=Paper=
{{Paper
|id=Vol-3074/paper19
|storemode=property
|title=Building intuitionistic fuzzy sets in machine learning
|pdfUrl=https://ceur-ws.org/Vol-3074/paper19.pdf
|volume=Vol-3074
|authors=Christophe Marsala
|dblpUrl=https://dblp.org/rec/conf/wilf/Marsala21
}}
==Building intuitionistic fuzzy sets in machine learning==
https://ceur-ws.org/Vol-3074/paper19.pdf
Building Intuitionistic Fuzzy Sets in Machine
Learning
Christophe Marsala
Sorbonne Université, CNRS, LIP6, F-75005 Paris, France
Abstract
The construction of intuitionistic fuzzy sets is a difficult task. Some approaches have been proposed
in the literature and they have been used successfully in some application domains. However, these
approaches do barely take into account the representativeness of the data used to build the intuitionistic
fuzzy set. In this paper, a new approach is proposed to build intuitionistic fuzzy sets (IFS). This approach
is based on the use of a representativeness degree of the data. This approach enables to build an IFS
with an intuitionistic fuzzy index that is a good indicator of the lack of knowledge associated with the
data that make it a good approach to be used in a Machine learning setting.
Keywords
Intuitionistic fuzzy sets, Machine learning, lack of knowledge, representativeness.
1. Introduction
(Atanassov) Intuitionistic Fuzzy Sets (IFS) [1] have been proposed to represent a kind of lack of
knowledge in the membership of elements to a set. This lack of knowledge is valued through
the margin of hesitancy. Similarly, such a representation of a lack of knowledge exists with the
Interval-Valued Fuzzy Sets that offer a representation close to the IFS one [2]. This capacity to
associate such an information about the lack of knowledge to data is a powerful property of
this representation model that find a natural application domain in Machine learning.
Indeed, in Machine learning the aim is to build a model from a given training set (the inductive
reasoning approach) with the underlying hypothesis that this training set is completely repre-
sentative of the whole universe of the data. However, it is well-known that such an induction
process is often not applicable with great success in real-world problems: ”From the early days,
theoreticians of machine learning have focused on the iid assumption, which states that the test
cases are expected to come from the same distribution as the training examples. Unfortunately, this
is not a realistic assumption in the real world. [. . .] As a practical consequence, the performance
of today’s best AI systems tends to take a hit when they go from the lab to the field.” [3].
Using IFS to tackled this problem in Machine learning could be a very promising approach
and obtained some success till today [4, 5]. However, a weakness of the IFS approach is still
the construction of the IFS from a dataset. Even if some approaches exists [6, 7], they have not
been defined expressly to be used in a Machine learning setting.
WILF’21: 13th International Workshop on Fuzzy Logic and Applications, Dec. 20–22, 2021, Vietri sul Mare, Italy
! Christophe.Marsala@lip6.fr (C. Marsala) - ORCID ID: 0000-0002-4022-9796
© 2021 Copyright for this paper by its authors. Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
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In this paper, we propose a new approach to construct IFS from a dataset that takes care of the
specificity of the Machine learning setting. Firstly, in Section 2 a recall of existing approaches
is done. Secondly, in Section 3, the proposed approach is introduced. Thirdly, in Section 4, an
experimental study is presented to bring out some main properties of the proposed approach.
Finally, a conclusion and some future work are presented.
2. Construction of intuitionistic fuzzy sets: existing approaches
In this section, we focus on existing approaches to build IFS. The following notations are used.
Let U = {u1 , . . . , un } be a discrete universe and let A be a subsetP
of U , and let P = {p1 , . . . , pn }
be a probability distribution on U with 0 ≤ pi+1 < pi ≤ 1 and ni=1 pi = 1.
The question that should be answered is “how to define the intuitionistic fuzzy set A of U
from P ?” There does not exist a lot of approaches in the literature to answer this question, we
present hereafter the 2 main ones. First of all we present the approach based on the definition
of a mass assignment [7], and afterwards we present the approach by [6] based on the use of an
intuitionistic fuzzy generator.
2.1. Basic recalls
Intuitionistic fuzzy sets have been introduced by Atanassov [1].
An IFS A of U is defined as: A = {(u, µA (u), νA (u)) | u ∈ U } with µA : U → [0, 1] and
νA : U → [0, 1] such that for all u ∈ U , 0 ≤ µA (u) + νA (u) ≤ 1. The values µA (u) and νA (u)
are, respectively, the membership degree and the non-membership degree of u to A.
The margin of hesitancy (or intuitionistic fuzzy index) of u to A is defined as πA (u) = 1 −
(µA (u) + νA (u)), it informs about the lack of knowledge about A when it exists. When the
margin of hesitancy according to A is equal to zero for all u, that is to say when µA (u)+νA (u) =
1, ∀u ∈ U , then A is a Zadeh’s fuzzy set.
2.2. The mass assignment approach
This approach has been introduced in [7] and detailed also in [8]. It has been used in Machine
learning applications presented in [9] and [5].
2.2.1. Mass assignment
From a probability distribution P , a mass assignment mA of a fuzzy set A of U can be defined
thanks to the mass assignment theorem introduced in [10] that enables such a construction:
mA (Fi ) = µi − µi+1 for i = 1, . . . , n − 1
mA (Fn ) = µn
where Fi = {u ∈ U | p(u) ≥ pi } and
X
n
µi = |Fi |pi + (|Fj | − |Fj−1 |)pj .
j=i+1
2.2.2. Algorithm to build an IFS from a mass assignment
In [7], the building of an IFS A = {(u, µA (u), νA (u)) | u ∈ U} could be done from two
independent probability distributions on U : P + the probability distribution connected to the
membership of the elements of U to the IFS A, and P − the probability distribution connected
to their non-membership to the IFS A. The process is done according to the following steps [7]:
1. On the one hand, the mass assignment m+ +
A is build from P with the mass assignment
theorem (see Section 2.2.1). As stated in [7]: m+A (u) is the possibility that u has the value
m+A (u) and thus, it is considered that m +
A (u) = µA (u) + πA (u).
2. On the other hand, the mass assignment m− −
A is build from P with the mass assignment
theorem. m− −
A (u) is the possibility that u has the value mA (u) and it is considered that
−
mA (u) = νA (u) + πA (u).
−
3. Finally, the aggregation of m+ A and mA enables to obtain µA and νA taking into account
−
that, for all u ∈ U , µA (u) + νA (u) + πA (u) = 1, it gives m+
A (u) + mA (u) = 1 + πA (u)
+ −
and thus for all u ∈ U , πA (u) = mA (u) + mA (u) − 1 that leads to the determination of
the values µA (u) and νA (u).
By means of the above process, the IFS A ⊆ U can be completely defined from P + and P − .
2.2.3. Discussion
This approach to define the IFS A could have the drawback to produce a negative value for
−
πA (u) as there is no guarantee that m+ A (u) + mA (u) ≥ 1 for all u ∈ U . Indeed, an IFS could
not necessarily be build from any probability distributions and IFS could not be a convenient
model in this case.
However, one solution when facing a negative value for πA (u) is to set it to 0, it is the solution
that has been used in the experimental part of this paper.
In a Machine learning setting, this approach to define the IFS A is interesting because it does
not need any hyper-parameter to be set.
2.3. The fuzzy generator approach
The fuzzy generator approach to build an IFS that is described in this section has been proposed
in [6]. This approach has been mainly used in image segmentation [11] [4], or in clustering
problems [12] to cite some examples of its use.
2.3.1. Intuitionistic fuzzy generator
The definition of an intuitionistic fuzzy generator (IFG) has been introduced in [6]: a function
Φ : [0, 1] → [0, 1] is called an intuitionistic fuzzy generator if Φ(x) ≤ 1 − x for all x ∈ [0, 1].
An example of such a generator is:
1−x
N (x) = , with λ > 0
1 + λx
where N (0) = 1 and N (1) = 0.
2.3.2. Construction of an IFS by means of an IFG
An IFS can be defined from a fuzzy set by means of an IFG [6]: let A = {(u, µA (u)) | u ∈
U} be a fuzzy sets on U , and let Φ be an intuitionistic fuzzy generator, then the set à =
{(u, µA (u), Φ(µA (u))) | u ∈ U} is an intuitionistic fuzzy set on U .
2.3.3. Discussion
In [4] and in [12], the IFG presented in Section 2.3.1 is used to build the IFS, but another IFG
can be used, see for instance [11].
In a Machine learning setting, this approach to build IFS is very promising, in particular
in non-supervised Machine learning problems. However, the choice of at least one hyper-
parameter, the IFG, should be done. Moreover, if the IFG given in Section 2.3.1 is used, it is also
mandatory to choose a good value for λ.
3. A new approach to build an IFS
The proposed approach is dedicated to the construction of an IFS in a machine learning setting
(but not limited to this kind of application domain). As a consequence, we consider that the
information provided to build the IFS is not only probability distributions but a complete set of
instances separated into two classes.
Let X be a universe of values and U = {u1 , . . . , un } ⊆ X be a discrete subset of X , and let
C = {+, −} be a set of classes over the elements of X . For i = 1, . . . , n, let n+ i be the number
of instances of ui that are associated with class +, and P let n− i be the number Pof instances of
n n
ui that are associated with class −. We denote by n = i=1 ni and n = i=1 n−
+ + −
i and we
−
assume in the following that n+ 6= 0 and n− 6= 0. Moreover, we denote by ni = n+ i + ni ,
∀i = 1, . . . , n and we assume that ni 6= 0, ∀i = 1, . . . , n.
− −
In the following, a training set is the tuple hU , C, (n+ +
1 , . . . , nn ), (n1 , . . . , nn )i.
+ − −
+
The set of values n1 , . . . , nn (resp. n1 , . . . , nn ) defines a probability distribution P +
n+ n−
(resp. P − ) over U such that P + (ui ) = ni+ (resp. P − (ui ) = ni− ).
These two probability distributions provide us with information about the elements of U and
their membership (P + ) or non-membership (P − ) to a set A ⊆ U that we want to build.
Our approach aims at defining A as an intuitionistic fuzzy set of U according to the knowledge
that is provided by a training set. This approach is composed of two main steps. First of all,
from the training set two corresponding weighted distributions are built taking into account
the representativeness of the training set. Secondly, the IFS A is built using these two weighted
distributions. The process is detailed in the following.
3.1. Representativeness and weighted distributions
− −
Given a training set T S = hU , C, (n+ +
1 , . . . , nn ), (n1 , . . . , nn )i, our approach aims at building
an IFS A over U . As usual in machine learning, the training set provides only a restricted view
about X , thus this brings out the question of its representativeness.
3.1.1. Representativeness degree
To highlight the representativeness of a training set, we introduce the use of a degree that
should be set either by a user that knows the problem under study, or automatically by means
of an objective decision process. This degree highlights how the training set can be considered
as sufficiently representative to infer knowledge that could be generalised to X .
− −
The representativeness degree ρ ∈ [0, 1] of the training set hU , C, (n+ +
1 , . . . , nn ), (n1 , . . . , nn )i
is such that:
• ρ = 0 when the training set is not representative of X . In this case, the knowledge it
provides are not usable for any elements of X not in U .
• ρ = 1 when the training set is completely representative of X . In this case, the knowledge
it provides is completely usable for any elements of X .
• the greater ρ, the more representative the training set.
The representativeness degree evaluates how much we could be confident in the fact that
−
the probability distributions induced by (n+i )i=1,..,n and (ni )i=1,..,n reflects the respective
probability distributions on X . This representativeness degree is either given by the user
that appreciate how the knowledge provided by U could be generalised to X , or it could be
determined automatically (to be studied in future work).
3.1.2. Lack-of-knowledge degree
The representativeness degree is an information about the training set. It could be used to
weight the information associated with any example u ∈ U provided in the training set.
− −
Definition 1. Let T S = hU , C, (n+ +
1 , . . . , nn ), (n1 , . . . , nn )i be a training set of X , and ρ ∈ [0, 1]
be the representativeness degree of T S. The lack-of-knowledge degree of ui ∈ U is defined as
ni
l(ui ) = ρ ∗ nmax with nmax = sup{i=1,...,n} ni .
The lack-of-knowledge degree takes into account not only the representativeness of the
training set, but also, for a given property (membership or non-membership) the number of
elements that possess this property. This is represented by the use of nmax = sup{i=1,...,n} ni to
weight the value ni . This degree enables us to take into account the representativeness of U to
reduce the influence of an example used for the construction of the IFS A from P + and P − .
ni
It is easy to see that, for all i = 1, . . . , n, l(ui ) ∈ [0, 1] as ρ ∈ [0, 1] and 0 ≤ nmax ≤ 1.
