Decomposition of time series

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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[edit]

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[edit]

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.


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]


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

See also[edit]


  1. ^ a b Dodge, Y. (2003). The Oxford Dictionary of Statistical Terms. New York: Oxford University Press. ISBN 0-19-920613-9. 
  2. ^ Kendall, M. G. (1976). Time-Series (Second ed.). Charles Griffin. (Fig. 5.1). ISBN 0-85264-241-5. 

Further reading[edit]

  • Enders, Walter (2004). "Models with Trend". Applied Econometric Time Series (Second ed.). New York: Wiley. pp. 156–238. ISBN 0-471-23065-0.