ORIGINAL ARTICLE
Concepts reduction in formal concept analysis with fuzzy settingusing Shannon entropy
Prem Kumar Singh • Aswani Kumar Cherukuri •
Jinhai Li
Received: 17 October 2013 / Accepted: 12 November 2014
� Springer-Verlag Berlin Heidelberg 2014
Abstract In this paper we propose a method for reducing
the number of formal concepts in formal concept analysis
of data with fuzzy attributes. We compute the weight of
fuzzy formal concepts based on Shannon entropy. Further,
the number of fuzzy formal concepts is reduced at chosen
granulation of their computed weight. We show that the
results obtained from the proposed method are in good
agreement with Levenshtein distance method and interval–
valued fuzzy formal concepts method but with less com-
putational complexity.
Keywords Formal concept analysis � Fuzzy concept
lattice � Fuzzy formal concept � Granulations � Levenshtein
distance � Shannon entropy
1 Introduction
Formal concept analysis (FCA) is a mathematical model
based on the applied lattice theory that offers conceptual
knowledge representation in a hierarchical order [51]. FCA
starts the analysis from a given incidence matrix which
comprises a set of objects, a set of attributes and a binary
relationship between them. The basic outputs of FCA are
formal concepts, concept lattice and attribute implications.
Formal concept is regarded as a basic unit of human
thought and plays a major role in knowledge processing
tasks [22]. It is a maximal pair of set of objects (extent) and
its properties (intent) closed with Galois connection.
Concept lattice provides a hierarchical order visualization
between the discovered formal concepts and has been
applied in various fields like knowledge discovery [4],
information retrieval [7], designing role based access
control [8], information sciences [46] and knowledge pro-
cessing tasks [40]. Attribute implications provide depen-
dency between given set of attributes and are applied in
fields like health care [3], clustering [6], formal context ([5,
50]) and, decision formal context [31].
FCA was incorporated in the fuzzy environment by
Burusco and Fuentes–Gonzales [15] to represent the
uncertainty and vagueness (in data) more precisely than
crisp setting. FCA with fuzzy setting represents the
knowledge in the form of fuzzy formal concepts and their
hierarchical order visualization using fuzzy concept lattice
[11]. The properties of fuzzy concept -lattice were applied
in domains like text mining [16], collaborative filtering
[47], semantic web [21] as well as in ontology [17]. The
complexity for generating the set of all formal concepts
from a given formal context and their visualization is
exponential [18]. Due to this fact proper analysis of
underlying knowledge using FCA becomes difficult, when
a large number of formal concepts are generated from a
given fuzzy formal context [20]. In general, reduction can
be applied on formal context [52], formal concepts [41]
and implications ([27, 49]). In this paper we have focused
on reducing the number of formal concepts. Several
approaches were proposed for reducing the number of
formal concepts generated from a given binary formal
context ([2, 26]), a decision formal context ([28, 29, 32]), a
P. K. Singh � A. K. Cherukuri (&)
School of Information Technology and Engineering, VIT
University, Vellore 632014, Tamilnadu, India
e-mail: [email protected]
P. K. Singh
e-mail: [email protected]
J. Li
Faculty of Science, Kunming University of Science and
Technology, Kunming 650500, People’s Republic of China
e-mail: [email protected]
123
Int. J. Mach. Learn. & Cyber.
DOI 10.1007/s13042-014-0313-6
fuzzy formal context ([12, 42, 44]), a real context ([30]) as
well as for a rough context ([53, 54]). Recently, few
researchers have concentrated on the selection of important
formal concepts. Babin and Kuznetsov [10] introduced
concepts stability method to measure the importance of
concepts. Belohlavek and Macko [13] investigated impor-
tant concepts using weights and discussed their applica-
tions [14]. Medina [37] and Ma et al. [36] reduced the
attributes based on object and attribute oriented concept
lattices. Mi et al. [38] reduced the attribute using axialities.
Dias and Viera [19] proposed junction based object simi-
larity (JBOS) method for reducing the number of formal
concepts. Li et al. [33] and Zhang et al. [56] discussed
weighted concept lattice and its applications. Kang et al.
[25] provided a method for reducing the size of fuzzy
concept lattice at different granulation. Li et al. [34] dis-
cussed K–medoids clustering method. Martin et al. [35]
introduced a method to measure the changes in fuzzy
concept lattice using Levenshtein distance. In this paper we
have focused on reducing the number of formal concepts in
FCA with fuzzy setting. For this purpose, we have used the
theory of Shannon entropy and granulation.
Shannon entropy provides the best possible measure-
ment of uncertainty in information theory [48] whereas
granulation provides a way to process the information into
granules [52]. In this paper we have proposed a method to
compute the weight of a given fuzzy formal concept using
Shannon entropy. Further, the number of fuzzy formal
concepts is reduced at chosen granulation of their com-
puted weight. Recently two methods have been introduced
in FCA with fuzzy setting to represent the formal concepts
in the given interval [0, 1]: One by Martin et al. [35] and
another by Prem Kumar and Aswani Kumar [41]. Hence,
the knowledge processed by the obtained fuzzy concepts
from the proposed method is compared with Levenshtein
distance method [35] and interval–valued fuzzy concepts
method [41].
Rest of this paper is organized as follows: Sect. 2 pro-
vides a brief background about FCA in the fuzzy setting.
Section 3 discusses the computation of Shannon entropy
based weighted concepts and their reduction. Section 4
provides an empirical analysis of the proposed method.
Section 5 provides conclusions followed by acknowl-
edgements and references.
2 Formal concept analysis in the fuzzy setting
FCA in the fuzzy setting starts analysis from a given
incidence matrix called fuzzy formal context. Let L be a
scale of truth degrees of some structure like complete re-
siduated lattice L=(L,...). Then, a fuzzy formal context (L–
context or L–context) is a triplet F = (O, P, ~R), where O is a
set of objects, P is a set of attributes and ~R is an L–relation
between O and P, ~R: O �P! L [11]. Each ~Rðo; pÞ 2 L
represents the membership value at which the object o 2 O
has the attribute p 2 P with a certain degree in [0, 1] where
L is a support set of some complete residuated lattice L.
A residuated lattice L=ðL;^;_;�;!; 0; 1Þ is the basic
structure of truth degrees. L is a complete residuated lattice
iff [11, 39]:
(1) ðL;^;_; 0; 1Þ is a complete lattice.
(2) ðL;�; 1Þ is commutative monoid.
(3) � and ! are adjoint operators (called as multipli-
cation and residuum, respectively), that is a� b� c
iff a� b! c; 8a; b; c 2 L.
The operators � and ! are defined distinctly by Lu-
kasiewicz, Godel, and Goguen t–norms and their residua as
described below [11, 39]:
Lukasiewicz:
• a� b = max (a?b-1, 0);
• a! b=min (1-a?b, 1).
Godel:
• a� b = min (a, b);
• a! b = 1 if a� b, otherwise b.
Goguen:
• a� b = a � b;
• a! b = 1 if a� b, otherwise b/a.
Classical logic is a special case of complete residuated
lattice which is represented as ð 0; 1f g;^;_;�;!, 0, 1).
For any L–set A 2 LO of objects, and B 2 LP of attri-
butes, we can define an L–set A " 2 LP of attributes and an
L–set B # 2 LO of objects as follows [11, 39]:
(1) A "ðpÞ ¼ ^o2OðAðoÞ ! ~Rðo; pÞÞ;(2) B #ðoÞ ¼ ^p2PðBðpÞ ! ~Rðo; pÞÞ.A "ðpÞ is interpreted as the L-set of all attributes p 2 P
shared by objects from A. Similarly, B #ðoÞ is interpreted as
the L-set of all objects o 2 O having the attributes from
B in common. The fuzzy formal concept is a pair of (A,
B)2 LO � LP satisfying A" ¼ B and B# ¼ A, where A is
called as extent and B is called as intent.
The pair ("; #) is known as a Galois connection [39].
When the operator (") is applied on a fuzzy set of objects, it
provides a fuzzy set of attributes with its membership value
being maximal with respect to integrating the information
from all the objects. Consequently, when the operator (#) is
applied on the fuzzy set constituted by these covered
attributes resulting from integrating the membership
Int. J. Mach. Learn. & Cyber.
123
information between objects and attributes. It takes a fuzzy
set of objects with its membership value being maximal
with respect to integrating the information from the attri-
butes. Since we consider the maximal membership value,
we cannot find any fuzzy set of objects (attributes) which
can make the membership value of the obtained fuzzy set
of attributes (objects) bigger, if the pair of the set of objects
and the set constituted by its covered attributes forms a
fuzzy formal concept.
The set of fuzzy formal concepts equipped with the
relation (� ) as follows: ðA1;B1Þ� ðA2;B2Þ ()A1 � A2ð() B2 � B1Þ. Together with this ordering, in the
complete lattice there exist an infimum and a supremum for
some formal concepts given by ([18, 22]):
• ^j2JðAj;BjÞ=ðT
j2J Aj; ðS
j2J BjÞ#"Þ,• _j2JðAj;BjÞ=ðð
Sj2J AjÞ"#;
Tj2J BjÞ.
Generating the formal concepts from the given formal
context and their hierarchical order visualization in the
concept lattice structure are an important issue for practical
applications of FCA [9, 11, 26]. In this process a major
issue is reducing the number of fuzzy formal concepts and
the size of fuzzy concept lattice [22, 52]. To deal with this
problem, in this paper we propose a novel method for
selecting the fuzzy formal concepts having higher weight
than chosen granulation and remove other concepts. The
weight of given fuzzy formal concepts is computed using
the Shannon entropy as illustrated in the next section.
3 Shannon entropy based weighted fuzzy concepts
and their reduction
In this section, we focus on introducing a method for
computing the weight of formal concepts in FCA with
fuzzy setting. The proposed method provides a possible
measurement of uncertainty in the formal concept. For this
purpose, the theory of Shannon entropy is utilized in this
paper. The computed degree of uncertainty in (fuzzy)
attributes is regarded as the weight value of the attributes.
Further, the proposed method provides a way to reduce the
number of fuzzy formal concepts at chosen granulation
[h1;h2] of their computed weight where 0 � h1� h2 � 1 .
Let us consider any object oi 2 O of a given fuzzy
formal context F. The probability (P) of oi possessing the
corresponding attribute pj can be computed by Pðpj=oiÞwhere pj and oi represent j-th attribute and i-th object,
respectively. For this purpose we do not consider the fuzzy
membership–value for each object. The EðpjÞ represents
the average information weight of the object ( oi) to pro-
vide the attribute pj 2 P followed by its weight wj.
WeightðkÞ represents the average weight of intent, based on
its attribute weight(wj) where k represents number of
attributes in the intent of given formal concepts i.e. k� jmj.Following is the details:
(1) EðpjÞ ¼ �Pm
j¼1 Pðpj=oiÞ log2(Pðpj=oiÞÞ, where m
represents the total number of attributes in the given
context F.
(2) wj ¼ EðpjÞ=Pm
j¼1 EðpjÞ.(3) WeightðkÞ ¼
Pkj¼1ðwjÞ=k.
In order to explore the deviation of wj from WeightðkÞ, we
can evaluate DðkÞ as follows:
DðkÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðPjwj �WeightðkÞjÞ=m
p, where DðkÞ denotes
the deviation of the multi–attribute intent value. It mea-
sures the absolute difference between any one two formal
concepts from the average information weight. Hence it
provides the deviation of each formal concept from the
avearge weight to decide their distances and importance. If
m=1 thenPjwj � weightðkÞj =0. Similarly, we can com-
pute for other values of m. The above steps can be used for
computing the weight of each extent as well.
For illustration of the proposed method we consider a
fuzzy formal context shown in Table 1 [35]. The concept
lattice of this context is generated using ConExp tool [24]
(http://conexp.sourceforge.net/) is shown in Table 2 and
the concept lattice is shown in Fig. 1.
Formal concept is basic unit of knowledge containing a
pair of set of objects (extent) and set of attributes(intent).
Formal concepts can be distinguished by their extent or
intent, independently [43, 51]. Hence, we can consider
either concept–extent or concept–intent for computing the
weight. In this paper, concept–extent is considered to
compute the weight of fuzzy formal concept generated
from Table 1. The extent of each fuzzy formal concept
shown in Fig. 1 is shown in Table 2. Then for each object
in Table 1, we compute the probability, average informa-
tion weight and overall weight. These values are shown in
Table 3. Further we calculate the weights for formal con-
cept based on the objects available in their extents that are
shown in Table 2. These values are summarized in Table 4.
Now we can remove some of the fuzzy formal concepts
shown in Fig. 1 at chosen granulation for their computed
weight (as shown in Table 5). Granulation is a computing
paradigm that is concerned in the processing of complex
information entities called information granules to derive
Table 1 A fuzzy formal
contextp1 p2 p3
o1 0.3 0.2 0.1
o2 0.5 1.0 0.6
o3 0.4 1.0 0.3
Int. J. Mach. Learn. & Cyber.
123
the knowledge [55] and applied in fuzzy concept lattice
representation [25], attribute reduction [27], ontology [33],
formal context decomposition [42], knowledge reduction
[52] and finding frequent formal concepts [56]. The prop-
erty of granulation helps us to remove some of the fuzzy
formal concepts whose intent weight are out of the interval
[h1; h2] as shown in Table 5.
The steps for the removal of fuzzy formal concepts at
chosen granulation are defined in Table 6. The proposed
algorithm first computes the weight of each attribute
(through steps 1–5). Then, it computes the weight of each
fuzzy formal concept by summation of weight of each
attribute available in the corresponding intent (extent) of the
given fuzzy concept (through steps 6–8). The weight of each
fuzzy formal concept is stored and arranged. We can observe
Table 2 Extent for each fuzzy formal concepts shown in Fig. 1
Extent of Fig. 1 Nomenclature
o1=1:0þ o2=1:0þ o3=1:0 C1
o1=0:2þ o2=1:0þ o3=1:0 C2
o1=0:3þ o2=0:5þ o3=0:4 C3
o1=0:2þ o2=0:5þ o3=0:4 C4
o1=0:1þ o2=0:5þ o3=0:3 C5
o1=0:1þ o2=0:6þ o3=0:3 C6
Fig. 1 Fuzzy concept lattice
generated from Table 1
Table 3 Computed weight–wi
for each object of Table 1Each
object
P(o) EðoÞ wi
o1=0:1 1.00 0.00 0.00
o1=0:2 0.66 0.3956 0.1428
o1=0:3 0.33 0.5278 0.1905
o1=1:0 0.00 0.00 0.00
o2=0:5 1.00 0.00 0.00
o2=0:6 0.66 0.3956 0.1428
o2=1:0 0.33 0.5278 0.1905
o3=0:3 1.00 0.00 0.00
o3=0:4 0.66 0.3956 0.1428
o3=1:0 0.33 0.5278 0.1905
Table 4 Computed WeightðkÞ for each extent shown in Table 2
Concept Extent shown in Table
2
Sum of wi WeightðkÞ
C1 o1=1:0þ o2=1:0þ o3=1:0 0.0 ? 0.0 ? 0.1905 0.0635
C2 o1=0:2þ o2=1:0þ o3=1:0 0.1428 ? 0.00 ? 0.1905 0.1111
C3 o1=0:3þ o2=0:5þ o3=0:4 0.1905 ? 0.00 ? 0.1428 0.1111
C4 o1=0:2þ o2=0:5þ o3=0:4 0.1428 ? 0.00 ? 0.1428 0.0952
C5 o1=0:1þ o2=0:5þ o3=0:3 0.00 ? 0.00 ? 0.00 0.0000
C6 o1=0:1þ o2=0:6þ o3=0:3 0.0 ? 0.1428 ? 0.00 0.0476
Table 5 Reduction of concepts from Fig. 1 using proposed method
Weight at granulation (WeightðkÞ) Removed fuzzy formal concepts
0:1111 C1, C2, C3, C4, C5, C6
0:1111\WeightðkÞ� 1 C1, C4, C5, C6
0:0952\WeightðkÞ� 0:1111 C1, C5, C6
0:0635\WeightðkÞ� 0:0952 C5, C6
0:0635\WeightðkÞ� 0:0476 C4
0\WeightðkÞ� 0:0476 ø
Table 6 Proposed algorithm for removing the concepts
Input: Array [1 : k] of formal concepts.
