6.4.3.2. Forecasting with Single Exponential Smoothing

6. Process or Product Monitoring and Control
6.4. Introduction to Time Series Analysis
6.4.3. What is Exponential Smoothing?

6.4.3.2. Forecasting with Single Exponential Smoothing

Forecasting Formula

Forecasting the next point

The forecasting formula is the basic equation S(t+1) = alpha*y(t) + (1-alpha)*S(t), 0 < alpha<= 1, t > 0

New forecast is previous forecast plus an error adjustment

This can be written as: S(t+1) = S(t) + alpha*(e(t))

where

_t 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

Bootstrapping 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: S(t+1) = alpha*y(orgin) + (1-alpha)*S(t)

S(t+1) = alpha*y(orgin) + (1-alpha)*S(t)

where y_origin remains constant. This technique is known as bootstrapping.

Example of Bootstrapping

Example

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 S(t+1) = alpha*y(orgin) + (1-alpha)*S(t)

= .1(70) + .9(71.7) = 71.5 ( alpha

= .1)

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

S_t₊₂=. 1(70) + .9(71.5 )= 71.35

Comparison between bootstrap and regular forecasting

Table comparing two methods

The following table displays the comparison between the two methods:

Period	Bootstrap forecast	Data	Single Smoothing Forecast

13	71.50	75	71.5
14	71.35	75	71.9
15	71.21	74	72.2
16	71.09	78	72.4
17	70.98	86	73.0

Single Exponential Smoothing with Trend

Single Smoothing (short for single exponential smoothing) is not very good when there is a trend. The single coefficient alpha

is not enough.

Sample data set with trend

Let us demonstrate this with the following data set smoothed with an alpha

of 0.3:

Data	Fit

6.4
5.6	6.4
7.8	6.2
8.8	6.7
11.0	7.3
11.6	8.4
16.7	9.4
15.3	11.6
21.6	12.7
22.4	15.4

Plot demonstrating inadequacy of single exponential smoothing when there is trend

The resulting graph looks like:

Plot showing raw data with single exponential smoothed values