0.1 Time Series Decomposition
In general, a time series is composed of a deterministic element (which we are interested in describing) and a random element (noise). The deterministic element of a time series can take numerous forms but in practice typically consists of a trend component (describing the tendency of the series to move consistently in a certain direction) and a seasonal component (describing the predictable oscillations of the series at a fixed period). This leaves the remainder/irregular component, ideally close to stationary white noise.
\[ \text{Additive:}\quad X_t = m_t + s_t + Y_t, \qquad \text{Multiplicative:}\quad X_t = m_t \cdot s_t \cdot Y_t, \] where \(m_t\) is the trend, \(s_t\) the seasonal component (\(s_{t+p}=s_t\) for period \(p\)), and \(Y_t\) the (stationary) remainder. The multiplicative form is used when seasonal/noise fluctuations scale with the trend level, and reduces to the additive form under a log transform.
0.2 Estimating the Trend and Seasonal Components
Moving average smoothing. The trend can be estimated by a symmetric moving average that averages out the seasonal and noise components, e.g. for period \(p\) even, \[ \hat m_t = \frac{1}{p}\left(\tfrac12 x_{t-p/2} + x_{t-p/2+1} + \dots + x_{t+p/2-1} + \tfrac12 x_{t+p/2}\right). \] This is closely related to the use of the backshift/differencing operators discussed for ARIMA models — seasonal differencing \((1-B^p)\) is an alternative, purely stochastic way to remove \(s_t\) without ever estimating it explicitly.
Seasonal estimate. Once detrended (e.g. \(X_t - \hat m_t\)), average the detrended values across each season/period position to obtain \(\hat s_t\), then mean-center so \(\sum_{t=1}^p \hat s_t = 0\).
STL (Seasonal-Trend decomposition using Loess). A more flexible, robust, iterative decomposition using local regression (Loess) smoothing to estimate trend and (a potentially time-varying) seasonal component, handling any period length and outlier-robust remainder estimation. X-11/X-13ARIMA-SEATS are official-statistics-agency alternatives combining moving-average filters with model-based (ARIMA) extrapolation at the series ends.
0.3 Relation to Other Approaches
- SARIMA models seasonality stochastically via seasonal differencing/ARMA terms rather than as a fixed deterministic component.
- Structural time-series models treat trend and seasonal components as their own latent stochastic processes (e.g. local level + seasonal), estimated jointly via the Kalman filter — a probabilistic generalization of classical decomposition.
- Wavelet Analysis offers a multiresolution decomposition when the notion of a single fixed “trend” and “seasonal” scale is inadequate (e.g. evolving cycles).