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# Decomposition of time series

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 Title: Decomposition of time series Author: World Heritage Encyclopedia Language: English Subject: Collection: Time Series Analysis Publisher: World Heritage Encyclopedia Publication Date:

### 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.

## Contents

• Decomposition based on rates of change 1
• Decomposition based on predictability 2
• Examples 3
• Software 4
• References 6

## 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 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
• 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.

## 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.

## References

1. ^ a b Dodge, Y. (2003). The Oxford Dictionary of Statistical Terms. New York: Oxford University Press.
2. ^ Kendall, M. G. (1976). Time-Series (Second ed.). Charles Griffin. (Fig. 5.1).