by abd el-monem m. kozea, mohamed m. e. abd el-monsef, soaad abd el-badie attia el-afify

WRSTA2006, 13 August 2006

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 ,

Tanta University, Egypt

Email: [email protected]


OutlineOutline Motivation / IntroductionMotivation / Introduction Basic Concepts of Rough SetsBasic Concepts of Rough Sets Rheumatic Fever Data: Rheumatic Fever Data: CharacteristicsCharacteristics New ThinkingNew Thinking

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

New Approaches for Data Reduction in GMDISNew Approaches for Data Reduction in GMDIS Non-equivalence Relations, Non-equivalence Relations, Topological Spaces and Topological Spaces and Degree of Dependencies in GMDISDegree of Dependencies in GMDIS

Reduct Algorithms based on GMDISReduct Algorithms based on GMDIS Rheumatic Fever GMDIS Reduction: Rheumatic Fever GMDIS Reduction: Worked exampleWorked example

Conclusion and Future WorkConclusion and Future Work


Motivation / Introduction (1)Motivation / Introduction (1) Rough set theoryRough set theory was developed by was developed by Zdzislaw PawlakZdzislaw Pawlak in the early 1982’s. in the early 1982’s.

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

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

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

Rough Set Features:Rough Set Features: It is applicable to problems with both numeric and descriptive attributesIt is applicable to problems with both numeric and descriptive attributes It is capable of finding all minimal knowledge representationIt is capable of finding all minimal knowledge representation It is highly automated based on strict rules.It is highly automated based on strict rules. A A multi-valued information systemmulti-valued information system (MIS) is a generalization of the idea of (MIS) is a generalization of the idea of

a single valued information system (SIS). a single valued information system (SIS). In a multi-valued information system, In a multi-valued information system,

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


Motivation / Introduction (2)Motivation / Introduction (2) Initiated a new approach for data reduction in GMDIS. Initiated a new approach for data reduction in GMDIS.

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

Two general relations are defined Two general relations are defined Constructing new classes using the general relations Constructing new classes using the general relations The measure of decision dependency on the condition attributes is studied To The measure of decision dependency on the condition attributes is studied To

evaluate the performance of the approach,evaluate the performance of the approach, An application of An application of Rheumatic Fever datasets.Rheumatic Fever datasets.


Rough Set Theory: Rough Set Theory: Basic ConceptsBasic Concepts

Information/Decision Systems (Tables)Information/Decision Systems (Tables) IndiscernibilityIndiscernibility Set ApproximationSet Approximation Reducts and CoreReducts and Core Rough MembershipRough Membership Dependency of AttributesDependency of Attributes


FactNo.

Driving Conditions

Weather Road Time

1 Misty Icy Day

2 Foggy Icy Night

3 MistyNot icy

Night

4 Sunny Icy Day

5 FoggyNot icy

Dusk

6 MistyNot icy

Night

Information Systems/TablesInformation Systems/Tables

IS is a pair (U, A)IS is a pair (U, A) U is a non-empty U is a non-empty

finite set of objects.finite set of objects. A is a non-empty finite A is a non-empty finite

set of attributes such set of attributes such that for that for every every

is called the value is called the value set of set of aa..

aVUa :.Aa

aV


Decision Systems/TablesDecision Systems/Tables

FactNo.

Driving Conditions Consequence

Weather Road Time Accident

1 Misty Icy Day Yes

2 Foggy Icy Night Yes

3 Misty Not icy Night Yes

4 Sunny Icy Day No

5 Foggy Not icy Dusk Yes

6 Misty Not icy Night No

The elements ofThe elements of AA are called the are called the condition condition attributes.attributes.

Ad DSDS: : is the is the decisiondecision

attributeattribute (instead of (instead of one we can consider one we can consider more decision more decision attributes).attributes).

