Parameter Definitions

After observing month t, we will have used all data observed through the end of month t to estimate the following quantities of interest:

• L_t=Level of series

• T_t=Trend of series

• S_t=Seasonal index for current month

The key to Winter’s method is the following three equations, which are used to update L_t, T_t, and S_t. In the following formulas, alp, bet, and gam are called smoothing parameters. The values of these parameters will be chosen to optimize our forecasts. In the following formulas, c equals the number of periods in a seasonal cycle (c=12 months for example) and x_t equals the observed value of the time series at time t.

• Formula 1: L_t=alp(x_t/s_t–c)+(1–alp)(L_t–1*T_t–1)

• Formula 2: T_t=bet(L_t/L_t–1)+(1–bet)T_t–1

• Formula 3: S_t=gam(x_t/L_t)+(1–gam)s_t–-c

Formula 1 indicates that our new base estimate is a weighted average of the current observation (deseasonalized) and last period’s base updated by our last trend estimate. Formula 2 indicates that our new trend estimate is a weighted average of the ratio of our current base to last period’s base (this is a current estimate of trend) and last period’s trend. Formula 3 indicates that we update our seasonal index estimate as a weighted average of the estimate of the seasonal index based on the current period and the previous estimate. Note that larger values of the smoothing parameters correspond to putting more weight on the current observation.

We define F_t,k as our forecast (F) after period t for the period t+k. This results in the formula F_t,k=L_t*(T_t)^ks_t+k–c.

This formula first uses the current trend estimate to update the base k periods forward. Then the resulting base estimate for period t+k is adjusted by the appropriate seasonal index.