Winters Method

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Winters Method

  1. 1. Chapter 2: Forecasting <ul><li>Methods for Seasonal Series </li></ul><ul><ul><ul><li>Methods for Stationary Series </li></ul></ul></ul><ul><ul><ul><ul><li>Seasonal factors </li></ul></ul></ul></ul><ul><ul><ul><ul><li>Seasonal decomposition using MA </li></ul></ul></ul></ul><ul><ul><ul><li>Methods for seasonal series with trend </li></ul></ul></ul><ul><ul><ul><ul><li>Winter’s Method </li></ul></ul></ul></ul>
  2. 2. A Seasonal Demand Series Fig. 2-8
  3. 3. Seasonal Series with Increasing Trend Fig. 2-10
  4. 4. Winter’s Method <ul><li>We assume a model of the form </li></ul><ul><li>μ : base signal or intercept at time 0 </li></ul><ul><li>G: trend or slope component </li></ul><ul><li>c t : multiplicative seasonal component </li></ul><ul><li>ε t : error term </li></ul><ul><li>This model assumes that the underlying series has a form similar to that in Figure 2-10. </li></ul>
  5. 5. <ul><li>Assumptions: </li></ul><ul><li>The season is exactly N periods </li></ul><ul><li>Seasonal factors are the same each period and </li></ul><ul><li>Σ c t = N </li></ul><ul><li>Three exponential smoothing equations are used each period to </li></ul><ul><li>update estimates of : </li></ul><ul><ul><ul><li>Deseasonalized series </li></ul></ul></ul><ul><ul><ul><li>Seasonal factors </li></ul></ul></ul><ul><ul><ul><li>Trend </li></ul></ul></ul><ul><li>These equations have different smoothing constants, α , β , and γ </li></ul>
  6. 6. <ul><li>The series: </li></ul><ul><li>The trend </li></ul><ul><li>The seasonal factors </li></ul>
  7. 7. Forecast made in period t for any future period t + τ
  8. 8. Initialization Procedure <ul><li>Suppose that current period is t=0 </li></ul><ul><li>Past observations are labeled D -2N+1 , D -2N+1 , … , D 0 </li></ul><ul><li>Calculate the sample means for the 2 seasons data: </li></ul><ul><li>2. Define the initial slope estimate </li></ul>
  9. 9. Initialization for Winters’s Method Fig. 2-11
  10. 10. <ul><li>3. Set the estimate of the value of the series at t=0 </li></ul><ul><li>4 (a). Initial SF are obtained by dividing each observation by the corresponding point on the line connecting V 1 and V 2 using the formula: </li></ul><ul><li>i=1,2 for the 1st , 2 nd season </li></ul><ul><li>j : period of the season </li></ul><ul><li>(b). Average the seasonal factors: </li></ul><ul><li>(c ). Normalize the SF: </li></ul>
  11. 11. 22/ [21.75 -(5/2-4)(.875) ] =.9539 30/ [21.75 -(5/2-3)(.875) ] =1.352 23/ [21.75 -(5/2-2)(.875) ] =1.079 12/ [21.75 -(5/2-1)(.875) ] =.5872 17/ [ 18.25-(5/2-4)(.875) ] =.869 26/ [ 18.25-(5/2-3)(.875) ] =1.391 20/ [ 18.25-(5/2-2)(.875) ] =1.123 10/ [ 18.25-(5/2-1)(.875) ] =.5904 c 0 c -1 c -2 c -3 c -4 c -5 c -6 c -7 So=21.75+(.875)(1.5) =23.06 Go=(21.75-18.25)/4 =.875 .9115 22 8 1.3720 30 7 1.1010 23 6 (.5904+.5872)/2 =.5888 21.75 12 5 17 4 26 3 20 2 18.25 10 1 V i D t Period
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