Classification and Clustering Applications - An Invariant Approach to Statistical Analysis of Shapes - page 248

measure based on the arithmetic differences between the two
matrices, D A , is given by:

K - 1
å

K
å

( FM ij ( X ) - FM ij ( M )) 2

D A ( X , M ) =

i = 1 j = i + 1

This measure is simple to calculate and does not depend on any cen-
tering protocol. However, it does not take into account the covariance
between the landmarks.

3. Dissimilarity measure based on the Euclidean Distance

Matrix representation using relative differences: A dis-
similarity measure based on the relative differences between
matrices representing the mean form and an individual is
given by:

2

K - 1
å

K
å

æ

FM ij ( X )
FM ij ( M ) - 1

ö

ç ÷

D R ( X , M ) =

è ø

i = 1 j = i + 1

This measure is also simple to calculate but weighs the differences dif-
ferently; e.g., the smaller distances receive more weight in the
calculation. This may or may not be desirable depending upon the
research problem.

4. Dissimilarity measure based on logarithmic differences:

The following dissimilarity measure is based on the differ-
ences between the logarithms of the distances:

K 1 K

D ( X M , ) (log FM ( X ) log FM ( M ))

2

LR ij ij

i 1 ji 1

= +

5. Dissimilarity measure based on the Euclidean

Distance Matrix representation of relative differences
in growth pattern: We have found that there are situations
in which a growth pattern can be useful in distinguishing
between groups (Richtsmeier and Lele, 1993; Richtsmeier et
al. 1993a, 1993b). There are situations in which organisms
achieve similar adult forms through differing growth pat-
terns. In this case a dissimilarity measure based on the
growth matrices (and the differences between them) can be
used. Growth data are required for each individual in ques-
tion and for each of the defined groups. The dissimilarity