TESI DOCTORAL - La Salle

6.2. Adapting consensus functions to soft cluster ensembles
Recall that the contents of each clustering matrix Λi enclosed in the soft cluster ensemble
E results from the execution of a fuzzy clustering process, and that, depending on its nature,
the interpretation of the scalar values that ultimately make up E may differ largely. Thus,
for conducting a consensus process on the soft cluster ensemble E it is necessary that such
values hold the same type of proportionality with repect to the degree of association between
objects and clusters (i.e. they all are either directly or inversely proportional).
This prerequisite becomes more evident if an analogy between soft clustering and voting
procedures is established. Such analogy is inspired by the parallelism between supervised
classification processes and voting drawn in (van Erp, Vuurpijl, and Schomaker, 2002).
According to this analogy, the process of fuzzily clustering an object can be regarded as an
election, in which the clusterer (regarded as a voter) casts its preference for each one of the
clusters (or candidates). Put that way, it becomes quite obvious that, when the results of
several fuzzy clustering processes are gathered into a soft cluster ensemble with the purpose
of building a consolidated clustering solution upon it, they should be straightly comparable
—possibly, after applying some scale normalization.
Regardless of the characteristics and nature of soft cluster ensembles, it is interesting
to evaluate how classic consensus functions (i.e. those originally designed to combine crisp
partitions) can be applied on the fuzzy consensus clustering problem. The next section
deals with this very issue.
6.2 Adapting consensus functions to soft cluster ensembles
The consensus functions employed so far in the experimental sections of this work (i.e.
CSPA, EAC, HGPA, MCLA, ALSAD, KMSAD and SLSAD) are originally designed to
operate on hard cluster ensembles (see appendix A.5 for a description). Nevertheless, they
can be easily adapted for combining fuzzy partitions. The key point is that all these
consensus functions base their clustering combination processes on object co-association
matrices (i.e. matrices the contents of which estimate the degree of similarity between
objects upon the partitions contained in the cluster ensemble). Fortunately, this type of
co-association matrices can be derived not only from hard cluster ensembles, but also from
their soft counterparts, which makes these consensus functions easily applicable on the
fuzzy consensus clustering problem. In the following paragraphs, we elaborate on this issue
resorting again to the previous toy example, continuously drawing parallelisms between the
hard and soft clustering scenarios —for brevity, such comparison will only be referred to
fuzzy clustering processes that codify object to cluster associations in terms of membership
probabilities, although an equivalent study could also be formulated in the case these were
expressed by means of magnitudes inversely proportional to the strength of object to cluster
associations, such as object to cluster centroids distances.
Consider the hard clustering solution of equation (6.3) corresponding to our clustering
toy example, that is:
λ =[222111333]
Notice that an equivalent representation of this partition can be also given by a k × n
incidence matrix Iλ (called binary membership indicator matrix in (Strehl and Ghosh,
2002)), the (i,j)th entry of which is equal to 1 in the case the jth object is assigned to
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