Onto.PT: Towards the Automatic Construction of a Lexical Ontology ...

82 Chapter 5. Synset Discovery
the membership degree of the word in vj to Fi, µFi (vj).
5. For each cluster Fi with all elements included in a larger cluster Fj (Fi ∪Fj = Fj
and Fi ∩Fj = Fi), Fi and Fj are merged, giving rise to a new cluster Fk with the
same elements of Fj, where the membership degrees of the common elements
are summed, µFk (vj) = µFi (vj) + µFj (vj).
Figure 5.2 is the normalised clustering matrix C for the network in figure 5.1,
where we present the resulting fuzzy clusters as well. Similarities are computed with
the cosine similarity measure, as follows:
sim(a, b) = cos(va, vb) = va.vb
| va||vb| =
|V |
i=0
|V |
i=0
vai × vbi
v 2 ai ×
|V |
v
i=0
2 bi
v1 v2 v3 v4 v5 v6 v7 v8 v9 v10
0.27 0.17 0.17 0.17 0.14 0.06 0.00 0.00 0.00 0.00
0.21 0.33 0.16 0.16 0.13 0.00 0.00 0.00 0.00 0.00
0.21 0.16 0.33 0.16 0.13 0.00 0.00 0.00 0.00 0.00
0.21 0.16 0.16 0.33 0.13 0.00 0.00 0.00 0.00 0.00
0.13 0.10 0.10 0.10 0.25 0.14 0.10 0.07 0.00 0.00
0.07 0.00 0.00 0.00 0.11 0.32 0.23 0.08 0.09 0.09
0.00 0.00 0.00 0.00 0.17 0.29 0.41 0.14 0.00 0.00
0.00 0.00 0.00 0.00 0.08 0.07 0.10 0.28 0.24 0.24
0.00 0.00 0.00 0.00 0.00 0.09 0.00 0.27 0.32 0.32
0.00 0.00 0.00 0.00 0.00 0.09 0.00 0.27 0.32 0.32
• F1,2,3,4,5 = {v1(0.94), v2(1), v3(1), v4(1), v5(0.68), v6(0.19), v7(0.17), v8(0.08)}
• F1,5,6,7,8,9,10 = {v1(0.06), v5(0.32), v6(0.81), v7(0.83), v8(0.92), v9(1), v10(1)}
Figure 5.2: Clustering matrix C after normalisation and resulting fuzzy sets.
(5.1)
If µFi (va) > 0, the word va has a sense with a common meaning to the other
words in Fi. The membership degree µFi (va) may be seen as the confidence on the
usage of the word va with the meaning of the synset Fi.
Also, step 3 of the algorithm is optional. In a normalised C, all membership
degrees of the same word sum up to 1, µFi (vj) = 1. Therefore, membership
degrees of a can also be interpreted as the possible senses of the word a and the
likelihood of the word a conveying their meanings. However, normalising C will
make highly connected words to have low memberships.
In order to obtain simple synsets from fuzzy synsets Fi, one has just to apply a
threshold θ to the membership degrees, so that all words a with membership lower
than θ, µFi (va) > θ, are excluded from the synset. In this case, attention should be
paid when C is normalised, as using the same θ for all fuzzy synsets might prevent
that highly connected words are included in any synset.