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Forecasting

Forecasting. Purpose is to forecast, not to explain the historical pattern Models for forecasting may not make sense as a description for ”physical” beaviour of the time series Common sense and mathematics in a good combination produces ”optimal” forecasts. Exponential smoothing.

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Forecasting

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  1. Forecasting • Purpose is to forecast, not to explain the historical pattern • Models for forecasting may not make sense as a description for ”physical” beaviour of the time series • Common sense and mathematics in a good combination produces ”optimal” forecasts

  2. Exponential smoothing • Use the historical data to forecast the future • Let different parts of the history have different impact on the forecasts • Forecast model is not developed from any statistical theory

  3. Single exponential smoothing • Assume historical values y1,y2,…yT • Assume data contains no trend, i.e.

  4. Forecasting scheme: where is a smoothing parameter between 0 and 1

  5. The forecast procedure is a recursion formula • How shall we choose α? • Where should we start, i.e. Which is the initial value l0 ?

  6. Use a part (usually half) of the historical data to estimate β0  Set l 0= Update the estimates of β0 for the rest of the historical data with the recursion formula  l Twhich can be used to forecastyT+τ

  7. Example: Sales of everyday commodities Year Sales values 1985 151 1986 151 1987 147 1988 149 1989 146 1990 142 1991 143 1992 145 1993 141 1994 143 1995 145 1996 138 1997 147 1998 151 1999 148 2000 148

  8. Time series graph

  9. Assume the model: Estimate β0by calculating the mean value of the first 8 observations of the series  Set l8 = =146.75

  10. Assume first that the sales are very stable, i.e. during the period the mean value β0 is assumed not to change Set α to be relatively small. This means that the latest observation plays a less role than the history in the forecasts. E.g. Set α=0.1 Update the estimates of β0using the next 8 values of the historical data

  11. Forecasts

  12. Alternative In Bowerman/O’Connell/Koehler instead the updates of estimates of β0are done on all historical data i.e. for T=1,…, n and l0=

  13. Analysis of example data with MINITAB year saleslTyT – lTforecasts 1985 151 146,750 4,25000 * 1986 151 147,175 3,82500 * 1987 147 147,558 -0,55750 * 1988 149 147,502 1,49825 * 1989 146 147,652 -1,65158 * 1990 142 147,486 -5,48642 * 1991 143 146,938 -3,93778 * 1992 145 146,544 -1,54400 * 1993 141 146,390 -5,38960 * 1994 143 145,851 -2,85064 * 1995 145 145,566 -0,56557 * 1996 138 145,509 -7,50902 * 1997 147 144,758 2,24188 * 1998 151 144,982 6,01770 * 1999 148 145,584 2,41593 * 2000 148 145,826 2,17433 * 146,043 146,043 146,043 146,043

  14. Assume now that the sales are less stable, i.e. during the period the mean value β0 is possibly changing Set α to be relatively large. This means that the latest observation becomes more important in the forecasts. E.g. Set α=0.5

  15. Analysis with MINITAB

  16. We can also use some adaptive procedure to continuosly evaluate the forecast ability and maybe change the smoothing parameter over time Alt. We can run the process with different alphas and choose the one that performs best. This can be done with the MINITAB procedure.

  17. Automatic selection of smoothing parameter with MINITAB

  18. Exponential smoothing for times series with trend and/or seasonal variation • Double exponential smoothing (one smoothing parameter) • Holt-Winter’s method (two smoothing parameters) • Multiplicative Winter’s method (three smoothing parameters) • Additive Winter’s method (three smoothing parameters)

  19. Example: Quarterly sales data year quarter sales 1991 1 124 1991 2 157 1991 3 163 1991 4 126 1992 1 119 1992 2 163 1992 3 176 1992 4 127 1993 1 126 1993 2 160 1993 3 181 1993 4 121 1994 1 131 1994 2 168 1994 3 189 1994 4 134 1995 1 133 1995 2 167 1995 3 195 1995 4 131

  20. Applying Winter’s multiplicative method with MINITAB

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