Multidimensional Scaling

Details

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

δ_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 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 x_i to obtain a visualization of the similarities between the objects. Note that the vectors x_i 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 x_i. 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.

Read more about this topic: Multidimensional Scaling

Famous quotes containing the word details:

“There was a time when the average reader read a novel simply for the moral he could get out of it, and however naïve that may have been, it was a good deal less naïve than some of the limited objectives he has now. Today novels are considered to be entirely concerned with the social or economic or psychological forces that they will by necessity exhibit, or with those details of daily life that are for the good novelist only means to some deeper end.”
—Flannery O’Connor (1925–1964)

“Working women today are trying to achieve in the work world what men have achieved all along—but men have always had the help of a woman at home who took care of all the other details of living! Today the working woman is also that woman at home, and without support services in the workplace and a respect for the work women do within and outside the home, the attempt to do both is taking its toll—on women, on men, and on our children.”
—Jeanne Elium (20th century)

“Then he told the news media
the strange details of his death
and they hammered him up in the marketplace
and sold him and sold him and sold him.
My death the same.”
—Anne Sexton (1928–1974)

Multidimensional Scaling - Details

Famous quotes containing the word details: