TESI DOCTORAL - La Salle

4.1. Description of the consensus self-refining procedure
computational efficiency criteria solely. While put forward in a hard clustering scenario, this
proposal could be exported to a fuzzy context by introducing several minor modifications.
This chapter is organized as follows: section 4.1 describes the proposed self-refining consensus
procedure. Next, several experiments regarding the application of the self-refining
process on the consensus clustering solutions output by the three types of consensus architectures
described in the previous chapter are presented in section 4.2. An alternative
procedure based on cluster ensemble component selection for obtaining refined consensus
clustering solutions upon a given cluster ensemble is described in section 4.3, and finally,
the discussion and conclusions presented in section 4.4 put the end to this chapter.
4.1 Description of the consensus self-refining procedure
The proposed approach for refining the quality of the consensus clustering solution λc is
pretty straightforward, and it is based on the notion of average normalized mutual information
φ (ANMI) (Strehl and Ghosh, 2002) between a cluster ensemble E and a consensus
clustering solution λc built upon it, as defined by equation (4.1).
φ (ANMI) (E, λc) = 1
l
l
φ (NMI) (λi, λc) (4.1)
where l represents the number of partitions (or components) contained in the cluster ensemble
E and λi is the ith of these components.
The higher φ (ANMI) (E, λc), the more information the consensus clustering solution λc
shares with all the clusterings in E, thus it can be considered to capture the information
contained in the ensemble to a larger extent. In fact, the computation of the φ (ANMI)
between a given cluster ensemble E and a set of consensus clustering solutions obtained
by means of different consensus functions is proposed in (Strehl and Ghosh, 2002) as a
means for choosing among them in a unsupervised fashion, giving rise to what is called a
supraconsensus function.
It is important to notice that each term of the summation in equation (4.1) measures
the resemblance between each cluster ensemble component and λc. As a consequence, those
cluster ensemble components more similar to the consensus clustering solution contribute
in a greater proportion to the sum in equation (4.1).
Assuming that the consensus function F applied for obtaining the consensus clustering
solution λc delivers a moderately good performance –in the sense that the quality of λc will
be reasonably higher than the one of the poorest components of the cluster ensemble E–,
then the normalized mutual information (φ (NMI) ) between λc and each cluster ensemble
component λi (∀i ∈ [1,...,l]), gives an approximate measure of the quality of the latter
(Fern and Lin, 2008).
Allowing for this fact, the proposed consensus self-refining procedure is based on ranking
the l components of the cluster ensemble E accordingtotheirφ (NMI) with respect the
consensus clustering solution λc. The results of this sorting process is represented by means
of an ordered list Oφ (NMI) = {λ
φ (NMI) 1, λ φ (NMI)2,...λφ (NMI)l}, the subindices of which refer
to the aforementioned φ (NMI) based ranking, i.e. λ
φ (NMI) 1 denotes the cluster ensemble
component that attains the highest normalized mutual information with respect to λc,
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i=1