Comprises a forecast equation and three smoothing equations
The smoothin equation are: * level using smoothing parameter * trend using smoothing parameter * seasonal using smoothing parameter
The seasonal component present two variations: * Additive method: when seasonal variation is constant through the series * Multiplicative method: when seasonal variation are changing proportionally to the level of the series
Single Exponential Smoothing (similar to 7.3. EWMA Theory)
y(0)&=&x(0)\\ y(t)&=&(1-\alpha)y(t-1)+\alpha x(t) \end{eqnarray}$$ Double Exponential Smoothing (Holt's method): $$\begin{eqnarray} l(t)&=&(1-\alpha)l(t-1)+\alpha x(t)\\ b(t)&=&(1-\beta)b(t-1)+\beta (l(t)-l(t-1))\\ y(t)&=&l(t) + b(t)\\ \hat{y}(t+h)&=&l(t)+hb(t)\;, \end{eqnarray}$$ where $y$ is the fitted model and $\hat{y}$ is the forecasting model with $h$ periods in the future. Can predict general level and trend, but not seasonality. Triple Exponential Smoothing (Holts-Winters' method): $$\begin{eqnarray} l(t)&=&(1-\alpha)l(t-1)+\alpha x(t)\\ b(t)&=&(1-\beta)b(t-1)+\beta (l(t)-l(t-1))\\ c(t)&=&(1-\gamma)c(t-L)+\gamma (x(t)-l(t-1)-b(t-1))\\ y(t)&=&(l(t) + b(t))c(t)\\ \hat{y}(t+h)&=&(l(t)+mb(t))c(t-L+1+(m-1)\mod L)\;, \end{eqnarray}$$ where $c$ is seasonality component, $y$ is the fitted model and $\hat{y}$ is the forecasting model with $h$ periods in the future, $L$ is divisions per cycle (e.g. in a year data, $L=12$ months).