RESULT COMPARISON OF TWO ROUGH SET BASED
DISCRETIZATION ALGORITHMS
Shanchan Wu, Wenyuan Wang
Department of Automation, Tsinghua University,Beijing 100084, P.R.C.
K
eywords: Rought set, Cuts, Discretization, Data mining
Abstract: The area of knowledge discovery and data mining is growing rapidly. A large number of methods are
employed to mine knowledge. Many of the methods rely of discrete data. However, most of the datasets
used in real application have attributes with continuous values. To make the data mining techniques useful
for such datasets, discretization is performed as a preprocessing step of the data mining. In this paper, we
discuss rough set based discretization. We use UCI data sets to do experiments to compare the quality of
Local discretization and Global discretization based on rough set. Our experiments show that Global
discretization and Local discretization are dataset sensitive. Neither of them is always better than the other,
though in some cases Global discretization generates far better results than Local discretization.
1 INTRODUCTION
Rough Set theory is a tool to tackle fuzzy and
uncertainty knowledge. It was put forward firstly by
Z.Pawlak (Pawlak Z, 1982). In decades, rough set
theory has been successfully implemented in Data
mining, artificial intelligence and pattern
recognition.
But rough set and many other methods used in
data mining can't deal with continuous attributes and
a very large proportion of real data sets include
continuous variables. One solution to this problem is
to partition numeric variables into a number of
intervals and treat each interval as a category. This
process is usually termed dicretization.
Several methods have been proposed to
discretize data as a preprocessing step for the data
mining process. Nguyen S. H. proposed the named
discretization approach based on rough set methods
and boolean reasoning (
Nguyen, 1995, 1997). The
main idea is to seek possibly minimum number of
discrete intervals, and at the same time it should not
weaken the indiscernibility. It has been proven that
Optimal Discretization Problem is NP-complete
(
Nguyen, 1995). In this paper, we examine two
discretization algorithms based on rough set, Local
Discretization and Global Discretization (Hung Son
Nguyen, 1996). We do experiments to compare the
results of the two algorithms.
This paper is organized as follows. In Section 2,
we describe discretization based on rough set. Then
we explain determination of candidate cuts and
calculating of discernibility of cuts in Section 3. In
Section 4, we describe the local discretization
algorithm and global discretization algorithm and in
section 5 we show the experiment results. Finally
Section 6 concludes this paper.
2 DESCRIPTION OF ROUGH SET
BASED DISCRETIZATION
An information system is defined as follows:
AaFVAUS
aa
∈= ),,,,( (1)
where
{}
n
xxxU ,,,
21
L= is a finite set of
objects(n is the number of objects), A is a finite set
of attributes,
a
aA
V
V
∈
=
U
, and
a
V is a domain of
attribute a,
aa
VAUF →×: is a total function
such that
ai
Vaxf ∈),( for each Aa ∈ , Ux
i
∈ .
An information system S in definition (1) is
called a decision system or decision table when the
attributes in S can be divided into condition
attributes C and decision attributes D. i.e.
DCA U= , and
φ
=DC I .
In information systems, each subset of attributes
AI ⊆ determines a binary relation as follows:
{
}
() , ,() ()
I
ND I x y U U a I a x a y=< >∈ × ∀∈ =
511
Wu S. and Wang W. (2004).
RESULT COMPARISON OF TWO ROUGH SET BASED DISCRETIZATION ALGORITHMS.
In Proceedings of the Sixth International Conference on Enterprise Information Systems, pages 511-514
DOI: 10.5220/0002611505110514
Copyright
c
 SciTePress

It is easily shown that
)(IIND is an
equivalence relation on the sets U and is called an
indiscernible relation. The partition of U as defined
by B will be denoted U/B and the equivalence
classes introduced by B will be denoted [u]
B.
In
particular, [u]
{d}
will be called the decision classes of
the decision system.
