Information Technology Reference
In-Depth Information
Multiplying by
1
n
k
and dividing by
n
k
to obtain the coordinates of the centroid of the
k
th interpolated group gives its coordinates as
&
&
)
&
)
)
D
11
D
22
.
D
KK
D
1
k
D
2
k
D
Kk
(
−
(
+
−
2
(
+
−
1
+
.
1
2
=
−
1
Y
y
k
K
1
2
Now, by definition
hk
=−
(
D
hh
+
D
kk
−
2
D
hk
)
, so the above may be written as
&
&
)
)
&
&
)
)
1
2
D
kk
1
k
+
1
k
2
k
.
Kk
(
(
+
−
1
+
=
−
1
Y
(
(
+
−
1
+
1
2
k
+
2
D
kk
=
−
1
Y
y
k
.
Kk
+
K
K
1
2
D
kk
where the constant column is eliminated because of the centring
Y
1
=
0
.
The dimensions of the centroids
y
k
are
r
×
1, and if we combine all the
K
interpolated
centroids into
Y
K
we obtain
=
−
1
Y
,
−
11
K
Y
K
which may be written, again because of the centring, as
=
−
1
Y
I
I
11
K
11
K
Y
K
−
−
.
Finally,
I
−
I
−
11
K
11
K
=
YY
,
therefore we have
=
−
1
Y
(
YY
)
=
−
1
(
Y
Y
)
Y
=
Y
.
Y
K
So the interpolated centroids coincide with the group average positions given by
Y
.
5.8.2 A simple example of analysis of distance
Note that in this example, where we use Pythagorean distance, the AoD reduces to a
PCA of the group means. In this example, similar to our introductory AoD example, we
now have five groups and so no exact two-dimensional group mean representation. The
AoD methodology is actually redundant, although we can still use it as a motivation for
performing the permutation testing procedure discussed below.
