# Decomposition of time series

The decomposition of time series is a statistical method that deconstructs a time series into notional components. There are two principal types of decomposition which are outlined below.

## Decomposition based on rates of change

This is an important technique for all types of time series analysis, especially for seasonal adjustment.[1] It seeks to construct, from an observed time series, a number of component series (that could be used to reconstruct the original by additions or multiplications) where each of these has a certain characteristic or type of behaviour. For example, time series are usually decomposed into:

• the Trend Component [itex]T_t[/itex] that reflects the long term progression of the series (secular variation)
• the Cyclical Component [itex]C_t[/itex] that describes repeated but non-periodic fluctuations
• the Seasonal Component [itex]S_t[/itex] reflecting seasonality (seasonal variation)
• the Irregular Component [itex]I_t[/itex] (or "noise") that describes random, irregular influences. It represents the residuals of the time series after the other components have been removed.

## Decomposition based on predictability

The theory of time series analysis makes use of the idea of decomposing a times series into deterministic and non-deterministic components (or predictable and unpredictable components).[1] See Wold's theorem and Wold decomposition.

## Examples

Kendall shows an example of a decomposition into smooth, seasonal and irregular factors for a set of data containing values of the monthly aircraft miles flown by UK airlines.[2]

## Software

An example of statistical software for this type of decomposition is the program BV4.1 that is based on the so-called Berlin procedure.