Probabilistic Distance Clustering: Algorithm and Applications - Clustering Challenges in Biological Network - page 33

and for K =3,

d 2 ( x ) d 3 ( x ) /q 2 q 3
d 1 ( x ) d 2 ( x ) /q 1 q 2 + d 1 ( x ) d 3 ( x ) /q 1 q 3 + d 2 ( x ) d 3 ( x ) /q 2 q 3 , (2.8)

p 1 ( x )=

etc.

2.2.2. The Joint Distance Function

We denote the constant in (2.5) by D ( x ), a function of x . Since the probabilities
p k ( x )= D ( x ) / d k ( x )
q k

,k =1 ,...,K,

j =1

K

d j ( x )
q j

addto1weget,

D ( x )= . (2.9)

i =1 j = i

K

d j ( x )
q j

D ( x ) is called the joint distance function (abbreviated JDF )of x ,andis,
up to a constant, the harmonic mean of the K weighted distances { d k ( x ) /q k } ,

see [1].
In particular, for K =2,

d 1 ( x ) d 2 ( x ) /q 1 q 2
d 1 ( x ) /q 1 + d 2 ( x ) /q 2 ,

D ( x )= (2.10)

and K =3,

d 1 ( x ) d 2 ( x ) d 3 ( x ) /q 1 q 2 q 3
d 1 ( x ) d 2 ( x ) /q 1 q 2 + d 1 ( x ) d 3 ( x ) /q 1 q 3 + d 2 ( x ) d 3 ( x ) /q 2 q 3 . (2.11)

D ( x )=

Example 2.2. Figure 2.2(a) shows level sets of the JDF (2.10) for the data of
Example 2.1.

2.2.3. An Extremal Principle

The principle (2.5) may be derived from an extremal principle. For notational
simplicity we consider here the case of 2 clusters, the results in the general case
are analogous.