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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 Wrapped distribution ： ウィキペディア英語版 Wrapped distribution In probability theory and directional statistics, a wrapped probability distribution is a continuous probability distribution that describes data points that lie on a unit ''n''-sphere. In one dimension, a wrapped distribution will consist of points on the unit circle. Any probability density function (pdf) $p(\backslash phi)$ on the line can be "wrapped" around the circumference of a circle of unit radius. That is, the pdf of the wrapped variable :$\backslash theta=\backslash phi\; \backslash mod\; 2\backslash pi$ in some interval of length $2\backslash pi$ is : $$ p_w(\theta)=\sum_^\infty . which is a periodic sum of period $2\backslash pi$. The preferred interval is generally for which $\backslash ln(e^)=\backslash arg(e^)=\backslash theta$ ==Theory== In most situations, a process involving circular statistics produces angles ($\backslash phi$) which lie in the interval from negative infinity to positive infinity, and are described by an "unwrapped" probability density function $p(\backslash phi)$. However, a measurement will yield a "measured" angle $\backslash theta$ which lies in some interval of length $2\backslash pi$ (for example $[0,2\backslash pi)$). In other words, a measurement cannot tell if the "true" angle $\backslash phi$ has been measured or whether a "wrapped" angle $\backslash phi+2\backslash pi\; a$ has been measured where ''a'' is some unknown integer. That is: :$\backslash theta=\backslash phi+2\backslash pi\; a.$ If we wish to calculate the expected value of some function of the measured angle it will be: :$\backslash langle\; f(\backslash theta)\backslash rangle=\backslash int\_^\backslash infty\; p(\backslash phi)f(\backslash phi+2\backslash pi\; a)d\backslash phi.$ We can express the integral as a sum of integrals over periods of $2\backslash pi$ (e.g. 0 to $2\backslash pi$): :$\backslash langle\; f(\backslash theta)\backslash rangle=\backslash sum\_^\backslash infty\; \backslash int\_^\; p(\backslash phi)f(\backslash phi+2\backslash pi\; a)d\backslash phi.$ Changing the variable of integration to $\backslash theta\text{'}=\backslash phi-2\backslash pi\; k$ and exchanging the order of integration and summation, we have :$\backslash langle\; f(\backslash theta)\backslash rangle=\; \backslash int\_0^\; p\_w(\backslash theta\text{'})f(\backslash theta\text{'}+2\backslash pi\; a\text{'})d\backslash theta\text{'}$ where $p\_w(\backslash theta\text{'})$ is the pdf of the "wrapped" distribution and ''a' '' is another unknown integer (a'=a+k). It can be seen that the unknown integer ''a' '' introduces an ambiguity into the expectation value of $f(\backslash theta)$. A particular instance of this problem is encountered when attempting to take the mean of a set of measured angles. If, instead of the measured angles, we introduce the parameter $z=e^$ it is seen that ''z'' has an unambiguous relationship to the "true" angle $\backslash phi$ since: :$z=e^=e^.$ Calculating the expectation value of a function of ''z'' will yield unambiguous answers: :$\backslash langle\; f(z)\backslash rangle=\; \backslash int\_0^\; p\_w(\backslash theta\text{'})f(e^)d\backslash theta\text{'}$ and it is for this reason that the ''z'' parameter is the preferred statistical variable to use in circular statistical analysis rather than the measured angles $\backslash theta$. This suggests, and it is shown below, that the wrapped distribution function may itself be expressed as a function of ''z'' so that: :$\backslash langle\; f(z)\backslash rangle=\; \backslash oint\; p\_w(z)f(z)\backslash ,dz$ where $p\_w(z)$ is defined such that $p\_w(\backslash theta)\backslash ,d\backslash theta=p\_w(z)\backslash ,dz$. This concept can be extended to the multivariate context by an extension of the simple sum to a number of $F$ sums that cover all dimensions in the feature space: : $$ p_w(\vec\theta)=\sum_^_F)} where $\backslash mathbf\_k=(0,\backslash dots,0,1,0,\backslash dots,0)^\{\backslash mathsf\{T\}\}$ is the $k$th Euclidean basis vector. 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「Wrapped distribution」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.