Let
S ),},{,( fVdAU ∪= be a decision table
where
}=
n
x,,x,x,{x U
321
L . Assuming that
RrI
aa
⊂= ),[V
a
for any Aa ∈ where R is the
set of real numbers.
Assume now that the
S is a consistent decision
table. Let
D
a
be a partition of V
a
(for Aa ∈ ) into
subintervals, i.e.
{
}
0112 1
a
D[ ,),[,),,[, ),
kk
aaa aa aa a
plppp pp r
+
== =L where
0112 1
a
V[ ,)[,) [, ),
kk
aaa aa aa a
p
lp pp pp r
+
== ∪ ∪∪ =L and
a
k
a
k
aaaaa
rpppprp =<<<<<=
+1210
L
Any D
a
is uniquely defined by the set of cuts on
},{:
21 k
aaaa
pppV L (empty if card(D
a
) = 1). The
set of cuts on V
a
defined by D
a
can be identified by
D
a
. A family }:{
aa
VaDD ∈= of partitions on S
can be represented in the form
a
Aa
Da ×
∈
U
}{
Any
a
Dv)(a, ∈ will be also called a cut on
a
V .
Then the family
}:{
aa
VaDD ∈= defines from
}){,( dAUS ∪= a new decision table
}){,( dAUS
pp
∪= , where }:{A
p
Aaa
p
∈=
and
),[)()(a
1p +
∈⇔=
i
a
i
a
ppxaix for any
Ux ∈ and },0{ ki L∈ .
After discretization, the original decision system
is replaced with the new one. And different sets of
cuts will construct different new decision systems. It
is obvious that discretization process is associated
with loss of information. Usually, the task of
discretization is to determine a minimal set of cuts
from a given decision system and keeping the
discernibility between objects and the rationality of
the selected cuts can be evaluated by the following
criteria(Nguyen H S, 1995, 1997): (1) Consistency
of P. For any objects
Uvu, ∈ , they are satisfying
if u, v are discerned by A, then u, v are discerned by
P;(2) Irreducibility. There is no
P
P
⊂
′
, satisfying
the consistency; (3) Optimality. For any
P
′
satisfying consistency, it follows
)P(card(P)
′
≤ card , then P is optimal cuts. It has
been proven that Optimal Discretization Problem is
NP-complete (Nguyen H S, 1995).
3 DETERMINATION OF
CANDIDATE CUTS AND
CALCULATION OF
DISCERNIBILITY OF CUTS
Let ),},{,(S fVdAU ∪= be a decision
system. An arbitrary condition attribute Aa ∈ ,
defines a sequence
a
n
a
2
a
1
a
vvv <<< L , where
{}{}
Uxxa ∈= :)(v,,v,v
a
n
a
2
a
1
a
L , Then the
set of all possible cuts on a is defined by:
⎪
⎭
⎪
⎬
⎫
⎪
⎩
⎪
⎨
⎧
+
++
=
−
)
2
vv
(a,,),
2
vv
(a,),
2
vv
(a,C
a
n
a
1n
a
3
a
2
a
2
a
1
a
aa
L .
The set of all possible cuts on all attributes is
denoted by:
U
Aa
a
C
∈
=
A
C .This method usually
generates a large set of candidate cuts. In order to
reduce the number of candidate cuts, we can use
bound cuts (Jian-Hua Dai, 2002).
Since we are only interested in separating objects
that have different decision values, each cut in our
representation is given information about how many
objects from each decision class are to the left and to
the right of the cut, i.e. how many pairs of objects
with different decision values that are discerned
from each other. The algorithm where this measure
is later used sequentially deals with each attribute
and the set of cuts that may be introduced on that
attribute. By assuming that we can totally order all
objects so that they primarily are sorted on the value
of the current attribute and secondly in some
arbitrary order, we use the algorithm 1 to calculate
Table 1: Discernibility Conventions the discernibility
value of a cut.