Outputs: Formal concepts WðkÞ at granulation 0� h1� h2� 1 .
1.for j ¼ 1; :::;m where m is number of attributes
2. Compute the probability of each attribute Pðpj=oiÞ3. EðpjÞ ¼ �
Pmj¼1 Pðpj=oiÞ log2(Pðpj=oiÞÞ//Average information
weight
4. wj ¼ EðpjÞ=Pm
j¼1 EðpjÞ // Computing weight
5. end for
6. for j ¼ 1; :::; k where k is number of concepts
7. Weight of attributes (objects) in the intent (extent)=Pm
j¼1ðwjÞ
8. Weight of the concept WðkÞ=Pk
j¼1ðwjÞ=k
9. Set the granulation 0� h1� h2� 1
10. if (WðkÞ� h1 or WðkÞ� h2)
11. remove the concept
12. end if
13. end for
Int. J. Mach. Learn. & Cyber.
123
that the proposed algorithm uses the collection of fuzzy
formal concepts as information granules that can be chosen
at the granulation h (0 � h� 1). Further the obtained fuzzy
concepts can be arranged together using their computed
weight. The (weighted) fuzzy formal concept can be
removed whose weight is out of the chosen granulation
0� h1� h2� 1 (through steps 9 to 11). In this process, the
proposed algorithm selects the concepts having computed
weight between the chosen granulation and discards the
remaining fuzzy concepts. Complexity of the proposed
algorithm is based on average weight of the intent or extent
i.e. O (m lnðmÞ) or O (n lnðnÞ) where m is number of attributes
and n is number of objects in the formal context.
This proposed method is useful for the researchers in
following tasks:
• Knowledge processing tasks([1, 40]).
• Selecting some of the important concepts ([13], [19,
56])
• Reducing the size of concept lattice ([2, 12], [41] ) and,
• Reducing the attribute–implications ([3, 5] [20, 49, 50])
4 Empirical analysis
Recently, several methods have been proposed for computing
the weight of given formal concepts [10, 12–14, 19, 23, 33,
35, 41, 52, 56]. Most of the available approaches are based on
selecting the concepts through a specific measurement of their
importance. The proposed method is different from all the
above methods mainly in two aspects as given below:
• The proposed method computes the probability of each
attribute (objects) of given fuzzy formal concepts
followed by their weight (using Shannon entropy).
Hence, it provides a possible measurement of uncer-
tainty in the fuzzy formal concept, and
• The proposed method gives priority to the fuzzy formal
concepts which are having the computed weight
between the chosen granulation. This fact provides
flexibility to the user in selecting some of the important
fuzzy formal concepts and building a corresponding
sub–lattice of desired size.
To compare the performance of the proposed method, we
have considered two recent methods that select the formal
concepts based on similarity ([35, 41]).
4.1 Comparison of the proposed method
with Levenshtein distance
We compare the proposed method with Levenshtein dis-
tance method introduced by Martin et al. [35] in FCA with
fuzzy setting. Levenshtein distance is a string metric for
measuring the difference between two words based on
number of single–character edits (i.e. insertions, deletions
or substitutions) required to change one word into the
other. To illustrate the comparison, we have considered the
fuzzy contexts discussed in Martin et al. [35] as shown in
Tables 7 and 8. The fuzzy concept lattices generated from
these two contexts are shown in Figs. 2 and 3, respectively.
The computed weight for each object of Table 7 is shown
in Table 9 using the proposed method. The weight of each
formal concept of Fig. 2 is depicted in Table 10 based on
the objects contained in their extent. Similarly, we can
compute the weight for the each fuzzy formal concept
shown in Fig. 3. The computed weight for each object of
Table 8 is shown in Table 11. Table 12 shows the com-
puted weight for each fuzzy formal concept shown in Fig. 3
based on the objects contained in their extent.
