Double exponential smoothing

Double exponential smoothing supplements a smoothed level with a smoothed slope so that forecasts can extrapolate a local linear trend. Two common versions are Holt's linear trend method, with separate level and slope gains, and Brown's double exponential smoothing, which applies the same smoothing gain twice. They are different algorithms, not two names for one recursion. Neither version by itself models seasonal variation.

For $0<\alpha <1$, set $s_t^{(1)}=\alpha X_t+(1-\alpha )s_{t-1}^{(1)}$ and $s_t^{(2)}=\alpha s_t^{(1)}+(1-\alpha )s_{t-1}^{(2)}$. Use the displayed corrected level and slope, then forecast $a_t+h{b}_t$. For an exact linear trend after initialization transients, the first and second smooths have lags $(1-\alpha )/\alpha$ and twice that lag; their difference recovers the slope and the corrected level removes the lag. This is distinct from Holt's linear trend method.

Update the level and slope by ${\ell }_t=\alpha X_t+(1-\alpha )({\ell }_{t-1}+b_{t-1})$ and $b_t=\beta ({\ell }_t-{\ell }_{t-1})+(1-\beta )b_{t-1}$, with gains in $(0,1)$. The first update compares the new observation with the previous level extrapolated one step; the second smooths the latest level increment. The displayed forecast extrapolates that local slope. This is a double exponential smoothing method for a nonseasonal trending time series.