2.2.4.3.2 Algorithm for Double Exponential Smoothing

NAG function nag_tsa_exp_smooth (g13amc) is used for double exponential smoothing[1].

Double Exponential Model

\[L_t = \alpha y_t + (1- \alpha)(L_{t-1}+T_{t-1})\]

\[T_t = \gamma (L_t - L_{t-1}) + (1- \gamma)T_{t-1}\]

\[\hat{y_t} = L_{t-1} + T_{t-1}\]

\[y'_t = L_{t}\]

where \(L_t\) is the level(mean), \(T_t\) is the trend component at time \(t\). The parameters, \(\alpha\) and \(\gamma\)control the weight of smoothing. \(y_t\), \(\hat{y_t}\) and \(y'_t\) are data value, fitted value and smoothed value at time \(t\).

Weights by Optimal ARIMA

Use an ARIMA (0,2,2) model to fit the data. With the parameters \(ma_1\) and \(ma_2\), calcualte \(\alpha\) and \(\gamma\).

\[\alpha = 1 + ma_2\]

\[\gamma = (2- \alpha -ma_1) / \alpha\]

Forecast

\[\hat{y}_{t+f} = L_t + f T_t \]

Reference

1. nag_tsa_exp_smooth (g13amc)