翻訳と辞書 Words near each other ・ Wolcottsburg, New York ・ Wolcottsville, New York ・ Wolcottville, Indiana ・ Wolcott–Rallison syndrome ・ Wold ・ WOLD (song) ・ Wold (surname) ・ Wold Cottage (meteorite) ・ Wold Newton ・ Wold Newton Cricket Club ・ Wold Newton family ・ Wold Newton, East Riding of Yorkshire ・ Wold Newton, Lincolnshire ・ Wold Nunatak ・ Wold's decomposition ・ Wold's theorem ・ WOLD-FM ・ Woldai Futur ・ Wolde ・ Wolde (disambiguation) ・ Wolde Giyorgis Wolde Yohannes ・ Wolde Harris ・ Wolde Selassie ・ Woldeab Woldemariam ・ Woldegk ・ Woldegk (Amt) ・ Woldemar ・ Woldemar Bargiel ・ Woldemar Brinkmann ・ Woldemar Friedrich Dictionary Lists mini英和辞書 mini和英辞書 Webster 1913 Latin-English FOLDOC Wikipedia English ウィキペディア

翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Wold's theorem ： ウィキペディア英語版 Wold's theorem 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 covariance-stationary time series $Y\_$ can be written as the sum of two time series, one ''deterministic'' and one ''stochastic''. Formally :$Y\_t=\backslash sum\_^\backslash infty\; b\_j\; \backslash varepsilon\_+\backslash eta\_t,$ where: : *$Y\_t\; \backslash ,$ is the time series being considered, : *$\backslash varepsilon\_t$ is an uncorrelated sequence which is the innovation process to the process $Y\_t\; \backslash ,$ – that is, a white noise process that is input to the linear filter $\backslash$. : *$b\; \backslash ,$ is the ''possibly'' infinite vector of moving average weights (coefficients or parameters) : *$\backslash eta\_t\; \backslash ,$ is a deterministic time series, such as one represented by a sine wave. Note that the moving average coefficients have these properties: # Stable, that is square summable $\backslash sum\_^|b\_|^2$ < $\backslash infty$ # Causal (i.e. there are no terms with ''j'' < 0) # Minimum delay # Constant ($b\_j$ independent of ''t'') # It is conventional to define $b\_0\; =\; 1$ 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 $b\_j$ 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 $Y\_$ to be approximated by a linear model. If the innovations $\backslash varepsilon\_$ are independent, then the linear model is the only possible representation relating the observed value of $Y\_$ to its past evolution. However, when $\backslash varepsilon\_$ 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 $Y\_$ 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. ==References== * * * * 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. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Wold's theorem」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.