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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 Central limit theorem for directional statistics ： ウィキペディア英語版 Central limit theorem for directional statistics In probability theory, the central limit theorem states conditions under which the average of a sufficiently large number of independent random variables, each with finite mean and variance, will be approximately normally distributed. Directional statistics is the subdiscipline of statistics that deals with directions (unit vectors in R''n''), axes (lines through the origin in R''n'') or rotations in R''n''. The means and variances of directional quantities are all finite, so that the central limit theorem may be applied to the particular case of directional statistics. This article will deal only with unit vectors in 2-dimensional space (R''2'') but the method described can be extended to the general case. == The central limit theorem == A sample of angles $\backslash theta\_i$ are measured, and since they are indefinite to within a factor of $2\backslash pi$, the complex definite quantity $z\_i=e^=\backslash cos(\backslash theta\_i)+i\backslash sin(\backslash theta\_i)$ is used as the random variate. The probability distribution from which the sample is drawn may be characterized by its moments, which may be expressed in Cartesian and polar form: :$m\_n=E(z^n)=\; C\_n\; +i\; S\_n\; =\; R\_n\; e^\backslash ,$ It follows that: :$C\_n=E(\backslash cos\; (n\backslash theta))\backslash ,$ :$S\_n=E(\backslash sin\; (n\backslash theta))\backslash ,$ :$R\_n=|E(z^n)|=\backslash sqrt\backslash ,$ :$\backslash theta\_n=\backslash arg(E(z^n))\backslash ,$ Sample moments for N trials are: :$\backslash overline=\backslash frac\backslash sum\_^N\; z\_i^n\; =\backslash overline\; +i\; \backslash overline\; =\; \backslash overline\; e^=\backslash frac\backslash sum\_^N\backslash cos(n\backslash theta\_i)$ :$\backslash overline=\backslash frac\backslash sum\_^N\backslash sin(n\backslash theta\_i)$ :$\backslash overline=\backslash frac\backslash sum\_^N\; |z\_i^n|$ :$\backslash overline=\backslash frac\backslash sum\_^N\; \backslash arg(z\_i^n)$ The vector (central limit theorem states that the joint probability distribution for $\backslash overline$ and $\backslash overline$ in the limit of a large number of samples is given by: :$()\; \backslash xrightarrow\; \backslash mathcal((),\backslash Sigma/N)$ where $\backslash mathcal()$ is the bivariate normal distribution and $\backslash Sigma$ is the covariance matrix for the circular distribution: :$$ \Sigma = \begin \sigma_ & \sigma_ \\ \sigma_ & \sigma_ \end \quad :$\backslash sigma\_=E(\backslash cos^2\backslash theta)-E(\backslash cos\backslash theta)^2\backslash ,$ :$\backslash sigma\_=\backslash sigma\_=E(\backslash cos\backslash theta\backslash sin\backslash theta)-E(\backslash cos\backslash theta)E(\backslash sin\backslash theta)\backslash ,$ :$\backslash sigma\_=E(\backslash sin^2\backslash theta)-E(\backslash sin\backslash theta)^2\backslash ,$ Note that the bivariate normal distribution is defined over the entire plane, while the mean is confined to be in the unit ball (on or inside the unit circle). This means that the integral of the limiting (bivariate normal) distribution over the unit ball will not be equal to unity, but rather approach unity as ''N'' approaches infinity. It is desired to state the limiting bivariate distribution in terms of the moments of the distribution. 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「Central limit theorem for directional statistics」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.