TESI DOCTORAL - La Salle

Chapter 3. Hierarchical consensus architectures
3.3 Deterministic hierarchical consensus architectures
This section is devoted to the description of deterministic hierarchical consensus architectures
(or DHCA). As in the previous section, we present a generic definition of this
architectural variant along with a study of its computational complexity.
3.3.1 Rationale and definition
As opposed to random HCA, this proposal drives the creation of the mini-ensembles by
a deterministic criterion. The main idea behind DHCA is to exploit the distinct ways of
introducing diversity in the cluster ensemble as the guiding principle for creating the miniensembles
upon which the intermediate consensus clustering solutions are built. That is,
a key differential factor between DHCA and RHCA is that the former type of architecture
is indirectly designed by the user when creating the cluster ensemble, whereas the latter
requires the user to fix an architectural defining factor (i.e. assign a value to the size of the
mini-ensembles b).
Enlarging on the relationship between the creation of the cluster ensemble and the
configuration of the DHCA, it is important to recall the strategies employed for introducing
diversity in cluster ensembles (see section 2.1).
For instance, heterogeneous cluster ensembles –whose components are generated by
the execution of multiple clustering algorithms on the data set– have a single diversity
factor, i.e. the set of distinct clustering algorithms employed. Meanwhile, when creating
homogeneous cluster ensembles (those compiling the outcomes of multiple runs of a single
clustering algorithm), a wider spectrum of diversity factors can be applied, such as the
random starting configuration of a stochastic algorithm, or the use of distinct attributes for
representing the objects in the data set, among others.
As aforementioned, in this work we combine both the homogeneous and heterogeneous
approaches for creating cluster ensembles, aiming not only to obtain highly diverse cluster
ensembles, but also to design a strategy for fighting against clustering indeterminacies. This
means that we employ several mutually crossed diversity factors (e.g. multiple clustering
algorithms are run on several data representations with varying dimensionalities), and this
will be the scenario where DHCA will be defined.
In general terms, let us denote the number of diversity factors employed in the cluster
ensemble creation process as f. Each diversity factor dfi ∀i ∈ [1,f] has a cardinality |dfi|
—e.g. |dfi| denotes the number of clustering algorithms employed for creating the cluster
ensemble in case that the ith diversity factor dfi represents the algorithmic diversity of the
ensemble.
Finally, notice that, if fully mutual crossing between all diversity factors is ensured (e.g.
each cluster ensemble component is the result of running each clustering algorithm on each
document representation of each distinct dimensionality), the cluster ensemble size l can be
expressed as:
f
l = |dfk| (3.12)
k=1
Let us see how the design of the cluster ensemble determines the topology of a deterministic
hierarchical consensus architecture. The guiding principle is that the consensus
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