*Intended learning outcomes: Identify forecast errors and their exponential weighting (mean deviation). Explain the tracking signal following Trigg and Leach. Describe the determination of the smoothing constant in first-order exponential smoothing.*

Adaptive smoothing is a form of exponential smoothing in which the smoothing constant is automatically adjusted as a function of forecast error measurement.

A good forecasting technique is not biased:

A (forecast) bias is a consistent deviation of the actual demand from the forecast in one direction, either high or low.

If forecast values exceed the control limits of, for example, +/– the standard deviation from the mean several consecutive times, we must alter either the parameters or the model. Trigg and Leach ([TrLe67]) suggest the following method for continuous adjustment of the exponential smoothing parameter:

The smoothing constant ℽ, or gamma factor smoothes forecast errors exponentially according to the formula in Figure 10.3.3.1.

**Fig. 10.3.3.1** Forecast errors and exponential weighting (mean deviation).

A mean calculated in this way is also referred to as *mean deviation*.

The formula in Figure 10.3.3.2 defines the *tracking signal* and its standard deviation.

**Fig. 10.3.3.2** Tracking signal following Trigg and Leach.

Lewandowski shows the nontrivial result of the standard deviation ([Lewa80], p. 128 ff.). According to that source, the deviation signal is a nondimensional, randomly distributed variable with a mean of 0 and the standard deviation described above. Because of the manner of its calculation, the absolute value of the deviation signal is always ≤1.

Trigg and Leach also developed forecasting techniques that use the deviation signal to adjust the smoothing constant a automatically. Particularly when the mean of the process to be measured changes, a large deviation signal results. In that case, we should choose a relatively large smoothing constant α, so that the mean adjusts rapidly.

In first-order exponential smoothing, it is reasonable to choose a smoothing constant that is equal to the absolute value of the deviation signal, as in Figure 10.3.3.3.

**Fig. 10.3.3.3** Determination of the smoothing constant in first-order exponential smoothing.

The result is a forecast formula with the variable smoothing constant α_{t}. The factor ℽ used to smooth forecast errors remains constant and is kept relatively small, between 0.05 and 0.1 for example. This forecasting technique is not only adaptive but also simple from a technical calculation standpoint.

## Course section 10.3: Subsections and their intended learning outcomes

##### 10.3 Historically Oriented Techniques with Trend-Shaped Behavior

Intended learning outcomes: Explain the regression analysis forecast and the second-order exponential smoothing forecast. Describe the Trigg and Leach adaptive smoothing technique. Produce an overview on seasonality.

##### 10.3.1 Regression Analysis Forecast

Intended learning outcomes: Explain mean, standard deviation, and forecast error in linear regression.

##### 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.

##### 10.3.3 Trigg and Leach Adaptive Smoothing Technique

Intended learning outcomes: Identify forecast errors and their exponential weighting (mean deviation). Explain the tracking signal following Trigg and Leach. Describe the determination of the smoothing constant in first-order exponential smoothing.

##### 10.3.4 Seasonality Forecast

Intended learning outcomes: Identify the seasonal index Sf. Explain forecasting that considers seasonality. Differentiate between “Additive seasonality” and “Multiplicative seasonality” formulation.