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Relevance of Classes in a Fuzzy Partition. A Study from a Group of Aggregation Operators

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 ÞÞފ8n [ 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; 1Šn ! ½0; 1Šgn [ 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; 1Š2 ! ½0; 1Š with

u2 lc xị; ld xịị ẳ N 1 ½/2 ½N ðlc ð xÞÞ; N ðld ð xÞފŠ;

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; 1Š2 ! ½0; 1Š, dened as:

r2 lc xị; ld xịị ẳ lc xị ỵ ld xị u2 lc xị; ld xịị; and

2. d2 : ẵ0; 12 ! ½0; 1Š, 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  A ! ½0; 1Š is defined in the following way:

Pm







iẳ1 ẵh2 lc xi ị; ld xi ịị k2 ðlc ðxi Þ; ld ðxi Þފ

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 Þފj

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 Þފ

m0 ð/2 ; r2 Þðc; d ị ẳ

m

 Pm









l



x





l



x





u



l



x

ị;

l



d i

2 c i

d xi ịị /2 lc xi ị; ld xi ịị

iẳ1 c i



m

 Pm









d



l



x

ị;

l



x







u



l

2

i

i

c

d

2

c xi ị; ld xi ịị

iẳ1

ẳ m0 d2 ; u2 ịc; d ị



m

Consider the complements of the images of m0 , i.e., the mapping m : ½0; 1Š ! ½0; 1Š,

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



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