Editing Multidimensional scaling (section)

===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"/> which is given by
<math display=block>\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> 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.

: '''Steps of a Classical MDS algorithm:''' 
: Classical MDS uses the fact that the coordinate matrix <math>X</math> can be derived by [[Eigendecomposition of a matrix|eigenvalue decomposition]] from <math display="inline">B=XX'</math>. And the matrix <math display="inline">B</math> can be computed from proximity matrix <math display="inline">D</math> by using double centering.<ref>Wickelmaier, Florian. "An introduction to MDS." ''Sound Quality Research Unit, Aalborg University, Denmark'' (2003): 46</ref>
:# Set up the squared proximity matrix <math display="inline">D^{(2)}=[d_{ij}^2]</math>
:# Apply double centering: <math display="inline">B=-\frac{1}{2}CD^{(2)}C</math> using the [[centering matrix]] <math display="inline">C=I-\frac{1}{n}J_n</math>, where <math display="inline">n</math> is the number of objects, <math display="inline">I</math> is the <math display="inline">n \times n</math> identity matrix, and <math display="inline">J_{n}</math> is an <math display="inline">n\times n</math> matrix of all ones.
:# Determine the <math display="inline">m</math> largest [[Eigenvalues and eigenvectors|eigenvalues]] <math display="inline">\lambda_1,\lambda_2,...,\lambda_m</math> and corresponding [[Eigenvalues and eigenvectors|eigenvectors]] <math display="inline">e_1,e_2,...,e_m</math> of <math display="inline">B</math> (where <math display="inline">m</math> is the number of dimensions desired for the output).
:# Now, <math display="inline">X=E_m\Lambda_m^{1/2}</math> , where <math display="inline">E_m</math> is the matrix of <math display="inline">m</math> eigenvectors and <math display="inline">\Lambda_m</math> is the [[diagonal matrix]] of <math display="inline">m</math> eigenvalues of <math display="inline">B</math>.
:Classical MDS assumes metric distances. So this is not applicable for direct dissimilarity ratings.