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1 TIME SERIES SIMPLE EXPONENTIAL SMOOTHING If the mean of y t remains constant over n time, each observation gets equal weight: y ˆ t1 yt 0 et, 0 y n t 1
2 If the mean of y t changes slowly over time, recent observations should have more weight than remote observations. Simple exponential smoothing can be used for forecasting such a series. NOTATIONS USED: yt = value of the time series at time level l( (mean) of fthe series at time t t t 2
3 SIMPLE EXPONENTIAL SMOOTHING: n/2 y t t1 (1) 0 (mean of half of the data) n (2) y (1 ), 0 1 (Smoothing equation) t t t1 (3) Compute forecasts of historical data as follows: yˆ ˆ (0) = = point forecast made at time 0 1 y0 1 1 for value of next period, y 0 1 yˆ yˆ (1) = = point forecast made at time for value of next period, y11 yˆ yˆ (1) = point forecast made at time for value of next period, y2 1 3
4 SIMPLE EXPONENTIAL SMOOTHING (3) ()(continued) For forecasts of historical data the general equation can be written as yˆ = point forecast made at time T T fo r value of next period, y T 1 T 0,1,2,..., n (historical data) = 123 1,2,3,... T 4
5 SIMPLE EXPONENTIAL SMOOTHING (4) When T n (last historical data) yˆ = point forecast made at time T T for value of next period, = 1,2, 3,... y T 1 The prediction interval, however, will get wider as is increased (forecast made further in future) A 95% prediction interval computed at time T for y T 2 is T 1.96 s 1 ( 1) where T s SSE n 1 n n 2 2 [ y ˆ t yt( t1)] [ yt t1] t 1 t 1 n1 n1 5
6 n alpha SSE ssquare s squared Observed smoothed Forecast Forecast Forecast cod catch estimate last period error error t y
7 t squared Observed smoothed Forecast Forecast Forecast cod catch estimate last period error error y
8 Holt s Simple Exponential Smoothing, Example on page 348 using Excel SOLVER to find optimum so that SSE is minimum In EXCEL: Data/solver, then enter the Solver Parameters as shown below then click on SOLVE to obtain the results on the next slide. obtained by fitting least squares line yt = b0 + b1*t fitted to first 26 data points of the total 52 points 8
9 Holt s Exponential Smoothing, Optimal for Example on page 348 obtained from Excel SOLVER 9
10 REMARKS ON SIMPLE EXPONENTIAL SMOOTHING 1) Large value of gives more weight to last observation y t and hence yields rapid changes in the fitted values. Small value of gives a smoother look in the fitted values. 10
11 2)Minitab : Stat/TimeSeries/Single Series/Single Exponential Smooting performs SES in a different way: it computes an optimal weight by fitting an ARIMA (0, 1, 1) model to the data. With this option, Minitab calculates the initial smoothed value by backcasting. 3) Minitab calculates the prediction intervals for forecasts using = 1, and hence the prediction interval width does not change (see next slide), which is not correct. The width of the prediction interval should increase as is incresaed. 11
12 12
13 Holt s Trend Corrected Exponential Smoothing or Double Exponential lsmoothing If y t exhibits a linear trend yt t 0 1 ie i.e., level (mean) changes at a constant rate 1, then SES should be used. When both thlevel land growth rate are changing, Holt s Trend Corrected Smoothing (DES) is appropriate. 13
14 NOTATIONS USED: T 1 estimate of level (mean) at time T -1 b T 1 estimate of growth rate of yt at time T -1 T yt (1 )[ T 1bT1] b [ ] (1 ) b T T T1 T1 Adjustment for changing growth rate where and are smoothing constants, t 0 1, Forecast made at time T for y t is yˆ T T bt ( T 0,1,2,..., n1; 1,2,3,..., n) for historical data (1 t n). 14
