2.2.3.5.2 Algorithm for Decompostion

Decompostion Model

• Multiplicative: \(y_t = T_t \times S_t \times E_t\)
• Additive: \(y_t = T_t + S_t + E_t\)

where \(y_t\) is the observation, \(T_t\) is the trend component, \(S_t\) is the seasonal component, and \(E_t\) is the error.

Model Fitting

• Calculate smoothed time series\(\bar{T_t}\)

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.

• Calculate the detrended series \( y_t - \bar{T_t}\)

• Calculate the seasonal component \(S_t\).

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

\[Residuals = \hat{y_t}- y_t\]

Forecast

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

• Trend component \(t_t'\)

Perform linear extrapolation on the fitted trend \(T_t\).

• Seasonal component \(s_t'\)

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

• Forecasts \(y_t'\)
  • Multiplicative: \(y_t' = t_t' \times s_t'\)
  • Additive: \(y_t' = t_t' + s_t'\)