Relevance of Classes in a Fuzzy Partition. A Study from a Group of Aggregation Operators
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Relevance of Classes in a Fuzzy Partition
97
of the concept of recursiveness. That is, from the classical approach of aggregation of
information by a single index (associative conjunctive rules), the notion of associativity
is extended through the establishment of recursive rules.
Such an approach proposes indexes that allow measuring the redundancy, relevance
and coverage of the classes obtained in a fuzzy partition; that is to say, a system is
constructed that in turn allows, evaluating the obtained classiﬁcation. However, the
evaluation of the quality of a partition is a problem that still needs to be addressed in
greater depth (see [5, 10]) with special emphasis on the proposed indexes.
In this line of research, some ﬁrst approaches led to the development of works such
as those presented in [11–13], and more recently in [14, 15] about a more in-depth
study of aggregation functions, which allow evaluating the redundancy and coverage of
a particular classiﬁcation by means of overlapping and grouping functions, respectively. Even studies on redundancy, based on other fuzzy partition concepts, have been
developed (see e.g. [16]).
An issue that is still open and with a broad ﬁeld of development, is the study of the
relevance property as initially explored in [2]. In [6] an alternative approach is proposed from a more statistical perspective. Here we propose some ﬁrst steps towards the
study of relevance, its characterization, and with it, a general study of a global quality
index for a fuzzy classiﬁcation system.
2 The Relevance Property
In the study of the intrinsic properties of a fuzzy partition, we can highlight the
covering and redundancy properties, respectively graded by the degree in which a
family of classes allows explaining the object’s main attributes and the degree of
overlapping between that family of classes. (see [5]).
Relevance, in general terms, is a fuzzy concept and from a more general and
intuitive perspective, people may be able to distinguish irrelevant information or, in
some cases, more relevant information from less relevant information. The fact that
there is a linguistic notion of relevance with a vague and variable meaning exposes the
complexity of the problem and reveals different ways of approaching it. Moreover,
intuitions of relevance are relative to contexts, and there is no way of controlling
exactly which context someone will have in mind at a given moment or how to
understand such a context [17].
By its nature, the concept of relevance requires a treatment beyond its etymological
meaning; the fundamental thing is to characterize when an object is relevant with
respect to a given context. Therefore, as a technical concept which can be suitable of
being measured by computational methods, relevance requires a characterization that
allows its formal understanding for computational use. Keeping this in mind, here we
propose a new approach over relevance (following [5] but also [6]), and the means for
evaluating and measuring it regarding a given fuzzy partition.
In particular, establishing that a proposition is relevant necessarily requires considering a space or context of reference, in such a way that the element or proposition
generates changes or modiﬁcations when it is removed or added from the context or
space. Therefore, relevance implies a comparison process, understanding relevance as a
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F. Castiblanco et al.
local property and not a global property. Hence, we evaluate the relevance of an object
in a context and not the relevance of the context or space in which the object is framed.
According to the above, in a ﬁrst stage, the relevance of an object in a context can
be established through the comparison of diverse information provided by the conformation of three sets: the information of the context with the object, the information
of the context without the object and the information provided by object in itself,
without any context.
The comparison process necessary to establish relevance may or may not reveal
changes in the context, however, in case of changes, these changes may be more or less
intense in the context. It is possible that the changes are signiﬁcant or not. Therefore,
establishing the relevance of an object in a context requires identifying the kind of
changes that occur in that context, and their degree of intensity. In this sense, it is
desirable to establish a threshold or admissible parameter of relevance, or in general, of
modiﬁcation in the context and kind of changes.
According to the above, the relevance of an object depends on two fundamental
aspects: on the one hand, a process of comparison between the object and the context
that allows determining changes by the inclusion or elimination of the object and, a
measure of the intensity of the changes.
In the framework of a fuzzy partition, let us assume a ﬁnite set of objects X. A fuzzy
classiﬁcation system is a ﬁnite family C of fuzzy sets or classes (each c 2 C with its
associated membership function lc xị : X ! ẵ0; 1ị, together with a recursive triplet1
ðu; /; NÞ, where:
1. / is a standard recursive rule such that /2 0; 1ị ẳ /2 1; 0ị ẳ 0
2. N : ẵ0; 1!ẵ0; 1 is a strict negation function2, i.e., a bijective strictly decreasing
function such that N N 1 l xịị ẳ l xị for all l xị 2 ẵ0; 1
3. u is a standard recursive rule such that un ðlðx1 Þ; . . .; lxn ịị ẳ N 1 ẵ/n N lx1 ịị;
. . .; N ðlðxn ÞÞÞ8n [ 1.
Notice that, un is a disjunctive recursive À rule,
Á in the sense that un lx1 ị;
. . .; lx2 ịị ẳ 1 whenever there is j such that l xj ¼ 1, while /n is a conjunctive
recursive
rule in the sense that /n ðlðx1 Þ; . . .; lx2 ịị ẳ 0 whenever there is j such that
À Á
l xj ¼ 0.
About the relevance property in [5] it is proposed to compare the behavior of each
family of non-empty classes A & C with the behavior of the remaining classes C À A,
taking into account the values obtained through the following expressions for each
object x 2 X.
1
2
Recursiveness is a property of a sequence of operators f/n gn [ 2 allowing the aggregation of any
number of items: /2 tells us how to aggregate two items, /3 tells how to aggregate three items and
so on. A recursive rule / is a family of aggregation functions f/n : ½0; 1n ! ½0; 1gn [ 1 allowing a
sequential reckoning by means of a successive application of binary operators, once data have been
properly ordered: the ordering rule assures that new data do not introduce modiﬁcations in the
relative position of items already ordered. For more details see [5, 18].
Here we refer to strict negations of the type N xị ẳ f À1 ðf ð1Þ À f ð xÞÞ with f : ẵ0; 1 ! ẵ0; 1
increasing, bijective, f 0ị ẳ 0, and 0\f 1ị 1. In particular, if N xị ẳ 1 x, then f xị ẳ x.
Relevance of Classes in a Fuzzy Partition
99
1. un flc ð xÞ=c 2 C g
2. un flk ð xÞ=k 2 Ag
3. un fld ð xÞ=d 2 C À Ag
The following criterion is established: when the value obtained through expression 1 above is signiﬁcantly greater than that obtained through expression 3, then A is a
family of relevant classes, as long as the value obtained through expression 2 is not
high. When expression 1 produces a value not signiﬁcantly different from that obtained
through expression 3, then A is a family of non-relevant classes, as long as expression 2 does not produce a low value.
The above criterion requires additional developments since,
1. There may be x 2 X such that some of the above situations do not appear clearly,
for instance, when the values of l1 ð xÞ; . . .; lc ð xÞ, for x 2 X, are in a highly uniform.
2. It is opportune to establish a global index for each of the properties studied, that is,
the aggregation of the degrees of coverage, relevance and overlap for all x 2 X for
each class c, in the perspective posed by [19].
According to the above, it would be desirable to establish one or several criteria for
evaluation of the relevance property. In general, it is sought to establish a set of criteria
that allows the evaluation of a fuzzy classiﬁcation system.
3 The Group of Aggregation Operators
From the fuzzy classiﬁcation system ðC; u; /; N Þ, with c; d 2 C and in particular
working on /2 : ½0; 12 ! ½0; 1 with
u2 lc xị; ld xịị ẳ N 1 ½/2 ½N ðlc ð xÞÞ; N ðld ð xÞÞ;
two new mappings are built for all aggregation operators /2 and u2 such that the
standard strict negation is N xị ẳ 1 À x (in this particular case, we have that
u2 lc xị; ld xịị ẳ 1 /2 ð1 À lc ð xÞ; 1 À ld ð xÞÞ. These mappings are:
1. r2 : ½0; 12 ! ½0; 1, dened as:
r2 lc xị; ld xịị ẳ lc xị ỵ ld xị u2 lc xị; ld xịị; and
2. d2 : ẵ0; 12 ! ½0; 1, deﬁned as:
d2 ðlc ð xÞ; ld ð xÞÞ ẳ lc xị ỵ ld xị /2 ðlc ð xÞ; ld ð xÞÞ
The proposed mappings can be generalized by (conjunctive, disjunctive or average)
aggregation operators, leaving its formal speciﬁcation for future research.