15 y (1 )[ b ] T t T1 T1 b [ ] (1 ) b T T T T 1 1 Forecast made at time T for is y t yˆ b ( 1,2,3,..., n) T T t T b 0: y (1 )[ b ] [ ] (1 ) b yˆ (0) 1 b growth rate b0 and level 0 can be obtained by fitting a straight line to 1-st half of data, b = slope 0 intercept 15
16 Forecast made at time T=1 for y : 2 b y (1 )[ b] [ ] (1 ) b yˆ (1) () b ( =1) Forecast made at time T=2 for y : b y (1 )[ b ] [ ] (1 ) b yˆ (2) b ( =1)
17 Prediction intervals for future observations: b y (1 )[ b ] [ ] (1 ) b yˆ ( T) b ( =1,2,3,...; T n) T T T If =1, a 95% prediction interval computed at time T( n) for y ( y ) is ( b ) 1.96s T T T 1 n 1 17
18 If =2, a 95% prediction interval computed at time T( n) for y ( y ) is T2 n2 ( 2 b ) 1.96s 1 (1 ) T T 2 2 If =3, a 95% prediction interval computed at time T( n) for y ( y ) is T3 n3 b s ) (12 ) 2 ( T 3 T) (1 In general: b s ( T T) (1 ) (1 2 )... [( ( 1) ] 18
19 Example 8.3/page 359 of Bowerman, O Connell, Koehler, 4 th Edition Forecasting, Time Series, and Regression: weekly thermostat sales data The regression equation fitted to first 26 data points is yt = t. To get betas to 4 th decimal, store coefficients: COEF To be used as initial estimates 19
20 Example 8.3/page 359 of, Bowerman, O Connell, Koehler, 4 th Edition Forecasting, Time Series, and Regression: weekly thermostat sales data n alpha gamma SSE ssquare s forecast observed made squared thermostat growth last forecast forecast t sales y level, LT rate, BT period error error STATS24x7.com ADI NV, INC
21 forecast observed made squared thermostat growth last forecast forecast t sales y level, LT rate, BT period error error
22 forecast observed made squared thermostat growth last forecast forecast t sales y level, LT rate, BT period error error
23 forecast observed made squared thermostat growth last forecast forecast t sales y level, LT rate, BT period error error
24 Holt s Trend Corrected Exponential Smoothing, Example on page 360 using Excel SOLVER to find optimum and so that SSE is minimum In EXCEL: Data/solver, then enter the Solver Parameters as shown below click on SOLVE to obtain results shown on the next slide. 24
25 Holt s Trend Corrected Exponential Smoothing, Optimal for Example on page 360 obtained from Excel SOLVER 25
26 MSD = SSE/(n 2) = MAD = Notice that results calculated in Excel are slightly better than the results from MINITAB (next slide). 26
27 27
28 Residual plots indicate good fit. 28
29 Autocorrelation plot of residual shows residuals are not correlated.. 29
30 Patterns based on Pegel's Classification No Seasonal Effect Additive Seasonal Multiplicative Seasonal NO TREND EFFECT 4 Index Index Index ADDITIVE TREND Index Index Index MULTIPLICATIVE TREND 4 Index Index Index
31 Seasonal Component Trend Component 1 (none) 2 (additive) 3 (multiplicative) A (none) A1 - SES A3 HW-Mult B (Additive) B1 - DES(Holt s) B2 - HW-Add B3 HW-Mult C (Multiplicative) C1 C2 C3 HW = Holt Winter s t MthdSES Method, = Simple Exponential lsmoothing, DES = Double Exponential Smoothing (Holt s Trend Corrected Exponential Smoothing) 31
32 HOLT WINTERS METHODS Additive H W method for time series with constant seasonal variation i Multiplicative H W method for time series with increasing seasonal variation i 32
33 If a time series has a linear trend with a constant growth rate 1, fixed seasonal pattern SN t with constant variation, then the time series can be described by the model y ( t) SN e t 0 1 t t If the level (mean), growth rate, and the seasonal pattern of the time series are changing, then H W method is appropriate. 33
