# 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.

`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 αt. 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.