When we use the strict negation N ðlð xÞÞ on d2 ðlc ð xÞ; ld ð xÞÞ or r2 ðlc ð xÞ; ld ð xÞÞ,
this can be interpreted as the complement of the set of aggregated classes
flc ð xÞ; ld ð xÞg. In particular, if u2 ðlc ð xÞ; ld ð xÞÞ represents the degree of coverage of
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F. Castiblanco et al.
the classes, then Nðu2 Þ represents the degree of non-coverage of the classes, understanding u2 as a proposition and Nðu2 Þ as the negation of such a proposition.
Notice that the mapping r2 can be understood as the degree of partial non-coverage
of the aggregated classes and d2 as the degree up to which the aggregated classes do not
partially overlap. In general, both mappings can be understood as partial complements
of the aggregated classes.
The idea that motivates this construction is based on the possibility of establishing a
relationship between the conjunctive, disjunctive operators and their partial complements, in such a way that a fuzzy partition can be evaluated taking into account a global
vision of the corresponding fuzzy classiﬁcation system. It seeks to compare the degrees
of coverage, overlap and partial complements of pairs of classes in fuzzy partitions with
different number of classes, and determine the partition with highest quality.
A close relationship is established between the set of mappings /2 ; u2 ; r2 and d2 .
Let A ¼ f/2 ; u2 ; r2 ; d2 g, the composition of the mappings (denoted by ) is deﬁned as
presented in Table 1.
Table 1. Composition
/2
r2
d2
u2
/2
/2
r2
d2
u2
r2
r2
/2
u2
d2
d2
d2
u2
/2
r2
u2
u2
d2
r2
/2
For instance, we have that: ðr2 u2 Þðlc ð xÞ; ld xịị ẳ lc xị ỵ ld xị 1 ỵ u2
1 lc xị; 1 ld xịị ẳ lc xị ỵ ld ð xÞ À /2 ðlc ð xÞ; ld ð xÞÞ ¼ d2 ðlc ð xÞ; ld ð xÞÞ
Clearly, ðA; Þ is a commutative group, where /2 is the neutral element and each
element is its own inverse. As mentioned above, r2 and d2 mappings can be formulated
under a general framework, considering the strict negation function N for each pair /2
and u2 and an adequate aggregation of the classes that maintains the group structure.
The composition of the deﬁned mappings obtains a particular structure for the
algebraic group and therefore, allows proposing a relation of similarity between the
functions of aggregation and their partial complements in the perspective presented by
[20]. Therefore, such a similarity relation allows a ﬁrst comparison process between the
information obtained from the aggregated classes. Based on [20], for each pair
ðlc ð xÞ; ld ð xÞÞ 2 C, the mapping m0 : A Â A ! ½0; 1 is deﬁned in the following way:
Pm
iẳ1 ẵh2 lc xi ị; ld xi ịị k2 ðlc ðxi Þ; ld ðxi ÞÞ
m0 ðh2 ðlc ð xÞ; ld ð xÞÞ; k2 ðlc ð xÞ; ld ð xÞÞÞ ¼
m
With h2 , k2 2 A, m ¼ j X j, lc ðxi Þ is the membership degree of the element xi in
class c. For simplicity, let us consider:
Relevance of Classes in a Fuzzy Partition
101
m0 ðh2 ðlc ð xÞ; ld xịị; k2 lc xị; ld xịịị ẳ m0 ðh2 ; k2 Þðc; d Þ:
Let us also denote:
Pm
j iẳ1 ẵr2 lc xi ị;ld xi ịị/2 lc xi ị;ld xi ịịj
ẳ b0
m
Pm
j iẳ1 ẵu2 lc xi ị;ld xi ịị/2 ðlc ðxi Þ;ld ðxi ÞÞj
2. m0 ð/2 ; u2 Þðc; d ị ẳ m0 u2 ; /2 ịc; d ị ¼
¼ a0
m
1. m0 ð/2 ; r2 Þðc; d Þ ¼ m0 r2 ; /2 ịc; d ị ẳ
Then, the relation for all the element of A Â A is represented in Table 2.