}){,( dAUT


Information Systems TypesInformation Systems Types The first concept of IS was developed by The first concept of IS was developed by

Grzymala-Busse (1988). There are many types of Grzymala-Busse (1988). There are many types of IS as follows:IS as follows:

Single valued Information System (SIS)Single valued Information System (SIS) The data takes a single value for each elementThe data takes a single value for each element

Single valued Decision Information System (SDIS)Single valued Decision Information System (SDIS)

A Multi-valued Information System (MIS)A Multi-valued Information System (MIS)

An ordinary information system which its values ore setsAn ordinary information system which its values ore sets

A Multi-valued Decision Information System (MDIS)A Multi-valued Decision Information System (MDIS)

))},},::{{,,,,(( aaaa ffAtAtaaVVAtAtUU

))},},::{{,,,,(( aaaa ffAtAtaaVVDDAtAtUU


Rheumatic Fever Data: Rheumatic Fever Data: CharacteristicsCharacteristics

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

All patients are between All patients are between 9-12 years9-12 years old with history old with history of Arthritis began from age 3-5 years. of Arthritis began from age 3-5 years.

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

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


Rheumatic Fever Data: Characteristics

Rheumatic Arthritis Rheumatic Arthritis and Carditisand Carditis

Carditis RheumaticCarditis RheumaticDiagnosisDiagnosis

Rheumatic ArthritisRheumatic Arthritis

NoNoHeadacheHeadacheYesYes

PresentPresentAbdonominal PainAbdonominal Pain

AbsentAbsent

HighHighESRESRNormalNormal

NoNoChoreaChoreaYesYes

Not affectedNot affectedCarditisCarditisAffectedAffected

Began in the ankleBegan in the ankle

Began in the kneeBegan in the kneeArthritisArthritisNo arthritisNo arthritis

NoNoPharyngitisPharyngitisYesYes

FemaleFemaleSexSexMaleMale

�ِ�ِAttribute SymbolAttribute Symbol �ً�ً to?Refersto?Refers �ِ�ِAttribute ValuesAttribute Values to?Refersto?Refers

1s2s1f

2f

2a

0a

1r2r

1k

2k1e

2e1p

2p

1h

2h

1d

2d

3d

1a

S

F

A

R

K

E

P

H

D


New Thinking (1)New Thinking (1)

A A multi-valued information systemmulti-valued information system ( (MISMIS) is a ) is a generalization of the idea of a generalization of the idea of a single valued single valued information systeminformation system ( (SISSIS).).

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

Covert the SDIS to a MDIS and vice Covert the SDIS to a MDIS and vice

versa?versa?


New Thinking (2)New Thinking (2)

Initiative two methods to:Initiative two methods to:

Covert the SIS to a MIS and vice Covert the SIS to a MIS and vice

versa!versa!

Covert the SDIS to a MDIS and vice Covert the SDIS to a MDIS and vice

versa!versa!

by ( Collecting of Attributes).by ( Collecting of Attributes).


Worked Example 1 (SDIS )):

Rheumatic Fever SDIS Data

x7 d3

x6 d3

x5 d2

x4 d1

x3 d3

x2 d3

x1 d3

S F A R K E P H D

ss11 ff11 aa22 rr11 kk22 ee11 pp11 hh22


ss11 ff22 aa00 kk22 ee11 pp22 hh22





rr11


Rheumatic arthritis and carditisd3

Rheumatic carditisd2

Rheumatic arthritisd1

DiagnosisD

{R,P,H}→{r2,p2,h2}

δ4

H→h1δ3

P→p1δ2

R → r1δ1

{R,P,H}δ

{F,A,E} →{f2,a0,e2}β5

E → e1β4

A →a2β3

A →a1β2

F → f1β1

{F,A,E}β

{S,K}→ {s2,k2}α3

K → k1α2

S → s1α1

{S,K}α

�ًRefers to ?�ًAttribute Values�ًRefers to ?Attribute Symbol

Worked Example 2 (MDIS ):

