Additive vs Multiplicative Models Time Series

Additive Model

In an additive model, the observed time series is expressed as the sum of its components:

$Y_t=T_t+S_t+R_t$

Where:

• $Y_t$: observed value at time $t$
• $T_t$: trend component
• $S_t$: seasonal component
• $R_t$: residual (random noise)

This assumes each component contributes independently and linearly to the series.

We typically use an additive model when:

• The seasonal effect is stable over time (e.g., summer–winter differences remain consistent).
• The trend does not distort or amplify seasonality.
• Seasonal fluctuations stay within a fixed range, even if minor noise exists.

For example, in daily minimum temperature data, the seasonal swings remain steady year after year. This stability indicates that the additive model is appropriate, since seasonality does not depend on the trend.

Multiplicative Model

In a multiplicative model, the components interact by scaling:

$Y_t=T_t\times S_t\times R_t$

Where:

• $Y_t$: observed value
• $T_t$: trend
• $S_t$: seasonal effect
• $R_t$: residual

This model is suited for series where the magnitude of seasonality changes with the trend. For example, when seasonal peaks and troughs grow larger as the trend increases.

Choosing Between Models

• Additive model →stable seasonal effect, constant magnitude, independent of trend.
• Multiplicative model → seasonal effect scales with trend.

The choice can often be made by visually inspecting the decomposition plot. See Decomposition in Time Series.

Example:

• Additive: seasonal effect is constant in magnitude (e.g., demand increases by +100 GWh every winter).
• Multiplicative: seasonal effect scales with the level (e.g., winter demand is 20% higher than the baseline).

Related

See Baseline Forecast for once the models type is selected.