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New Approaches for Data Reduction in Generalized Multi-valued Decision Information System (GMDIS): Case Study of Rheumatic Fever Patients. By Abd El-Monem M. Kozea, Mohamed M. E. Abd El-Monsef, Soaad Abd El-Badie Attia El-Afify Mathematics Department, Faculty of Science,

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New Approaches for Data Reduction in Generalized Multi-valued Decision Information System (GMDIS):Case Study ofRheumatic Fever Patients

WRSTA2006, 13 August 2006


By Multi-valued Decision Information System (GMDIS):

Abd El-Monem M. Kozea,

Mohamed M. E. Abd El-Monsef,Soaad Abd El-Badie Attia El-Afify

Mathematics Department, Faculty of Science,

Tanta University, Egypt

Email:[email protected]

WRSTA2006, 13 August 2006


Outline
Outline Multi-valued Decision Information System (GMDIS):

  • Motivation / Introduction

  • Basic Concepts of Rough Sets

  • Rheumatic Fever Data: Characteristics

  • New Thinking

    • Generalized Multi-Valued Decision Information System (GMDIS)

    • New Approaches for Data Reduction in GMDIS

      • Non-equivalence Relations,

      • Topological Spaces and

      • Degree of Dependencies in GMDIS

  • Reduct Algorithms based on GMDIS

    • Rheumatic Fever GMDIS Reduction: Worked example

  • Conclusion and Future Work

WRSTA2006, 13 August 2006


Motivation introduction 1
Motivation / Introduction (1) Multi-valued Decision Information System (GMDIS):

  • Rough set theory was developed by Zdzislaw Pawlak in the early 1982’s.

    • RS is based on the idea of equivalence relations which partition the domain into different classes.

    • It is a mathematical tool for dealing with incomplete data for induction of approximations of concepts and for discovering patterns hidden in data.

    • It can be used for feature selection, data reduction, identifies partial/total dependencies in data, gives approach to null values and missing data, and decision rule generation.

  • Rough Set Features:

    • It is applicable to problems with both numeric and descriptive attributes

    • It is capable of finding all minimal knowledge representation

    • It is highly automated based on strict rules.

    • A multi-valued information system (MIS) is a generalization of the idea of a single valued information system (SIS).

    • In a multi-valued information system,

      • Attribute functions are allowed to map elements to sets of attribute values.

WRSTA2006, 13 August 2006


Motivation introduction 2
Motivation / Introduction (2) Multi-valued Decision Information System (GMDIS):

  • Initiated a new approach for data reduction in GMDIS.

    • By converting the Single-Valued Decision Information System (SDIS) to a GMDIS.

  • Two general relations are defined

  • Constructing new classes using the general relations

  • The measure of decision dependency on the condition attributes is studied To evaluate the performance of the approach,

    • An application of Rheumatic Fever datasets.

WRSTA2006, 13 August 2006


Rough set theory basic concepts
Rough Set Theory: Multi-valued Decision Information System (GMDIS):Basic Concepts

  • Information/Decision Systems (Tables)

  • Indiscernibility

  • Set Approximation

  • Reducts and Core

  • Rough Membership

  • Dependency of Attributes

WRSTA2006, 13 August 2006


Information Systems/Tables Multi-valued Decision Information System (GMDIS):

  • IS is a pair (U, A)

  • U is a non-empty finite set of objects.

  • A is a non-empty finite set of attributes such that for every

  • is called the value set of a.

WRSTA2006, 13 August 2006


Decision Systems/Tables Multi-valued Decision Information System (GMDIS):

  • DS:

  • is the decisionattribute(instead of one we can consider more decision attributes).

  • The elements ofAare called the condition attributes.

WRSTA2006, 13 August 2006


Information systems types

= Multi-valued Decision Information System (GMDIS):

Î

(

U

,

At

,

{

V

:

a

At

},

f

)

a

a

=

Î

(

U

,

At

D

,

{

V

:

a

At

},

f

)

U

a

a

Information Systems Types

  • The first concept of IS was developed by Grzymala-Busse (1988). There are many types of IS as follows:

    • Single valued Information System (SIS)

      • The data takes a single value for each element

    • Single valued Decision Information System (SDIS)

    • A Multi-valued Information System (MIS)

      • An ordinary information system which its values ore sets

    • A Multi-valued Decision Information System (MDIS)

WRSTA2006, 13 August 2006


Rheumatic fever data characteristics
Rheumatic Fever Data: Characteristics Multi-valued Decision Information System (GMDIS):

  • We obtained the used Rheumatic Fever patients data from Tanta University Hospital, Egypt.

  • All patients are between 9-12 years old with history of Arthritis began from age 3-5 years.

  • This disease has many symptoms and it is usually started in young age and still with the patient along his life.

  • The following table shows seven patients characterized by 8 symptoms (attributes) using them to decide the diagnosis for each patient (decision attribute).

WRSTA2006, 13 August 2006


Rheumatic Fever Data: Characteristics Multi-valued Decision Information System (GMDIS):

ِAttribute Symbol

ً to?Refers

ِAttribute Values

to?Refers

Sex

Male

Female

Pharyngitis

Yes

No

Arthritis

No arthritis

Began in the knee

Began in the ankle

Carditis

Affected

Not affected

Chorea

Yes

No

ESR

Normal

High

Abdonominal Pain

Absent

Present

Headache

Yes

No

Diagnosis

Rheumatic Arthritis

CarditisRheumatic

Rheumatic Arthritis and Carditis

WRSTA2006, 13 August 2006


New thinking 1
New Thinking (1) Multi-valued Decision Information System (GMDIS):

  • A multi-valued information system (MIS) is a generalization of the idea of a single valued information system (SIS).

  • In a multi-valued information system, attribute functions are allowed to map elements to sets of attribute values

  • Covert the SDIS to a MDIS and vice versa?

WRSTA2006, 13 August 2006


New thinking 2
New Thinking (2) Multi-valued Decision Information System (GMDIS):

  • Initiative two methods to:

    • Covert the SIS to a MIS and vice versa!

    • Covert the SDIS to a MDIS and vice versa!

      by ( Collecting of Attributes).

WRSTA2006, 13 August 2006


S Multi-valued Decision Information System (GMDIS):

F

A

R

K

E

P

H

D

x1

s2

f1

a1

r1

k1

e1

p1

h2

d3

x2

s1

f1

a1

r1

k1

e2

p1

h1

d3

x3

s2

f1

a2

r1

k2

e1

p1

h2

d3

x4

s1

f1

a1

r2

k2

e1

p1

h2

d1

r1

x5

s1

f2

a0

k2

e1

p2

h2

d2

x6

s1

f1

a1

r1

k2

e2

p1

h2

d3

x7

s1

f1

a2

r1

k2

e1

p1

h2

d3

Worked Example 1 (SDIS):

Rheumatic Fever SDIS Data

WRSTA2006, 13 August 2006


Worked Example 2 ( Multi-valued Decision Information System (GMDIS):MDIS ):

Converted Data Description (MDIS)

Attribute Symbol

ًRefers to ?

ِAttribute Values

ًRefers to ?

α

{S,K}

α1

S → s1

α2

K → k1

α3

{S,K}→ {s2,k2}

β

{F,A,E}

β1

F → f1

β2

A →a1

β3

A →a2

β4

E → e1

β5

{F,A,E} →{f2,a0,e2}

δ

{R,P,H}

δ1

R → r1

δ2

P→p1

δ3

H→h1

δ4

{R,P,H}→

{r2,p2,h2}

D

Diagnosis

d1

Rheumatic arthritis

d2

Rheumatic carditis

d3

Rheumatic arthritis and carditis

WRSTA2006, 13 August 2006


α Multi-valued Decision Information System (GMDIS):

β

δ

D

x1

{α2}

{β1,β2,β4}

{δ1,δ2,}

{d3 }

x2

{α1,α2}

{β1, β2,}

{δ1,δ2,δ3}

{d3 }

x3

{α3}

{β1, β2, β4}

{δ1,δ2}

{d3 }

x4

{α1}

{β1,β2,β4}

{δ2 }

{d1 }

x5

{α1}

{β4}

{δ1 }

{d2 }

x6

{α1}

{β1,β2}

{δ1,δ2}

{d3 }

x7

{α1}

{β1, β3, β4}

{δ1,δ2,δ3}

{d3 }

Worked Example 3 (MDIS ):

Rheumatic Fever MDIS Data

WRSTA2006, 13 August 2006


Generalized Multi-Valued Decision Information System (GMDIS) Multi-valued Decision Information System (GMDIS):

WRSTA2006, 13 August 2006


Initiated a new approach
Initiated a New Approach Multi-valued Decision Information System (GMDIS):

  • Initiate a new approach for data reduction in Generalized Multi–Valued Decision Information System (GMDIS).

    • Convert the SDIS to GMDIS.

    • Two general relations are defined on condition attributes and decision attribute.

    • Construct new classes using the general relations which are used for data reduction.

    • Study The measure of decision dependency on the condition attributes

    • Evaluate the performance of the approach,

      • an application of, rheumatic fever datasets has been chosen and the reduct approach have been applied to see their ability and accuracy.

WRSTA2006, 13 August 2006


Generalized multi valued decision information system

= Multi-valued Decision Information System (GMDIS):

y

Î

h

Í

GMIS

(

U

,

At

,

{

:

a

At

},

f

,

{

:

B

At

})

(1)

a

a

B

=

y

Î

h

Í

GMDIS

(

U

,

At

D

,

{

:

a

At

},

f

,

{

:

B

At

})

U

a

a

B

(2)

Generalized Multi-valued Decision Information System

A Generalized Multi-valued Information System can be defined as follows.

