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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 Stress majorization ： ウィキペディア英語版 Stress majorization Stress majorization is an optimization strategy used in multidimensional scaling (MDS) where, for a set of ''n'' ''m''-dimensional data items, a configuration ''X'' of ''n'' points in ''r(<\sigma(X). Usually ''r'' is 2 or 3, i.e. the ''(r'' x ''n)'' matrix ''X'' lists points in 2- or 3-dimensional Euclidean space so that the result may be visualised (i.e. an MDS plot). The function $\backslash sigma$ is a cost or loss function that measures the squared differences between ideal ($m$-dimensional) distances and actual distances in ''r''-dimensional space. It is defined as: : $\backslash sigma(X)=\backslash sum\_w\_(d\_(X)-\backslash delta\_)^2$ where $w\_\backslash ge\; 0$ is a weight for the measurement between a pair of points $(i,j)$, $d\_(X)$ is the euclidean distance between $i$ and $j$ and $\backslash delta\_$ is the ideal distance between the points (their separation) in the $m$-dimensional data space. Note that $w\_$ can be used to specify a degree of confidence in the similarity between points (e.g. 0 can be specified if there is no information for a particular pair). A configuration $X$ which minimizes $\backslash sigma(X)$ gives a plot in which points that are close together correspond to points that are also close together in the original $m$-dimensional data space. There are many ways that $\backslash sigma(X)$ could be minimized. For example, Kruskal〔.〕 recommended an iterative steepest descent approach. However, a significantly better (in terms of guarantees on, and rate of, convergence) method for minimizing stress was introduced by Jan de Leeuw.〔.〕 De Leeuw's ''iterative majorization'' method at each step minimizes a simple convex function which both bounds $\backslash sigma$ from above and touches the surface of $\backslash sigma$ at a point $Z$, called the ''supporting point''. In convex analysis such a function is called a ''majorizing'' function. This iterative majorization process is also referred to as the SMACOF algorithm ("Scaling by majorizing a convex function"). == The SMACOF algorithm == The stress function $\backslash sigma$ can be expanded as follows: : $$ \sigma(X)=\sum_w_(d_(X)-\delta_)^2 =\sum_w_\delta_^2 + \sum_w_d_^2(X)-2\sum_w_\delta_d_(X) Note that the first term is a constant $C$ and the second term is quadratic in X (i.e. for the Hessian matrix V the second term is equivalent to tr$X\text{'}VX$) and therefore relatively easily solved. The third term is bounded by: : $$ \sum_w_\delta_d_(X)=\,\operatorname\, X'B(X)X \ge \,\operatorname\, X'B(Z)Z where $B(Z)$ has: : $b\_=-\backslash frac\}$ for $d\_(Z)\backslash ne\; 0,\; i\; \backslash ne\; j$ and $b\_=0$ for $d\_(Z)=0,\; i\backslash ne\; j$ and $b\_=-\backslash sum\_^n\; b\_$. Proof of this inequality is by the Cauchy-Schwarz inequality, see Borg〔.〕 (pp. 152–153). Thus, we have a simple quadratic function $\backslash tau(X,Z)$ that majorizes stress: : $\backslash sigma(X)=C+\backslash ,\backslash operatorname\backslash ,\; X\text{'}VX\; -\; 2\; \backslash ,\backslash operatorname\backslash ,\; X\text{'}B(X)X$ : $\backslash le\; C+\backslash ,\backslash operatorname\backslash ,\; X\text{'}\; V\; X\; -\; 2\; \backslash ,\backslash operatorname\backslash ,\; X\text{'}B(Z)Z\; =\; \backslash tau(X,Z)$ The iterative minimization procedure is then: * at the kth step we set $Z\backslash leftarrow\; X^$ * $X^k\backslash leftarrow\; \backslash min\_X\; \backslash tau(X,Z)$ * stop if otherwise repeat. This algorithm has been shown to decrease stress monotonically (see de Leeuw〔). 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「Stress majorization」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.