An enhanced fuzzy rough set based clustering algorithm for categorical data

IJRET: International Journal of Research in Engineering and Technology eISSN:2319-1163 | pISSN: 2321-7308
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Volume: 03 Issue: 11 | Nov-2014, Available @ http://www.ijret.org 184
AN ENHANCED FUZZY ROUGH SET-BASED CLUSTERING
ALGORITHM FOR CATEGORICAL DATA
Raval Dhwani Jayant1
, Vashi Viral Mineshkumar2
1
M.Tech (Software Tech.), VIT University, India
2
M.Tech (Software Tech.), VIT University, India
Abstract
In today’s world everything is done digitally and so we have lots of data raw data. This data are useful to predict future events if
we proper use it. Clustering is such a technique where we put closely related data together. Furthermore we have types of data
sequential, interval, categorical etc. In this paper we have shown what is the problem with clustering categorical data with rough
set and who we can overcome with improvement.
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1. INTRODUCTION
In database Categorical variable is variable which have to
take one of the values from set of finite values attached to it.
Value of this categorical variable relies to values of other
related variables this as a whole is known as categorical data
[1]. Now clustering based on what category data falls is
issues here. Because unlike categorical data of blood group
when we know that data will definitely fall in only one
group i.e. A, B, AB e.t.c. this are crisp categorical data but
we also have some categorical data which are not crisp and
have may-be scenario e.g. for giving loan there will be some
situation when it is tough to decide if its “risky” or “safe”
and thus sometimes same set of data will fall in “risky”
category where as sometimes in “safe”. Another example is
disease treatment when sometime we go for treatment A
sometimes for B for same medical data. Clustering this kind
of data to get correct cluster set we cannot apply existing
rough set clustering technique as it only work with data
which have at least one crisp relation using which clustering
is done. With little modification in lower approximation
formulation and use of fuzzy logic we can improve
conventional rough set technique to better cluster the data so
that all closely related data fall in same cluster.
In following section we will go through basics of rough set
and fuzzy logic and then with example we will show how
rough set clustering works problem in rough set clustering
and Improved fuzzy rough set clustering all with proper
example.
2. LITERATURE SURVEY
Rough Set Theory: - It was first proposed by Pawlak[6] for
processing incomplete information in information systems.
It is defined by Pawlak[7] as Formal approximation of a set
which have strong one to one relation or as called are crisp
set based on its low and high set of approximation is roughs
set.
As defined by Kumar and its colleague [2] in Rough cluster
lower approximation contains objects that belong to unique
cluster and it has no membership to other cluster and upper
approximation contains objects which are in this cluster as
well as in other clusters.
Rough Set uses following defined Nomenclature and
formula to do clustering as defined by Darshit Parmar [9]
U universe or the set of all objects (x1, x2..)
X subset of the set of all objects, (X ⊆ U)
xi object belonging to the subset of the set of all objects, xi
∈ X
A the set of all attributes (features or variables) ai attribute
belonging to the set of all attributes, ai ∈ A
V(ai) set of values of attribute ai (or called domain of ai)
B non-empty subset of A(B ⊆ A)
XB lower approximation of X with respect to B
XB upper approximation of X with respect to B
Rai(X) roughness with respect to {ai}
Ind(B) indiscernibility relation
[xi]Ind(B) equivalence class of xi in relation Ind(B),
Indiscernibility relation Ind(B): - Objects, xi, xj ∈ U are
indiscernible by the attributes set B in A,that is xi, xj ∈
Ind(B) if and only if a ∈ B where B⊆A, a(xi) = a(xj).
Equivalence class [xi]Ind(B): - Set of objects xi with
respect to indiscrmibility realtion on B i.e. Ind(B) Having
similar values for the set of attributes in B consists of an
equivalence classes, [xi]Ind(B).

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Lower approximation: - For attributes set of B in A the
objects X in U, the lower approximation of X is defined as
XB=U {xi|[xi]Ind(B) ⊆ X}
Upper approximation: - For attributes set of B in A the
objects X in U, the upper approximation of X is defined as
XB=U {xi|[xi]Ind(B) ∩ X ≠ ∅}
Roughness: - The ratio of the lower approximation and the
upper approximation is called roughness it is measured as
RB(X) = 1 - |XB|/|XB|
If RB(X) = 0, X is crisp or precise with B. If RB(X) < 1, X
is rough or vague with B.
