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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 Monte Carlo integration ： ウィキペディア英語版 Monte Carlo integration In mathematics, Monte Carlo integration is a technique for numerical integration using random numbers. It is a particular Monte Carlo method that numerically computes a definite integral. While other algorithms usually evaluate the integrand at a regular grid,〔Press et al, 2007, Chap. 4.〕 Monte Carlo randomly choose points at which the integrand is evaluated.〔Press et al, 2007, Chap. 7.〕 This method is particularly useful for higher-dimensional integrals.〔 There are different methods to perform a Monte Carlo integration, such as uniform sampling, stratified sampling, importance sampling, Sequential Monte Carlo (a.k.a. particle filter), and mean field particle methods. == Overview == In numerical integration, methods such as the Trapezoidal rule use a deterministic approach. Monte Carlo integration, on the other hand, employs a non-deterministic approach: each realization provides a different outcome. In Monte Carlo, the final outcome is an approximation of the correct value with respective error bars, and the correct value is within those error bars. The problem Monte Carlo integration addresses is the computation of a multidimensional definite integral :$I\; =\; \backslash int\_f(\backslash overline\}$ where Ω, a subset of R''m'', has volume :$V\; =\; \backslash int\_d\backslash overline\}\_1,\; \backslash cdots,\; \backslash overline\; \backslash sum\_^N\; f(\backslash overline\; Q\_N\; =\; I$. Given the estimation of ''I'' from ''QN'', the error bars of ''QN'' can be estimated by the sample variance using the unbiased estimate of the variance: :$\backslash mathrm(f)\backslash equiv\backslash sigma\_N^2\; =\; \backslash frac\; \backslash sum\_^N\; \backslash left\; (f(\backslash overline(Q\_N)\; =\; \backslash frac\; \backslash sum\_^N\; \backslash mathrm(f)\; =\; V^2\backslash frac\; =\; V^2\backslash frac$. As long as the sequence :$\backslash left\; \backslash$ is bounded, this variance decreases asymptotically to zero as 1/''N''. The estimation of the error of ''QN'' is thus :$\backslash delta\; Q\_N\backslash approx\backslash sqrt=V\backslash frac\{\backslash sqrt\{N\}\}$. This result does not depend on the number of dimensions of the integral, which is the promised advantage of Monte Carlo integration against most deterministic methods that depend exponentially on the dimension.〔Press et al, 2007〕 It is important to notice that, like in deterministic methods, the estimate of the error is not a strict error bound; random sampling may not uncover all the important features of the integrand that can result in an underestimate of the error. While the naive Monte Carlo works for simple examples, this is not the case in most problems. A large part of the Monte Carlo literature is dedicated in developing strategies to improve the error estimates. In particular, stratified sampling - dividing the region in sub-domains -, and importance sampling - sampling from non-uniform distributions - are two of such techniques. 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「Monte Carlo integration」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.