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variables but has only one degree of freedom). The simplest approach to this problem is to
use PCA as a dimensionality reduction tool, and carry out the CVA on some subset of the
PC scores, rather than on the actual data. The issue that then arises is how to determine
the number of PC axes to use. If all PC axes with non-zero eigenvalues are included, then
there is no loss of information, but most software has some level of rounding error pres-
ent, which can make it difficult to determine if small eigenvalues are zero or not. Some
workers will simply use a set of PC axes comprising 95 or 99% of the variance. One study
showed decreased overfitting of the CVA when fewer PC axes were used ( Sheets et al.
2006 ). The Between Groups PCA ( Mitteroecker and Bookstein, 2011 ) may be another viable
approach to this issue.
A radically different approach to working with rank-deficient data is to use a
machine learning approach to specimen classification such as the Weka system ( Witten
and Frank, 2005 ). These approaches attempt to build computer-based classification rule
systems, with reference to parametric statistical models. The performance of these meth-
ods is typically assessed using cross-validation methods, and these machine learning
methods appear to do as well as CVA, at least with some data sets ( Van Bocxlaer and
Schultheiß, 2010 ).
BETW EEN GROUPS PRINCIPAL COMPONENTS ANA LYSIS
To avoid the problems engendered by the rescaling step in CVA, Mitteroecker and
Bookstein (2011) suggest Between Groups Principal Components Analysis (BGPCA). This
method analyzes differences between means without regard to the magnitude or pattern
of within group variation. It is, simply, a PCA of the means. Absence of the rescaling step
eliminates distortion of the distances between the means. It also obviates concerns about
the artificiality of discriminators based on trivial distinctions, although this may not be
advantageous in all applications. Restricting the analysis to the subspace defined by the
means reduces the dimensionality of the space from the number of variables (2k
4, for a
set of two-dimensional landmarks) to a maximum of 1 less than the number of groups.
BGPCA finds the principal components of this subspace.
Figure 6.19A shows the mean shapes of the jaw in the three squirrel samples analyzed
previously. Again, the position of landmark 13 differs greatly between the mean shapes;
differences in other regions are also apparent. The first PC of this subspace is approxi-
mately the axis between the two most different means
those of the western Michigan
and southern samples ( Figure 6.19B ). The second axis, as in other PCAs, is constrained to
be orthogonal to the first. In this example, it describes the divergence of the eastern
Michigan sample from the axis connecting the other two means. Were the differences
between means less neatly balanced, or if there were more groups, the axes might not be
so conveniently aligned with the means. As might be predicted from the shape compari-
son in Figure 6.19A , the deformation illustrating the shape differences associated with PC1
scores shows that the mean shapes differ primarily in the relative position of landmark 13,
but also in shape of the angular process and space between the other posterior processes
and the molars ( Figure 6.19C ). In this particular data set, the mean shapes happen to have
differed primarily in the direction of highest within-sample variation. This might not have
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