# 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, monthly or quarterly economic time series are usually decomposed into:

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

An example of statistical software for this type of decomposition is the progr₣Second Edition, Charles Griffin & Co.. ISBN 0-85264-241-5 (Fig. 5.1)</ref> 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.

## 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.