2.2.3.5.2 Algorithm for Decompostion

Decompostion Model

where $y_t$ is the observation, $T_t$ is the trend component, $S_t$ is the seasonal component, and $E_t$ is the error.

If seasonal length $m$ is an even number, compute $\bar{T_t}$ using $2 \times m$ -MA. if $m$ is an odd number, compute $\bar{T_t}$ using $m$ -MA.

For each season, calculate the median of the detrended series for that season.And then the median is replicated in each season of $S_t$ .

For multiplicative model, adjust $S_t$ to average of 1. For additive model, adjust $S_t$ to average of 0.

Seasonal adjusted data $S_{adj} = y_t / S_t$ for multiplicative model, $S_{adj} = y_t - S_t$ for additive model.

Trend component $T_t$ : If model includes trend, perform linear regression on $S_{adj}$ to compute $T_t$ . Otherwise calculate the mean of $S_{adj}$ as $T_t$ .

Fitted data $\hat{y_t} = T_t \times S_t$ for multiplicative model, $\hat{y_t}= T_t + S_t$ for additive model.

Detrended data $D_t = y_t / T_t$ for multiplicative model, $D_t = y_t - T_t$ for additive model.

$Residuals = \hat{y_t}- y_t$

The forecasts are calculated by computing the trend and seasonal component separately.

Perform linear extrapolation on the fitted trend $T_t$ .

The forecasts begin at the end of $S_t$ . Replicate the values of the same season in $S_t$ to get $s_t'$ .

Forecasts
- Multiplicative: $y_t' = t_t' \times s_t'$
- Additive: $y_t' = t_t' + s_t'$