Biology Reference

In-Depth Information

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

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