Seasonality Estimation and Elimination
Classical Decomposition (4 steps)
Given $X_t = m_t + s_t + Y_t$, period $d$, $\sum_{j=1}^d s_j = 0$:
Step 1: Estimate trend via d-point MA filter (cancels seasonal component).
Step 2: Estimate seasonal component. For each season $k$:
- $w_k$ = average of $\{x_{k+jd} - \hat{m}_{k+jd}\}$ over complete cycles
- $\hat{s}_k = w_k - \frac{1}{d}\sum_{i=1}^d w_i$
Step 3: Re-estimate trend from deseasonalized data $d_t = x_t - \hat{s}_t$.
Step 4: Estimate noise $\hat{Y}_t = x_t - \hat{m}_t - \hat{s}_t$.
Lag-d Difference Operator
$$\nabla_d X_t = X_t - X_{t-d} = (1 - B^d)X_t$$Applying: $\nabla_d X_t = (m_t - m_{t-d}) + (Y_t - Y_{t-d})$. Seasonal component vanishes. If trend remains, apply $\nabla$ afterward.