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STEP 2:Calculate whereD is
the dimension of the observations.
The estimator obtained above, although square and symmetric, is
not guaranteed to be positive semi-definite. One can obtain a positive
semi-definite version using the following steps.
STEP 3: Calculate the spectral decomposition of the matrix
K * . That is, find its eigenvectors and eigenvalues and write it
as ˆ
PDP T where matrix D is a diagonal matrix with the
diagonal elements corresponding to the eigenvalues of K * .If
there are any negative elements in D , replace them by zero
and call this modified matrix ~ D . Obtain a new matrix K *
K *
P ~ DP T . This matrix is guaranteed to be square, symmetric, and
positive semi-definite.
Algorithm 2
In this section, we provide an algorithm for the estimation of the mean
form and the covariance structure when we expect that the perturba-
tions along the x, y, and z axes are correlated with each other. We
provide the algorithm for the two-dimensional three landmarks case in
detail. More general situations can be developed using the description
in Part 2 . The reader should, however, be aware that to fit more com-
plex models, larger sample sizes are needed to obtain reasonable
estimates. Thus, although more realistic, in practice, this general
model might be more difficult to fit.
Define a matrix
Let X c
LX denote the centered matrices. This centering is slightly
different than the one used in the previous algorithm.
Let K
K L T and D denote a diagonal matrix consisting of the
eigenvalues of D .
It was proven in Part 2 that these are the only estimable quantities
related to covariance.
Define a new matrix
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