214. Classical

Decompose a time series ${𝑦}_{𝑡}$ into four components that add up (or multiply) back to the original:

• Level ${𝐿}_{𝑡}$: baseline value around which the series fluctuates
• Trend ${𝑇}_{𝑡}$: long-term progression or direction
• Seasonal ${𝑆}_{𝑡}$: regular, repeating patterns over a fixed period $𝑚$
• Noise ${𝑁}_{𝑡}$: irregular fluctuations not explained by the other components (residuals)

Two functional forms — additive and multiplicative — depending on whether the seasonal swing is constant in absolute units or proportional to the level.

214.0.1. Additive form

Use when the amplitude of seasonal swings is roughly constant over time, regardless of the level. Example: temperature (summer is  10°C above average whether the long-run average is 15°C or 25°C).

214.0.2. Multiplicative form

Use when the seasonal amplitude scales with the level. Example: retail sales (December peak is 30% above the trend, whether the trend is at $1M or $10M).

A multiplicative decomposition can always be converted to an additive one via $log{𝑦}_{𝑡}=log{𝐿}_{𝑡}+log{𝑇}_{𝑡}+log{𝑆}_{𝑡}+log{𝑁}_{𝑡}$ — useful trick if your tools support only additive.

214.0.3. How to choose

Plot the series. If the seasonal swing grows with the level → multiplicative. If it stays roughly constant → additive. When unsure, try both and check which has more uniform residuals.

214.0.4. Procedure (additive case, integer period $𝑚$)

The classic step-by-step:

1. Estimate trend via centered moving average of size $𝑚$. For even $𝑚$ (typical with monthly data, $𝑚=12$), use a 2-step MA: first an $𝑚$-MA, then a 2-MA. Result: ${\widehat{𝑇}}_{𝑡}$.
2. Detrend: ${𝑦}_{𝑡}-{\widehat{𝑇}}_{𝑡}$.
3. Estimate seasonal indices: average the detrended values within each seasonal slot (e.g., all Januaries, all Februaries, etc.). Adjust so the seasonal indices sum to zero (additive) or average to 1 (multiplicative). Result: ${\widehat{𝑆}}_{𝑡}$ (depends only on $𝑡mod𝑚$).
4. Residual / noise: ${\widehat{𝑁}}_{𝑡}={𝑦}_{𝑡}-{\widehat{𝑇}}_{𝑡}-{\widehat{𝑆}}_{𝑡}$.

214.0.5. Limitations of classical decomposition

• Constant seasonal pattern: classical decomposition assumes the seasonal effect is the same in every cycle. Real series often have evolving seasonality (Christmas effect grows over decades). Use STL ([stl.typ](stl.typ)) for those.
• Endpoint problem: the centered MA has gaps at the start and end (size $𝑚/2$ each). Modern methods (STL, X-13) handle this better.
• Sensitivity to outliers: a single anomalous month can distort the seasonal index for that slot. Robust methods (STL with robust loess) address this.

For a quick first pass, classical decomposition is fine. For production forecasting, prefer STL.