翻訳と辞書 Words near each other ・ Covenant Christian High School (Michigan) ・ Covenant Christian School ・ Covenant Christian School (Canberra) ・ Covaresa ・ Covariance ・ Covariance (disambiguation) ・ Covariance and contravariance ・ Covariance and contravariance (computer science) ・ Covariance and contravariance of vectors ・ Covariance and correlation ・ Covariance function ・ Covariance group ・ Covariance intersection ・ Covariance mapping ・ Covariance matrix ・ Covariance operator ・ Covariant classical field theory ・ Covariant derivative ・ Covariant formulation of classical electromagnetism ・ Covariant Hamiltonian field theory ・ Covariant return type ・ Covariant transformation ・ Covariate ・ Covariation model ・ Covario ・ Covarion ・ Covarrubias ・ Covarrubias (surname) ・ Covarrubias, Province of Burgos ・ Covas Dictionary Lists mini英和辞書 mini和英辞書 Webster 1913 Latin-English FOLDOC Wikipedia English ウィキペディア

翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Covariance operator ： ウィキペディア英語版 Covariance operator In probability theory, for a probability measure P on a Hilbert space ''H'' with inner product $\backslash langle\; \backslash cdot,\backslash cdot\backslash rangle$, the covariance of P is the bilinear form Cov: ''H'' × ''H'' → R given by :$\backslash mathrm(x,\; y)\; =\; \backslash int\_\; \backslash langle\; x,\; z\; \backslash rangle\; \backslash langle\; y,\; z\; \backslash rangle\; \backslash ,\; \backslash mathrm\; \backslash mathbf\; (z)$ for all ''x'' and ''y'' in ''H''. The covariance operator ''C'' is then defined by :$\backslash mathrm(x,\; y)\; =\; \backslash langle\; Cx,\; y\; \backslash rangle$ (from the Riesz representation theorem, such operator exists if Cov is bounded). Since Cov is symmetric in its arguments, the covariance operator is self-adjoint (the infinite-dimensional analogy of the transposition symmetry in the finite-dimensional case). When P is a centred Gaussian measure, ''C'' is also a nuclear operator. In particular, it is a compact operator of trace class, that is, it has finite trace. Even more generally, for a probability measure P on a Banach space ''B'', the covariance of P is the bilinear form on the algebraic dual ''B''#, defined by :$\backslash mathrm(x,\; y)\; =\; \backslash int\_\; \backslash langle\; x,\; z\; \backslash rangle\; \backslash langle\; y,\; z\; \backslash rangle\; \backslash ,\; \backslash mathrm\; \backslash mathbf\; (z)$ where $\backslash langle\; x,\; z\; \backslash rangle$ is now the value of the linear functional ''x'' on the element ''z''. Quite similarly, the covariance function of a function-valued random element (in special cases called random process or random field) ''z'' is :$\backslash mathrm(x,\; y)\; =\; \backslash int\; z(x)\; z(y)\; \backslash ,\; \backslash mathrm\; \backslash mathbf\; (z)\; =\; E(z(x)\; z(y))$ where ''z''(''x'') is now the value of the function ''z'' at the point ''x'', i.e., the value of the linear functional $u\; \backslash mapsto\; u(x)$ evaluated at ''z''. 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「Covariance operator」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.