Classification and Clustering Applications - An Invariant Approach to Statistical Analysis of Shapes - page 249

measured based on differences between growth patterns is
defined as:

2

K - 1
å

K
å

æ

GM ij ( X 1 , X 2 )
GM ij ( M 1 , M 2 ) - 1

ö

ç ÷

D G ( X , M ) =

è ø

i = 1 j = i + 1

The dissimilarity measures provided above are only a sample of possi-
ble dissimilarity measures that may be used in classification.

Description of the classification rule

All of the dissimilarity measures given above can be used in devising
a classification rule. The steps of the procedure are the same regard-
less of which measure is used. Suppose, for example, we choose the
Procrustes distance to use as the dissimilarity measure. Then, the
steps are as follows:

STEP 1: Collect samples from each of the classes that are of
interest. Estimate the mean form matrix and the variance-
covariance matrix for each sample using these data. Let M 1
and denote the estimated mean form and the variance
covariance matrix for the first sample. Do this for the samples
representing each class.

( ˆ S K )

STEP 2: Given a new individual represented by a landmark
coordinate matrix, calculate the Procrustes distance between
the individual and each of the classes.

STEP 3: Classify the new individual into that class that it
resembles the most; i.e., the smallest Procrustes distance.

In the next section, we compare the performance of the dissimilar-
ity measures described earlier using data from individuals diagnosed
with differing craniofacial malformations.

6.4 A classification example

As discussed in Chapter 1 , the developing neurocranium is made up of
a number of roughly shell-shaped bony plates that align with one
another at joints or articulations called sutures. The typical shape of