Concepts reduction in formal concept analysis with fuzzy setting using Shannon entropy

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


J. Li

Faculty of Science, Kunming University of Science and

Technology, Kunming 650500, People’s Republic of China


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


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


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http://conexp.sourceforge.net/

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


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


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].


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


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


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)


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

References

1. Annapurna J, Aswani Kumar Ch (2013) Exploring attribute with

domain knowledge in formal concept analysis. J Comput Inf

Technol 21(2):109–123

2. Aswani Kumar Ch, Srinivas S (2010) Concept lattice reduction

from fuzzy K-means clustering. Expert Syst Appl

37(3):2696–2704

3. Aswani Kumar Ch, Srinivas S (2010) Mining associations in

health care data using formal concept analysis and singular value

decomposition. J Biol Syst 18(4):787–807

4. Aswani Kumar Ch (2011) Knowledge discovery in data using

formal concept analysis and random projections. Int J Appl Math

Comput Sci 21(4):745–756

5. Aswani Kumar Ch (2011) Communications in computer and

information science. Springer, Berlin

6. Aswani Kumar Ch (2012) Fuzzy clustering based formal concept

analysis for association rules mining. Appl Artif Intell

26(3):274–301

7. Aswani Kumar Ch, Radavansky M, Annapurna J (2012) Analysis

of vector space model, latent semantic indexing and formal

concept analysis for information retrieval. Cybern Inf Technol

12(1):34–48

8. Aswani Kumar Ch (2013) Designing role-based access control

using formal concept analysis. Secur Commun Netw 6:373–383

9. Aswani Kumar Ch, Singh PK (2014) In: Tripathy BK, Acharjya

DP (eds) Knowledge representation using formal concept ana-

lysis: a study on concept generation., Global trends in knowledge

representation and computational intelligenceIGI Global Pub-

lishers, Hershey, pp 306–336

10. Babin MA, Kuznetsov SO (2012) Approximating concept sta-

bility, Lecture Notes in Computer Science (including subseries.

