Statistical Models for Landmark Coordinate Data - An Invariant Approach to Statistical Analysis of Shapes - page 100

transformations involved in the analysis of landmark coordinates con-
sists of translation, rotation and possibly reflection. Thus gX

XR 1 t .It

K , R T

is easy to see that if X d

N ( M , K , D ) , then gX d N ( MR 1 t , D R ) ,

where d means equal in distribution.

Let us now consider the parameter space used by the Matrix
Normal distribution and the partition of this parameter space induced
by the group - G .
Let

D matrix, K is a K x K real, positive
definite, symmetric matrix and D is a D x D real, positive definite,
symmetric matrix.}.
The matrix Normal distribution defines equivalency sets on so that
o / , ~ o /

{( M , K , D ): M K

are equivalent if and only if
Notice that under this model, the parameter combinations ( M ,

D )
and for any c >0 are equivalent. The equivalency sets
defined under the group → G

K ,

( M , c S K , c S D )

are such that o / , ~ o /

are equivalent if
and only if Notice again that
here again the parameter combinations ( M ,

K , D ) and

(, M KD

c

, )

for any c>0 are equivalent.
We now consider one particular maximal invariant under this
group and derive the parameters that are obtained under it, T (·) and
v ( 0 - ) in the notation used before. We then study the partition induced by
v ( 0 - ) of the parameter space and discuss the identifiability of various
parameters. Since all maximal invariants are equivalent, the identifi-
ability issues will remain the same for all of them.
Let L be a ( K

K matrix whose first column consists of -1's and
the rest of the matrix is an identity matrix of dimension ( K

1)

1) ( K 1) .

LXX T L T . Since L 1

Now define T ( X )

0 and because R is an orthogonal

matrix i.e., RR T

1 t ) and therefore is
invariant. To show that it is a maximal invariant, we need to show
that, given T ( X ), one can map it back to a unique orbit in the original
sample space. This can be proved by using the fact that T ( X ) is a cen-
tered inner product matrix and that there exists a unique mapping (up
to rotation, translation and reflection) from the centered inner product
matrix to a coordinate matrix, see Lele (1991, 1993). This is also a stan-
dard result in the multidimensional scaling literature (Young and
Householder, 1938). For an elementary discussion of multidimensional
scaling analysis, see Mardia et al., (1979, Chapter 14). Next, we derive
the distribution of this maximal invariant in order to determine which
parameters are identifiable.
We introduce some additional notation. Let D =diag{ 1 , 2 ,..., D }
denote the diagonal matrix of the eigenvalues of D and let K *

I , it is easy to see that T ( X ) T ( XR

K L T .

L