Simple exponential smoothing is a weighted average between the most recent
observation and the most recent forecasting, with weights \(\alpha\) and
\(1 - \alpha\), respectively. To be precise, the smoothing equation of single exponential
smoothing (constant model, trend = 1) is given by
$$level[t] = \alpha *x[t] + (1 - \alpha)*level[t-1],$$
and the forecasting equation is
$$hat{x}[t+1|t] = level[t],$$
for \(t = 1,...,n\).
The initial value \(level[0] = x[1]\). For example, \(hat{x}[1|0] = level[0]\),
\(hat{x}[2|1] = level[1]\),..., etc.
Let \(x1[t]\) be the smoothed values of single exponential smoothing. The double
exponential smoothing (trend = 2, a linear model) is to apply a single
exponential smoothing again to the smoothed sequence \(x1[t]\), with a new smoothing
parameter beta. Similarly, we denote the smoothed values of double
exponential smoothing to be \(x2[t]\). The triple exponential smoothing
(trend = 3, a quadratic model) is to apply the single exponential smoothing
to the smoothed sequence \(x2[t]\) with a new smoothing parameter gamma. The
default smoothing parameters (weights) alpha, beta, gamma are
taken from the equation 1 - 0.8^{1/trend} respectively, which is similar
to the FORECAST procedure in SAS.