A Generalized Multi-valued Decision Information System can be defined as follows.

WRSTA2006, 13 August 2006


(1) Multi-valued Decision Information System (GMDIS):

c

h

=

Í

"

Î

Í

{(

x

,

y

)

:

f

(

x

)

f

(

y

)

,

a

B

,

B

At

}

(4)

m

=

m

=

m

¹

Î

¹

{

A

A

,

A

,

A

,

A

,

A

A

,

i

j

}

U

I

h

=

{(

x

,

y

)

:

f

(

x

)

depends

on

f

(

y

)}

B

a

a

h

a

l

i

j

l

k

i

j

k

a

D

D

k

D

=

Í

{(

x

,

y

)

:

f

(

x

)

f

(

y

)}

(3)

D

D

h

=

Í

"

Î

Í

{(

x

,

y

)

:

f

(

y

)

f

(

x

)

,

a

B

,

B

At

}

B

a

a

(2)

Set Approximations in GMDIS (1)

Define the set of all intersections of members of as the Meeting Point Relation (MPR) can be written as:

WRSTA2006, 13 August 2006


D Multi-valued Decision Information System (GMDIS):

=

Í

(5)

POS

(

D

)

X

,

B

At

U

h

B

B

Î

X

A

h

D

Where, for any subset the lower and upper approximations are defined by,

Í

X

U

=

h

h

Í

Í

X

{

:

X

},

B

At

U

h

Bx

Bx

(6)

B

=

h

h

¹

F

Í

X

{

:

X

},

B

At

U

I

h

Bx

Bx

B

Set Approximations in GMDIS (2)

WRSTA2006, 13 August 2006


Suggested new technique consideration 1

2. Multi-valued Decision Information System (GMDIS):The attribute is called the principal attribute (PA) if , and if then both a and b are principal attributes.

Î

a

At

t

£

t

B

D

Í

B

At

t

£

t

"

Î

t

$

Î

t

Ì

¹

iff

G

,

G

'

s

.

t

.

G

G

'

,

G

,

G

'

U

B

D

B

D

t

=

t

t

t

"

Î

¹

,

a

,

b

At

,

b

a

f

a

b

a

b

Suggested New Technique : Consideration (1)

1.The set of attributes is called a reduct if and B is minimal, where

WRSTA2006, 13 August 2006


Suggested new technique consideration 2

Y Multi-valued Decision Information System (GMDIS):

=

{

R

,

R

,

,

R

}

L

1

2

n

Y

=

'

{

R

'

,

R

'

,

,

R

'

}

L

1

2

n

"

Î

Y

$

Î

Y

Í

R

'

'

,

R

s

.

t

.

R

'

R

i

i

i

i

Suggested New Technique : Consideration (2)

  • The set of attributes of equal highest degree of dependency is the PA of the GMDIS.

If the set of all reducts of any SDIS is ,

, and the set of reducts for the GMDIS system using tha new approach is, . .Then, it can be said that Y’ is more refinement than, Y if .

WRSTA2006, 13 August 2006


Simplified Reducts Multi-valued Decision Information System (GMDIS):

  • Is the set of all reducts, after omitted the supersets of each reduct in the set RED (At), and we denote it by SRED (At).

WRSTA2006, 13 August 2006


Gmdis reduction algorithms
GMDIS Reduction Algorithms Multi-valued Decision Information System (GMDIS):

  • Algorithm 1: GMDIS Reduct

  • Algorithm 2: GMDIS PA Algorithm

WRSTA2006, 13 August 2006


Í Multi-valued Decision Information System (GMDIS):

R

At

A GMDIS =

y

Î

h

Í

(

U

,

At

D

,

{

:

a

At

},

f

,

{

:

B

At

})

U

a

a

B

(1)

¬

R

GMDIS

¬

(6)

R

{}

¬

(3)

GMDIS

R

t

£

t

(7)

R

D

R

Î

-

Loop

a

(

At

R

)

(9) Return

¬

GMDIS

R

{

a

}

º

U

Where

R

Reduct

GMDIS Reduction Algorithms:

GMDIS Reduct

(2) Do

(4)

(8) Until

t

£

t

(5) If

{

}

R

a

D

U

R : A set of minimum attribute subset;

WRSTA2006, 13 August 2006


Í Multi-valued Decision Information System (GMDIS):

PA: A set of principal attribute subset,

PA

At

A GMDIS =

y

Î

h

Í

(

U

,

At

D

,

{

:

a

At

},

f

,

{

:

B

At

})

(1)

U

(6)