Table 13 compares Figs. 2 and 3 using the computed
weight of their fuzzy formal concepts and concludes that:
• The computed weight for the (fuzzy) extent of Fig. 2
and Fig. 3 are same.
Table 7 A fuzzy formal con-
text–1p1 p2 p3
o1 0.1 0.2 0.1
o2 0.5 1.0 0.6
o3 0.4 1.0 0.3
Table 8 A fuzzy formal con-
text–2p1 p2 p3
o1 0.1 0.2 0.1
o2 0.4 1.0 0.6
o3 0.4 1.0 0.3
Fig. 2 Fuzzy concept lattice generated from Table 7
Int. J. Mach. Learn. & Cyber.
123
• Only the memberships of C22–C32 and C23–C33 are
differing as given in Table 10 and Table 12.
The Levenshtein distance 0.2 of both the lattices are
already computed by Martin et al. [35] with following
conclusions: The extents of Fig. 2 and Fig. 3 are nearly
identical, differing in only the membership values of the
fuzzy formal concepts– C22–C32 and C23–C33. We can
observe that the conclusions obtained from the proposed
method for Fig. 2 and Fig. 3 are in good agreement with
Martin et al. [35].
Table 14 shows the comparison between Levenshtein
distance and the proposed method which provides the
following observations:
• Both Levenshtein distance and the proposed method
measure the information content in the fuzzy formal
concept. However, the proposed method provides a set
of fuzzy formal concepts at the chosen granulation
which helps in reducing the size of concept lattice.
• Levenshtein distance may provide infinite cost for the
given formal concepts while the proposed method
provides the computed weight of formal concepts in the
interval [0, 1].
• Levenshtein distance method is based on the optimization
of three operations: insertion(i), deletion(d) and replace-
ment (r) while the proposed method is based on the
probability of each attribute and their average weight.
Hence, computing the weight of fuzzy formal concepts is
easier than their edit cost using Levenshtein distance.
4.2 Comparison of the proposed method with interval–
valued fuzzy concepts
Recently, Prem Kumar and Aswani Kumar [41] discussed a
method to reduce the number of fuzzy formal concepts
Fig. 3 Fuzzy concept lattice generated from Table 8
Table 9 Computed weight–wi for single object of Table 7
Each object P(o) EðoÞ wi
o1=0:1 1.00 0.00 0.00
o1=0:2 0.33 0.52 0.22
o2=0:5 1.00 0.00 0.00
o2=0:6 0.66 0.39 0.16
o2=1:0 0.33 0.52 0.22
o3=0:3 1.00 0.00 0.00
o3=0:4 0.66 0.39 0.16
o3=1:0 0.33 0.52 0.22
Table 10 Computed WeightðkÞ for each extent in Fig. 2
Concept Extent Sum of weight wi WeightðkÞ
C21 o1=0:2þ o2=1:0þ o3=1:0 0.22 ? 0.22 ? 0.22 0.22
C22 o1=0:1þ o2=0:5þ o3=0:4 0.00 ? 0.00 ? 0.16 0.053
C23 o1=0:1þ o2=0:5þ o3=0:3 0.00 ? 0.00 ? 0.00 0.00
C24 o1=0:1þ o2=0:6þ o3=0:3 0.00 ? 0.16 ? 0.00 0.053
Table 11 Computed weight–wi
for each object of Table 8Single
object
P(o) EðoÞ wi
o1=0:1 1.00 0.00 0.00
o1=0:2 0.33 0.52 0.22
o2=0:4 1.00 0.00 0.00
o2=0:6 0.66 0.39 0.16
o2=1:0 0.33 0.52 0.22
o3=0:3 1.00 0.00 0.00
o3=0:4 0.66 0.39 0.16
o3=1:0 0.33 0.52 0.22
Table 12 Computed (WeightðkÞ) for each extent for Fig. 3
Concept Extent Sum of Weight wi WeightðkÞ
C31 o1=0:2þ o2=1:0þ o3=1:0 0.22 ? 0.22 ? 0.22 0.22
C32 o1=0:1þ o2=0:4þ o3=0:4 0.00 ? 0.00 ? 0.16 0.053
C33 o1=0:1þ o2=0:4þ o3=0:3 0.00 ? 0.00 ? 0.00 0.00
C34 o1=0:1þ o2=0:6þ o3=0:3 0.00 ? 0.16 ? 0.00 0.053
Table 13 Comparison of Fig. 2 and Fig. 3 using weight of concept/
intent and distance
WeightðkÞ in Fig. 2 WeightðkÞ in Fig. 3 Levenshtein distance
C21–0.22 C31–0.22 Levenshtein distance
C22–0.053 C32–0.053 between Fig. 2 and Fig. 3
C23–0.00 C33–0.00 is 0.2, which includes
C24– 0.053 C34– 0.053 they are very similar [35].
Int. J. Mach. Learn. & Cyber.
