Time series decomposition involves separating a time series into several distinct components. There are three components that are typically of interest:
Tt, a deterministic, nonseasonal secular trend component. This component is sometimes restricted to being a linear trend, though higher-degree polynomials are also used.
St, a deterministic seasonal component with known periodicity. This component captures level shifts that repeat systematically within the same period (e.g., month or quarter) between successive years. It is often considered to be a nuisance component, and seasonal adjustment is a process for eliminating it.
It, a stochastic irregular component. This component is not necessarily a white noise process. It can exhibit autocorrelation and cycles of unpredictable duration. For this reason, it is often thought to contain information about the business cycle, and is usually the most interesting component.
There are three functional forms that are most often used for representing a time series yt as a function of its trend, seasonal, and irregular components:
Additive decomposition, where
Multiplicative decomposition, where
Log-additive decomposition, where
You can estimate the trend and seasonal components by using filters (moving averages) or parametric regression models. Given estimates and , the irregular component is estimated as
Similarly, the series (or ) is called a deseasonalized series.