¬

¬

PA

PA

{

a

}

PA

{}

U

a

a

B

(7)

End Loop

(2)

Do

Î

(8)

a

At

End Loop

(9)

Î

PA

b

At

Return

t

t

f

a

b

GMDIS Reduction Algorithms: GMDIS PA Algorithm

(3)

Loop

(4)

Loop

(5)

If

WRSTA2006, 13 August 2006


c Multi-valued Decision Information System (GMDIS):

h

=

Í

"

Î

Í

{(

x

,

y

)

:

f

(

x

)

f

(

y

)

,

a

B

,

B

At

}

B

a

a

a

{

}

=

a

=

RED

(

At

)

{

}

{

S

,

K

}

Rheumatic Fever GMDIS Reduction: Worked Example

  • Applying the new approach on MDIS Rheumatic Fever data to be a GMDIS by using the relation

  • So we conclude that is the reduct and it is the PA of the GMDIS and this is the same result obtained using the second consideration.

WRSTA2006, 13 August 2006


x Multi-valued Decision Information System (GMDIS):1

x2

x3

x4

x5

x6

x7

x1

Ф

x2

Ф

Ф

x3

Ф

Ф

Ф

x4

{S,R,K}

{R,K,E,H}

{S,A,R}

Ф

x5

{S,F,A,K,P}

{F,A,K,E,P,H}

{S,F,A,P}

{F,A,R,P}

Ф

x6

Ф

Ф

Ф

{R,E}

{F,A,E,P}

Ф

x7

Ф

Ф

Ф

{A,R,H}

{F,A,P,H}

Ф

Ф

Discernibility Matrix versus GMDIS

Rheumatic Fever Data Discernibility Matrix

WRSTA2006, 13 August 2006


= Multi-valued Decision Information System (GMDIS):

Ú

Ú

Ù

Ú

Ú

Ú

Ù

Ú

Ú

Ù

Ú

Ú

Ú

Ú

f

{

S

R

K

}

{

R

K

E

H

}

{

S

A

R

}

{

S

F

A

K

P

}

At

Ù

Ú

Ú

Ú

Ú

Ú

Ù

Ú

Ú

Ú

Ù

Ú

Ú

Ú

Ú

Ù

Ú

{

F

A

K

E

P

H

}

{

S

F

A

P

}

{

F

A

R

K

P

}

{

R

E

}

Ù

Ú

Ú

Ú

Ù

Ú

Ú

Ù

Ú

Ú

Ú

{

F

A

E

P

}

{

A

R

H

}

{

F

A

P

H

}

=

Ú

Ú

Ú

Ú

Ú

Ú

Ú

Ú

Ú

Ú

Ú

Ú

Re

d

(

At

)

{{

S

R

K

},

{

S

A

R

},

{

S

F

A

P

},

{

F

A

R

K

P

},

{

R

E

}

Ú

Ú

Ú

Ú

Ú

Ú

Ú

Ú

,

{

F

A

E

P

},

{

A

R

H

},

{

F

A

P

H

}}

=

a

=

RED

(

At

)

{

}

{

S

,

K

}

The discernibility function

WRSTA2006, 13 August 2006


Final Note Multi-valued Decision Information System (GMDIS):

Reducts obtained by GMDIS is contained in the reducts obtained on SDIS using the discernibility matrix, that means that the new approach gives more reduction.

WRSTA2006, 13 August 2006


Conclusion
Conclusion Multi-valued Decision Information System (GMDIS):

  • New approach for data reduction in GMDIS is considered as a generalization in the case of MDIS.

    • This approach extended to Pawlak approach if the system is single-valued and the relations are equivalence.

  • Opens the way for other approaches of data reduction

    • if we use the general topological recent concepts such as Pre-open sets, Semi-open sets, etc.

  • In many real life situations, the use of attributes in a single fashion is not represetable for the actual effect of attributes. So, it is necessary to consider subsets of the attributes as a multi criteria.

  • An application of, Rheumatic Fever datasets has been chosen and the reduct approach has been applied to see their ability and accuracy.

WRSTA2006, 13 August 2006


Acknowledgment
Acknowledgment Multi-valued Decision Information System (GMDIS):

  • The authors greatly appreciate and thanks many peoples for their valuable comments and advices:

    • Dr. K. E. Sturtz, , Air Force Research Laboratory, Wright Patterson Air Force Base, Ohio;

    • Prof. Aboul Ella Hassanien, Cairo University

    • Prof. E. Rady,, I.S.S.R., Cairo University.

    • Dr. A. S. Salama. Pure Mathematics Dept., Faculty of Science, Tanta University.

WRSTA2006, 13 August 2006


شكرا Multi-valued Decision Information System (GMDIS):

لحسن استماعكم

WRSTA2006, 13 August 2006


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