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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.
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| ==Decomposition based on rates of change==
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| This is an important technique for all types of [[time series analysis]], especially for [[seasonal adjustment]].<ref name='Dodge'>{{cite book |last=Dodge |first=Y. |year=2003 |title=The Oxford Dictionary of Statistical Terms |location=New York |publisher=Oxford University Press |isbn=0-19-920613-9 }}</ref> 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:
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| *the [[Trend estimation|Trend Component]] <math>T_t</math> that reflects the long term progression of the series ([[secular variation]])
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| *the Cyclical Component <math>C_t</math> that describes repeated but non-periodic fluctuations
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| *the Seasonal Component <math>S_t</math> reflecting [[seasonality]] (seasonal variation)
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| *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.
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| ==Decomposition based on predictability==
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| 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).<ref name="Dodge"/> See [[Wold's theorem]] and [[Wold decomposition]].
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| ==Examples==
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| 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 [[List of airlines of the United Kingdom|UK airlines]].<ref>{{cite book |last=Kendall |first=M. G. |year=1976 |title=Time-Series |edition=Second |publisher=Charles Griffin |isbn=0-85264-241-5 |at=(Fig. 5.1) }}</ref>
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| ==Software==
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| An example of statistical software for this type of decomposition is the program [[BV4.1]] that is based on the so-called [[Berlin procedure]].
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| ==See also==
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| *[[Hilbert–Huang transform]]
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| *[[Stochastic drift]]
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| ==References==
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| {{Reflist}}
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| {{Statistics|analysis}}
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| [[Category:Time series analysis]]
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