Table 2. Mapping m0
m0
/2
r2
d2
u2
/2
0
b0
b0 ỵ a0
a0
r2
b0
0
a0
jb0 a0 j
d2
b0 ỵ a0
a0
0
b0
u2
a0
jb0 a0 j
b0
0
For instance, if m0 /2 ; r2 ịc; d ị ẳ m0 r2 ; /2 Þðc; d Þ ¼ b0 ; then we have that:
Pm
iẳ1 ẵr2 lc xi ị; ld xi ịị /2 ðlc ðxi Þ; ld ðxi ÞÞ
m0 ð/2 ; r2 Þðc; d ị ẳ
m
Pm
ẵ
l
x
ị
ỵ
l
x
ị
u
l
x
ị;
l
d i
2 c i
d xi ịị /2 lc xi ị; ld xi ịị
iẳ1 c i
ẳ
m
Pm
ẵ
d
l
x
ị;
l
x
ị
ị
u
l
2
i
i
c
d
2
c xi ị; ld xi ịị
iẳ1
ẳ m0 d2 ; u2 ịc; d ị
ẳ
m
Consider the complements of the images of m0 , i.e., the mapping m : ½0; 1 ! ½0; 1,
such that:
mb0 ị ẳ 1 b0 ẳ bc;d
ma0 ị ẳ 1 a0 ẳ ac;d
mb0 ỵ a0 ị ẳ 1 b0 ỵ a0 ị ẳ pc;d
mjb0 a0 jị ¼ 1 À jb0 À a0 j ¼ cc;d
Therefore, for this mapping m it is possible to establish the relationships shown in
Table 3.
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F. Castiblanco et al.
Table 3. Mapping m
m
/2
r2
d2
u2
/2
1
b
p
a
r2
b
1
a
c
d2
p
a
1
b
u2
a
c
b
1
Proposition 1. Let ðC; u; /; N Þ be a fuzzy classiﬁcation system. If /2 lc xị; ld xịị ẳ
r2 ðlc ð xÞ; ld ð xÞÞ for all x 2 X then, ac;d ¼ pc;d ¼ cc;d \bc;d ¼ 1:
Proof. If /2 lc xị; ld xịị ẳ r2 lc xị; ld xịị then b0 ẳ 0 wherewith bc;d ¼ 1 and
pc;d ¼ 1 À a0 ¼ ac;d . Similarly, cc;d ¼ 1 À jÀa0 j ¼ cc;d .
As established in [8], a fuzzy partition is of a higher quality, if the aggregated
classes have high degrees of coverage and low degrees of overlap. Therefore, if classes
are considered in pairs for comparison, a similar result would be expected. That is, if
the pairs of classes (c, d), are analyzed, it is expected that:
• The degree of coverage and non-overlap of the classes studied are high, so that their
difference must be small, and, bc;d must be high.
• The degree of overlap is less than its partial complement and the degree of coverage
is greater than its partial complement, expecting that pc;d and cc;d are low.
Deﬁnition 1. Given a fuzzy classiﬁcation system ðC; u; /; N Þ it is said that the pair of
classes {c, d} are relevant in C if cc;d \bc;d . The following cases are established:
1. If /2 ðlc xị; ld xịị ẳ r2 lc xị; ld ð xÞÞ, the relevance of {c, d} is established from a
parameter t 2 ẵ0; 1, such that, t ẳ ac;d ¼ pc;d ¼ cc;d : The lower, the better.
2. If /2 lc xị; ld xịị 6ẳ r2 ðlc ð xÞ; ld ð xÞÞ, the relevance of {c, d} is established from a
parameter t, such that, t ¼ cc;d \bc;d . The lower, the better.
3. If /2 ðlc xị; ld xịị 6ẳ r2 lc xị; ld xịị and ẳ cc;d [ bc;d then, the classes {c, d} are
not relevant.
In this sense, the coverage and overlapping of the classes analyzed by pairs, allow
estimating the degree of relevance of such pair of classes (comparing the information
obtained from the degree of grouping, the degree of partial non-coverage, the degree of
overlap and the degree of partial overlap). Therefore, in a ﬁrst stage the relevance of
any pair of classes offers information on their usefulness, or relative meaning regarding
their signiﬁcance with respect to the already considered set of classes i.e., there is a
signiﬁcative lose if the classes are deleted.
