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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 Łukaszyk–Karmowski metric ： ウィキペディア英語版 Łukaszyk–Karmowski metric In mathematics, the Łukaszyk–Karmowski metric is a function defining a distance between two random variables or two random vectors.〔(Metryka Pomiarowa, przykłady zastosowań aproksymacyjnych w mechanice doświadczalnej (Measurement metric, examples of approximation applications in experimental mechanics) ), (PhD thesis ), Szymon Łukaszyk (author), Wojciech Karmowski (supervisor), Tadeusz Kościuszko Cracow University of Technology, submitted December 31, 2001, completed March 31, 2004〕〔(A new concept of probability metric and its applications in approximation of scattered data sets ), Łukaszyk Szymon, Computational Mechanics Volume 33, Number 4, 299–304, Springer-Verlag 2003 〕 This function is not a metric as it does not satisfy the identity of indiscernibles condition of the metric, that is for two identical arguments its value is greater than zero. The concept is named after Szymon Łukaszyk and Wojciech Karmowski. ==Continuous random variables== The Łukaszyk–Karmowski metric ''D'' between two continuous independent random variables ''X'' and ''Y'' is defined as: :$D(X,\; Y)\; =\; \backslash int\_^\backslash infty\; \backslash int\_^\backslash infty\; |x-y|f(x)g(y)\; \backslash ,\; dx\backslash ,\; dy$ where ''f''(''x'') and ''g''(''y'') are the probability density functions of ''X'' and ''Y'' respectively. One may easily show that such ''metrics'' above do not satisfy the identity of indiscernibles condition required to be satisfied by the metric of the metric space. In fact they satisfy this condition if and only if both arguments ''X'', ''Y'' are certain events described by Dirac delta density probability distribution functions. In such a case: :$D\_(X,\; Y)\; =\; \backslash int\_^\backslash infty\; \backslash int\_^\backslash infty\; |x-y|\backslash delta(x-\backslash mu\_x)\backslash delta(y-\backslash mu\_y)\; \backslash ,\; dx\backslash ,\; dy\; =\; |\backslash mu\_x-\backslash mu\_y|$ the Łukaszyk–Karmowski metric simply transforms into the metric between expected values $\backslash mu\_x$, $\backslash mu\_y$ of the variables ''X'' and ''Y'' and obviously: :$D\_(X,\; X)\; =\; |\backslash mu\_x-\backslash mu\_x|\; =\; 0.$ For all the other cases however: :$D\backslash left(X,\; X\backslash right)\; >\; 0.\; \backslash ,$ The Łukaszyk–Karmowski metric satisfies the remaining non-negativity and symmetry conditions of metric directly from its definition (symmetry of modulus), as well as subadditivity/triangle inequality condition: :$\backslash begin$ &^\infty |x-z|f(x)h(z) \, dx\, dz\ = \int_^\infty \int_^\infty |x-z|f(x)h(z) \, dx\, dz \int_^\infty g(y) dy\ \\ &^\infty \int_^\infty |(x-y)+(y-z)|f(x)g(y)h(z) \, dx\, dy\, dz\ \\ &^\infty \int_^\infty (|x-y|+|y-z|)f(x)g(y)h(z) \, dx\, dy\, dz\ \\ &^\infty \int_^\infty |x-y|f(x)g(y)h(z) \, dx\, dy\, dz\ + \int_^\infty \int_^\infty \int_^\infty |y-z|f(x)g(y)h(z) \, dx\, dy\, dz\ \\ &^\infty |x-y|f(x)g(y) \, dx\, dy\ + \int_^\infty \int_^\infty |y-z|g(y)h(z) \, dy\, dz\ \\ &^\infty \int_^\infty |x-y| f(x, y) \, dx\, dy. 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「Łukaszyk–Karmowski metric」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.