TESI DOCTORAL - La Salle

Positional voting
Chapter 6. Voting based consensus functions for soft cluster ensembles
Positional (aka ranking) voting methods rank the candidates according to the confidence
scores emitted by the voters. Thus, fine-grain information regarding preference differences
between candidates is ignored, although problems in scaling the voters confidence scores
are avoided —that is, positional voting is useful in situations where confidence values are
hard to scale correctly (van Erp, Vuurpijl, and Schomaker, 2002).
As an aid for describing the positional voting methods that constitute the core of our
consensus functions, equation (6.38) defines Λi (the ith component of the soft cluster ensemble
E) in terms of its columns, represented by vectors λij (∀j =1,...,n).
⎛ ⎞
Λ1
⎜
⎜Λ2
⎟
E = ⎜ ⎟
⎝ . ⎠ where Λi =
λi1 λi2 ... λin
Λl
(6.38)
In this work, we propose employing two positional voting strategies (namely the Borda
and the Condorcet voting methods) for deriving the consensus clustering solution, which
gives rise to the following consensus functions:
– BordaConsensus (BC): the Borda voting method (Borda, 1781) computes the mean
rank of each candidate over all voters, reranking them according to their mean rank.
This process results in a grading of all the n objects with respect to each of the k
clusters, which is embodied in a k × n Borda voting matrix BE. Such grading process
is conducted as follows: firstly, for each object (election), clusters (candidates) are
ranked according to their degree of association with respect to it (from the most to
the least strongly associated). Then, the top ranked candidate receives k points, the
second ranked cluster receives k − 1 points, and so on. After iterating this procedure
across the l cluster ensemble components, the grading matrix BE is obtained. The
whole process is described in algorithm 6.3. Notice that the Rank procedure orders
the clusters from the most to the least strongly associated to each object, yielding a
ranking vector r which is a list of the k clusters ordered according to their degree of
association with respect to the object under consideration (i.e. its first component
(r(1)) identifies the most strongly associated cluster, and so on. Thus, the Rank
procedure must take into account whether the scalar values contained in λab are
directly or inversely proportional to the strength of association between objects and
clusters.
When applied on our toy example, the resulting Borda grading matrix is the one
presented in equation (6.39).
⎛
6 6 5 4 4 4 4 4
⎞
4
BE = ⎝3
3 2 2 2 2 4 6 6⎠
(6.39)
3 3 5 6 6 6 4 2 2
According to Borda voting, the higher the value of the (i,j)th entry of BE, themore
likely the jth object belongs to the ith cluster. Thus, again, dividing each column of
matrix BE by its L1-norm transforms it into a cluster membership probability matrix,
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