34 At time T, let l T b T estimated level estimated growth sn T estimated seasonal factor L = number of seasons in a year L 4 for quarterly data, L 12 for monthlly data 34
35 H W smoothing equations are: l ( y sn ) (1 )( l b ) T T TL T1 T b ( l l ) (1 ) b T T T1 T1 sn ( y l ) (1 ) sn T T T TL 0 1,0 1,0 1 35
36 A point forecast in time period T for y is y T yˆ l ) b sn T T T T L most recent estimate of the seasonal factor for the season corresponding to time T+ NOTE: For historical data (1 T n) T 0,1,2,3,... and =1 as we predict for next time period using the previous values. 36
37 Once we have reached T = n, and are ready to forecast into future (when T = n) ˆ ( ), 1,,3,... y T b sn T T T TL A 95% prediction interval computed at time T ( n) is yˆ ( T) 1.96s c T where SSE s T 3 1, = c [1 ( i j) ], 2 L j ( i j) ] d jl, (1 ), L j1 and d jl, 1 if j is an integer multiple of L 0 otherwise 37
38 Example 8.4 (Mountain Bike Sales data from Chapter 6, Table 6.8) NOTE: Figure 8.9/page 370 has data errors in the last 8 lines (correct data is given in Table 6.8/page 318 of text) 38
39 Example 8.4 (Mountain Bike Sales data from Chapter 6, Table 6.8) NOTE: Figure 8.9/page 370 has data errors in the last 8 lines (correct data is given in Table 6.8/page 318 of text) InExcel Excel, DATA/SOLVER, then add constraints to get following figure: 39
40 Example 8.4 (Mountain Bike Sales data from Chapter 6, Table 6.8) NOTE: Figure 8.9/page 370 has data errors in the last 8 lines (correct data is given in Table 6.8/page 318 of text) OPTIMAL solution 40
41 Autocorrelation function for RESID = Yt Yt_hat from Holt Winters Method 41
42 Prediction Intervals for t = 17, 18, 19 using T = 16 tau c_tau t Yt level growth rate SNt Forecast PL95 PU
43 MULTIPLICATIVE HOLT WINTERS METHOD The multiplicative Holt Winters method is appropriate for a times series which has: Linear trend, and a multiplicative seasonal factor which are not fixed but change with time y TR SN IR [ t ( t)] SN IR t t t t 0 1 t t level l t 43
44 MULTIPLICATIVE HOLT-WINTERS METHOD: T (deseasonalized y ) (1 ) (adjusted level) ( y / sn ) (1 ) ( + b ) T T TL T1 T1 t b ( ) (1 ) b T T T1 T1 sn T (current estimate of SN )+(1- ) ( previous estimate of SN ) sn ( y / )+(1- ) sn T T T T L T T 44
45 An approximate 95% prediction interval computed at time T for is yt yˆ ( T) 1.96 s c sn where s c T r TL r relative standard error = 2 ( T bt), 1 c (1 ) ( T bt ) T t1 y ˆ ( 1) t yt t ˆ yt ( t1) T 3 2 ( T 2 bt ), 2 c (12 ) ( b ) (1 ) ( 2 b ) ( 3 b ), T T T T T T 2 45
46 Example 85/page /page of text: Yt = quarterly sales (1000 cases) of Tiger Sports Drink Since seasonal factor appears to be increasing, multiplicative Holt Winters method is appropriate. 46
47 From regression line from 1 st 8 points From regression estimates (see next slide) n alpha gamma delta SSE ssquare s SSRE sr Squared Squared Forecast Forecast Forecast Relative t Yt Level Growth SNt made error error error Rate last period E E
48 t Yt Regression Detrended SN_bar estimates
49 Squared Squared Forecast Forecast Forecast Relative t Yt Level Growth SNt made error error error Rate last period E E
50 Squared Squared Forecast Forecast Forecast Relative t Yt Level Growth SNt made error error error Rate last period E E E E E E-05 50
51 Optimal,, obtained by minimizing SSE (using Excel SOLVER) under the constraints 0 < <1, 0<<1, 0< <1 n alpha gamma delta SSE ssquare s SSRE sr Squared Squared Forecast Forecast Forecast Relative t Yt Level Growth SNt made error error error Rate last period E E
52 Prediction Intervals from Holt Winters Method PL95 PU95 tau c_tau sqrt(c_tau)
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