Statistical Models for Landmark Coordinate Data - An Invariant Approach to Statistical Analysis of Shapes - page 117

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
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.

L

Define a new matrix