Preface to First Edition - lib

MULTIDIMENSIONAL SCALING 303from X; multidimensional scaling is essentially concerned with the reverseproblem, given the distances how do we find X?An n × n inner products matrix B is first calculated as B = XX ⊤ , theelements of B are given byq∑b ij = x ik x jk . (17.1)k=1It is easy to see that the squared Euclidean distances between the rows of Xcan be written in terms of the elements of B asd 2 ij = b ii + b jj − 2b ij . (17.2)If the bs could be found in terms of the ds as in the equation above, then therequired coordinate value could be derived by factoring B = XX ⊤ .No unique solution exists unless a location constraint is introduced; usuallythe centre of the points ¯x is set at the origin, so that ∑ ni=1 x ik = 0 for all k.These constraints and the relationship given in (17.1) imply that the sumof the terms in any row of B must be zero.Consequently, summing the relationship given in (17.2) over i, over j andfinally over both i and j, leads to the following series of equations:n∑d 2 ij = trace(B) + nb jjn∑i=1n∑d 2 ij = trace(B) + nb iij=1i=1 j=1n∑d 2 ij = 2n × trace(B)where trace(B) is the trace of the matrix B. The elements of B can now befound in terms of squared Euclidean distances as⎛⎞b ij = − 1 ∑n ∑n ∑n n∑⎝d 2 ij − n −1 d 2 ij − n −1 d 2 ij + n −2 d 2 ⎠ij .2j=1i=1i=1 j=1Having now derived the elements of B in terms of Euclidean distances, itremains to factor it to give the coordinate values. In terms of its singular valuedecomposition B can be written asB = VΛV ⊤where Λ = diag(λ 1 , ...,λ n ) is the diagonal matrix of eigenvalues of B andV the corresponding matrix of eigenvectors, normalised so that the sum ofsquares of their elements is unity, that is, V ⊤ V = I n . The eigenvalues areassumed labeled such that λ 1 ≥ λ 2 ≥ · · · ≥ λ n .When the matrix of Euclidian distances D arises from an n×k matrix of fullcolumn rank, then the rank of B is k, so that the last n − k of its eigenvalues© 2010 by Taylor and Francis Group, LLC