Forecasting Theory - Exponential Smoothing

As for the moving average, this method assumes that the time series follows a constant model.

The value of b is estimated as the weighted average of the last observation and the last estimate. Here is a parameter in the interval [0, 1].

Rearranging, obtains an alternative form.

The new estimate is the old estimate plus a proportion of the observed error.
Because we are supposing a constant model, the forecast is the same as the estimate.

We illustrate the method using the parameter value = 0.2 and the data below.

The first 10 observations are used to warm up the procedure with the average of the observations providing the estimate for time 10. The average is 11.1. The value of the estimate for time 11 is computed below.

Subsequent estimates are computed with the exponential smoothing formula and are shown in the table.

Only two data elements are required to compute each estimate, the observed data and the old estimate. This contrasts with the moving average which requires the previous m observations for the computation.

Replacing with its equivalent, we find that the estimate is

Continuing in this fashion, we discover that the estimate is really a weighted sum of all past data.

The larger values of provide relatively greater weight to more recent data than smaller values of . With the value of 1, is the last data point. With the value 0, is the same as . The figure shows the parameter estimates obtained for three different values of together with the mean of the time series. Although the model for this method is a constant, we illustrate the response to a time series with a trend. The simulated example includes a trend of 1 from 20 to 30.

A lag characteristic, similar to the one associated with the moving average estimate, can also be seen in the figure. The lag and bias for the exponential smoothing estimate can be expressed as a function of . The quantity a in the expression is the linear trend value.

For smaller values of we obtain a greater lag in response to the trend.

The error is the difference between the actual data and the forecasted value. If the time series is truly a constant value, the expected value of the error is zero and the variance of the error is comprised of a term that is a function of and a second term that is the variance of the noise, .

The variance of the error increases as increases. To minimize the effect of noise, we would like to make as small as possible (0), but this makes the forecast unresponsive to a change in the underlying time series. To make the forecast responsive to changes, we want as large as possible (1), but this increases the error variance. Practical forecasting requires an intermediate value.

We equate the approximating error for the moving average and exponential smoothing methods.

Solving for , we find the value providing the same approximation error as the moving average.

Using this relation between the parameters of the two methods, we find that the lag and bias introduced by the trend will also be the same.

The parameters used in the moving average illustrations of the last page (m = 5, 10, 20) are roughly comparable to the parameters used for exponential smoothing in figure above ( = 0.4, 0.2, 0.1).

Forecasting with Excel

The Forecasting add-in implements the exponential smoothing formulas. The example below shows the analysis provided by the add-in for the sample data. The first 10 observations are indexed -9 through 0. Compared to the table above, the period indices are shifted by -10.

The first ten observations provide the startup values for the estimate. The EXP column (C) shows the computed estimates. The Fore(1) column (D) shows a forecast for one period into the future. The forecast interval is in cell D3. When the forecast interval is changed to a larger number the numbers in the forecast column are shifted down. The value of is in cell C3. When this cell is changed, all the computed cells automatically adjust.

The Err(1) column (E) shows the error between the observation and the forecast. The standard deviation and Mean Average Deviation (MAD) are computed in cells E6 and E7. The value in C3 can be used as the optimization variable for the Excel Solver to minimize the error standard deviation or the MAD.

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by Paul A. Jensen