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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 <math>T_t</math> that reflects the long term progression of the series (secular variation)
  • the Cyclical Component <math>C_t</math> that describes repeated but non-periodic fluctuations
  • the Seasonal Component <math>S_t</math> reflecting seasonality (seasonal variation)
  • the Irregular Component <math>I_t</math> (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.


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


  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

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