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

With time series regression models, forecasting (prediction) is a natural step and forecasting limits (intervals) can be constructed With Classical decomposition, forecasting may be done, but estimation of accuracy lacks and no forecasting limits are produced Classical decomposition is usually combined with Exponential smoothing methods

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

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. Thumb rule: 0.05 < α < 0.3 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 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 

MTB > Name c3 "FORE1" c4 "UPPE1" c5 "LOWE1" MTB > SES 'Sales values'; SUBC> Weight 0.1; SUBC> Initial 8; SUBC> Forecasts 3; SUBC> Fstore 'FORE1'; SUBC> Upper 'UPPE1'; SUBC> Lower 'LOWE1'; SUBC> Title "SES alpha=0.1". Single Exponential Smoothing for Sales values Data Sales values Length 16 Smoothing Constant Alpha 0.1

Accuracy Measures MAPE MAD MSD Forecasts Period Forecast Lower Upper

MINITAB uses smoothing from 1st value!

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 (A bit exaggerated)

Single Exponential Smoothing for Sales values Data Sales values Length 16 Smoothing Constant Alpha 0.5 Accuracy Measures MAPE MAD MSD Forecasts Period Forecast Lower Upper

Slightly wider prediction intervals

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.

Single Exponential Smoothing for Sales values --- Smoothing Constant Alpha Accuracy Measures MAPE MAD MSD Forecasts Period Forecast Lower Upper Yet, wider prediction intervals

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

Example: Real Estate Price Index for Weekend Cottages in Sweden YearREPI_C Trend but no seasonal variation

Applying Holt’s method with MINITAB (denoted Double exponential smoothing in Minitab)

2 smoothing parameters, one for level and one for trend. Option to let Minitab calculate optimal parameters. Smoothing parameters should still be kept low (0.05,0.3)

Double Exponential Smoothing for REPI_C Data REPI_C Length 13 Smoothing Constants Alpha (level) 0.2 Gamma (trend) 0.2 Accuracy Measures MAPE 9.78 MAD MSD Forecasts Period Forecast Lower Upper

Example: Quarterly sales data yearquartersales

Applying Winter’s multiplicative method with MINITAB

3 smoothing parameters, one for level, one for trend an one for seasonal variation. No option to calculate optimal parameters. Choices have do be based on visual inspection of the times series

Winters' Method for sales Multiplicative Method Data sales Length 20 Smoothing Constants Alpha (level) 0.2 Gamma (trend) 0.2 Delta (seasonal) 0.2 Accuracy Measures MAPE MAD MSD Forecasts Period Forecast Lower Upper Q Q Q Q