Preface to First Edition - lib

304 MULTIDIMENSIONAL SCALINGwill be zero. So B can be written as B = V 1 Λ 1 V1 ⊤ , where V 1 contains thefirst k eigenvectors and Λ 1 the q non-zero eigenvalues. The required coordinatevalues are thus X = V 1 Λ 1/21 , where Λ 1/21 = diag( √ λ 1 , ..., √ λ k ).The best fitting k-dimensional representation is given by the k eigenvectorsof B corresponding to the k largest eigenvalues. The adequacy of thek-dimensional representation can be judged by the size of the criterionP k =k∑λ ii=1n−1.∑λ ii=1Values of P k of the order of 0.8 suggest a reasonable fit.When the observed dissimilarity matrix is not Euclidean, the matrix B is notpositive-definite. In such cases some of the eigenvalues of B will be negative;corresponding, some coordinate values will be complex numbers. If, however,B has only a small number of small negative eigenvalues, a useful representationof the proximity matrix may still be possible using the eigenvectorsassociated with the k largest positive eigenvalues.The adequacy of the resulting solution might be assessed using one of thefollowing two criteria suggested by Mardia et al. (1979); namelyk∑|λ i |i=1n∑|λ i |i=1ork∑λ 2 ii=1n∑λ 2 ii=1Alternatively, Sibson (1979) recommends the following:1. Trace criterion: Choose the number of coordinates so that the sum of theirpositive eigenvalues is approximately equal to the sum of all the eigenvalues.2. Magnitude criterion: Accept as genuinely positive only those eigenvalueswhose magnitude substantially exceeds that of the largest negative eigenvalue..17.2.2 Non-metric Multidimensional ScalingIn classical scaling the goodness-of-fit measure is based on a direct numericalcomparison of observed proximities and fitted distances. In many situationshowever, it might be believed that the observed proximities contain little reliableinformation beyond that implied by their rank order. In psychologicalexperiments, for example, proximity matrices frequently arise from asking subjectsto make judgements about the similarity or dissimilarity of the stimuliof interest; in many such experiments the investigator may feel that, realistically,subjects can give only ‘ordinal’ judgements. For example, in comparinga range of colours they might be able to specify that one was say ‘brighter’than another without being able to attach any realistic value to the extent© 2010 by Taylor and Francis Group, LLC