Exponential Smoothing Tool

Figure 8-21Exponential Smoothing Tool Dialog

The Exponential Smoothing tool performs the exponential smoothing for the given set or sets of values. It provides the choice of 5 different exponential smoothing methods:

• Simple exponential smoothing according to (Hunter, 1968).
• Simple exponential smoothing according to (Roberts, 1959).
• Holt's trend corrected exponential smoothing (occasionally also referred to as double exponential smoothing)
• Additive Holt-Winters exponential smoothing
• Multiplicative Holt-Winters exponential smoothing (occasionally also referred to as triple exponential smoothing)

Since the kind of options available depend on the type of exponential smoothing desired, you can choose the type on the Input page.

8.7.1.1. Common Options of the 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.

8.7.1.2. Exponential Smoothing According to Hunter

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. yt is the tth value in the original data set and lt the corresponding smoothed value.

Figure 8-22Exponential 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.

If you choose to have the tool enter formulas rather than values into the output region, then you can modify the damping factor α even after you executed the tool.

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

If you check the Include chart check box, a line graph showing the observations yt and the predicted values lt will also be created.

Example 8-6Using the Exponential Smoothing Tool

Figure 8-24 shows some example data, Figure 8-25 the selected options and Figure 8-26 the corresponding output.

Figure 8-24Some Example Data for the Exponential Smoothing Tool
Figure 8-25The Options for the Exponential Smoothing Tool
Figure 8-26Exponential Smoothing Tool Output (Hunter)

8.7.1.3. Exponential Smoothing According to Roberts

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. yt is the tth value in the original data set and lt 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.

If you choose to have the tool enter formulas rather than values into the output region, then you can modify the damping factor α and the estimated value at time 0 after executing the tool.

Figure 8-27Exponential 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 8-28The Standard Error Formula For Exponential Smoothing According To Roberts

If you check the Include chart check box, a line graph showing the observations yt and the predicted values lt will also be created.

Example 8-7Using the Exponential Smoothing Tool

Figure 8-29 shows example output for the exponential smoothing tool using the formula according to Roberts. Cell A4 contains the estimated level at time 0. If you requested to have formulas rather than values entered into the sheet, then changing the estimate in A4 and/or the value for α in A2 will result in an immediate change to the predicted values.

Figure 8-29Exponential Smoothing Tool Output (Roberts)

8.7.1.4. Holt's Trend Corrected Exponential Smoothing

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.)

yt is the true value at time t, lt is the estimated level at time t and bt 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 8-30Formulae Of Holt's Trend Corrected Exponential Smoothing

If you choose to have the tool enter formulas rather than values into the output region, then you can modify the damping factors α and γ as well as the estimated level and growth rate at time 0 after executing the tool.

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

If you check the Include chart check box, a line graph showing the observations yt and the estimated level values lt will also be created.

Example 8-8Using the Exponential Smoothing Tool

Figure 8-32 shows example output for Holt's trend corrected exponential smoothing. Cell A4 contains the estimated level at time 0 and B4 the estimated growth rate at time 0. If you requested to have formulas rather than values entered into the sheet, then changing the estimates in A4, B4, the values for α in A2 and/or for γ in B2 will result in an immediate change to the predicted values.

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

8.7.1.5. Additive Holt-Winters Method

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.)

yt is the true value at time t, lt is the estimated level at time t, bt is the estimated growth rate at time t and st 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 8-33Exponential Smoothing Formulae Of The Additive Holt-Winters Method

If you choose to have the tool enter formulas rather than values into the output region, then you can modify the damping factors α, γ and δ as well as all estimates after executing the tool.

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

If you check the Include chart check box, a line graph showing the observations yt and the estimated level values lt will also be created.

Example 8-9Using the Exponential Smoothing Tool

Figure 8-35 shows the options' tab of the exponential smoothing tool for the additive Holt-Winters method. The data is expected to have a seasonal period of 4 (this would for example happen if we have a data value for each quarter of a year). Figure 8-36 shows the corresponding example output for the additive Holt-Winters method. Cell C7 contains the estimated level at time 0, D7 the estimated growth rate at time 0, and E4 to E7 the initial seasonal adjustments for each of the 4 seasons preceding our data time period. If you requested to have formulas rather than values entered into the sheet, then changing any of these estimates, the values for α in A2, for γ in B2 and/or for δ in C2 will result in an immediate change to the estimated values.

Figure 8-35Exponential Smoothing Tool Options (Additive Holt-Winters))
Figure 8-36Exponential Smoothing Tool Output (Additive Holt-Winters)

8.7.1.6. Multiplicative Holt-Winters Method

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.)

yt is the true value at time t, lt is the estimated level at time t, bt is the estimated growth rate at time t and st 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 8-37Exponential Smoothing Formulae Of The Multiplicative Holt-Winters Method

If you choose to have the tool enter formulas rather than values into the output region, then you can modify the damping factors α, γ and δ as well as all estimates after executing the tool.

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

If you check the Include chart check box, a line graph showing the observations yt and the estimated level values lt will also be created.

Example 8-10Using the Exponential Smoothing Tool

Figure 8-39 shows the example output for the multiplicative Holt-Winters method, assuming 4 seasons. Cell C7 contains the estimated level at time 0, D7 the estimated growth rate at time 0, and E4 to E7 the initial seasonal adjustments for each of the 4 seasons preceding our data time period. If you requested to have formulas rather than values entered into the sheet, then changing any of these estimates, the values for α in A2, for γ in B2 and/or for δ in C2 will result in an immediate change to the estimated values.

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