Probabilistic Distance Clustering: Algorithm and Applications - Clustering Challenges in Biological Network - page 30

where · is a norm. In particular, the Mahalanobis distance with the norm,

u , Σ − 1

1 / 2 ,

u = u (2.2)

where Σ is the covariance matrix of the data involved.

2 with N = 1100 data points is shown in Fig. 2.1.
The data on the left was simulated from a normal distribution N (

Example 2.1. A data set in R

µ , Σ), with:

µ 1 =(2 , 0) , Σ 1 = 0 . 0005 0

, (100 points) ,

0 . 05

and the data on the right consist of 1000 points, simulated in a circle of diameter
1 centered at µ 2 =(3 , 0) according to a radially symmetric distribution with

1
2 .
This data will serve as illustration in Examples 2.2-2.3 below.

Prob { x − µ 2 ≤ r } =2 r , 0 ≤ r ≤

0.8

0.6

0.4

0.2

0

−0.2

−0.4

−0.6

−0.8

1.8 2 2.2 2.4 2.6 2.8 3 3.2 3.4 3.6 3.8

R 2 Fig. 2.1. A data set in

In d-clustering each point is typically assigned to the cluster with the near-
est center. After each assignment, the cluster centers may change, requiring a
re-classification of the data points. A d-clustering algorithm will therefore it-
erate between centers and re-assignments. The best known such method is the
k -means clustering algorithm , see Hartigan [5].