Relevance of Classes in a Fuzzy Partition
103
4 Application
In order to apply the deﬁned criterion, the image presented in Fig. 1 has been selected,
and the unsupervised classiﬁcation problem of obtaining a set of classes such that
similar pixels are assignedQto the same class is considered. The conjunctive operator
3
n
lk x ị
kẳ1
Q
/n l1 xị; . . .; ln xịị ẳ
and the disjunctive operator un l1 xị; . . .;
n
l xị
1ỵ2
kẳ1 k
Qn
1 kẳ1 1lk xịị
Qn
whose negation is N l xịị ẳ 1 l xị have been selected.
lc xịị ẳ
1ỵ2
kẳ1
1lk xịị
The fuzzy c-means algorithm has been applied for c ¼ 3.
Fig. 1. Aurora Borealis
The values ac;d ; pc;d ; cc;d and bc;d are presented for each pair of classes (class 1, 2
and 3) in Tables 4, 5 and 6, and the classes are presented in Fig. 2.
Table 4. mðm0 ðh2 ; k2 ị1; 2ịị
m
/2
r2
d2
u2
/2
1
0; 93
0; 6
0; 67
a ẳ 0; 67
r2
0; 93
1
0; 67
0; 74
c ¼ 0; 74
d2
0; 6
0; 67
1
0; 93
p ¼ 0; 6
u2
0; 67
0,74
0; 93
1
b ¼ 0; 93
Table 5. mðm0 ðh2 ; k2 ị1; 3ịị
m
/2
r2
d2
u2
/2
1
0; 8
0; 4
0; 5
a ẳ 0; 5
r2
0; 8
1
0; 5
0; 7
c ¼ 0; 7
d2
0; 4
0; 5
1
0; 8
p ¼ 0; 4
u2
0; 5
0; 7
0; 8
1
b ¼ 0; 8
104
F. Castiblanco et al.
Table 6. mm0 h2 ; k2 ị2; 3ịị
m
/2
r2
d2
u2
/2
1
0; 93
0:63
0; 69
a ẳ 0; 69
r2
0; 93
1
0; 69
0; 76
c ¼ 0; 76
d2
0; 63
0; 69
1
0; 93
p ¼ 0; 63
u2
0:69
0; 76
0; 93
1
b ¼ 0; 93
Fig. 2. Classes applying fuzzy 3-means algorithm (top left: class 1, top right: class 2, bottom:
class 3). The gray scale represents the membership degree of each pixel to each class, where
black = 0 and white = 1
In the case of the fuzzy c-means algorithm for c ¼ 4, the results are presented in
Tables 7, 8, 9, 10, 11 and 12, and the classes are presented in Fig. 3
Table 7. mðm0 ðh2 ; k2 Þð1; 2ÞÞ
Table 8. mðm0 ðh2 ; k2 Þð1; 3ÞÞ
m
/2
r2
d2
u2
m
/2
r2
d2
u2
/2
r2
d2
u2
1
0; 92
0; 71
0; 79
a ¼ 0; 79
0; 92
1
0; 79
0; 86
c ¼ 0; 8
0; 71
0; 79
1
0; 92
p ¼ 0; 71
0; 79
0; 86
0; 92
1
b ¼ 0; 9
/2
r2
d2
u2
1
0; 87
0; 57
0; 7
a ¼ 0; 7
0; 87
1
0; 7
0; 82
c ¼ 0; 82
0; 57
0; 7
1
0; 87
p ¼ 0; 57
0; 7
0; 82
0; 87
1
b ¼ 0; 87
Relevance of Classes in a Fuzzy Partition
Table 9. mðm0 ðh2 ; k2 Þð1; 4ÞÞ
105
Table 10. mðm0 ðh2 ; k2 Þð2; 3ÞÞ
m
/2
r2
d2
u2
m
/2
r2
d2
u2
/2
r2
d2
u2
1
0; 861
0; 58
0; 72
a ¼ 0; 7
0; 861
1
0; 72
0; 864
c ¼ 0; 864
0; 58
0; 72
1
0; 861
p ¼ 0; 5
0; 72
0; 864
0; 861
1
b ¼ 0; 861
/2
r2
d2
u2
1
0; 86
0; 6
0; 74
a ¼ 0; 74
0; 86
1
0; 74
0; 88
c ¼ 0; 88
0; 6
0; 74
1
0; 86
p ¼ 0; 6
0; 74
0; 88
0; 86
1
b ¼ 0; 86
Table 11. mðm0 ðh2 ; k2 Þð2; 4ÞÞ
Table 12. mðm0 ðh2 ; k2 Þð3; 4ÞÞ
m
/2
r2
d2
u2
m
/2
r2
d2
u2
/2
r2
d2
u2
1
0; 9
0; 72
0; 82
a ¼ 0; 82
0; 9
1
0; 82
0; 91
c ¼ 0; 91
0; 72
0; 82
1
0; 9
p ¼ 0; 72
0; 82
0; 91
0; 9
1
b ¼ 0; 9
/2
r2
d2
u2
1
0; 93
0; 75
0; 81
a ¼ 0; 81
0; 93
1
0; 81
0; 88
c ¼ 0; 88
0; 75
0; 81
1
0; 93
p ¼ 0; 75
0; 81
0; 88
0; 93
1
b ¼ 0; 93
Fig. 3. Classes applying fuzzy 4-means algorithm (top left: class 1, top right: class 2, bottom
left: class 3, bottom right: class 4) The gray scale represents the membership degree of each pixel
to each class, where black = 0 and white = 1
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F. Castiblanco et al.
