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| {{Otheruses4|the theorem as used in time series analysis|an abstract mathematical statement|Wold decomposition}}
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| In [[statistics]], '''World's decomposition''' or the '''Wold representation theorem''' (not to be confused with the Wold theorem that is the discrete-time analog of the [[Wiener–Khinchine theorem]]) named after [[Herman Wold]], says that every [[stationary process|covariance-stationary]] [[time series]] <math>Y_{t}</math> can be written as the sum of two time series, one ''deterministic'' and one ''stochastic''.
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| Formally
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| :<math>Y_t=\sum_{j=0}^\infty b_j \varepsilon_{t-j}+\eta_t,</math>
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| where:
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| :*<math>Y_t \,</math> is the [[time series]] being considered,
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| :*<math>\varepsilon_t</math> is an uncorrelated sequence which is the [[Innovation (signal processing)|innovation process]] to the process <math>Y_t \,</math> – that is, a white noise process that is input to the linear filter <math>\{b_j \} </math>.
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| :*<math>b \,</math> is the ''possibly'' infinite vector of moving average weights (coefficients or parameters)
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| :*<math>\eta_t \,</math> is a deterministic time series, such as one represented by a sine wave.
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| Note that the moving average coefficients have these properties:
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| # Stable, that is square summable <math>\sum_{j=1}^{\infty}|b_{j}|^2</math> < <math>\infty</math>
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| # Causal (i.e. there are no terms with ''j'' < 0)
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| # Minimum delay
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| # Constant (<math> b_j </math> independent of ''t'')
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| # It is conventional to define <math>b_0 = 1</math>
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| This theorem can be considered as an existence theorem: any stationary process has this seemingly special representation. Not only is the existence of such a simple linear and exact representation remarkable, but even more so is the special nature of the moving average model. Imagine creating a process that is a moving average but not satisfying these properties 1–4. For example, the coefficients <math>b_j</math> could define an acausal and non-minimum delay model. Nevertheless the theorem assures the existence of a causal minimum delay moving average that exactly represents this process. How this all works for the case of causality and the minimum delay property is discussed in Scargle (1981), where an extension of the Wold Decomposition is discussed.
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| The usefulness of the Wold Theorem is that it allows the [[dynamical system|dynamic]] evolution of a variable <math>Y_{t}</math> to be approximated by a [[linear model]]. If the innovations <math>\varepsilon_{t}</math> are [[statistical independence|independent]], then the linear model is the only possible representation relating the observed value of <math>Y_{t}</math> to its past evolution. However, when <math>\varepsilon_{t}</math> is merely an [[uncorrelated]] but not independent sequence, then the linear model exists but it is not the only representation of the dynamic dependence of the series. In this latter case, it is possible that the linear model may not be very useful, and there would be a nonlinear model relating the observed value of <math>Y_{t}</math> to its past evolution. However, in practical [[time series analysis]], it often the case that only linear predictors are considered, partly on the grounds of simplicity, in which case the Wold decomposition is directly relevant.
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| The Wold representation depends on an infinite number of parameters, although in practice they usually decay rapidly. The [[autoregressive model]] is an alternative that may have only a few coefficients if the corresponding moving average has many. These two models can be combined into an [[Autoregressive moving average model|autoregressive-moving average (ARMA) model]], or an autoregressive-integrated-moving average (ARIMA) model if non-stationarity is involved. See Scargle(1981) and references there.
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| ==References== | |
| * [[Theodore Wilbur Anderson|Anderson, T. W.]] (1971) ''The Statistical Analysis of Time Series''. Wiley.
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| * [[Herman Wold|Wold, H.]] (1954) ''A Study in the Analysis of Stationary Time Series'', Second revised edition, with an Appendix on "Recent Developments in Time Series Analysis" by [[Peter Whittle]]. Almqvist and Wiksell Book Co., Uppsala.
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| * [[Jeffrey Scargle|Scargle, J. D.]] (1981) ''Studies in astronomical time series analysis. I – Modeling random processes in the time domain,'' 1981, ''Astrophysical Journal Supplement Series'', 45, pp. 1–71.
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| {{Statistics|analysis}}
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| [[Category:Statistical theorems]]
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| [[Category:Time series analysis]]
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| [[Category:Multivariate statistics]]
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| [[Category:Stochastic processes]]
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