Multidimensional scaling: Difference between revisions

Revision as of 02:14, 30 July 2023

Revision as of 02:19, 30 July 2023

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===Non-metric multidimensional scaling (NMDS)===

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. For NMDS, it is unnecessary to

Let <math>d_{ij}</math> be the dissimilarity between points <math>i, j</math>. Let <math>\hat d_{ij} = \| x_i - x_j\|</math> be the Euclidean distance between embedded points <math>x_i, x_j</math>.

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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 MDSNMDS algorithm is a twofold optimization process. First theThe optimal monotonic transformation of the proximities has to be found. Secondly, theThe points of a configuration have to be optimally arranged, so that their distances match the scaled proximities as closely as possible. This is usually done iteratively:

NMDS needs to optimize two objectives simultaneously. This is usually done iteratively:

:# Initialize <math>x_i</math> randomly, e. g. by sampling from a normal distribution.

:# Do until a stopping criterion (for example, <math>S < \epsilon</math>)