From these tables it is observed that for the case of fuzzy 3-means, for all pairs of
classes c, d holds that cc;d \bc;d : By contrast, in fuzzy 4-means the inequality is not
met, i.e., c1;4 [ b1;4 , c2;3 [ b2;3 and c2;4 [ b2;4 . Therefore, it is established that there
are pairs of classes for which their degree of non-coverage is greater than their degree
of coverage without considering their overlap. In principle, class 2 and class 4 are those
that most affect the relevance of the classes analyzed by pairs. Therefore, the fuzzy
3-means algorithm application obtains greater relevance, illustrating the proposed
criterion for identifying the partition with highest quality.
5 Final Comments
Through this work, some of the fundamental elements that allow characterizing the
property of relevance in the framework of the evaluation of a fuzzy classiﬁcation
system are given. Three determining aspects are considered for the study of relevance:
(1) a process of comparison between classes and the way they cover the objects under
consideration, (2) degrees of intensity in the changes generated by the elements in the
space and (3) a stopping criterion for inclusion of classes in a fuzzy partition.
The complement of two ratios b0 and c0 for each pair of classes {c, d} have been
established as elements for comparison. The ratio b0 expresses the global degree
(aggregation of the degree for all items x 2 X) in which the overlap of the two classes
covers the objects under consideration, while c0 expresses the global degree in which
the coverage of the two classes differs in relation to their partial complement. Comparing the complements of these two ratios it is expected that b ¼ 1 À b0 is greater
than c ¼ 1 À c0 .
According to the above, the stopping criterion corresponds to a comparison process
in which a pair of classes is relevant up to a degree t, provided that the degree of
coverage of two classes without considering their overlap is greater than the degree of
non-coverage of the classes in relation to the objects under consideration. In this sense,
the class pair {c, d} will be non-relevant when c [ b.
As future work, it is proposed to build a model that allows generalizing the
mappings together with the stopping criterion, while maintaining the group structure.
Such a model should be general enough to include cases that do not meet the Ruspini´s
partition.
The characterization of the relevance of classes in a fuzzy partition still requires
further developments and as future research, we propose to study the kinds of changes
that the inclusion or elimination of a class can generate in a partition. Although the
changes are measured in degrees of intensity, such changes can also be of a different
nature, for example, affecting both the grouping and overlapping of each element, as
there may be changes that affect only one of the properties. Likewise, a more in-depth
study is necessary to relate the degrees of coverage and overlap of the partition, with
the degree of relevance for every pair of classes.