3.1.3. Weighted probability distributions
The lack-of-knowledge degree is used to weight the influence of the information related to an
example u with regard to the global information brought out by the training set.
− −
Definition 2. Let T S = hU , C, (n+ +
1 , . . . , nn ), (n1 , . . . , nn )i be a training set of X , and ρ ∈ [0, 1]
be the representativeness degree of T S. The weighted probability distributions over U related to
n+ n−
the classes C are defined ∀i = 1, . . . , n as: Pw+ (ui ) = l(ui ) nii and Pw− (ui ) = l(ui ) nii .
n+ n−
It is easy to see that, for all i = 1, . . . , n, as l(ui ) ∈ [0, 1] and nii ∈ [0, 1] (resp. nii ∈ [0, 1]),
we have Pw+ (ui ) ∈ [0, 1] (resp. Pw− (ui ) ∈ [0, 1]).
3.2. Construction of an IFS
From a training set, associated with a representativeness degree, it is possible to define an
intuitionistic fuzzy set taking into account the information related to the classes: class + repre-
senting the information related to the membership of elements to A and class − representing
the information related to their non-membership to A.
− −
Definition 3. Let T S = hU , C, (n+ +
1 , . . . , nn ), (n1 , . . . , nn )i be a training set of X , and ρ ∈ [0, 1]
be the representativeness degree of T S. The intuitionistic fuzzy set A is defined as
A = {(u, Pw+ (u), Pw− (u)) | u ∈ U}.
Indeed, it is easy to see that A is an IFS as we have for all ui ∈ U:
• Pw+ (ui ) ∈ [0, 1] and Pw− (ui ) ∈ [0, 1] (see Section 3.1.3);
n+ n−
• Pw+ (ui ) + Pw− (ui ) = l(ui ) nii + l(ui ) nii = l(ui ) and 0 ≤ l(ui ) ≤ 1 (see Section 3.1.2).
4. Experimental study
A preliminary experimental study is presented in this section to study the influence of the
representativeness degree, and to compare the IFS built by means of each of the presented
approaches.
The experiments are conducted on a training set proposed in [7]. The set U contains 10
elements associated with different sizes ni . In [7], two probability distributions on U are
given to represent the membership and the non-membership to the IFS A to build. In our
experiments, these probability distributions have been used to generate the values (n+ +
1 , . . . , nn ),
− −
and (n1 , . . . , nn ) by considering that the total size of the set + (resp. −) is 1000 elements. The
resulting training set is presented in the 2 first columns in Table 1.
A global analysis of this training set leads us to consider that, among the elements u, there
are some elements that are more representative than others. For instance, u1 has only a size of
125 where u5 is associated with a size of 351. Indeed, this highlights the fact that we could be
less confident in the decision that could be drawn from this training set for u1 than for u5 .
4.1. Comparison of approaches
In Table 1, we present the IFS built by each of the 3 approaches presented in this paper: our
proposed approach with ρ = 1 (complete representativeness of the training set), the approach
by [7], and the approach by [6] with λ = 0.5. Concerning this last approach, as it takes into
account only one probability distribution, the given IFS is obtained with only the information
related to +.
The three IFS built by these three approaches are very different. It should be seen that
• the IFS built by [6] does not provide a great hesitancy (intuitionistic indices π) unlike the
other 2 approaches. In this approach, this hesitancy is related to the choice of λ, but it
takes into account only one information from the training set: the probability distribution
associated with the membership of the elements.
Size Proposed app. (ρ = 1) Approach by [7] Approach by [6] (λ = 0.5)
+ - µA νA πA µS νS πS µB νB πB
u1 0 125 0.0 0.356 0.644 0.0 0.994 0.006 0.0 1.0 0.0
u2 0 128 0.0 0.365 0.635 0.0 1.0 0.0 0.0 1.0 0.0
u3 34 117 0.097 0.333 0.57 0.038 0.796 0.166 0.034 0.934 0.032
u4 165 80 0.47 0.228 0.302 0.26 0.272 0.468 0.165 0.717 0.118
u5 301 50 0.858 0.142 0.0 0.5 0.0 0.5 0.301 0.537 0.162
u6 301 50 0.858 0.142 0.0 0.5 0.0 0.5 0.301 0.537 0.162
u7 165 80 0.47 0.228 0.302 0.26 0.272 0.468 0.165 0.717 0.118
u8 34 117 0.097 0.333 0.57 0.038 0.796 0.166 0.034 0.934 0.032
u9 0 128 0.0 0.365 0.635 0.0 1.0 0.0 0.0 1.0 0.0
u10 0 125 0.0 0.356 0.644 0.0 0.994 0.006 0.0 1.0 0.0
Table 1
Comparison of the 3 approaches.
