Biology Reference
In-Depth Information
P
i
5
1
y
ti
k
a
y
5
(14.7)
which is simply the requirement that the target specimen also has a centroid at the origin.
The rotation angle becomes:
!
arctangent
P
i
5
1
ð
y
ri
ð
x
ti
2
a
x
Þ
2
x
ri
ð
y
ti
2
a
y
ÞÞ
(14.8)
θ5
P
i
5
1
ð
x
ri
ð
x
ti
a
x
Þ
1
y
ri
ð
y
ti
2
a
y
ÞÞ
2
which is identical to the expression for the angle of rotation found for a Procrustes super-
imposition using translation, rotation and scaling.
When one specimen is superimposed on a reference using this Procrustes-SP superim-
position, the minimized squared distance between the target and reference is the squared
Procrustes-SP distance between them. The translated landmark coordinates produced by
this method may then be analyzed using multivariate statistical methods; bearing in mind
the issues posed by the degrees of freedom and the lack of an underlying statistical model
of the distribution of this type of variable. These limitations on this superimposition
method and the resulting coordinates and distance measure mean that statistical studies
that use the data obtained by Procrustes-SP methods should use tests based on permuta-
tion or other re-sampling methods, not analytic statistical models.
WHAT DOES IT MEAN FOR SHAPES TO “MATCH”?
One issue that arises in the analysis of pattern evidence is to determine whether one
item of such evidence (e.g., a bitemark, a tool-mark or footwear pattern) “matches”
another, so that the two patterns could have been produced by a common source (
Bunch
et al., 2009
). Careful thought is needed to decide what it means for two measurements to
match, whether those be traditional morphometric measurements, outlines, or landmarks
superimposed by Procrustes or Procrustes-SP methods. One approach is to start from the
premise that:
If the observed difference between two measured specimens is no larger than the difference observed
in repeated measurements of a single specimen, then there is no evidence that the two measured speci-
mens are different, and thus they may be said to match.
In all continuous measurements made of physical objects, there is always some level of
error or uncertainty, no matter how carefully it is measured. If we ask two people their
heights, and both answer honestly that they are 5 foot 10 inches (178 cm) tall, then we
would reasonably state that their heights are equal. However neither individual is likely to
be exactly 5 foot 10 inches. They are probably within a span of 1 or
2
inch (2.54 or
1.27 cm) of 5 foot 10 inches at any given time. The claim that their heights are equal is due
to our inability to measure a difference between them, given our measurement instru-
ments, not to there being no difference at all in their heights. The issue of individuation in
forensic science is surprisingly contentious (
Saks and Koehler, 2008; Bunch et al., 2009;
1
