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- Die Woldsche Zerlegung bezeichnet eine spezielle Zerlegung in der Zeitreihenanalyse, einem Teilgebiet der mathematischen Statistik. (de)
- Le théorème de représentation de Granger et le théorème de décomposition de Wold sont deux résultats majeurs en économétrie des séries temporelles. Le théorème de représentation de Granger consiste à mettre en relation les modèles de cointégration et les (en). Introduit initialement par l'économiste anglais Clive Granger, le théorème de représentation constitue ainsi un cadre où sont analysés à la fois les concepts de stationnarité et d'intégration des séries temporelles, de corrélation, de causalité, de cointégration des processus aléatoires, avant le cryptage de la connexion entre les modèles de cointégration et les modèles à correction d'erreur. (fr)
- In statistics, Wold's decomposition or the Wold representation theorem (not to be confused with the Wold theorem that is the discrete-time analog of the Wiener–Khinchin theorem), named after Herman Wold, says that every covariance-stationary time series can be written as the sum of two time series, one deterministic and one stochastic. Formally where:
* is the time series being considered,
* is an uncorrelated sequence which is the innovation process to the process – that is, a white noise process that is input to the linear filter .
* is the possibly infinite vector of moving average weights (coefficients or parameters)
* is a deterministic time series, such as one represented by a sine wave. The moving average coefficients have these properties: 1.
* Stable, that is square summable < 2.
* Causal (i.e. there are no terms with j < 0) 3.
* Minimum delay 4.
* Constant ( independent of t) 5.
* It is conventional to define 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 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. The usefulness of the Wold Theorem is that it allows the dynamic evolution of a variable to be approximated by a linear model. If the innovations are independent, then the linear model is the only possible representation relating the observed value of to its past evolution. However, when 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 to its past evolution. However, in practical time series analysis, it is often the case that only linear predictors are considered, partly on the grounds of simplicity, in which case the Wold decomposition is directly relevant. 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 (ARMA) model, or an autoregressive-integrated-moving average (ARIMA) model if non-stationarity is involved. See and references there; in addition this paper gives an extension of the Wold Theorem that allows more generality for the moving average (not necessarily stable, causal, or minimum delay) accompanied by a sharper characterization of the innovation (identically and independently distributed, not just uncorrelated). This extension allows the possibility of models that are more faithful to physical or astrophysical processes, and in particular can sense ″the arrow of time.″ (en)
- Теорема Волда — утверждение математической статистики, согласно которому каждый слабо стационарный временной ряд можно представить в виде скользящего среднего бесконечного порядка . Такое представление называют представлением скользящим средним для временных рядов. Установлена Херманом Волдом. Формально: , где:
* — рассматриваемый временной ряд,
* — белый шум на входе линейного фильтра ; также применяется термин «инновация» (англ. innovation)
* — последовательность коэффициентов скользящего среднего (параметров или весов)
* — детерминированная компонента; равна нулю, если у нет трендов. Коэффициенты удовлетворяют следующим условиям: 1.
* 2.
* ряд сходится абсолютно: 3.
* отсутствуют члены с 4.
* постоянны (не зависят от ) (ru)
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- minimum delay moving average (en)
- non-minimum delay (en)
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- Die Woldsche Zerlegung bezeichnet eine spezielle Zerlegung in der Zeitreihenanalyse, einem Teilgebiet der mathematischen Statistik. (de)
- Le théorème de représentation de Granger et le théorème de décomposition de Wold sont deux résultats majeurs en économétrie des séries temporelles. Le théorème de représentation de Granger consiste à mettre en relation les modèles de cointégration et les (en). (fr)
- In statistics, Wold's decomposition or the Wold representation theorem (not to be confused with the Wold theorem that is the discrete-time analog of the Wiener–Khinchin theorem), named after Herman Wold, says that every covariance-stationary time series can be written as the sum of two time series, one deterministic and one stochastic. Formally where: The moving average coefficients have these properties: 1.
* Stable, that is square summable < 2.
* Causal (i.e. there are no terms with j < 0) 3.
* Minimum delay 4.
* Constant ( independent of t) 5.
* It is conventional to define (en)
- Теорема Волда — утверждение математической статистики, согласно которому каждый слабо стационарный временной ряд можно представить в виде скользящего среднего бесконечного порядка . Такое представление называют представлением скользящим средним для временных рядов. Установлена Херманом Волдом. Формально: , где:
* — рассматриваемый временной ряд,
* — белый шум на входе линейного фильтра ; также применяется термин «инновация» (англ. innovation)
* — последовательность коэффициентов скользящего среднего (параметров или весов)
* — детерминированная компонента; равна нулю, если у нет трендов. (ru)
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- Woldsche Zerlegung (de)
- Théorème de représentation de Granger (fr)
- Теорема Волда (ru)
- Wold's theorem (en)
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