Forecasting with Single Exponential Smoothing

Forecasting Formula

The forecasting formula is the basic equation

This can be written as:

where ϵt is the forecast error (actual – forecast) for period t.
In other words, the new forecast is the old one plus an adjustment for the error that occurred in the last forecast.

Bootstrapping of Forecasts
What happens if you wish to forecast from some origin, usually the last data point, and no actual observations are available? In this situation we have to modify the formula to become:

where y origin remains constant. This technique is known as bootstrapping.

Example of Bootstrapping
The last data point in the previous example was 70 and its forecast (smoothed value S) was 71.7. Since we do have the data point and the forecast available, we can calculate the next forecast using the regular formula with α=0.1 as

But for the next forecast we have no data point (observation). So now we compute:

Comparison between bootstrap and regular forecasting
The following table displays the comparison between the two methods:

Single Exponential Smoothing with Trend
Single Smoothing (short for single exponential smoothing) is not very good when there is a trend. The single coefficient α is not enough.

Let us demonstrate this with the following data set smoothed with an α of 0.3:

The resulting graph looks like: