7.1.3 Fuzzy Analysis (Fanny)

Fuzzy clustering is a generalization of partitioning. In a partition, each object of the data set is assigned to one and only one cluster. Pam and Clara produce a hard clustering, because they make a clear-cut decision for each object. On the other hand, a fuzzy clustering method allows for some ambiguity in the data, which often occurs in practice.

In fuzzy clustering, each object is ‘spread over’ various clusters and the degree of belonging of an object to different clusters is quantified by means of membership coefficients, which range from 0 to 1, with the stipulation that the sum of their values is one. This is called a fuzzification of the cluster configuration. It has the advantage that it does not force every object into a specific cluster. It has the disadvantage that there is much more information to be interpreted.

Fanny aims at the minimization of the following objective function:

Objective function =

where d ( i , j ) represents the given distances (or dissimilarities) between objects i and j, whereas uiv is the unknown membership of object i to cluster n . The membership functions are subject to the constraints.

  1. uiv0 for all i =1,-.,n and all v =1,-,k.
  2. for all i=1,-.,n.

These constraints imply that membership cannot be negative and that each object has a certain total membership distributed over different clusters. By convention, this total membership is normalized to 1.

The objective function is minimized numerically by means of an iterative algorithm, taking into account the above constraints.

When each object has equal membership in all clusters, the clustering is entirely fuzzy. On the other hand, when each object has a membership of 1 in some cluster and zero membership in all other clusters; the clustering is entirely hard. To have an idea how hard or fuzzy the clustering is, Dunn’s partition coefficient is computed:

Fk =

which always lies in the range [,1].

Dunn’s coefficient attains its extreme values in the following situations.

  1. Entirely fuzzy clustering; all
  2. Entirely hard clustering; all or

The normalized version of the Dunn’s coefficient is given by

which always lies in the range [0, 1]