Holt's linear trend and Holt–Winters seasonal methods

Exponential smoothing extended with a smoothed trend (Holt) and seasonal terms (Holt–Winters, additive or multiplicative), using the error-correction recursions of FPP3.

\begin{aligned} \ell_t &= \alpha(y_t - s_{t-m}) + (1-\alpha)(\ell_{t-1} + \phi b_{t-1}) \\ b_t &= \beta(\ell_t - \ell_{t-1}) + (1-\beta)\phi b_{t-1} \\ s_t &= \gamma(y_t - \ell_{t-1} - \phi b_{t-1}) + (1-\gamma)s_{t-m} \\ \hat y_{T+h \given T} &= \ell_T + (\phi + \phi^2 + \dots + \phi^h)b_T + s_{T+h-m(k+1)} \end{aligned}

Additive Holt–Winters (drop s for Holt; phi = 1 when not damped)

$\ell_t, b_t, s_t$: level, trend (slope per period) and seasonal state at time $t$
$\alpha, \beta, \gamma$: smoothing parameters for level, trend and season, each in (0, 1)
$\phi$: damping parameter (0.8 to 0.98 when damped; 1 otherwise)
$m$: seasonal period (e.g. 12 for monthly data)
$k$: $\lfloor (h-1)/m \rfloor$ , so the forecast reuses the latest estimate of the matching season

When to use

Holt: a series with a trend but no seasonality. Damped Holt: the trend is expected to flatten (often more accurate for long horizons).
Holt–Winters additive: seasonal swings of roughly constant size. Multiplicative: swings proportional to the level (seasonal terms are factors: $y_t/s_{t-m}$ in the level update, forecasts multiplied by $s$ ).

Assumptions

Equally spaced data with no gaps; at least two full seasons for the seasonal "simple" initialisation ( $\ell_0$ = mean of the first season, $b_0$ = average season-to-season change divided by $m$ , $s_j = y_{j+1} - \ell_0$ or $y_{j+1}/\ell_0$ ). Without seasonality $\ell_0 = y_1$ , $b_0 = y_2 - y_1$ .
Parameters not supplied are chosen by minimising in-sample one-step SSE.

How to interpret

β controls how quickly the slope adapts; γ how quickly the seasonal pattern can change.
An undamped trend is extrapolated forever in a straight line — check whether that is plausible over the forecast horizon.

Common mistakes

Using additive seasonality when the seasonal amplitude grows with the level (or vice versa).
Long-horizon forecasts from an undamped trend presented without caveats.
Reading β here as the slope itself: it is a smoothing weight; the slope is $b_t$ .

Example

Holt on 10, 12, 15, 16, 19, 21 with α = 0.5, β = 0.3, $\ell_0 = 10$ , $b_0 = 2$ : first update $\ell_1 = 0.5 \cdot 10 + 0.5(10 + 2) = 11$ , $b_1 = 0.3(1) + 0.7(2) = 1.7$ . Final $\ell_6 = 20.66$ , $b_6 = 2.00$ ; forecasts 22.65 and 24.65 for the next two periods.