Double Exponential Smoothing

Like the regression forecast, the double exponential smoothing forecast is based on the assumption of a model consisting of a constant plus a linear trend. For the purposes of a forecast where the parameters of the model may change, it is more convenient to express the model as a function of , where is the positive displacement from a reference time T. The estimate a and b at time T are based in the observation at time T and the estimates for the previous period, T -1. Here we have both the constant and trend coefficients estimated by exponential smoothing. The forecasting parameters, for the constant term and for the trend term can be set independently. Both paremeters must be between 0 and 1. The forecast for the expected value for future periods is the constant plus a linear term that depends on the number of periods into the future. With a linear term as part of the forecast, this method will track trends in the time series. We use the same data as for the other forecasting methods for illustration. We repeat the data below. Recall that the simulated data begins with a constant mean of 10. At time 11 the mean increases with a trend of 1 until time 20 when the mean becomes a constant again with value 20. The noise is simulated using a normal distribution with mean 0 and standard deviation 3. The values are rounded to the nearest integer. At any time T, only three pieces of information are necessary to compute the estimates, , , and . We illustrate the computations for time 20, using the estimated coefficients for time 19 and the data for time 20.
Forecasts
	We investigate three different forecasts. For simplicity we base the forecasting parameters on a single parameter, . Of course the parameters need not be related in this way. (5) The parameters are set with three different values of as in the table below. The estimates of the model for three cases are shown together with the mean of the time series in the figure below. The figure shows the estimate of the mean at each time and not the forecast. The estimate with the larger value of follows the trend more accurately but has more variability. The forecast with the smaller value of is considerably smoother, but never corrects entirely for the trend. Compared to the regression model, the exponential smoothing method never entirely forgets any part of its past. Thus it may take longer to recover in the event of a perturbation in the underlying mean. This is illustrated in the figure below where the variance of the noise is set to 0.
Forecasting with Excel
	The Forecasting add-in implements the double exponential smoothing formulas. The example below shows the analysis provided by the add-in for the sample data in column B. We use the parameters of the second case. 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 forecast. The values for the coefficients at time 0 are determined by the linear regression method. The remainder of the coefficient estimates in columns C and D are computed with double exponential smoothing. The Fore(1) column (E) shows a forecast for one period into the future. The the values of and are in cells C3 and D3 respectively. The forecast interval is in cell E3. When the forecast interval is changed to a larger number, the values in the Fore column are shifted down. The Err(1) column (F) shows the difference between the observation and the forecast. The standard deviation and Mean Average Deviation (MAD) are computed in cells F6 and F7 respectively.