Biology Reference
In-Depth Information
a muskrat to a dolphin (which is not surprising, given the extraordinary reorganization of
the cetacean head). This result suggests that most biologists are unlikely to encounter any
cases in which the differences among specimens are large enough to worry about the ade-
quacy of the linear approximations. It is unlikely that distances in the tangent space will
poorly approximate distances in shape space. Even so, using the average shape of all spe-
cimens in the data minimizes the risk that such a problem will occur. The use of any other
reference carries the responsibility to ensure that Euclidean distances in the tangent space
are accurate approximations of the distances in shape space.
Dimensions and Degrees of Freedom
The issue of degrees of freedom (or the number of independent measurements in a sys-
tem) is important for statistical analyses, but it can be confusing, especially when talking
about shape. To clarify it, we can consider a simple example. Suppose we wish to describe
the location of a notebook in a room. We could give its location in terms of three distances
from a reference point (such as the corner of the door of the room), and this is equivalent
to defining its position by three Cartesian coordinates relative to that reference point. In
this example, there are three degrees of freedom for the location of the notebook because
three variables are required to describe it. Knowing those variables and the reference suf-
fices to find the notebook. However, if the notebook is on a chair, and all chairs are known
to be the same height, specifying the height conveys no more information than saying that
the notebook is on a chair. Knowing what we do about the chairs, we only need two addi-
tional pieces of information, the
X
- and
Y
-coordinates, to specify the location of the note-
book in the room. Thus, by specifying the constraint that the notebook is on a chair of
fixed height, we have removed one of the three degrees of freedom.
We can take this example a step further by specifying that all the chairs are located
along walls of the room, with every chair touching the wall. Now, the
X
- and
Y
-coordi-
nates can be replaced by the distance (
L
) around the perimeter of the room from the door
to the notebook, and the direction of the measurement (clockwise or counter-clockwise). If
we agree that distances around a perimeter are always measured in the same direction,
then the value of
L
is sufficient to describe the location of the notebook. The additional
constraints (chairs against the wall, perimeter measured in clockwise direction) have
reduced the degrees of freedom from two (
X
and
Y
) to one (
L
). We have not actually elimi-
nated either
X
or
Y
; rather, we have merely replaced that pair by
L
. Nor have we lost any
information; given
L
, and the direction in which
L
is measured, as well as the height of the
chairs, we can reconstruct the original three Cartesian coordinates (
X
,
Y
, and
Z
) of the
notebook.
In the case of two-dimensional shapes, we start out with
K
landmarks in two dimen-
sions, so we have
2K
coordinates, which constitute
2K
independent measurements
(because each coordinate is independent of the others, in principle). In the course of super-
imposing the shapes on the reference form, we perform three operations: (1) we center the
matrix on the centroid, thereby losing two degrees of freedom; (2) we set centroid size to
one, thereby losing another; and (3) we compute the angle through which to rotate the
specimen, thereby losing one more. By the end, we have lost four degrees of freedom as a