123
using interval–valued fuzzy set based on their specific
measurement of importance in the interval [0, 1]. The
proposed method in this paper also measures the impor-
tance of the fuzzy formal concepts in the interval [0, 1].
Hence, in this section we analyze the knowledge reduction
from both of the methods. For the illustration, a fuzzy
formal context shown in Table 15 is considered [23]. Prem
Kumar and Aswani Kumar [41] have already discussed the
interval–valued fuzzy formal concepts and its extension to
bipolar fuzzy formal concepts [45].
The fuzzy formal concepts generated from the fuzzy
formal context shown in Table 15 are:
1. fø; 1:0=p1 þ 1:0=p2 þ 1:0=p3 þ 1:0=p4 þ 1:0=p5
þ1:0=p6g2. f0:5=o1; 1:0=p2 þ 1:0=p3 þ 1:0=p4 þ 1:0=p5g3. f1:0=o2; 1:0=p1 þ 1:0=p2 þ 1:0=p3g4. f0:5=o3; 1:0=p1 þ 1:0=p2 þ 1:0=p6g5. f0:5=o1 þ 0:5=o5; 1:0=p3 þ 1:0=p4g6. f0:5=o1 þ 0:5=o4; 1:0=p4 þ 1:0=p5g7. f1:0=o1; 1:0=p2 þ 0:5=p3 þ 0:5=p4 þ 1:0=p5g8. f0:5=o1 þ 1:0=o2; 1:0=p2 þ 1:0=p3g9. f1:0=o2 þ 0:5=o3; 1:0=p1 þ 1:0=p2g
10. f1:0=o3; 0:5=p1 þ 0:5=p2 þ 1:0=p6g11. f0:5=o1 þ 1:0=o5; 1:0=p3 þ 0:5=p4g12. f0:5=o1 þ 1:0=o4; 1:0=p4 þ 0:5=p5g
13. f1:0=o1 þ 0:5=o4; 0:5=p4 þ 1:0=p5g14. f1:0=o1 þ 1:0=o5; 0:5=p3 þ 0:5=p4g15. f0:5=o1 þ 1:0=o2 þ 1:0=o5; 1:0=p3g16. f1:0=o1 þ 1:0=o2; 1:0=p2 þ 0:5=p3g17. f1:0=o2 þ 0:5=o3 þ 0:5=o6; 1:0=p1g18. f1:0=o2 þ 1:0=o3; 0:5=p1 þ 0:5=p2g19. f0:5=o1 þ 1:0=o4 þ 0:5=o5; 1:0=p4g20. f1:0=o1 þ 1:0=o4; 0:5=p4 þ 0:5=p5g21. f1:0=o1 þ 1:0=o2 þ 1:0=o5; 0:5=p3g22. f1:0=o1 þ 1:0=o2 þ 0:5=o3; 1:0=p2g23. f1:0=o1 þ 1:0=o4 þ 1:0=o5; 0:5=p4g24. f1:0=o1 þ 1:0=o2 þ 1:0=o3; 0:5=p2g25. f1:0=o2 þ 1:0=o3 þ 1:0=o6; 0:5=p1g26. f1:0=o1 þ 1:0=o2 þ 1:0=o3 þ 1:0=o4 þ 1:0=o5
þ1:0=o6; øgwhere ø represents null set. The fuzzy concept lattice for
above generated concepts is shown in Fig. 4. Table 16
shows the computed weight value for all the attributes of
given fuzzy context shown in Table 15. Table 17 shows the
Table 14 Comparison of Levenshtein distance and proposed method
Levenshtein distance[35] Proposed algorithm
1. Finds similar concepts at chosen granulation Finds similar concepts at chosen granulation
2. Operation:insertion (i) deletion (d), replacement(r) Computes Probability of each attribute in the given fuzzy formal
context
3. Measure the information between the concepts using above
operation
Measure the information between the concepts using their computed
weight
4. Minimizing the weight using above operations vary from expert to
expert
Computing weight is always in between [0, 1] for each expert
5. Threshold used for finding the concepts Define threshold for finding the concepts
6. Applied in similar attributes (objects) context Applied for any context
7. Do not analyze the deviation Discuss the deviation analysis
8. Complexity O (m n) Complexity O (m lnðmÞ) or O (n lnðnÞ)
Table 15 A fuzzy formal context
p1 p2 p3 p4 p5 p6
o1 0.0 1.0 0.5 0.5 1.0 0.0
o2 1.0 1.0 1.0 0.0 0.0 0.0
o3 0.5 0.5 0.0 0.0 0.0 1.0
o4 0.0 0.0 0.0 1.0 0.5 0.0
o5 0.0 0.0 1.0 0.5 0.0 0.0
o6 0.5 0.0 0.0 0.0 0.0 0.0
Fig. 4 Fuzzy concept lattice for the context of Table 15
Int. J. Mach. Learn. & Cyber.
123
computed weight for each of the fuzzy formal concepts
(based on their intent) shown in Fig. 4. The reduction of
fuzzy formal concepts at chosen granulation of their weight
is shown in Table 18.
