Classical Decomposition
Classical decomposition breaks a time series into trend, seasonality, and residual noise so the remaining series can be modeled with stationary-process tools.
1. Additive Model
$$X_t = m_t + s_t + Y_t$$| Component | Description | Properties |
|---|---|---|
| $m_t$ | Deterministic trend | e.g., linear $a + bt$, polynomial $\sum_{k=0}^p a_k t^k$ |
| $s_t$ | Deterministic seasonal component | Periodic: $s_t = s_{t+d}$ with period $d$ |
| $Y_t$ | Random (noise) process | Zero-mean, often modeled as Gaussian |
When seasonality is recorded over one full period, it is often normalized by
$$\sum_{j=1}^d s_j = 0$$so the average level is not counted twice in both $m_t$ and $s_t$.
2. Multiplicative View
If seasonal amplitude grows with the level of the series, a multiplicative structure is more natural. A common workaround is to log-transform:
$$\log(X_t) = m_t + s_t + Y_t$$which converts multiplicative behavior into an additive decomposition.
3. Trend Estimation
Polynomial Regression
Fit $\hat{m}_t = \hat{a} + \hat{b} t$ (or higher degree) by minimizing
$$\sum_{t=1}^n (X_t - a - bt)^2$$Moving Average Filter
$$\hat{m}_t = \frac{X_{t-q} + \cdots + X_t + \cdots + X_{t+q}}{2q+1}, \quad q+1 \leq t \leq n-q$$Limitations:
- All past observations have equal weight (but recent observations usually matter more)
- Cannot estimate $m_t$ near the endpoints — cannot be used for Forecasting
4. Trend Elimination
Use the difference operator $\nabla X_t = X_t - X_{t-1}$.
If $m_t$ is a polynomial of degree $p$, then $p$ applications of $\nabla$ eliminate the trend:
$$\nabla^p m_t = \text{constant}$$5. Seasonality Elimination
Use the lag-$d$ difference operator:
$$\nabla_d X_t = X_t - X_{t-d} = (1 - B^d)X_t$$If $s_t$ has period $d$ (i.e., $s_t = s_{t+d}$), then $\nabla_d$ eliminates $s_t$:
$$\nabla_d s_t = s_t - s_{t-d} = 0$$For multiple seasonal components with periods $d_1, \dots, d_k$, the smallest $d$ such that $\nabla_d$ eliminates all seasonality is $d = \text{lcm}(d_1, \dots, d_k)$.
6. Course Strategy
In STA457, decomposition is usually a preprocessing step:
- Identify trend and seasonal structure.
- Remove them using regression, smoothing, or differencing.
- Treat the residual component as approximately stationary.
- Fit AR, MA, or ARMA structure to the residual series.