Graphics Reference
In-Depth Information
A measure of the discrepancy between the two configurations is given by the sum of
squared distances, R
, between corresponding points in the two spaces, i.e.
n
r
=
R
T
=
(
y
r
−
x
r
)
(
y
r
−
x
r
)
(
.
)
T
,
T
and
x
r
and
y
r
are the coordinate vectors
where
X=[
x
,...,
x
n
]
Y=[
y
,...,
y
n
]
of the rth point in the two spaces.
he points in the X space are dilated, translated, rotated and reflected to new co-
ordinates,
x
′
,where
′
r
T
(
x
x
=
ρ
A
)
+
b
,
(
.
)
r
ρ is a dilation,
is an orthogonal matrix giving a rotation and possibly a reflection
and
b
isa translation. heoptimal values of these that minimizes R
are summarized
in the following procedure:
. (Optimum translation) Place the centroids of the two configurations at the ori-
gin.
. (Optimum rotation) Find
A
T
T
T
−
and rotate
to
.
A=(X
YY
X)
(Y
X)
X
XA
. (Optimum scaling) Scale the
configuration bymultiplying each coordinate by
X
T
T
T
.
. Calculate the Procrustes statistic
ρ
=
tr
(X
YY
X)
tr
(X
X)
R
T
T
T
T
=
−
tr
(X
YY
X)
tr
(X
X)
tr
(Y
Y)
(
.
)
he value of R
can be between
and
, where
implies a perfect matching of the
configurations. he larger the value of R
, the worse the match.
Procrustes analysis was used on the cost of rail travel data. Figure
.
shows the
non-metric scaling result (Fig.
.
) matched to the metric (Fig.
.
). In this case the
Procrustesstatisticis
.
,showingthatthepointconfigurationsareremarkablysim-
ilar.
Extensions to basic Procrustes analysis of matching one configuration to another
includeweighting ofpoints andaxes, the allowance of oblique axes and thematching
of more than two configurations; see Cox and Cox (
)or Gower and Dijksterhuis
(
) for a detailed account of the area.
Unidimensional Scaling
3.6
When the space in which the points representing objects or individuals has only one
dimension, the scaling technique becomes that of unidimensional scaling. he loss
function to be minimized is
.
S
=
r
<
s
(
δ
rs
−
x
r
−
x
s
)
(
.
)
