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instance, and that no particular rule should be discarded universally.
6.3 Dissimilarity measures for landmark coordinate data
In the following paragraphs, we present a few dissimilarity measures
that are intuitive, simple to calculate, and useful for classification
based on landmark coordinate data. An example of their use in classi-
fication is illustrated in the following section.
Let
X
denote the landmark coordinate matrix for the new observa-
tion. In this description, let
M
denote the true mean form for one of the
classes and
K
denote the true variance. Recall however from the dis-
cussion in
Chapter 3
, that these values are non-estimable. Only the
centered and rotated version of the mean form,
M
C
, the form matrix,
FM
(
M
), and the singular version of
K
are estimable. Next we provide
various dissimilarity measures for determining the classification of
X
into one of several known classes. Notice that in all the measures
described below only these estimable parameters are used. These dis-
similarity measures can therefore be computed in practice.
1.
Unweighted Procrustes dissimilarity measure:
This is a
Procrustes distance between the centered landmark coordi-
nate matrix,
X
C
= HX
, and the centered mean form matrix,
M
C
= H M
, where
H
is the centering matrix defined in
Chapter 3
.
The distance is given by:
D
P
(X
C
,M
C
) = tr (X
C
T
X
C
)
tr(M
C
T
M
C
) - 2tr(X
C
M
C
T
M
C
X
C
T
)
1/ 2
.
The last component of this expression corresponds to the
square root of a matrix. This dissimilarity measure is simple
to calculate, however the value of the measure depends on the
specific centering adopted during analysis and does not take
correlations between landmarks into consideration.
2.
Dissimilarity measure based on the Euclidean Distance
Matrix representation using arithmetic differences:
Suppose we calculate the form matrices corresponding to the
observation
X
and the mean form matrix
M
. A dissimilarity
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