Time series often have a tendency of some growth or decrease. When such series are smoothed using simple exponential smoothing, the value of exponential average lags behind the value of the time series. The less is α, the greater is the lag. Such cases require models that take into account rates of change (decrease) in the level of the time series at each step.
The generalized model of exponential smoothing
has the component d2 to take into account growth or decrease.
If each next value of a source series differs from the previous value by a rather stable value, this series tends to linearly increase (decrease). In this case, the exponential smoothing model has an additional coefficient taking into account this specificity:
where:
. Linear growth component.
However, with the course of time the growth (decrease) rate of the series may decrease and the series levels tend to some asymptote. This feature (saturated growth or decrease) is taken into account in the damped growth model that assumes the following:
where:
. Damped growth component.
If the frequency of a source series grows or decreases proportionally (that is, each next series value is the result of the previous value multiplied by a constant coefficient), this type of growth is named exponential, and the model reflecting the properties of this process is named the exponential growth model, for which:
where:
. Exponential growth component.
Thus, if growth component is included into exponential smoothing model, it is recalculated at each step depending on its previous value, and adapts to changes of the smoothed series at this step. Values of the parameter y belong to the range [0; 1]. If the parameter γ is zero, this coefficient does not respond to changes in the smoothed series (and changes in the source series), in case of linear or exponential growth it is constant for each time moment; in case of damped growth it decreases depending on the parameter ϕ. If γ is 1, the growth coefficient depends only on the current change of the smoothed series. In case of damped growth the parameter ϕ shows to what degree the current growth coefficient is weakened compared with the previous one. Values of the parameter ϕ belong to the range (0; 1).
Suppose that:
E. The last point of the sample period.
τ. The number of forecast points.
The table below shows models of forecasting time series values depending on the growth and seasonal effect:
| Seasonal effect | ||||
| None | Additive | Multiplicative | ||
| Growth type | None |  |
 |
 |
| Linear |  |
 |
 |
|
| Exponential |  |
 |
 |
|
| Damped |  |
 |
 |
Generally, the calculated level of the smoothed series at the last point of the sample period is adjusted for growth coefficients for τ points forward; after which the seasonal component is applied to the forecast.
See also:
Exponential Smoothing | Exponential smoothing: Seasonal effects model | Best Trial Method