It is also defined as the accuracy of estimation.
Fuzzy Logic: - Fuzzy logic is multi value logic where we
don’t have exact value instead we have approximate value.
All Fuzzy logic variables are assigned degree to which they
are associated with these degree ranges 0 and 1. It deals with
partial truth.[4] We then use linguistic variables to describe
this fuzzy logic.
Fuzzy Rough Set: - I. Jagielska, C. Matthews, T.
Whitfort,[8] stats that Knowledge, Information or data
hidden ie which creates pattern in information systems can
also be found by describing them data in form of some set of
decision rules using application of neural networks, fuzzy
logic, genetic algorithms, and rough sets and thus
automating the process of information gathering.
Chris Cornelis, Martine De Cock, Anna Maria
Radzikowska[5] has given Fuzzy Rough set also known as
fuzzy hybridization of Rough Set where we use rough set to
find dependency which help to cluster the universe where as
fuzzy logic is use to represent the pattern in data through
linguistic variables. This combination of fuzzy with rough is
also called as soft computing or hybrid intelligent system.
3. WORKING OF ROUGH SET
In table 1 we have 2 attribute A1 and A2 and 10 objects now
to show rough set clustering works we take here A1 as base
attribute to cluster 10 objects set
A1 A2
1 Big Blue
2 Medium Red
3 Small Yellow
4 Medium Blue
5 Small Yellow
6 Big Green
7 Small Yellow
8 Small Yellow
9 Big Green
10 Medium Green
Steps
1. Find elementary set of A1 and A2
X(A1=Small) = {3,5,7,8}
X(A1=Medium)={2,4,10}
X(A1=Big)={1,6,9}
X(A2=Blue)={1,4}
X(A2=Red)={2}
X(A2=Yellow)={3,5,7,8}
X(A2=Green)={6,9,10}
2. Find Lower and Upper Approximation of A1 sets
with Respect to A2 sets
Lower Approximation of X(A1=Small)={3,5,7,8}
Upper Approximation of X(A1=Small) ={3,5,7,8}
Lower Approximation of X(A1=Medium) ={2}
Upper Approximation of X(A1=Medium)
={1,2,4,6,9,10}
Lower Approximation of X(A1=Big) ={No Lower
Approximation}
Upper Approximation of X(A1=Big) ={As No
Lower Thus No Upper Approximation}
3. Find Roughness of each elementary set of A1
Roughness X(A1=Small)=1-(4/4)=0
Roughness X(A1=Medium)=1 (1/6)=0.833
Roughness X(A1=Big)=N/A
4. Cluster objects based on elementary set
As X(A1=Small) have lower Roughness we will
split into two Cluster one with Small and Other with
Big, Medium
4. PROBLEM WITH ROUGH SET
With traditional rough set when we have no crisp relation in
data we cannot do clustering let us take table 2 as example
A1 A2
1 Big Blue
2 Medium Pink
3 Small Red
4 Medium Orange
5 Small Blue
6 Big Red
7 Small Yellow
8 Small Pink
9 Big Orange
10 Medium Yellow
X(A1=Small) = {3,5,7,8}
X(A1=Medium)={2,4,10}
X(A1=Big)={1,6,9}
X(A2=Blue)={1,5}
X(A2=Pink)={8,2}

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X(A2=Orange)={9,4}
X(A2=Yellow)={10,7}
X(A2=Red)={3,6}
Lower Approximation of X(A1=Small)={ No Lower
Approximation }
Upper Approximation of X(A1=Small) ={ As No Lower
Thus No Upper Approximation }
Lower Approximation of X(A1=Medium) ={ No Lower
Approximation }
Upper Approximation of X(A1=Medium) ={ As No Lower
Thus No Upper Approximation }
Lower Approximation of X(A1=Big) ={No Lower
Approximation}
Upper Approximation of X(A1=Big) ={As No Lower Thus
No Upper Approximation}
As we don’t have lower and upper approximation for any X
we cannot find Roughness
5. PROPOSED WORK
In our proposed clustering method we have new lower
approximation formulation through which we will find
lower approximation with this we will use fuzziness degree
of each attribute value to decide which set will be taken as
lower approximation if we don’t have crisp set and more
than one partial set which cannot be distinguished by given
lower approximation formula. Once we have clustered data
we can write Fuzzy linguistic rule base for future reference.
