Multidimensional Scaling - Details

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

\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 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, rotated, and reflected, 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.

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