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===Classical multidimensional scaling===
It is also known as '''Principal Coordinates Analysis''' (PCoA), Torgerson Scaling or Torgerson–Gower scaling. It takes an input matrix giving dissimilarities between pairs of items and outputs a coordinate matrix whose configuration minimizes a [[loss function]] called ''strain.''<ref name="borg"/> For example, given the Euclidean aerial distances <math display="inline">d_{ij}</math> between various cities indexed by ''i'' and ''j'', you want to find the coordinates <math display="inline">(x_i, y_i)</math> of the cities such that <math display="inline">d_{ij}=\sqrt{(x_i-x_j)^2+(y_i-y_j)^2}</math>. In this example, an exact solution is possible (assuming the Euclidean distances are exact). In practice, this is usually not the case, and MDS therefore seeks to approximate the lower-dimensional representation by minimising a loss function. General forms of loss functions are called ''stress'' in distance MDS and ''strain'' in classical MDS. The strain is given by:
<math>\text{Strain}_D(x_1,x_2,...,x_N)=\Biggl(\frac{ \sum_{i,j} \bigl( b_{ij} - x_i^T x_j \bigr)^2}{\sum_{i,j}b_{ij}^2} \Biggr)^{1/2}</math>,
where <math>x_{i}</math> now denote vectors in ''N''-dimensional space, <math>x_i^T x_j </math> denotes the scalar product between <math>x_{i}</math> and <math>x_{j}</math>, and <math>b_{ij}</math> are the elements of the matrix <math>B</math> defined on step 2 of the following algorithm, which are computed from the distances.
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