(a,c) A cut point c on an attribute a dividing all
objects in a decision system in two parts
})(:{ cuaUu <∈ and })(:{ cuaUu >∈
D A set of cuts (a,c)
AllCuts All possible cuts on the decision system
L {} DcaU ∈),( or ABBU ⊆,
),(),,( carcal
XX
number of elements that are to the left/right of
the cut (a, c) in the equivalence class X
),(),,( carcal
X
i
X
i
number of elements with
decision value i in equivalence class X that
are to the left/right of the cut (a, c)
j
c a value indicating where the cut is made
ICEIS 2004 - ARTIFICIAL INTELLIGENCE AND DECISION SUPPORT SYSTEMS
512

Before turning to the details of algorithm 1,
internal notations used in the algorithm are explained
in table 1. These notations are also used in the next
section.
Algorithm 1: Calculate the discernibility value of
a cut
Input: information system S }){,( dAU ∪= ,
candidate cut (a, c
j
) on condition attribute Aa ∈
Output: the discernibility value of the cut (a, c
j
)
Method:
0N ←
for each
L
X
∈ do
()
1
N N ( , ) ( , ) ( , ) ( , )
d
XX XX
ii
i
l
acrac lacra
c
=
←+ ⋅ − ⋅
∑
return N
The discernibility value returned by algorithm 1
is equal to the number of pairs of objects from S
discerned by cut (a,c
j
). The proof that this algorithm
is correct can be found in (Hung Son Nguyen, 1996).
4 LOCAL DISCRETIZATION AND
GLOBAL DISCRETIZATION
In this section, we describe Local Discretization and
Global Discretization algorithms (Hung Son Nguyen,
1996). Local Discretization algorithm works by
finding a maximally discerning cut (see algorithm 1)
from the set of all possible cuts (AllCuts) and then
dividing the dataset into two subsets as long as there
are objects with different decision values.
Algorithm 2. Local discretization
Input: information system
S }){,( dAU ∪= ,
all candidate cuts in
S
Output: new information system after being
discretized
Method:
N
UM
C
LASSES
(S)
Return number of decision classes in S
T
RAVERSE
(S)
If N
UM
C
LASSES
(S) > 1 Then
from AllCuts select cut point
),(
**
ca
which has maximal discernibility value
using algorithm 1
{} ),(
**
caDD ∪←
{} ),(\
**
caAllCutsAllCuts ←
{} cxaUxU <∈← )(:
*
1
{} cxaUxU ≥∈← )(:
*
2
T
RAVERSE
(U
1
)
T
RAVERSE
(U
2
)
L
OCAL
D
ISCRETIZATION
(S)
AllCuts
← the set of all possible cuts in S
φ
←D
T
RAVERSE
(U)
Discretize S using the cuts in D
In algorithm 3 (Hung Son Nguyen, 1996), it
works with decision classes and check each
consecutive cut that is added to the final set D
against all objects that are not completely separated
into equivalence classes uniform w.r.t. decision
value by the current set of cuts. It splits the decision
classes into smaller and smaller parts until they are
uniform with respect to the decision values of the
objects.
Algorithm 3. Global Discretization
Input: information system
S }){,( dAU ∪= ,
all candidate cuts in
S
Output: new information system after being
discretized
Method:
N
UM
C
LASSES
(S)
Return number of decision classes in S
G
LOBAL
D
ISCRETIZATION
(S)
AllCuts
← the set of all possible cuts in S
φ
←D
BUL /← , B is the set of attributes that will
not be discretized
repeat
from AllCuts select cut point ),(
**
ca
which has maximal discernibility value
using algorithm 1
{} ),(
**
caDD ∪←
{} ),(\
**
caAllCutsAllCuts ←
for each
L
X
∈ do
}{\ XLL ←
if N
UM
C
LASSES
(X) > 1 then
})(:{
**
1
cxaXxX ≤∈←
})(:{
**
2
cxaXxX >∈←
},{
21
XXLL ∪←
until
φ
=L
5 EXPERIMENTS AND ANALYSIS
We do our experiments on three data sets from UCI
named abalone and iris and liver disorders
respectively, which can be downloaded from the
website (MLR). Some information about the data
sets is shown in table 2.
