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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>
The factor of <math>\sum d^2</math> in the denominator is necessary to prevent a "collapse". Suppose we define instead <math>\text{Stress}=\sqrt{\sum\bigl(f(x)-d\bigr)^2}</math>, then it can be trivially minimized by setting <math>f = 0</math>, then collapse every point to the same point.
A few variants of this cost function exist. MDS programs automatically minimize stress in order to obtain the MDS solution.
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