Lecture Notes in Artificial Intelligence and Lecture Notes in

Bioinformatics) 7278 LNAI, pp 7–15

11. Belohlavek R, Vychodil V (2005) What is fuzzy concept lattice.

In: Proceedings of CLA Olomuc, Czech Republic, pp 34–45

12. Belohlavek R, Vychodil V (2005) Reducing the size of fuzzy

concept lattice by hedges. In: Proceedings of 14th IEEE inter-

national conference on fuzzy systems, pp 663–668

13. Belohlavek R, Macko J (2011) Selecting important concepts

using weights. Lecture Notes in Computer Science (Including

Subseries Lecture Notes in Artificial Intelligence and Lecture

Notes in Bioinformatics) 6628 LNAI, pp 65–80

14. Belohlavek R, Trnecka M (2012) Basic level of concepts in

formal concept analysis, Lecture Notes in Computer Science

Including Subseries Lecture Notes in Artificial Intelligence and

Lecture Notes in Bioinformatics) 7278 LNAI, pp 28–44

15. Burusco A, Fuentes-Gonzales R (1994) The study of L-fuzzy

concept lattice. Math Soft Comput 3:209–218

16. Cimiano P, Hotho A, Staab S (2005) Learning concept hierar-

chies from text corpora using formal concept analysis. J Artif Int

Res 24(1):305–339

17. Cross V, Kandasamy M (2011) Fuzzy concept lattice construc-

tion: a basis for building fuzzy ontologies. FUZZ-IEEE

2011:1743–1750

18. Davey BA, Priestley HA (2002) Introduction to lattices and order.

Cambridge University Press, Cambridge

19. Dias SM, Viera NJ (2013) Applying the JBOS reduction method

for relevant knowledge extraction. Experts Syst Appl

40(5):1880–1887

20. Elloumi S, Jaam J, Hasnah A, Jaoua A (2004) A multi-level

conceptual data reduction approach based on the Lukasiewicz

implication. Inf Sci 163(4):253–262

21. Fenza G, Loia V, Senatore S (2008) Concept mining of semantic

web services by means of extended fuzzy formal concept analysis

(FFCA), IEEE international conference on systems, man and

cybernetics, SMC, pp 240–245

22. Ganter B, Wille R (1999) Formal concept analysis: mathematical

foundation. Springer, Berlin

23. Ghosh P, Kundu K, Sarkar D (2010) Fuzzy graph representation

of a fuzzy concept lattice. Fuzzy Sets Syst 161:1669–1675

24. http://conexp.sourceforge.net/

25. Kang X, Li D, Wang S, Qu K (2012) Formal concept analysis

based on fuzzy granularity base for different granulation. Fuzzy

Sets Syst 203:33–48


123

http://conexp.sourceforge.net/

26. Kuznetsov SO, Obiedkov SA (2002) Comparing performance of

algorithms for generating concept lattices. J Exp Theor Artif Int

14:189–216

27. Li L, Jhang J (2010) Attribute reduction in fuzzy concept lattices

based on the T-implication. Knowl-Based Syst 23:497–503

28. Li J, Mei C, Lv Y (2011) A heuristic knowledge-reduction

method for decision formal contexts. Comput Math Appl

61(4):1096–1106

29. Li J, Mei C, Lv Y (2011) Knowledge reduction in decision formal

contexts. Knowl-Based Syst 24(5):709–715

30. Li J, Mei C, Lv Y (2012) Knowledge reduction in real decision

formal contexts. Inf Sci 189:191–207

31. Li J, Mei C, Zhang X (2013) On rule acquisition in decision

formal contexts. Int J Mach Learn Cybern 4(6):721–731

32. Li J, Mei C, Lv Y (2013) Incomplete decision contexts:

approximate concept construction, rule acquisition and knowl-

edge reduction. Int J Approx Reason 54(1):149–165

33. Li J, He Z, Zhu Q (2013) An entropy-based weighted concept

lattice for merging multi-source geo-ontologies. Entropy

15:2303–2318

34. Li C, Li J, He M (2014) Concept lattice compression in incom-

plete contexts based on K-medoids clustering. Int J Mach Learn

Cybern. doi:10.1007/s13042-014-02883

35. Martin TP, Abd Rahim NH (2013) A general approach to the

measurement of change in fuzzy concept lattices. Soft Comput

17(12):2223–2234

36. Ma JM, Leung Y, Zhang WX (2014) Attribute reductions in

object-oriented concept lattices. Int J Mach Learn Cybern

5(5):789–813

37. Medina J (2012) Relating attribute reduction in formal, object-

oriented and property-oriented concept lattices. Comput Math

Appl 208:95–110

38. Mi JS, Leung Y, Wu WZ (2010) Approaches to attributes

reduction in concept lattices induced by axialities. Knowl-Based

Syst 23:504–511

39. Pocs J (2012) Note on generating fuzzy concept lattices via

Galois connections. Inf Sci 185(1):128–136

40. Poelmans J, Kuznetsov SO, Ignatov DI, Dedene G (2013) Formal

concept analysis in knowledge processing : a survey on appli-

cations. Expert Syst Appl 40(16):6538–6560

41. Singh PK, Aswani Kumar Ch (2012) Interval-valued fuzzy graph

representation of concept lattice. In: Proceedings of 12th inter-

national conference on intelligent system design and application,

pp 604–609

42. Singh PK, Aswani Kumar Ch (2012) A method for decomposi-

tion of fuzzy formal context. Proceedings of international con-

ference on modelling optimization and computing. Procedia Eng

38:1852–1857

43. Singh PK, Aswani Kumar Ch (2012) A method for reduction of

fuzzy relation in a fuzzy formal context. Proceedings of inter-

national conference on mathematical, modelling and scientific

computation, CCIS, Springer, Berlin, 283:343–350

44. Singh PK, Aswani Kumar Ch (2014) A note on constructing

fuzzy homomorphism map for a given fuzzy formal context. In:

Proceeding of 3rd international conference on SocPros 2013,

Springer. Adv Intell Syst Comput 258:845–855

45. Singh PK, Aswani Kumar Ch (2014) Bipolar fuzzy graph rep-

resentation of concept lattice. Inf Sci 288:437–448

46. Priss U (2006) Formal concept analysis in information science.

Annu Rev Inf Sci Technol 40:521–543

47. Senatore S, Pasi G, Lattice navigation for collaborative filtering

by means of (fuzzy) formal concept analysis. In: Proceedings of

the 28th annual ACM symposium on applied computing (SAC

’13). ACM, New York, pp 920–926

48. Shannon CE (1948) A mathematical theory of communication.

Bell Syst Tech J 27: 379–423, 623–656.

49. Shao MW, Leung Y, Wu WZ (2014) Rule acquisition and

complexity reduction in formal decision contexts. Int J Approx

Reason 55(1):259–274

50. Sumangali K, Aswani Kumar Ch (2014) Determination of

interesting rules in FCA using information gain. 1st international

conference on networks and soft computing (ICNSC)

2014:304–308

51. Wille R (1982) Restructuring lattice theory: an approach based on

hierarchies of concepts. In: Rival I (ed) Ordered Sets. Reidel

Dordrect, Boston, pp 445–470

52. Wu WZ, Leung Y, Mi JS (2009) Granular computing and

knowledge reduction in formal context. IEEE Trans Knowl Data

Eng 21(10):1461–1474

53. Yao YY (2004) A comparative study of formal concept analysis

and rough set theory in data analysis. In: Proceedings of 4th

international conference on rough sets and current trends in

computing (RSCTC), Sweden, pp 59–68.

54. Yao YY (2004) Concept lattices in rough set theory. Proceedings

of 2004 annual meeting of the North American fuzzy information

processing society. IEEE Computer Society, Washington D.C.,

pp 796–801

55. Yao YY (2004) Granular computing. In: Proceedings of the 4th

Chinese national conference on rough sets and soft computing,

Computer Science (Ji Suan Ji Ke Xue)vol. 31. pp. 1–5

56. Zhang S, Guo P, Zhang J, Wang X, Pedrycz W (2012) A com-

pleteness analysis of frequent weighted concept lattices and their

algebraic properties. Data Knowl Eng 81–82:104–117


123

Concepts reduction in formal concept analysis with fuzzy setting using Shannon entropy

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