Editing Multidimensional scaling (section)

=== Metric multidimensional scaling (mMDS) ===

It is a superset of classical MDS that generalizes the optimization procedure to a variety of loss functions and input matrices of known distances with weights and so on.  A useful loss function in this context is called ''stress'', which is often minimized using a procedure called [[stress majorization]]. Metric MDS minimizes the cost function called “stress” which is a residual sum of squares:<blockquote><math>\text{Stress}_D(x_1,x_2,...,x_n)=\sqrt{\sum_{i\ne j=1,...,n}\bigl(d_{ij}-\|x_i-x_j\|\bigr)^2}.</math></blockquote>
Metric scaling uses a power transformation with a user-controlled exponent <math display="inline">p</math>: <math display="inline">d_{ij}^p</math> and <math display="inline">-d_{ij}^{2p}</math> for distance. In classical scaling <math display="inline">p=1.</math> Non-metric scaling is defined by the use of isotonic regression to nonparametrically estimate a transformation of the dissimilarities.