• the IFS built by [7] provides an hesitancy, however, it does not take into account the
representativeness of each element.
• the IFS built by the proposed approach highlights the importance of the information
provided on each element. This leads to an hesitancy more important for u1 than for u5
for instance.
4.2. Influence of the representativeness degree
In order to study the influence of the representativeness degree, the proposed approach has
been used with different values of this degree for the same training set. The training is similar
to the one detailed in the previous section. The resulting IFS obtained with the approach used
with increasing values of the representativeness degree ρ is presented in Table 2. This table
should be completed with the result for ρ = 1 given in Table 1.
These results show the great influence of the representativeness degree in the resulting IFS,
this influence can be highlighted by the values of the intuitionistic indices π of the resulting
IFS: the greater the representativeness degree, the lower the values of the intuitionistic indices.
This perfectly highlights the behaviour of the approach to build an IFS that is needed in
a Machine learning setting: the intuitionistic fuzzy index is a representation of the lack of
knowledge that is associated with the membership of an element u to the IFS A [13], therefore
the representativeness of the training set induces the lack of knowledge that is associated with
the data belonging to this training set.
A main question arises here on ”how to choose a good value for ρ?”. This value could be set by
the user if he/she has sufficient knowledge to evaluate the representativeness of the training set.
Otherwise, ρ could also be set by means of an automatic approach, for instance, by considering
the size of the training set with regard to the dimension of X .
Size Proposed app. (ρ = 0.1) Proposed app. (ρ = 0.25) Proposed app. (ρ = 0.75)
+ - µ0.1 ν0.1 π0.1 µ0.25 ν0.25 π0.25 µ0.75 ν0.75 π0.75
u1 0 125 0.0 0.036 0.964 0.0 0.089 0.911 0.0 0.267 0.733
u2 0 128 0.0 0.036 0.964 0.0 0.091 0.909 0.0 0.274 0.726
u3 34 117 0.01 0.033 0.957 0.024 0.083 0.892 0.073 0.25 0.677
u4 165 80 0.047 0.023 0.93 0.118 0.057 0.825 0.353 0.171 0.476
u5 301 50 0.086 0.014 0.9 0.214 0.036 0.75 0.643 0.107 0.25
u6 301 50 0.086 0.014 0.9 0.214 0.036 0.75 0.643 0.107 0.25
u7 165 80 0.047 0.023 0.93 0.118 0.057 0.825 0.353 0.171 0.476
u8 34 117 0.01 0.033 0.957 0.024 0.083 0.892 0.073 0.25 0.677
u9 0 128 0.0 0.036 0.964 0.0 0.091 0.909 0.0 0.274 0.726
u10 0 125 0.0 0.036 0.964 0.0 0.089 0.911 0.0 0.267 0.733
Table 2
Results with various representativeness degrees.
5. Conclusion
In this paper, the problem of building intuitionistic fuzzy sets is tackled. After a survey of the
two main existing approaches from the literature to build IFS from probability distributions, a
new approach is proposed. The aim of this new approach is to be used in a Machine learning
setting where it is important to take into account the representativeness of the training set.
The proposed approach is based on the use of a representativeness degree of the given training
set and the determination of a lack-of-knowledge degree to weight the importance of each
element of the training set. This approach enables to build an IFS where the intuitionistic fuzzy
index is a good indicator of the lack of knowledge associated with the given training set.
In future work, even if a good choice could be to set ρ to 1, a first study to conduct will be to
find a way to make the choice of a good value for ρ, for instance by means of some automatic
ways to set this value. Secondly, the resulting IFS will be used in a complete Machine learning
approach, to build a classification model from a given training set. Moreover, future work
will also deepen the study on the proposed approach in term of convergence property and the
complexity
Acknowledgments
The author would like to thank the anonymous reviewers for their valuable suggestions.
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