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from level h - 1toh the clusters C
h
1
u
;
C
h
1
v
are merged in cluster C
w
. Then, the
density for the merged cluster at level h is:
p
h
1
x
=
C
h
1
u
þ
p
h
1
C
h
1
v
p
h
1
x
=
C
h
1
v
¼
p
h
1
C
h
1
p
h
x
=
C
w
u
þ
C
h
1
v
ð
4
:
17
Þ
p
h
1
C
h
1
u
v
are the priors or proportions of the clusters u, v at
level h - 1. The rest of the terms are the same in the mixture model at level h as at
level h - 1. The only difference from one level to the next one in the hierarchy is
that there is one cluster less and the prior for the new cluster is the sum of the
priors of its components and the density the weighted average of the densities that
are merged to form it. Therefore, the estimation of the distance at level h can be
done easily starting from the distances at level h - 1 and so on until level h =1.
Consequently, we can calculate the distances at level h from a cluster C
z
to a
merged cluster C
w
that was obtained by the agglomeration of clusters C
h
u
;
C
v
at
level h - 1 as the distance to its components weighted by the mixing proportions:
;
p
h
1
C
h
1
where p
h
1
C
h
1
u
==
p
h
x/C
z
¼
D
h
p
h
x/C
w
D
h
1
p
h
1
x/C
h
1
u
==
p
h
1
x/C
h
1
z
þ
p
h
1
C
h
1
v
D
h
1
p
h
1
x/C
h
1
v
==
p
h
1
x/C
h
1
z
p
h
1
C
h
1
u
þ
p
h
1
C
h
1
v
p
h
1
C
h
1
u
ð
4
:
18
Þ
As at level 1, we can obtain the decision rule to assign a new data to a cluster in the
hierarchy at level h by applying Bayes' theorem:
Þ¼
p
h
x
=
C
k
ð
Þ
p
h
C
ðÞ
arg max
C
k
p
h
C
k
=
x
ð
ð
4
:
19
Þ
K
h
þ
1
P
p
h
x
=
C
i
ð
Þ
p
h
C
ðÞ
i
¼
1
4.3.1 ICAMM-Based Hierarchical Clustering Algorithm
The summary of the ICAMM-based hierarchical clustering algorithm is presented
in Table
4.1
.
4.4 Simulations
The algorithm above was tested with several simulated data of ICA mixtures with
the following source distributions: Laplacian, uniform, K-type, and Rayleigh. The
mixtures were generated for two independent variables and bias terms for a
number of 200 observation vectors. Figure
4.1
shows some of the source distri-
butions used in the ICA mixtures.
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