MDS
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 objects (colors, faces, stocks, . . .) on which a distance function is defined,
- δi,j := distance between i th and j th objects.
These distances are the entries of the dissimilarity matrix
The goal of MDS is, given Δ, to find vectors
such that
for all
,
where 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 objects into RN such that distances are preserved. If the dimension N is chosen to be 2 or 3, we may plot the vectors xi to obtain a visualization of the similarities between the
objects. Note that the vectors xi are not unique: With the Euclidean distance, they may be arbitrarily translated and rotated, since these transformations do not change the pairwise distances
.
There are various approaches to determining the vectors xi. Usually, MDS is formulated as an optimization problem, where is found as a minimizer of some cost function, for example,
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.
posted on 2012-08-29 15:02 fire_fuxm_USTC 阅读(307) 评论(0) 编辑 收藏 举报