Holt-Winter Seasonal Method

Formula

Smoothing method for data with seasonal patterns with period length $p$.

Additive

$$\begin{array}{rl}{S}_t& =\alpha (X_t-M_{t-p})+(1-\alpha )(S_{t-1}+T_{t-1})\\ T_t& =\gamma (S_t-S_{t-1})+(1-\gamma )T_{t-1}\\ M_t& =\delta (X_t-S_t)+(1-\delta )M_{t-p}\\ F_{t+h}& =S_t+h{T}_t+M_{t-p+h}\end{array}$$

Tip

Used when the seasonal component is constant relative to the level.

Multiplicative

$$\begin{array}{rl}{S}_t& =\alpha (X_t/M_{t-p})+(1-\alpha )(S_{t-1}+T_{t-1})\\ T_t& =\gamma (S_t-S_{t-1})+(1-\gamma )T_{t-1}\\ M_t& =\delta (X_t/S_t)+(1-\delta )M_{t-p}\\ F_{t+h}& =(S_t+h{T}_t)M_{t-p+h}\end{array}$$

Tip

Used when seasonal variation changes along with the data level.