TESI DOCTORAL - La Salle

3.2. Random hierarchical consensus architectures
where ⌊x⌉ denotes the operation of rounding x to the nearest integer (Hastad et al., 1988).
The second option in equation (3.1) reduces the number of stages by one in the case that
the penultimate RHCA stage already yields one consensus clustering, whereas the third one
adds a supplementary stage so as to ensure the obtention of a single consensus solution at
the output of the RHCA.
Furthermore, the number of consensus solutions computed at the ith stage of the RHCA
(where i ∈ [1,s]) is determined by the expression in equation (3.2).
l
Ki =max
bi
, 1
(3.2)
where ⌊x⌋ stands for the greatest integer less than or equal to x (i.e. the result of applying
the floor function on number x).
However, it is important to notice that it will only be possible to keep the mini-ensembles
size constant all along the hierarchy (i.e. bij = b, ∀i ∈ [1,s]and∀j∈ [1,Ki]) when l is
an integer power of b. In the likely case that this condition is not met, in the current
implementation of RHCA we choose, for simplicity, integrating the spare clusterings2 of the
ith RHCA stage into its last (i.e. the Kith) mini-ensemble, thus introducing a bounded
increase of its size, as b ≤ biKi < 2b. Moreover, the size of the mini-ensemble input to the
sth stage is set to be equal to the number of halfway consensus output by the penultimate
RHCA stage, as defined in equation (3.3).
⎧
⎪⎨
b if i