We can observe that the following fuzzy formal con-
cepts are obtained from Fig. 4 at chosen granulation
0:0946\w� 0:0974: 1, 2, 3, 4, 5, 8, 9, 11, 12, 13, 14, 17,
18, 19, 20, 21, 22, 23, 24, 25, 26 (as shown in Table 18).
Subsequently, Prem Kumar and Aswani Kumar [41] dis-
cussed that the 26–fuzzy formal concepts generated from
Table 15 can be represented into 14–interval valued fuzzy
concepts as follows:
1. fø; ½1:0; 1:0=p1 þ ½1:0; 1:0=p2 þ ½1:0; 1:0=p3
þ½1:0; 1:0=p4 þ ½1:0; 1:0=p5 þ ½1:0; 1:0=p6g2. f½0:5; 1:0=o1; ½1:0; 1:0=p2 þ ½0:5; 1:0=p3
þ½0:5; 1:0=p4 þ ½1:0; 1:0=p5g3. f½1:0; 1:0=o2; ½1:0; 1:0=p1 þ ½1:0; 1:0=p2
þ½1:0; 1:0=p3g4. f½0:5; 1:0=o3; ½0:5; 1:0=p1 þ ½0:5; 1:0=p2
þ½0:1; 1:0=p6g5. f½0:5; 1:0=o1 þ ½0:5; 1:0=o2; ½0:5; 1:0=p2
þ½0:5; 1:0=p3g6. f½0:5; 1:0=o1 þ ½0:5; 1:0=o4; ½0:5; 1:0=p4
þ½0:5; 1:0=p5g7. f½0:5; 1:0=o2 þ ½0:5; 1:0=o3; ½0:5; 1:0=p1
þ½0:5; 1:0=p2g8. f½0:5; 0:5=o1 þ ½1:0; 1:0=o5; ½1:0; 1:0=p3
þ½0:5; 1:0=p4g9. f½0:5; 1:0=o1 þ ½0:5; 1:0=o5; ½0:5; 1:0=p3
þ½0:5; 1:0=p4g
Table 16 Computed weight value for single attribute of Table 15
Attribute P(p) EðpÞ wi
0:5=p1 0.25 0.346 0.101
1:0=p1 0.5 0.346 0.101
0:5=p2 0.2 0.3218 0.0946
1:0=p2 0.4 0.366 0.107
0:5=p3 0.2 0.3218 0.0946
1:0=p3 0.4 0.366 0.107
0:5=p4 0.25 0.346 0.101
1:0=p4 0.5 0.346 0.101
0:5=p5 0.33 0.365 0.107
1:0=p5 0.66 0.274 0.0806
1:0=p6 1.0 0.0 0.0
Table 17 The intent weight
value of each node of fuzzy
concept lattice shown in Fig. 4
Node Intent Average weight wðpÞ
1 1:0=p1 þ 1:0=p2 þ 1:0=p3 þ 1:0=p4 þ 1:0=p5 þ 1:0=p6 1 1
2 1:0=p2 þ 1:0=p3 þ 1:0=p4 þ 1:0=p5 0.0974 0.0974
3 1:0=p1 þ 1:0=p2 þ 1:0=p3 0.105 0.105
4 1:0=p1 þ 1:0=p2 þ 1:0=p6 0.104 0.104
5 1:0=p3 þ 1:0=p4 0.104 0.104
6 1:0=p4 þ 1:0=p5 0.0908 0.0908
7 1:0=p2 þ 0:5=p3 þ 0:5=p4 þ 1:0=p5 0.0958 0.0958
8 1:0=p2 þ 1:0=p3 0.107 0.107
9 1:0=p1 þ 1:0=p2 0.104 0.104
10 0:5=p1 þ 0:5=p2 þ 1:0=p6 0.0652 0.0652
11 1:0=p3 þ 0:5=p4 0.104 0.104
12 1:0=p4 þ 0:5=p5 0.104 0.104
13 0:5=p4 þ 1:0=p5 0.104 0.104
14 0:5=p3 þ 0:5=p4 0.0978 0.0978
15 1:0=p3 0.0 0.0
16 1:0=p2 þ 0:5=p3 0.058 0.058
17 1:0=p1 0.101 0.101
18 0:5=p1 þ 0:5=p2 0.0978 0.0978
19 1:0=p4 0.101 0.101
20 0:5=p4 þ 0:5=p5 0.104 0.104
21 0:5=p3 0.0946 0.0946
22 1:0=p2 0.0107 0.107
23 0:5=p4 0.101 0.101
24 0:5=p2 0.0946 0.0946
25 0:5=p1 0.101 0.101
26 ø 1 1
Int. J. Mach. Learn. & Cyber.
123
10. f½0:5; 1:0=o1 þ ½1:0; 1:0=o2 þ ½1:0; 1:0=o5;
½0:5; 1:0=p3g11. f½0:5; 1:0=o1 þ ½1:0; 1:0=o4 þ ½0:5; 1:0=o5;
½0:5; 1:0=p4g12. f½1:0; 1:0=o1 þ ½1:0; 1:0=o2 þ ½0:5; 1:0=o3;
½0:5; 1:0=p2g
13. f½1:0; 1:0=o2 þ ½0:5; 1:0=o3 þ ½0:5; 1:0=o6;
½0:5; 1:0=p1g14. f½1:0; 1:0=o1 þ ½1:0; 1:0=o2 þ ½1:0; 1:0=o3
þ½1:0; 1:0=o4 þ ½1:0; 1:0=o5 þ ½1:0; 1:0=o6; øgNow we compare the knowledge represented by the pro-
posed method with the above 14–interval–valued fuzzy
formal concepts. Table 19 summarizes this comparison.
