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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 Functional principal component analysis ： ウィキペディア英語版 Functional principal component analysis Functional principal component analysis (FPCA) is a statistical method for investigating the dominant modes of variation of functional data. Using this method, a random function is represented in the eigenbasis, which is an orthonormal basis of the Hilbert space ''L''2 that consists of the eigenfunctions of the autocovariance operator. FPCA represents functional data in the most parsimonious way, in the sense that when using a fixed number of basis functions, the eigenfunction basis explains more variation than any other basis expansion. FPCA can be applied for representing random functions, or functional regression and classification. ==Formulation== For a square-integrable stochastic process ''X''(''t''), ''t'' ∈ 𝒯, let :$\backslash mu(t)\; =\; \backslash text(X(t))$ and :$G(s,\; t)\; =\; \backslash text(X(s),\; X(t))\; =\; \backslash sum\_^\backslash infty\; \backslash lambda\_k\; \backslash varphi\_k(s)\; \backslash varphi\_k(t),$ where ''λ''1 ≥ ''λ''2 ≥ ··· ≥ 0 are the eigenvalues and ''φ''1, ''φ''2, ... are the orthonormal eigenfunctions of the linear Hilbert–Schmidt operator :$G:\; L^2(\backslash mathcal)\; \backslash rightarrow\; L^2(\backslash mathcal),\backslash ,\; G(f)\; =\; \backslash int\_\backslash mathcal\; G(s,\; t)\; f(s)\; ds.$ By the Karhunen–Loève theorem, one can express the centered process in the eigenbasis, :$X(t)\; -\; \backslash mu(t)\; =\; \backslash sum\_^\backslash infty\; \backslash xi\_k\; \backslash varphi\_k(t),$ where :$\backslash xi\_k\; =\; \backslash int\_\backslash mathcal\; (X(t)\; -\; \backslash mu(t))\; \backslash varphi\_k(t)\; dt$ is the principal component associated with the ''k''-th eigenfunction ''φ''''k'', with the properties :$\backslash text(\backslash xi\_k)\; =\; 0,\; \backslash text(\backslash xi\_k)\; =\; \backslash lambda\_k\; \backslash text\; \backslash text(\backslash xi\_k\; \backslash xi\_l)\; =\; 0\; \backslash text\; k\; \backslash ne\; l.$ The centered process is then equivalent to ''ξ''1, ''ξ''2, .... A common assumption is that ''X'' can be represented by only the first few eigenfunctions (after subtracting the mean function), i.e. :$X(t)\; \backslash approx\; X\_m(t)\; =\; \backslash mu(t)\; +\; \backslash sum\_^m\; \backslash xi\_k\; \backslash varphi\_k(t),$ where :$\backslash mathrm\backslash left(\backslash int\_\; \backslash lambda\_j\; \backslash rightarrow\; 0\; \backslash text\; m\; \backslash rightarrow\; \backslash infty\; .$ 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「Functional principal component analysis」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.