Biology Reference
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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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