# 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.