Exponential Smoothing Tool

Specify the cells containing the datasets in the “Input Range” entry. The entered range or ranges are grouped into datasets either by rows or by columns.

If you have labels in the first cell of each data set, select the “Labels” option.

If you select the “Include chart” option, Gnumeric will also create a chart showing both the data and corresponding smoothed values.

Each value in the smoothed set is predicted based on the forecast for the prior period. The formula is given in Figure 8-22. α is the value given as “Damping factor”. y_t is the tth value in the original data set and l_t the corresponding smoothed value.

Figure 8-22 Exponential Smoothing Formula According To Hunter

For example, a value for α between 0.2 and 0.3 represents 20 to 30 percent error adjustment in the prior forecast.

To have the standard errors output as well, check the “Standard error” check box. The formula used is given in Figure 8-23. The denominator can be adjusted by selecting the appropriate radio button. Since there are t−1 terms in the sum of the denominator, selecting “n−1” means that the denominator will be t−2.

Figure 8-23 The Standard Error Formula For Exponential Smoothing According To Hunter

If you check the “Include chart” check box, a line graph showing the observations y_t and the predicted values l_t will also be created.

Figure 8-24 Some Example Data for the Exponential Smoothing Tool

Figure 8-25 The Options for the Exponential Smoothing Tool

Figure 8-26 Exponential Smoothing Tool Output (Hunter)

The simple exponential smoothing method according to Roberts is used for forecasting a time series without a trend or seasonal pattern, but for which the level is nevertheless slowly changing over time. The predicted values are calculated according to the formula given in Figure 8-27. α is the value given as “Damping factor”. y_t is the tth value in the original data set and l_t the predicted value. l₀ is the predicted value at time 0 and must be estimated. This tool uses the average value of the first 5 observations as estimate.

Figure 8-27 Exponential Smoothing Formula According To Roberts

To have the standard errors output as well, check the “Standard error” check box. The formula used is given in Figure 8-28. The denominator can be adjusted by selecting the appropriate radio button.

Figure 8-28 The Standard Error Formula For Exponential Smoothing According To Roberts

If you check the “Include chart” check box, a line graph showing the observations y_t and the predicted values l_t will also be created.

Figure 8-29 Exponential Smoothing Tool Output (Roberts)

Holt's trend corrected exponential smoothing is appropriate when both the level and the growth rate of a time series are changing. (If the time series has a fixed growth rate and therefore exhibits a linear trend, a linear regression model is more appropriate.)

y_t is the true value at time t, l_t is the estimated level at time t and b_t is the estimated growth rate at time t. We use the two smoothing equations given in Figure 8-30 to update our estimates. α is the value given as “Damping factor” and γ is the value given as “Growth damping factor”.

This tool obtains initial (time 0) estimates for the level and growth rate by performing a linear regression using the first 5 data values.

Figure 8-30 Formulae Of Holt's Trend Corrected Exponential Smoothing

To have the standard errors output as well, check the “Standard error” check box. The formula used is given in Figure 8-31. The denominator can be adjusted by selecting the appropriate radio button.

Figure 8-31 The Standard Error Formula For Holt's Trend Corrected Exponential Smoothing

If you check the “Include chart” check box, a line graph showing the observations y_t and the estimated level values l_t will also be created.

Figure 8-32 Exponential Smoothing Tool Output (Holt's)

The additive Holt-Winters method of exponential smoothing is appropriate when a time series with a linear trend has an additive seasonal pattern for which the level, the growth rate and the seasonal pattern may be changing. An additive seasonal pattern is a pattern in which the seasonal variation can be explained by the addition of a seasonal constant (although we allow for this constant to change slowly.)

y_t is the true value at time t, l_t is the estimated level at time t, b_t is the estimated growth rate at time t and s_t is the estimated seasonal adjustment for time t. We use the three smoothing equations given in Figure 8-33 to update our estimates. α is the value given as “Damping factor”, γ is the value given as “Growth damping factor” and δ is the value given as “Seasonal damping factor”. L is the value given as “Seasonal period”. If your data consist of monthly values, then L should be 12, if it consist of quarterly values then L should be 4.

This tool obtains initial (time 0) estimates for the level and growth rate by performing a linear regression using all data values. It obtains estimates for the seasonal adjustments by averaging the appropriate seasonal differences from values predicted by linear regression alone.

Figure 8-33 Exponential Smoothing Formulae Of The Additive Holt-Winters Method

To have the standard errors output as well, check the “Standard error” check box. The formula used is given in Figure 8-34. The denominator can be adjusted by selecting the appropriate radio button.

Figure 8-34 The Standard Error Formula Of The Additive Holt-Winters Method

If you check the “Include chart” check box, a line graph showing the observations y_t and the estimated level values l_t will also be created.

Figure 8-35 Exponential Smoothing Tool Options (Additive Holt-Winters))

Figure 8-36 Exponential Smoothing Tool Output (Additive Holt-Winters)

The multiplicative Holt-Winters method of exponential smoothing is appropriate when a time series with a linear trend has a multiplicative seasonal pattern for which the level, the growth rate and the seasonal pattern may be changing. A multiplicative seasonal pattern is a pattern in which the seasonal variation can be explained by the multiplication of a seasonal constant (although we allow for this constant to change slowly.)

y_t is the true value at time t, l_t is the estimated level at time t, b_t is the estimated growth rate at time t and s_t is the estimated seasonal adjustment for time t. We use the three smoothing equations given in Figure 8-37 to update our estimates. α is the value given as “Damping factor”, γ is the value given as “Growth damping factor” and δ is the value given as “Seasonal damping factor”. L is the value given as “Seasonal period”. If your data consist of monthly values, then L should be 12, if it consist of quarterly values then L should be 4.

This tool obtains initial (time 0) estimates for the level and growth rate by performing a linear regression using the data values of the first 4 seasonal periods. It obtains estimates for the seasonal adjustments by averaging the appropriate seasonal differences from values predicted by linear regression alone during the first 4 seasonal periods.

Figure 8-37 Exponential Smoothing Formulae Of The Multiplicative Holt-Winters Method

To have the standard errors output as well, check the “Standard error” check box. The formula used is given in Figure 8-38. The denominator can be adjusted by selecting the appropriate radio button.

Figure 8-38 The Standard Error Formula Of The Multiplicative Holt-Winters Method

If you check the “Include chart” check box, a line graph showing the observations y_t and the estimated level values l_t will also be created.

Figure 8-39 Exponential Smoothing Tool Output (Multiplicative Holt-Winters)