2.2.4.3.2 Algorithm for Double Exponential Smoothing

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

1 Double Exponential Model
2 Weights by Optimal ARIMA
3 Forecast
4 Reference

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

nag_tsa_exp_smooth (g13amc)

2.2.4.3.2 Algorithm for Double Exponential Smoothing

Contents

Double Exponential Model

Weights by Optimal ARIMA

Forecast

Reference