Acknowledgements. This research has been partially supported by the Government of Spain
(grant TIN2015-66471-P), the Government of Madrid (grant S2013/ICE-2845), and
Complutense University (UCM Research Group 910149).
Relevance of Classes in a Fuzzy Partition
107
References
1. Zadeh, L.: Fuzzy sets. Inf. Control 8, 338–353 (1965)
2. Bezdek, J., Harris, J.: Fuzzy partitions and relations: an axiomatic basis for clustering. Fuzzy
Sets Syst. 1, 111–127 (1978)
3. Bellman, R., Kalaba, R., Zadeh, L.: Abstraction and pattern classiﬁcation. J. Math. Anal.
Appl. 13, 1–7 (1966)
4. Pedrycz, W.: Fuzzy sets in pattern recognition: methodology and methods. Pattern Recogn.
23, 121–146 (1990)
5. Del Amo, A., Montero, J., Biging, G., Cutello, V.: Fuzzy classiﬁcation systems. Eur. J. Oper.
Res. 156, 459–507 (2004)
6. Del Amo, A., Gómez, D., Montero, J., Biging, G.: Relevance and redundancy in fuzzy
classiﬁcation systems. Mathw. Soft Comput. 8, 203–216 (2001)
7. Ruspini, E.: A new approach to clustering. Inf. Control 15, 22–32 (1969)
8. Dombi, J.: Basic concepts for a theory of evaluation: the aggregative operator. Eur. J. Oper.
Res. 10, 282–293 (1982)
9. Dombi, J.: A general class of fuzzy operators, the DeMorgan class of fuzzy operators and
fuzziness measures induced by fuzzy operators. Fuzzy Sets Syst. 8, 149–163 (1982)
10. Matsakis, P., Andrefouet, P., Capolsini, P.: Evaluation of fuzzy partition. Remote Sens.
Environ. 74, 516–533 (2000)
11. Bustince, H., Fernández, J., Mesiar, R., Montero, J., Orduna, R.: Overlap functions.
Nonlinear Anal. Theory Methods Appl. 72, 1488–1499 (2010)
12. Bustince, H., Barrenechea, E., Pagola, M., Fernández, J.: The notions of overlap and
grouping functions. In: Saminger-Platz, S., Mesiar, R. (eds.) On Logical, Algebraic, and
Probabilistic Aspects of Fuzzy Set Theory, Studies in Fuzziness and Soft Computing, vol.
336, pp. 137–156. Springer, Switzerland (2016). https://doi.org/10.1007/978-3-319-288086_8
13. Gómez, D., Rodríguez, J., Bustince, H., Barrenechea, E., Montero, J.: n-dimensional overlap
functions. Fuzzy Sets Syst. 287, 57–75 (2016)
14. Qiao, J., Hu, B.Q.: On interval additive generators of interval overlap functions and interval
grouping functions. Fuzzy Sets Syst. 323, 19–55 (2017)
15. Qiao, J., Hu, B.Q.: On the migrativity of uninorms and nullnorms over overlap and grouping
functions. Fuzzy Sets Syst. (2017). http://doi.org/10.1016/j.fss.2017.11.012
16. Klement, E., Moser, B.: On the redundancy of fuzzy partitions. Fuzzy Sets Syst. 85, 195–201
(1997)
17. Sperber, D., Wilson, D.: Relevance: Communication and Cognition, 2nd edn. Blackwell
Publishers Inc., Cambridge (1995)
18. Cutello, V., Montero, J.: Recursive connective rules. Int. J. Intell. Syst. 14, 3–20 (1999)
19. Castiblanco, F., Gómez, D., Montero, J., Rodríguez, J.: Aggregation tools for the evaluation
of classiﬁcations. In: 2017 Joint 17th World Congress of International Fuzzy Systems
Association and 9th International Conference on Soft Computing and Intelligent Systems
(IFSA-SCIS). IEEE, Otsu, pp. 1–5 (2017)
20. Mordeson, J., Bhutani, K., Rosenfeld, A.: Fuzzy Group Theory. Springer, Heidelberg
(2005). https://doi.org/10.1007/b12359