Forecasting Purpose is to forecast, not to explain the historical pattern Models for forecasting may not make sense as a description for ”physical” beaviour.

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Presentation transcript:

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 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

Single exponential smoothing Assume historical values y 1,y 2,…y T Assume data contains no trend, i.e.

Forecasting scheme: whereis a smoothing parameter between 0 and 1

The forecast procedure is a recursion formula How shall we choose α? Where should we start, i.e. Which is the initial value l 0 ?

Use a part (usually half) of the historical data to estimate β 0  Set l 0 = Update the estimates of β 0 for the rest of the historical data with the recursion formula  l T which can be used to forecast y T+τ

Example: Sales of everyday commodities Year Sales values

Time series graph

Assume the model: Estimate β 0 by calculating the mean value of the first 8 observations of the series  Set l 8 = =146.75

Assume first that the sales are very stable, i.e. during the period the mean value β 0 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 β 0 using the next 8 values of the historical data

Forecasts

Alternative In Bowerman/O’Connell/Koehler instead the updates of estimates of β 0 are done on all historical data i.e. for T=1,…, n and l 0 =

Analysis of example data with MINITAB yearsales l T y T – l T forecasts ,7504,25000* ,1753,82500* ,558-0,55750* ,5021,49825* ,652-1,65158* ,486-5,48642* ,938-3,93778* ,544-1,54400* ,390-5,38960* ,851-2,85064* ,566-0,56557* ,509-7,50902* ,7582,24188* ,9826,01770* ,5842,41593* ,8262,17433* 146,043

Assume now that the sales are less stable, i.e. during the period the mean value β 0 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

Analysis with MINITAB

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.

Automatic selection of smoothing parameter with MINITAB

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)

Example: Quarterly sales data yearquartersales

Applying Winter’s multiplicative method with MINITAB