Information Technology Reference
In-Depth Information
Figure 5.7
Euclidean embedded representation of the matrix
X
in (5.8).
a
k
th Cartesian axis may be regarded as the locus of the
pseudo-sample
µ
e
k
as
µ
varies.
It follows that if we can place
µ
e
k
in the MDS space then we have defined a
k
th 'axis'.
To emphasize that there is no guarantee that this 'axis' is linear, we shall refer to it as
a
trajectory
.All
p
trajectories must have a common point of concurrency at
µ
=
0. We
will denote this point of concurrency by O.
To embed a new point
x
∗
into
R
, the vector
d
n
+
1
={−
2
d
n
+
1,i
}
of
ddistances
between
the samples
x
i
and the new point
x
∗
must be calculated according to the chosen Euclidean
embeddable distance measure. Gower (1968) shows that for embedding this sample into
the
m
-dimensional Euclidean space
R
, a further (
m
1
1)th dimension is needed where
m
is generally defined as
n
−
1. The (nonzero) coordinates are given by
+
y
∗
=
(
y
:1
×
m
,
y
m
+
1
)
,
with
y
=
−
1
Y
d
n
+
1
−
n
D1
1
(5.9)
