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5 Multidimensional scaling
and nonlinear biplots
5.1 Introduction
So far we have been concerned with linear biplots and the use of Euclidean distances.
With PCA the distance is Pythagorean (Chapter 3) and with CVA (Chapter 4) it is Maha-
lanobis distance, which is Euclidean embeddable. Later (Chapter 7) we shall encounter
chi-squared distance, which again is Euclidean embeddable. The analyses accompanying
these distances are all concerned with finding sets of points in a few dimensions whose
inter-sample distances generate approximations to the data, generally using orthogonal
projection in some form. There is a vast literature on a different approach, termed multidi-
mensional scaling (MDS), which attempts to find points in a low-dimensional Euclidean
space that generate distances that approximate distances given in an ( n × n ) symmetric
matrix D . Approximation may be measured in several ways, of which the two most
important are:
n
i < j ( d ij δ ij )
2 ,
stress =
(5.1)
n
i < j ( d ij δ
2
2
ij
sstress =
)
,
(5.2)
where d ij are the distances given in D and δ ij are the fitted distances. Here we use
the notation D to distinguish this matrix form the matrix D containing ddistances used
elsewhere in the text.
Minimizing these criteria is known as metric MDS and needs specialized algorithms
and associated software. In nonmetric MDS, more sophisticated versions of these cri-
teria exist in which the
δ ij are fitted to monotonic transformations of the observed d ij
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