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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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