Statistical Models for Landmark Coordinate Data - An Invariant Approach to Statistical Analysis of Shapes - page 106

Estimating the covariance matrix K *

Having obtained the estimator of the mean form matrix M described
above, it is fairly simple to obtain the estimator of the covariance K * .
Notice that Y ( LX )( LX ) T ~Wishart (( LM )( LM ) T , L K L T ) with D degrees of

freedom.
The mean of the non-central Wishart distribution is given by
(Arnold, 1981) E ( Y )

( LM )( LM ) T K L T ) . Hence, the moment estima-

D ( L

tor of K * K L T is given by:

L

where M is the estimator of the mean form as obtained at the end of
Step 4.
The estimator of K * obtained above, although square and symmet-
ric, is not guaranteed to be positive semi-definite. One can obtain a
positive semi-definite version using a procedure, sometimes known as
Principal Coordinate Analysis. Consider the spectral decomposition of
the matrix K * , namely, ˆ

ˆ

PDP T where matrix D is a diagonal
matrix with the diagonal elements corresponding to the eigenvalues of
K * . Replace the negative elements in D by zero and call this modified
matrix ~ D . Obtain a new matrix ˆ

K *

P ~ DP T . This matrix is guaranteed
to be square, symmetric, and positive semi-definite.
The estimator given above is slightly different than the one
described in Lele (1993). In that paper, instead of L

K *

K L T , an estimator

for H K H T was provided. In this monograph, we use the centering

matrix L , instead of H , to remain consistent with the rest of the chap-
ter. Replacing L in the above description by H retrieves the formulae
and description in Lele (1993).
In the above discussion, we assumed that D I . This imposes some

restrictions on the applicability of this model. We now consider a more
general situation where the covariance structure is given by K D .
For notational simplicity, let Y LX ( LX ) T . Let Y [ Y lm ] where T ( X )

1,2,…, K denoting the individual elements of the
matrix Y . From the previous section, we know that Y is distributed as
a linear combination of non-central Wishart random variables with
parameters given by (

l 1,2,…, K ; m

K L T , D D ) . Let

D L 1 1 , 2 2 ,…,

K * lm j ] . It follows from their definition that the

matrices j 's are symmetric and that

lm ] and j [ [

d l j = d l j

j

.

d mm