TESI DOCTORAL - La Salle

Chapter 3. Hierarchical consensus architectures
In this chapter, we propose two strategies for constructing hierarchical consensus architectures,
which differ in i) the way mini-ensembles are created, and ii) which HCA
parameters are tuned by the user. As a result, two HCA implementation alternatives are
put forward:
– random hierarchical consensus architectures, whose tunable parameter is the size of
the mini-ensembles –the components of which are selected at random–, which eventually
determines the HCA topology (i.e. its number of stages).
– deterministic hierarchical consensus architectures, where the construction of the miniensembles
is driven by the cluster ensemble creation process —in particular, the diversity
factors used in creating the ensemble determine the number of HCA stages
and the mini-ensembles components.
The following sections are devoted to describing the rationale and implementation details
of both HCA variants, specifying how the number of stages, the number of consensus
processes per stage and the size of the mini-ensembles are determined in each case. This
description is completed by an analysis of their computational complexity.
3.2 Random hierarchical consensus architectures
In this section, we introduce random hierarchical consensus architectures (RHCA for short),
we define their topology from a generic perspective, making a brief description of their
foundations, followed by an analysis of their computational complexity.
3.2.1 Rationale and definition
The idea behind random hierarchical consensus architectures is to construct a regular pyramidal
structure of intermediate consensus processes that delivers, at its top, the final consensus
clustering solution λc. The term random refers not to the consensus architecture
itself, but to the way mini-ensembles are created. In particular, the randomness of RHCA
lies in the fact that the clusterings input to each stage of the hierarchical architecture are
shuffled randomly.
Besides this fact, the most distinctive feature of RHCA is that the user determines
the size of the mini-ensembles, setting it to b, keeping it constant across the stages of
the consensus architecture. Therefore, given a cluster ensemble containing l component
clusterings and a mini-ensemble size set equal to b by the user, the number of stages s of
the resulting RHCA is computed by equation (3.1).
⎧
⎪⎩
⌊log b (l)⌉ if
⎪⎨
s = ⌊logb (l)⌉−1 if
⌊log b (l)⌉ +1 if
l
b ⌊log b (l)⌉
l
b ⌊log b (l)⌉
l
b ⌊log b (l)⌉
49
≤ 1and
≤ 1and
> 1
l
b ⌊log b (l)⌉−1
l
b ⌊log b (l)⌉−1
> 1
=1
(3.1)