# 10.3.2 Second-Order Exponential Smoothing Forecast

### Intended learning outcomes: Disclose the determination of trend lines in second-order exponential smoothing. Explain the formulas for calculation of the trend line and forecast error in second-order exponential smoothing. Present an example of determination of forecast value using second-order exponential smoothing.

Second-order exponential smoothing forecast technique extends first-order exponential smoothing to create a technique capable of capturing linear trend.

Second-order exponential smoothing starts out from:

• The mean, calculated using first-order smoothing
• The mean of this first-order means, calculated according to the same recursion formula

These two means are the estimated values for two points on the trend line. Figure 10.3.2.1 shows an overview of this technique, which is elaborated in the following discussion. The exact derivations can be found in [Gahs71], p. 60 ff., and in [Lewa80], p. 66 ff.
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Fig. 10.3.2.1      Determination of trend lines in second-order exponential smoothing.

Figure 10.3.2.2 shows the formulas necessary for calculating the trend line; this gives us the second-order forecast value for subsequent periods as well as the corresponding forecast error. See also the definitions in Figure 10.1.3.4.

The following numbered explanations correspond to those presented in Figure 10.3.2.2:

1. The previous formula to determine first-order mean.
2. The new formula to determine the second-order mean, as the mean of the first-order means. The second-order mean lies at the same distance from the first-order mean as does the latter from the current period.
3. Slope of the trend line to time t, when two means are given.
4. Starting value Tt for the forecast at time t.
5. Forecast for subsequent periods.
6. Forecast error for the next period t + 1. Because a linear trend entails that the forecast error is dependent on k, the same formula does not automatically hold for period t + k, although it is often used.
7. The determination of the starting value that can be calculated, for example, by means of regression analysis.

Fig. 10.3.2.2      Trend line and forecast error in second-order exponential smoothing.

Figure 10.3.2.3 provides an example of the determination of the forecast value using second-order exponential smoothing for the smoothing constant α = 0.2. We calculated the same demand value as the one in linear re­gression for the first 14 periods in order to obtain the same starting values.

Fig. 10.3.2.3      Determination of forecast value using second-order exponential smoothing (α = 0.2).