Probabilistic Distance Clustering: Algorithm and Applications - Clustering Challenges in Biological Network - page 36

2.2.5. Centers

Dealing first with the case of 2 clusters, we rewrite (2.15) as a function of the
cluster centers,

d 1 ( x i , c 1 ) p 1 ( x i ) 2
q 1

N

+ d 2 ( x i , c 2 ) p 2 ( x i ) 2
q 2

f ( c 1 , c 2 )= (2.21)

i =1

and look for centers c 1 , c 2 minimizing f .

Theorem 2.2. Let the distance functions d 1 ,d 2 in (2.21) be elliptic,
d ( x , c k )= ( x − c k ) ,Q k ( x − c k ) 1 / 2 ,k =1 , 2 , (2.22)

where Q 1 ,Q 2 are positive definite, so that

( x i − c 1 ) ,Q 1 ( x i − c 1 ) p 1 ( x i ) 2
q 1
+ ( x i − c 2 ) ,Q 2 ( x i − c 2 ) p 2 ( x i ) 2
q 2

N

f ( c 1 , c 2 )=

i =1

,

(2.23)

and let the probabilities p k ( x i ) and cluster sizes q k be given. If the minimizers
c 1 , c 2 of (2.23) do not coincide with any of the data points x i , they are given by

N N

x i , x i ,

u 1 ( x i ) u 2 ( x i )

N

c 1 = c 2 =

(2.24)

t =1 u 1 ( x t ) t =1 u 2 ( x t )

N

i =1 i =1

where

d 2 ( x i , c 2 )
q 2

2

1
d 1 ( x i , c 1 )

u 1 ( x i )=

2 , d 1 ( x i , c 1 )

q 1

+ d 2 ( x i , c 2 )
q 2

(2.25)

d 1 ( x i , c 1 )
q 1

2

1
d 2 ( x i , c 2 )

u 2 ( x i )=

2 , d 1 ( x i , c 1 )

q 1

+ d 2 ( x i , c 2 )
q 2

or equivalently, in terms of the probabilities (2.7),

u 1 ( x i )= p 1 ( x i ) 2

d 1 ( x i , c 1 ) , u 2 ( x i )= p 2 ( x i ) 2 d 2 ( x i , c 2 ) .

(2.26)