Biology Reference
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unknowable. If the path for each transparency were known, the
inverse of these paths could be used to reorient them to their original
positions. The mean form,
M
, is lost as well. This mimics the situation
encountered by a biologist while collecting data from a sample of bio-
logical organisms. The mean form that represents the population
average is unknown. Likewise, the random perturbation among speci-
mens,
E
i
, is unknown and the rotation and translation associated with
each form is unknown and unknowable.
Suppose the next day we return to our office and decide to create a
few more objects on additional transparencies. We realize that the
mean form template (the red transparency) is lost, and we did not
record the exact perturbation structure (the covariance matrix of
E
)
that was used when we created the original black triangles. Because
our transparencies are circular and of different diameters, there is no
particular edge or other outside frame of reference that can be used to
put the black transparencies together in a way that may bring us to
the original arrangement. We have an informed idea of what the mean
form might look like, but no information other than the black trans-
parencies that detail its configuration. This is exactly the situation a
scientist faces when examining a set of observations. The scientist has
observations but no fundamental frame of reference within which to
work. In addition, the mean form and the perturbation structures are
unknown.
Questions relating to samples or populations cannot be answered
without tools for summarizing the data. The mean form is one sum-
mary parameter and the perturbation pattern is yet another. The
biological and statistical relevance of the mean form is apparent, while
knowledge of the perturbation pattern is the keystone for studying
variability. Our primary questions include whether or not we can esti-
mate the mean form from the available observations and whether or
not we can estimate the perturbation pattern used to generate the
observations.
Estimation of the mean form and the covariance structure in the
situation described above is complicated. The complexity comes from
the arbitrary and
unknown
parameters of translation and rotation. We
have only the displaced black pen transparencies and no knowledge of
the original frame of reference (coordinate system) in which they were
created. It is up to the investigator to decide if these parameters are
critical or even of scientific interest for a specific problem. When study-
ing biological organisms, important biological considerations may
enter into this decision. If knowledge of rotation and translation is not
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