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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 Elastic map ： ウィキペディア英語版 Elastic map Elastic maps provide a tool for nonlinear dimensionality reduction. By their construction, they are a system of elastic springs embedded in the data space.〔 This system approximates a low-dimensional manifold. The elastic coefficients of this system allow the switch from completely unstructured k-means clustering (zero elasticity) to the estimators located closely to linear PCA manifolds (for high bending and low stretching modules). With some intermediate values of the elasticity coefficients, this system effectively approximates non-linear principal manifolds. This approach is based on a mechanical analogy between principal manifolds, that are passing through "the middle" of the data distribution, and elastic membranes and plates. The method was developed by A.N. Gorban, (A.Y. Zinovyev ) and A.A. Pitenko in 1996–1998. == Energy of elastic map == Let data set be a set of vectors $S$ in a finite-dimensional Euclidean space. The ''elastic map'' is represented by a set of nodes $W\_j$ in the same space. Each datapoint $s\; \backslash in\; S$ has a ''host node'', namely the closest node $W\_j$ (if there are several closest nodes then one takes the node with the smallest number). The data set $S$ is divided on classes $K\_j=\backslash$. The ''approximation energy'' D is the distortion : $D=\backslash frac\backslash sum\_^k\; \backslash sum\_\backslash |s-W\_j\backslash |^2$, this is the energy of the springs with unit elasticity which connect each data point with its host node. It is possible to apply weighting factors to the terms of this sum, for example to reflect the standard deviation of the probability density function of any subset of data points $\backslash$. On the set of nodes an additional structure is defined. Some pairs of nodes, $(W\_i,W\_j)$, are connected by ''elastic edges''. Call this set of pairs $E$. Some triplets of nodes, $(W\_i,W\_j,W\_k)$, form ''bending ribs''. Call this set of triplets $G$. : The stretching energy is $U\_=\backslash frac\backslash lambda\; \backslash sum\_\; \backslash |W\_i\; -W\_j\backslash |^2$, : The bending energy is $U\_G=\backslash frac\backslash mu\; \backslash sum\_\; \backslash |W\_i\; -2W\_j+W\_l\backslash |^2$, where $\backslash lambda$ and $\backslash mu$ are the stretching and bending moduli respectively. The stretching energy is sometimes referred to as the "membrane" term, while the bending energy is referred to as the "thin plate" term.〔Michael Kass, Andrew Witkin, Demetri Terzopoulos, Snakes: Active contour models, Int.J. Computer Vision, 1988 vol 1-4 pp.321-331〕 For example, on the 2D rectangular grid the elastic edges are just vertical and horizontal edges (pairs of closest vertices) and the bending ribs are the vertical or horizontal triplets of consecutive (closest) vertices. : The total energy of the elastic map is thus $U=D+U\_E+U\_G.$ The position of the nodes $\backslash$ is determined by the mechanical equilibrium of the elastic map, i.e. its location is such that it minimizes the total energy $U$. 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「Elastic map」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.