TESI DOCTORAL - La Salle

Chapter 4. Self-refining consensus architectures
λ φ (NMI) 2 is the component with the second highest φ(NMI) respect the consensus clustering
solution, and so on.
Subsequently, a percentage p of the highest ranked l individual partitions is selected so
as to form a select cluster ensemble Ep –see equation (4.2)–, upon which a refined consensus
clustering solution λcp will be derived through the application of the consensus function F.
Notice that, the larger the percentage p, the more components are included in the select
cluster ensemble Ep —ultimately, Ep = E if p = 100.
⎛
⎜
Ep = ⎜
⎝
λ
φ (NMI) 1
λ
φ (NMI) 2
.
λ p
φ (NMI) ⌊
100 l⌉
⎞
⎟
⎠
(4.2)
Following the rationale of the proposed self-refining procedure, it can be assumed that,
with a high probability, the worst components of the initial cluster ensemble E will have been
excluded from Ep. Therefore, the self-refined consensus clustering solution λcp obtained
through the application of the consensus function F on Ep will probably improve the initial
consensus labeling λc, as we will experimentally demonstrate in later sections.
Finally, three additional remarks so as to conclude this description: firstly, notice that
the consensus process run on the select cluster ensemble Ep can be conducted following
either a flat or a hierarchical approach, depending on the consensus function applied, the
characteristics of the data set and the value of p, which, as aforementioned, determines
thesizeofEp. As reported in the previous chapter, the proposed running time estimation
methodologies constitute an efficient means for deciding whether the self-refined consensus
solution λcp should be derived following either a flat or a hierarhical consensus architecture.
Secondly, notice that the proposed consensus self-refining process is entirely automatic
and unsupervised (hence its name), as it is solely based on the cluster ensemble E, the
consensus clustering solution λc and a similarity measure –φ (NMI) – that requires no external
knowledge for its computation. The only user-driven decision refers to the selection of the
value of the percentage p used for creating the select cluster ensemble Ep.
And the third remark deals just with this latter issue. Quite obviously, the selection
of the percentage p is made blindly. So as to avoid the negative consequences of choosing
a suboptimal value of p at random, our consensus self-refining proposal is completed by
the (possibly parallelized) creation of multiple refined consensus clustering solutions using
P distinct percentage values p = {p1,p2,...,pP }, i.e. λc p i ,fori = {1, 2,...,P}, selecting
as the final refined consensus clustering solution λ final
c
the one maximizing φ (ANMI) with
respect to the cluster ensemble E, as defined by equation (4.3) —in fact, this unsupervised a
posteriori clustering selection process is equivalent to the supraconsensus function proposed
in (Strehl and Ghosh, 2002).
λ final
c
= λ max φ (ANMI)
(E, λ) , λ ∈{λc, λcp1 ,...,λcpP } (4.3)
For summarization purposes, table 4.1 describes the steps that constitute the proposed
consensus self-refining procedure.
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