Statistical Models for Landmark Coordinate Data - An Invariant Approach to Statistical Analysis of Shapes - page 101

Consider a spectral decomposition of the matrix

D -1 ( LM ) T , that is,

where d j 's are the scaled eigenvectors of ( LM )

d j j ,

D -1 ( LM ) T . Let d j d j T

the j -th eigenvalue of ( LM )

j and note that j is of

rank one.
Theorem 1: The maximal invariant T ( X )

LXX T L T is distributed as a
linear combination of non-central Wishart matrices. More precisely

j , K * ). a non-central Wishart

stated, where Z j

d

W K (1,

matrix of dimension K

K with non-centrality parameter j and scale

parameter K * .
Proof: Notice that LX~N ( LM , L

D ) . The above result follows by
taking A to be an identity matrix in Theorem 2 presented by deGunst
(1987, 248-249).
The partition induced on the parameter space by T ( X ) is thus given
by the inverse image of (

K L T ,

K L T ,

D L 1 1 , 2 2 ,…, D D ) . The equiva-

lency sets defined by v ( - ) are such that / , o / are equivalent if and only

and ~

~
K = R D R T

M

1 t T

L K L T . Notice that similar to the
bivariate case discussed earlier, K D itself is not identifiable but
only D

if

MR D

K L T is identifiable. It will be shown in the next section,
that the 's along with D can be used to construct the mean form M
up to rotation, reflection, and translation. This will be used mainly for
representing the mean form graphically. All the inferences would and
should be based only on the identifiable parameters. That is, any infer-
ence should be the same regardless of which member of the partition
induced by T ( X ) is utilized.

L

Independence of the partition of the choice of c and L

K L T ) .

a) Choice of c : Let v ( M ,

K , D ) ( 1 1 , 2 2 ,…, D D , D L

Let ~ j and L ~ D and ~ K . Then ~ j , ~ c -1 c -1 c -1 L K L T

D c K j c j K L T

v ( M , ~ K , ~ L , ~ ( ~ 1 ~ 1 , ~ 2 ~ 2 ,…, ~ D ~ D , ~ Then K L T ) D ) .

D ) D v ( M , K ,

The partitions therfore are the same for any c> 0 .
b) Choice of the centering matrix L : Let L 1 and L 2 be two different
centering matrices. We can write L 2 L 2 L 1 T ( L 1 L 1 T ) -1 .

To show the invariance of the partition to the choice of the center-
ing matrix, we need to show that for ( M ,

AL 1 with A

D ) and ( M , ~ K , ~ D ) in the

K ,