Classical decomposition

Splitting a seasonal series into trend (centred moving average), a fixed seasonal pattern and a remainder, in additive or multiplicative form.

y_t = T_t + S_t + R_t \quad \text{(additive)}, \qquad y_t = T_t \times S_t \times R_t \quad \text{(multiplicative)}

Additive and multiplicative models

$T_t$: trend-cycle, estimated by the centred moving average of order $m$ (2×m-MA for even $m$ )
$S_t$: seasonal component: the same index every cycle, averaged from the detrended values; indices sum to 0 (additive) or average 1 (multiplicative)
$R_t$: remainder: what is left after removing trend and season

When to use

Describing a seasonal series and producing seasonally adjusted values ( $y_t - S_t$ or $y_t/S_t$ ).
Additive when the seasonal swings are roughly constant in size; multiplicative when they grow with the level.

Assumptions

A known period $m$ and at least two full cycles; the seasonal pattern is assumed identical every cycle.
The trend (and so the remainder) is missing for the first and last $\lfloor m/2 \rfloor$ observations.

How to interpret

An additive seasonal index of +3 means that season is typically 3 units above the trend; a multiplicative index of 1.08 means 8% above.
A remainder with structure (runs, spikes) suggests the model is missing something — a changing season, outliers, calendar effects.

Common mistakes

Forgetting that classical decomposition cannot let the seasonal pattern evolve (more modern methods such as STL can).
Reading the trend at the ends: it is not estimated there.
Using additive decomposition on data whose seasonal swings scale with the level.

Example

Quarterly series 10, 14, 8, 12, 11, 15, 9, 13, 12, 16, 10, 14 ( $m = 4$ , additive). The 2×4-MA trend rises from 11.125 to 12.875 by 0.25 per quarter; seasonal indices are −0.625, +3.125, −3.125, +0.625 for quarters 1–4 (sum 0); the remainder is 0 here because these illustrative data are exactly trend + season.