Multidimensional scaling: Difference between revisions

Revision as of 16:55, 10 April 2022

Revision as of 16:57, 10 April 2022

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In contrast to metric MDS, non-metric MDS finds both a [[non-parametric]] [[monotonic]] relationship between the dissimilarities in the item-item matrix and the Euclidean distances between items, and the ___location of each item in the low-dimensional space. The relationship is typically found using [[isotonic regression]]: let <math display="inline">x</math> denote the vector of proximities, <math display="inline">f(x)</math> a monotonic transformation of <math display="inline">x</math>, and <math display="inline">d</math> the point distances; then coordinates have to be found, that minimize the so-called stress,

:<blockquote><math>\text{Stress}=\sqrt{\frac{\sum\bigl(f(x)-d\bigr)^2}{\sum d^2}}.</math></blockquote>

: A few variants of this cost function exist. MDS programs automatically minimize stress in order to obtain the MDS solution.

: The core of a non-metric MDS algorithm is a twofold optimization process. First the optimal monotonic transformation of the proximities has to be found. Secondly, the points of a configuration have to be optimally arranged, so that their distances match the scaled proximities as closely as possible. The basic steps in a non-metric MDS algorithm are:

:# Find a random configuration of points, e. g. by sampling from a normal distribution.

:# Calculate the distances d between the points.