Package clusterCrit for R 7
The total number of pairs of distinct points in the data set is
N
T
=
N(N −1)
2
(27)
Since N =
P
K
k=1
n
k
, one can write :
N
T
=
N(N −1)
2
=
1
2
K
X
k=1
n
k
!
2
−
1
2
K
X
k=1
n
k
=
1
2
K
X
k=1
n
2
k
+ 2
X
k<k
0
n
k
n
k
0
!
−
1
2
K
X
k=1
n
k
= N
W
+
X
k<k
0
n
k
n
k
0
(28)
Let us denote by N
B
the number of pairs constituted of points which do not
belong to the same cluster, one has N
T
= N
W
+ N
B
and consequently:
N
B
=
X
k<k
0
n
k
n
k
0
. (29)
In the remainder, I
B
will denote the set of the N
B
pairs of between-cluster
indices and I
W
the set of the N
W
pairs of within-cluster indices.
1.2 Internal indices
The following sections provide the precise definitions of the various internal qual-
ity indices which have been proposed by various authors in order to determine
an optimal clustering. They are sorted in alphabetical order. These indices, also
called quality indices, are all denoted by the same letter C. Let us also denote
by d the distance function between two points (usually the ordinary euclidean
distance).
Table 1 summarizes the existing indices, their name in the package cluster-
Crit for R, the bibliographic reference and the date of the article where they
were originally defined.
1.2.1 The Ball-Hall index
The mean dispersion of a cluster is the mean of the squared distances of the
points of the cluster with respect to their barycenter. The Ball-Hall index is the
mean, through all the clusters, of their mean dispersion:
C =
1
K
K
X
k=1
1
n
k
X
i∈I
k
||M
{k}
i
− G
{k}
||
2
(30)
In the particular case where all the clusters have the same size N/K, this
sum reduces to
1
N
W GSS.
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