RESULT COMPARISON OF TWO ROUGH SET BASED DISCRETIZATION ALGORITHMS
513

Table 2: The data sets for experiments
Name #objects #continuous
attributes
#decision
classes
Iris 150 4 3
liver-
disorders
345 6 2
Abalone 4177 7 29
We make comparative experiments between local
discretization algorithm and global algorithm,
comparing the number of result cuts discretizing
continuous attributes. The results are shown in table
3, table 4, and table 5 respectively. In the tables,
#cuts L denotes the number of result cuts generated
by local discretization algorithm and #cuts G by
global discretization algorithm.
As the two algorithms are both applied on
consistent information systems and maintain the
original indiscernibility, the smaller number of the
result cuts, the better the algorithm is. From the
comparisons we know that for liver disorders dataset
and abalone dataset, the number of result cuts
generated by global algorithm is far smaller than by
local algorithm. But it is larger for liver iris dataset.
So we can’t say that global algorithm is always
better than local algorithm.
For liver iris data set, the number of result cuts of
attribute sepal_length generated by global algorithm
is far larger than by local algorithm, and the number
Table 3: Comparison of the results on liver disorders.
Attribute Mcv alkphos sgpt Sgo Gam- drinks total
magt
#cuts L 20 22 20 25 30 23 140
#cuts G 3 4 3 2 5 3 20
Table 4: Comparison of the results on iris.
Attribute sepal_ sepal_ petal_ petal_ Total
Length width length width
#cuts L 3 3 6 1 13
#cuts G 34 2 4 2 42
Table 5: Comparison of the results on abalone
Attri- len- diam- hei- Whole shucked viscera shell total
bute gth eter ght weight weight weight weight
#cuts L 421 389 419 539 564 674 555 3561
#cuts G 20 21 30 7 32 32 30 172
of result cuts of other attributes is almost equal. But
for two other data sets, the number of result cuts for
all attributes generated by global algorithm is far
smaller than by local algorithm. Hence, we can say
that the two algorithms are data set sensitive, and we
can conjecture that their quality depends on the
distributions of the values of the attributes and their
decision classes.
6 CONCLUSIONS
For discretization based on rough set, we should
seek possible minimum number of discrete internals,
and at the same time it should not weaken the
indiscernibility ability. This paper examines two
algorithms (Hung Son Nguyen,1996), local
discretization and global discretization. Our
experiments show that the discretization algorithms
are dataset sensitive. Neither of them always
generates smaller number of result cuts. On some
datasets, one algorithm generates fewer result cuts,
but on other datasets it is contrary. We can
conjecture that the quality of the two algorithms
depends on the distributions of the values of the
continuous dataset attributes and their decision
classes. How the distributions affect the results is
what we will study further. With that, we can use
some methods to improve the algorithms.
REFERENCES
Pawlak Z (1982, November 5). Rough Sets. Int'l J.
Computer & Science [J], 11, 341-356.
Nguyen H S, Skowron A (1995). Quantization of real
value attributes. Proceedings of Second Joint Annual
Conf. on Information Science, Wrightsville Beach,
North Carolina, 34-37.
Nguyen H S (1997). Discretization of Real Value
Attributes: Boolean reasoning Approach [PhD
Dissertation]. Warsaw University Warsaw, Poland.
Hung Son Nguyen, Sinh Hoa Nguyen (1996). Some
efficient algorithms for rough set methods. In 6th
International conference on Information Processing
and Management of Uncertainty in Knowledge-Based
Systems, 1451-1456.
Jian-Hua Dai, Yuan-Xiang Li (2002, November 4-5).
Study on discretization based on rough set theory.
Proceedings of the First International Conference on
Machine Learning and Cybernetics, 3, 1371-1373.
MLR, http://www.ics.uci.edu/~mlearn/MLRepository.html
.
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