We can observe that the analysis derived from the pro-
posed method is in good agreement with interval–valued
fuzzy formal concepts. Moreover, the proposed method pro-
vides a way to reduce the fuzzy formal concepts at chosen
granulation but with less complexity. For better understanding
Table 20 compares both the methods.
5 Conclusions
In this paper we aim at reducing the number of fuzzy
formal concepts. For this purpose we have introduced a
method to compute the weight of given fuzzy formal
concepts using Shannon entropy. The changes between the
obtained fuzzy formal concepts using the proposed method
are compared with Levenshtein distance[35] and interval–
valued fuzzy concept method [41]. We have observed that
the proposed method provides similar results when com-
pared to Levenshtein distance as well as interval–valued
fuzzy concepts. Further, the proposed method also provides
a way to select the fuzzy formal concepts at chosen gran-
ulation of their computed weight with less complexity.
Acknowledgments Authors sincerely acknowledge the financial
support from National Board of Higher Mathematics, Dept. of Atomic
Energy, Govt. of India under the grant number 2/48(11)/2010-R&D
II/10806.
Table 18 The reduced fuzzy formal concepts shown in Table 17 at
chosen granulation
Weight for fuzzy
concepts (wðpÞ)Obtained concepts
0:101\w� 1 1, 3, 4, 5, 8, 9, 11, 12, 13, 17, 19, 20, 22,
23, 25, 26
0:0974\w� 0:101 1, 2, 3, 4, 5, 8, 9, 11, 12, 13, 14, 17, 18, 19,
20, 22,
23, 25, 26
0:0946\w� 0:0974 1, 2, 3, 4, 5, 7, 8, 9, 11, 12, 13, 14,
17, 18, 19,
20, 21, 22, 23, 24, 25, 26
0:058\w� 0:0946 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14,
16, 17, 18, 19,
20, 21, 22, 23, 24, 25, 26
0\w� 0:058 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14,
15, 16, 17, 18,
19, 20, 21, 22, 23, 24, 25, 26
Table 19 Intent obtained using interval–valued and entropy based
approach
Intent of weighted
concepts in Fig. 4
Similar interval–valued
intent [41]
1. (p1; p2; p3; p4; p5; p6) 1. (p1; p2; p3; p4; p5; p6)
2. (p2; p3; p4; p5) 2. (p2; p3; p4; p5)
3. (p1; p2; p3) 3. (p1; p2; p3)
4. (p1; p2; p6) 4. (p1; p2; p6)
5. (p3; p4) 9. (p3; p4)
8. (p2; p3) 5. (p2; p3)
9. (p1; p2) 7. (p1; p2)
11. (p3; p4) 8. (p3; p4)
12. (p4; p5) 6. (p4; p5)
13. (p4; p5) 6. (p4; p5)
14. (p3; p4) 9. (p3; p4)
17. (p1) 13. (p1)
18. (p1) 13. (p1)
19. (p4) 11. (p4)
20. (p4; p5) 6. (p4; p5)
21. (p3) 10. (p3)
22. (p2) 12. (p2)
23. (p4) 11. (p4)
24. (p2) 12. (p2)
25. (p1) 13. (p1)
26. (ø) 14. (ø)
Table 20 Comparison of interval–valued fuzzy concepts and the
proposed method
Interval–valued concepts [41] Proposed algorithm
1. Finds similarity concepts Finds similar concepts
in the interval [0,1] in the interval [0,1]
2. Computes power set Computes Probability
of attributes of each attributes
3. Measure the information between the
concepts using interval
Measure the
information between
the
concepts using their
computed weight
4. Applied in similar attributes (objects)
context
Applied for any context
5. Do not analyze the deviation Discuss the deviation
analysis
6. Complexity O (2m n) Complexity O (m lnðmÞ)or O (n lnðnÞ)
Int. J. Mach. Learn. & Cyber.
123
Appendix
Nomenclature Meaning
L Scale of truth degree
L Residuated lattice
F Fuzzy formal context
O Set of objects
o An object
P Set of attributes
p An attribute
P Probability
~R L–relation between O and P
� Multiplication
! Residuum
a; b; c Elements in L
("; #) Galois connection
A Extent
B Intent
LO L–set of objects
LP L–set of attributesS
UnionT
Intersection
^ Infimum
_ Supremum
h; h1; h2 Granulation
E Average information weightP
Summation
m Total number of attributes
wj Weight of attribute
WeightðkÞ Weight of k–th formal concept
D Deviation
jj Absolute difference
C Single fuzzy formal concept
FC F Set of fuzzy formal concepts
oi i–th objects
pj j–th attibute
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