 2 +  t Time Series Plot of CANEMP, Residual2, Fitted 2 120 20 Variable 110 CANEMP 15 Residual2 Fitted 2 100 10 90 Data 80 5 Residual2 70 0 60 -5 50 40 -10 1 13 26 39 52 65 78 91 104 117 Time 25 Department of Logistics Management AR(2) Model Residual Sample Autocorrelation Autocorrelation Function for RESAR(2) (with 5% significance limits for the autocorrelations) 1.0 0.8 0.6 0.4 Autocorrelation 0.2 0.0 -0.2 -0.4 -0.6 -0.8 -1.0 1 5 10 15 20 25 30 Lag • Residuals 沒有明顯的 autocorrelations ， AR(2) 為合宜模式 26 13