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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

ˆ

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

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.

L

Define a new matrix

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