14.3 Holt and Holt–Winters methods

Simple exponential smoothing lags behind a trending series, as the example showed. Holt’s method smooths a level \(L_t\) and a trend (slope) \(T_t\) separately, with two constants \(\alpha \) and \(\gamma \): \begin {align*} L_t &= \alpha x_t + (1 - \alpha )(L_{t-1} + T_{t-1}) , \\ T_t &= \gamma (L_t - L_{t-1}) + (1 - \gamma )T_{t-1} , \end {align*}

and forecasts \(\hat x(n, k) = L_n + kT_n\), a straight line from the current level with the current slope. Each equation is a weighted average of a new estimate and the previous one, as in simple smoothing: the level averages the new observation with where the old level and slope predicted it would be, and the slope averages the latest change in level with the old slope.

For seasonal series, the Holt–Winters method adds a seasonal index \(I_t\) with period \(s\) and a third constant \(\delta \). In the additive form, \begin {align*} L_t &= \alpha (x_t - I_{t-s}) + (1 - \alpha )(L_{t-1} + T_{t-1}) , \\ T_t &= \gamma (L_t - L_{t-1}) + (1 - \gamma )T_{t-1} , \\ I_t &= \delta (x_t - L_t) + (1 - \delta )I_{t-s} , \end {align*}

with forecasts \(\hat x(n, k) = L_n + kT_n + I_{n+k-s}\) for \(k \le s\). The level is updated from the seasonally adjusted observation \(x_t - I_{t-s}\), and the index from the detrended observation \(x_t - L_t\), exactly the quantities of the decomposition in Example 2.4.3. In the multiplicative form, subtraction is replaced by division and the forecast is \((L_n + kT_n)I_{n+k-s}\). Starting values come from the first year or two of data, for instance a decomposition of them.

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