翻訳と辞書 Words near each other ・ ITerating ・ Iteration ・ Iteration (disambiguation) ・ Iteration mark ・ Iterations of I ・ Iterative and incremental development ・ Iterative aspect ・ Iterative closest point ・ Iterative compression ・ Iterative deepening A* ・ Iterative deepening depth-first search ・ Iterative design ・ Iterative impedance ・ Iterative learning control ・ Iterative method ・ Iterative proportional fitting ・ Iterative Receiver Design ・ Iterative reconstruction ・ Iterative refinement ・ Iterative Template Library ・ Iterative Viterbi decoding ・ Iteratively reweighted least squares ・ Iterator ・ Iterator pattern ・ Iteri language ・ Iteris ・ ITerm2 ・ Itero de la Vega ・ Itero del Castillo ・ Iteron 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 Iterative proportional fitting ： ウィキペディア英語版 Iterative proportional fitting The iterative proportional fitting procedure (IPFP, also known as biproportional fitting in statistics, RAS algorithm in economics and matrix raking or matrix scaling in computer science) is an iterative algorithm for estimating cell values of a contingency table such that the marginal totals remain fixed and the estimated table decomposes into an outer product. First introduced by Deming and Stephan in 1940 (they proposed IPFP as an algorithm leading to a minimizer of the Pearson X-squared statistic, which it ''does not'', and even failed to prove convergence), it has seen various extensions and related research. A rigorous proof of convergence by means of differential geometry is due to Fienberg (1970). He interpreted the family of contingency tables of constant crossproduct ratios as a particular (''IJ'' − 1)-dimensional manifold of constant interaction and showed that the IPFP is a fixed-point iteration on that manifold. Nevertheless, he assumed strictly positive observations. Generalization to tables with zero entries is still considered a hard and only partly solved problem. An exhaustive treatment of the algorithm and its mathematical foundations can be found in the book of Bishop et al. (1975). The first general proof of convergence, built on non-trivial measure theoretic theorems and entropy minimization, is due to Csiszár (1975). Relatively new results on convergence and error behavior have been published by Pukelsheim and Simeone (2009) . They proved simple necessary and sufficient conditions for the convergence of the IPFP for arbitrary two-way tables (i.e. tables with zero entries) by analysing an $L\_1$-error function. Other general algorithms can be modified to yield the same limit as the IPFP, for instance the Newton–Raphson method and the EM algorithm. In most cases, IPFP is preferred due to its computational speed, numerical stability and algebraic simplicity. == Algorithm 1 (classical IPFP) == Given a two-way (''I'' × ''J'')-table of counts $(x\_)$, where the cell values are assumed to be Poisson or multinomially distributed, we wish to estimate a decomposition $\backslash hat\_\; =\; a\_i\; b\_j$ for all ''i'' and ''j'' such that $(\backslash hat\_)$ is the maximum likelihood estimate (MLE) of the expected values $(m\_)$ leaving the marginals $\backslash textstyle\; x\_\; =\; \backslash sum\_j\; x\_\backslash ,$ and $\backslash textstyle\; x\_\; =\; \backslash sum\_i\; x\_\backslash ,$ fixed. The assumption that the table factorizes in such a manner is known as the ''model of independence'' (I-model). Written in terms of a log-linear model, we can write this assumption as $\backslash log\backslash \; m\_\; =\; u\; +\; v\_i\; +\; w\_j\; +\; z\_$, where $m\_\; :=\; \backslash mathbb(x\_)$, $\backslash sum\_i\; v\_i\; =\; \backslash sum\_j\; w\_j\; =\; 0$ and the interaction term vanishes, that is $z\_\; =\; 0$ for all ''i'' and ''j''. Choose initial values $\backslash hat\_^\; :=\; 1$ (different choices of initial values may lead to changes in convergence behavior), and for $\backslash eta\; \backslash geq\; 1$ set : $\backslash hat\_^\; =\; \backslash frac^x\_\}\_^\}$ : $\backslash hat\_^\; =\; \backslash frac^x\_\}\_^\}.$ Notes: * Convergence does not depend on the actual distribution. Distributional assumptions are necessary for inferring that the limit $(\backslash hat\_)\; :=\; \backslash lim\_\; (\backslash hat^\_)$ is an MLE indeed. * IPFP can be manipulated to generate any positive marginals be replacing $x\_$ by the desired row marginal $u\_i$ (analogously for the column marginals). * IPFP can be extended to fit the ''model of quasi-independence'' (Q-model), where $m\_\; =\; 0$ is known a priori for $(i,j)\backslash in\; S$. Only the initial values have to be changed: Set $\backslash hat\_^\; =\; 0$ if $(i,j)\backslash in\; S$ and 1 otherwise. 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「Iterative proportional fitting」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.