MDS-白红宇

MDS

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发布时间：2019-06-23

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MDS（multidimensional scaling）多维尺度分析:MDS利用的是成对样本间相似性，目的是利用这个信息去构建合适的低维空间，是的样本在此空间的距离和在高维空间中的样本间的相似性尽可能的保持一致。MDS试图找到一个子空间Rn，I个物体嵌入在这个子空间中，而彼此的相似度被尽可能的保留。如果这个子空间的维数N选择为2或者3，可以画出向量xj获得一个I个物体相似性的一个可视化的结果。

Details

The data to be analyzed is a collection of $I$ objects (colors, faces, stocks, . . .) on which a distance function is defined,

_i,j := distance between

i th and

j th objects.

These distances are the entries of the dissimilarity matrix

$\Delta := \begin{pmatrix} \delta_{1,1} & \delta_{1,2} & \cdots & \delta_{1,I} \\ \delta_{2,1} & \delta_{2,2} & \cdots & \delta_{2,I} \\ \vdots & \vdots & & \vdots \\ \delta_{I,1} & \delta_{I,2} & \cdots & \delta_{I,I} \end{pmatrix}.$

The goal of MDS is, given Δ, to find $I$ vectors $x_1,\ldots,x_I \in \mathbb{R}^N$ such that

$\|x_i - x_j\| \approx \delta_{i,j}$ for all

$i,j\in I$ ,

where $\|\cdot\|$ is a vector norm. In classical MDS, this norm is the Euclidean distance, but, in a broader sense, it may be a metric or arbitrary distance function.

In other words, MDS attempts to find an embedding from the $I$ objects into R^N such that distances are preserved. If the dimension N is chosen to be 2 or 3, we may plot the vectors x_i to obtain a visualization of the similarities between the $I$ objects. Note that the vectors x_i are not unique: With the Euclidean distance, they may be arbitrarily translated and rotated, since these transformations do not change the pairwise distances $\|x_i - x_j\|$ .

There are various approaches to determining the vectors x_i. Usually, MDS is formulated as an optimization problem, where $(x_1,\ldots,x_I)$ is found as a minimizer of some cost function, for example,

$\min_{x_1,\ldots,x_I} \sum_{i<j} ( \|x_i - x_j\| - \delta_{i,j} )^2. \,$

A solution may then be found by numerical optimization techniques. For some particularly chosen cost functions, minimizers can be stated analytically in terms of matrix eigendecompositions.

转载于:https://www.cnblogs.com/USTC-fuxm/archive/2012/08/29/2662101.html

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