An Asymmetric Criterion for Cluster Validation - Developing Concepts in Applied Intelligence - page 8

Cluster2 Cluster3 Cluster1 Cluster4

Cluster2 Cluster1 Cluster3 Cluster4

Fig. 5. Providing data for evaluating the APMM criterion. (a) Deleting all other clusters except
C 1 from P a . (b) deriving P b* , the corresponding samples of C 1 in P b .

Now, the entropy between remained clusters in two partitions P a and P b is
computed (see Fig. 6). On account of the other involved samples are eliminated, this
criterion is not symmetric.
All the previous works are based on the NMI definition as equation 1. Even for
evaluating the occurrence of a cluster in a partition, the problem is modified in some
way to become the comparing problem between two partitions and then the NMI
equation is used. In this paper, the problem is not changed according to definition of
NMI; instead, the NMI equation is modified so that the occurrence of a cluster in a
partition is computed. It is done by evaluating the entropy between the considered
cluster and other pseudo clusters in the corresponding partition. In this paper the
Alizadeh-Parvin-Moshki-Minaei criterion, APMM, is defined between a cluster C i
from P a and the partition P b* from P b , as below equation:

k

⎛ ⎞

n

b *

∑

⎜ ⎜ ⎟ ⎟

b
j

*

− 2 log n

a
i

n

⎝ ⎠

j = 1

a
i

b *

APMM ( C , P ) =

(3)

⎛

b
j

*

⎞

⎛

a
i

⎞

k

n

n

b *

∑

⎜ ⎜
⎝

⎟ ⎟
⎠

n

a
i

log

⎜ ⎜
⎝

⎟ ⎟
⎠

+ n

b
j

*

log

n n

j = 1