 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.