Time series data can exhibit a huge variety of temporal patterns. In many cases it is useful to categorize these patterns, and further, split them up into distinct components. In general many time series can be decomposed into three parts:

• Trend: A trend exists when there is a long-term increase or decrease in the data.
• Seasonal pattern: A seasonal pattern exists when a series is influenced by seasonal factors. Seasonality is always of a fixed and known period.
• Residual: The remainder, after accounting for the trend and seasonality; also known as noise.

Referring to the three terms from above we can think of a time series $$y_t$$ as consisting of these three components: a seasonal component, a trend-cycle component (containing both trend and cycle), and a remainder component (containing anything else in the time series).

We can principally model such a time series in two ways: as an additive model and as a multiplicative model.

The additive model can be written as

$y_t = T_t + S_t + E_t\text{,}$ and the multiplicative model can be written as

$y_t = T_t \times S_t \times E_t\text{,}$

where $$y_t$$ is the data at period $$t$$, $$T_t$$ is the trend-cycle component, $$S_t$$ is the seasonal component and $$E_t$$ is the remainder component at period $$t$$.

The additive model is most appropriate if the magnitude of the seasonal fluctuations or the variation around the trend-cycle does not vary with the level of the time series. When the variation in the seasonal pattern, or the variation around the trend-cycle, appears to be proportional to the level of the time series, then a multiplicative model is more appropriate.