Obtaining forecasts

In this page, we consider the details of forecasting with multiplicative models of the form

Data   =   (Seasonal effect)   ×   Trend   ×   Cyclical   ×   Residual

You have learned that this model is equivalent to an additive model for the log data.

log(Data)   =   Seasonal*   +   Trend*   +   Cyclical*   +   Residual*

where the new components are the logarithms of those in the earlier equation.

The model is therefore fitted by initially taking logs of the original data values and fitting an additive model in the usual way. However this model will give forecasts of the log data.

forecast of log(Data)   =   (forecast Seasonal*)  +  (forecast Trend*)  +  (forecast Cyclical*)

To obtain forecasts in the original scale, an inverse transformation must be applied. If logarithms to the base 10 have been used (as in the diagrams in CAST), the appropriate inverse transformation is

forecast of Data   =   10 forecast of log(Data)
 

If natural logarithms are used, the exponential function should be used to return the forecast based on the log data to a forecast in the original units. (Both types of logarithms result in identical forecasts.)

Forecasting New Zealand visitors

The time series plot below again shows the visitor arrivals on a log scale.

Click the checkboxes Seasonal, Trend and Cyclical to display the sum of these components. The components are also forecast 3 years into the future in the diagram. Click on any forecast to display it numerically, transformed back to the original scale.

Now use the slider to adjust the scale back to a plot of the original data. Observe that the multiplicative model extends the upward trend in visitor arrivals and also forecasts an increasing difference between summer and winter numbers.