TESI DOCTORAL - La Salle

λ 1
λ 2
= λ
= λ
λ3 = λ
λ 4
= λ
λ5 = λ
λ 6
λ 7
= λ
= λ
λ8 = λ
λ 9
λ 10
λ 11
λ 12
λ 13
λ 14
λ 15
λ 16
= λ
= λ
= λ
= λ
= λ
= λ
= λ
= λ
λ17 = λ
λ 18
= λ
1,
1,
1
2,
1,
1
3,
1,
1
1,
1,
2
2,
1,
2
3,
1,
2
1,
1,
3
2,
1,
3
3,
1,
3
1,
2,
1
2,
2,
1
3,
2,
1
1,
2,
2
2,
2,
2
3,
2,
2
1,
2,
3
2,
2,
3
3,
2,
3
Consensus
function
Consensus
function
Consensus
function
Consensus
function
Consensus
function
Consensus
function
λ
λ
λ
λ
λ
λ
A,
1,
1
A,
1,
2
A,
1,
3
A,
2,
1
A,
2,
2
A,
2,
3
Chapter 3. Hierarchical consensus architectures
Consensus
function
Consensus
function
λ
λ
A, 1 1,
D
A, 2 2,
D
Consensus
function c λ
Figure 3.9: An example of a deterministic hierarchical consensus architecture operating on a
cluster ensemble created using three diversity factors: three clustering algorithms |dfA| =3,
two object representations |dfR| = 2 of three dimensionalities each |dfD| = 3. The cluster
ensemble component obtained by running the ith clustering algorithm on the jth object
representation and the kth dimensionality is denoted as λi,j,k. Consensus are sequentially
created across the algorithmic, dimensional and representational diversity factors (dfA, dfD
and dfR, respectively).
Therefore, a total of K2 = |dfR| = 2 consensus processes are run, each on a mini-ensemble
of size b2j = |dfD| =3, ∀j ∈ [1, 2]. The halfway consensus clustering solutions obtained
after this second stage are designated as λA,j,D.
And finally, the last DHCA stage combines the clusterings output by the previous one,
which only differ in its original object representation. Being the final stage of the hierarchy,
a single consensus process is executed (K3 = 1), and the size of the mini-ensemble coincides
with the cardinality of the representation diversity factor, i.e. b3j = |dfR| =2.
3.3.2 Computational complexity
The maximum and minimum time complexities of DHCA –corresponding to the serial and
parallel execution of the consensus processes of each stage, respectively– are estimated in
the following paragraphs. In this case, the goal is to express these complexities in terms of
the cardinality and number of diversity factors employed in the cluster ensemble creation
process. Recall that the time complexity of consensus functions typically grows linearly or
quadratically with the cluster ensemble size, i.e. it is O (l w ), where w ∈{1, 2}.
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