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d
(d ii , d ii )
d ii
(d ii , d ii )
d
d ii
d ii
Figure 10.2 The vertical axis gives observed distances d ii and the horizontal axis
distances δ ii fitted in r dimensions. These determine the points ( δ ii , d ii ) indicated by
red filled circles. The rectangular boxes enclose sets of points that are monotonically
related; that is, all the points in any given box have δ ii either greater or lesser than any
value of the δ ii in the remaining boxes. A monotonic regression function is shown as
a thick vertical line within boxes and an inclined dotted line between successive boxes.
The projection (indicated by the horizontal green arrow) of ( δ ii , d ii ) onto the monotonic
regression function determines the point ( δ ii , ˆ
δ ii ) indicated in green, with the associated
ˆ
residual length of ( δ ii
δ ii ) . The monotonic stress criterion is the sum of squares of
these residuals. Notice also that all the red points in any given block of points project to
the same value of ˆ
δ ii .
in their metric or nonmetric forms, requires iterative methods. The crucial step in the
iteration is how to update a current setting of Z to give a convergent process. Quite
sophisticated algorithms have been developed that allow different types of monotonic
regression, different treatment of ties and efficient calculation. These are available in, for
example, ALSCAL, KYST and SMACOF (see Cox and Cox, 2001; Borg and Groenen, 2005).
Whatever metric or nonmetric multidimensional scaling method is used, the end
result is a map of n points, with coordinates given as the rows of Z , the distances between
which approximate observed or derived distances d ii . These are manifestly monoplots.
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