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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Propagation of uncertainty ： ウィキペディア英語版 Propagation of uncertainty :''For the propagation of uncertainty through time, see Chaos theory#Sensitivity to initial conditions.'' In statistics, propagation of uncertainty (or propagation of error) is the effect of variables' uncertainties (or errors) on the uncertainty of a function based on them. When the variables are the values of experimental measurements they have uncertainties due to measurement limitations (e.g., instrument precision) which propagate to the combination of variables in the function. The uncertainty is usually defined by the absolute error . Uncertainties can also be defined by the relative error , which is usually written as a percentage. Most commonly, the error on a quantity, , is given as the standard deviation, . Standard deviation is the positive square root of variance, . The value of a quantity and its error are often expressed as an interval . If the statistical probability distribution of the variable is known or can be assumed, it is possible to derive confidence limits to describe the region within which the true value of the variable may be found. For example, the 68% confidence limits for a one-dimensional variable belonging to a normal distribution are ± one standard deviation from the value, that is, there is approximately a 68% probability that the true value lies in the region . If the variables are correlated, then covariance must be taken into account. ==Linear combinations== Let $\backslash$ be a set of ''m'' functions which are linear combinations of $n$ variables $x\_1,x\_2,\backslash dots,x\_n$ with combination coefficients $A\_,A\_,\backslash dots,A\_,\; (k=1\backslash dots\; m)$. :$f\_k=\backslash sum\_i^n\; A\_\; x\_i$ or $\backslash mathrm=\backslash mathrm\backslash ,$ and let the variance-covariance matrix on x be denoted by $\backslash mathrm\backslash ,$. :$\backslash mathrm\; =$ \begin \sigma^2_1 & \sigma_ & \sigma_ & \cdots \\ \sigma_ & \sigma^2_2 & \sigma_ & \cdots\\ \sigma_ & \sigma_ & \sigma^2_3 & \cdots \\ \vdots & \vdots & \vdots & \ddots \\ \end = \begin \mathit^x_1 & \mathit^x_ & \mathit^x_ & \cdots \\ \mathit^x_ & \mathit^x_2 & \mathit^x_ & \cdots\\ \mathit^x_ & \mathit^x_ & \mathit^x_3 & \cdots \\ \vdots & \vdots & \vdots & \ddots \\ \end Then, the variance-covariance matrix $\backslash mathrm\backslash ,$ of ''f'' is given by :$\backslash mathit^f\_=\; \backslash sum\_k^n\; \backslash sum\_\backslash ell^n\; A\_\; \backslash mathit^x\_\; A\_:\; \backslash mathrm$. This is the most general expression for the propagation of error from one set of variables onto another. When the errors on ''x'' are uncorrelated the general expression simplifies to :$\backslash mathit^f\_=\; \backslash sum\_k^n\; A\_\; \backslash mathit^x\_k\; A\_.$ where $\backslash mathit^x\_k\; =\; \backslash sigma^2\_$ is the variance of ''k''-th element of the ''x'' vector. Note that even though the errors on ''x'' may be uncorrelated, the errors on ''f'' are in general correlated; in other words, even if $\backslash mathrm$ is a diagonal matrix, $\backslash mathrm$ is in general a full matrix. The general expressions for a scalar-valued function, ''f'', are a little simpler. :$f=\backslash sum\_i^n\; a\_i\; x\_i:\; f=\backslash mathrm\backslash ,$ :$\backslash sigma^2\_f=\; \backslash sum\_i^n\; \backslash sum\_j^n\; a\_i\; \backslash mathit^x\_\; a\_j=\; \backslash mathrm$ (where a is a row-vector). Each covariance term, $\backslash sigma\_$ can be expressed in terms of the correlation coefficient $\backslash rho\_\backslash ,$ by $\backslash sigma\_=\backslash rho\_\backslash sigma\_i\backslash sigma\_j\backslash ,$, so that an alternative expression for the variance of ''f'' is :$\backslash sigma^2\_f=\; \backslash sum\_i^n\; a\_i^2\backslash sigma^2\_i+\backslash sum\_i^n\; \backslash sum\_^n\; a\_i\; a\_j\backslash rho\_\; \backslash sigma\_i\backslash sigma\_j.$ In the case that the variables in ''x'' are uncorrelated this simplifies further to :$\backslash sigma^\_=\; \backslash sum\_i^n\; a\_^\backslash sigma^\_.$ In the simplest case of identical coefficients and variances, we find :$\backslash sigma\_=\; \backslash sqrt\; a\; \backslash sigma.$ 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「Propagation of uncertainty」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.