This type of clustering helps to deal with problem in
categorical data which we have mentioned in problem
statement.
A1 A2
1 Big Blue
2 Medium Pink
3 Small Red
4 Medium Orange
5 Small Blue
6 Big Red
7 Small Yellow
8 Small Pink
9 Big Orange
10 Medium Yellow
Steps
1. Assign Weight to each Attribute value of A1 and
A2
Weight
Small=0.1
Medium=0.2
Big=0.3
Blue=0.7
Pink=0.6
Red=0.8
Yellow=0.4
Orange=0.5
2. Find Elementary set of A1 and A2
X(A1=Small) = {3,5,7,8}
X(A1=Medium)={2,4,10}
X(A1=Big)={1,6,9}
X(A2=Blue)={1,5}
X(A2=Pink)={8,2}
X(A2=Orange)={9,4}
X(A2=Yellow)={10,7}
X(A2=Red)={3,6}
3. Find Lower and Upper Approximation for Set of
A1 with Respect to A2 using near match
fomulation
To find lower approximation we find set nearest to
base set X with formulation (No. of Match with base
set/total no. of element in set) for all set which are
having subset of base set we find this. Out of this we
will take one with minimum value.
Lower Approximation of
X(A1=Small)={3,6}=1/2=0.50
{1,5}=1/2=0.50
{10,7}=1/2=0.50
{8,2}=1/2=0.50
Now here as all set have same value we will look in
fuzzy weight allocated and select one with higher
rate.
Here we have Red having higher weight than other
attribute value and thus we select {3,6} as lower
approximation
Upper Approximation of X(A1=Small)
={3,5,7,8,1,6,10,2}
Lower Approximation of X(A1=Medium) ={8,2}
Upper Approximation of X(A1=Medium)
={2,4,10,8,9,7}
Lower Approximation of X(A1=Big) ={3,6}
Upper Approximation of X(A1=Big) ={1,6,9,4,3,5}
4. Find Roughness
Roughness X(A1=Small)=1-(2/8)=0.75
Roughness X(A1=Medium)=1-(2/6)=0.66
Roughness X(A1=Big)=1-(2/6)=0.66
5. Cluster Data based on roughness
As X(A1=Medium)and X(A1=Big) have similar
lower Roughness we will split into two Cluster

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based on weight as X(A1=Big) have higher weight
we will split into one with Big and Other with
Small,Medium
6. Fuzzy Linguistic Rule Base
Where Color is A2=Red A1=Big
6. CONCLUSION
Rough set can only be applied where we have at least one
crisp relation of the data. Thus in this paper we proposed
clustering technique with mixture of rough set with
improved lower approximation formulation and fuzzy logic
which works on any kind of categorical data whether data is
having crisp relationship between its attribute or not we can
do clustering. As shown with example proposed technique
works better than traditional technique.
REFERENCES
[1]. http://en.wikipedia.org/wiki/Categorical_variable
[2]. Pradeep Kumar, P. Radha Krishna, Raju. S. Bapi,
Supriya Kumar D “Rough clustering of sequential data”
[3]. Duo Chen, Du-Wu Cui, Chao-Xue Wang, Zhu-Rong
Wang “A Rough Set-Based Hierarchical Clustering
Algorithm for Categorical Data” School of Computer
Science and Engineering, Xi’an University of Technology,
Xi’an, 710048, China
[4]. http://en.wikipedia.org/wiki/Fuzzy_logic
[5]. Chris Cornelis, Martine De Cock, Anna Maria
Radzikowska, “Fuzzy Rough Sets: from Theory into
Practice” Computational Web Intelligence Dept. of Applied
Mathematics and Computer Science Ghent University,
Krijgslaan 281 (S9), 9000 Gent, Belgium
[6]. Z. Pawlak, “Rough sets”, International Journal of
Computer and Information Science (1982).
[7]. Z. Pawlak, “Rough Sets—Theoretical Aspects of
Reasoning about Data”, Kluwer Aca. Pub. (1991)
[8]. I. Jagielska, C. Matthews, T. Whitfort, “An
investigation into the application of neural networks, fuzzy
logic, genetic algorithms, and rough sets to automated
knowledge acquisition for classification problems”,
Neurocomputing (1999)
[9]. Darshit Parmar, Teresa Wu *, Jennifer Blackhurst,
“MMR: An algorithm for clustering categorical data using
Rough Set Theory"
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