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 \gammacontrol 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)
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