Biology Reference
In-Depth Information
Algorithm 1
This algorithm describes the estimation procedure when the covari-
ance structure is
K
I
D
, similar to the covariance structure depicted
in
Figure 3.1d
.
The estimation procedure is particularly simple in this
situation. We suggest that the reader first try to program this algo-
rithm and then try the more general algorithm (Algorithm 2) that is
applicable when the covariance structure is of the form
K
D
. All the
data analyses carried out in this topic and various papers using
Euclidean Distance Matrix Analysis (EDMA) use Algorithm 1 as pre-
sented by Lele (1993). The parameters estimated are the form matrix,
FM
(
M
)
, corresponding to the mean form
M
and the covariance matrix
K
*
, the singular version of
K
.
Let
X
1
,
X
2
,…,
X
n
denote the landmark coordinate matrices of n indi-
viduals in the sample.
a) Algorithm for estimation of mean form:
STEP 1: Calculate the form matrix corresponding to each of
the landmark coordinate matrices
X
1
,
X
2
,…,
X
n
. For simplicity
of notation let us denote the form matrices by
F
i
instead of the
clumsier (albeit clearer) notation
FM
i
or
FM
(
X
i
)
. Thus
F
i
is a
matrix of Euclidean distances between all pairs of landmarks
for the
i
-th individual.
STEP 2: Calculate the squared Euclidean distance matrices
E
1
,
E
2
,…,
E
n
corresponding to each individual in the sample,
where
E
i
is obtained by squaring each element in the matrix
F
i
. Let
e
lm,i
denote the squared Euclidean distance between
landmarks
l
and
m
in the individual “
i
”.
STEP 3: Calculate the average of the squared
Euclidean distance between landmarks
l
and
m
in n individuals.
STEP 4: Calculate the variance of
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