TESI DOCTORAL - La Salle

Chapter 6. Voting based consensus functions for soft cluster ensembles
However, if object to cluster centroid distances are inverted and normalized using a
softmax normalization, so they can be interpreted as membership probabilities are
obtained (i.e. the k-means clustering solutions are fuzzified). For the sake of greater
algorithmic diversity, variants of k-means using the Euclidean, city block, cosine and
correlation distance measures have been employed. Thus, the cardinality of the algorithmic
diversity factor is |dfA| = 5. Applying all these clustering algorithms on each
and every one of the distinct object representations created by the mutually crossed
application of the representational and dimensional diversity factors of each data set,
gives rise to soft cluster ensembles of the sizes l presented in table 6.1. In order to obtain
a representative analysis of the aforementioned consensus functions performance,
we have conducted multiple experiments on distinct diversity scenarios. To do so,
besides using the cluster ensemble of size l, we have also generated cluster ensembles
of sizes ⌊ l l l
l
20⌋, ⌊ 10⌋, ⌊ 5⌋ and ⌊ 2⌋, which are created by randomly picking a subset
of the original cluster ensemble components. For each distinct cluster ensemble, ten
independent runs of each consensus function are executed.
– How are results presented? The performances of the nine soft consensus functions
are summarized by means of a quality (φ (NMI) with respect to the ground truth) versus
time complexity (CPU time measured in seconds) diagram that describes, in a pretty
summarized manner, the qualities of the consensus functions compared. For each
consensus function, the depicted scatterplot corresponds to the region limited by the
mean ± 2-standard deviation curves corresponding to the two associated magnitudes
(i.e. φ (NMI) and CPU time) computed throughout all the experiments conducted
on each data collection —ten independent experiment runs on each one of the five
cluster ensemble sizes. In order to determine whether the differences between the
compared consensus functions are significant or not, we have conducted a pairwise
comparison (both in CPU time and φ (NMI) terms) among them applying a t-paired
test, measuring the significance level p at which the null hypothesis (equal means with
possibly unequal variances) is rejected. If the typical 95% confidence interval for true
difference in means is taken as a reference, significance level values of p