Converted Data Description (MDIS)


x7{α1} {β1, β3, β4} {δ1,δ2,δ3} {d3 }

x6{α1} {β1,β2} {δ1,δ2} {d3 }

x5{α1} {β4} {δ1 } {d2 }

x4{α1} {β1,β2,β4} {δ2 } {d1 }

x3{α3} {β1, β2, β4} {δ1,δ2} {d3 }

x2{α1,α2} {β1, β2,} {δ1,δ2,δ3} {d3 }

x1{α2} {β1,β2,β4} {δ1,δ2,} {d3 }

α β δ D

Worked Example 3 (MDIS ):

Rheumatic Fever MDIS Data


Generalized Multi-Valued Decision

Information System (GMDIS)


Initiated a New ApproachInitiated a New Approach Initiate a new approach for Initiate a new approach for data reductiondata reduction in in Generalized Multi–Valued Decision Generalized Multi–Valued Decision

Information System (GMDIS).Information System (GMDIS).

Convert the SDIS to GMDIS. Convert the SDIS to GMDIS. Two general relations Two general relations are definedare defined on condition attributes and decision on condition attributes and decision

attribute. attribute. Construct Construct new classesnew classes using the using the general relationsgeneral relations which are used for data which are used for data

reduction. reduction. Study The measure of decision dependency on the condition attributes Study The measure of decision dependency on the condition attributes Evaluate the performance of the approach, Evaluate the performance of the approach,

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


A Generalized Multi-valued Information SystemA Generalized Multi-valued Information System can be defined as can be defined as

follows.follows.

A Generalized Multi-valued Decision Information SystemA Generalized Multi-valued Decision Information System can be can be

defined as follows.defined as follows.

})})::{{,,},},::{{,,,,(( AtAtBBffAtAtaaAtAtUUGMISGMIS BBaaaa (1)(1)

})})::{{,,},},::{{,,,,(( AtAtBBffAtAtaaDDAtAtUUGMDISGMDISBBaaaa

(2)(2)

Generalized Multi-valued Generalized Multi-valued Decision Information SystemDecision Information System


Set Approximations in GMDIS (1)

}},,,,))(())((::)),,{({( AtAtBBBBaayyffxxffyyxx aacc

aaBB (1)(1)

}},,,,))(())((::)),,{({( AtAtBBBBaaxxffyyffyyxx aaaaBB (2)(2)

)})}(())((::)),,{({()})}(())((::)),,{({(

yyffxxffyyxxyyffonondependsdependsxxffyyxx

DDDDDDDDDD

(3)(3)

}},,,,,,,,,,{{ jjiiAAAAAAAAAAAAAAaakkjjiikk kklljjiillaa

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


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

UUXX

DD

AtBXDPOS

D

B

AXB

,)(

(5)(5)

AtAtBBXXXX

AtAtBBXXXX

BxBxBxBx

BxBxBxBx

BB

BB

},},::{{

},},::{{

(6)(6)

Set Approximations in GMDIS (2)


UUGGGGGGGGttssGGGGiffiff DDBBDDBB '',,,,''....'',,

1.1.The set of attributes is called a The set of attributes is called a

reduct if and B is minimal, where reduct if and B is minimal, where DB AtB

Suggested New Technique : Suggested New Technique : Consideration (1)Consideration (1)

2.The attribute is called the principal attribute (PA) if , and if

then both a and b are principal attributes.

Ata

abAtbaba ,,, ba


Suggested New Technique : Suggested New Technique : Consideration (2)Consideration (2)

3. 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 ’’ is more refinement than, if .

}},,,,,,{{ 2211 nnRRRRRR

}}'',,,,'',,''{{'' 2211 nnRRRRRR

iiiiiiii RRRRttssRRRR ''....,,''''


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).

Simplified Reducts


GMDIS Reduction AlgorithmsGMDIS Reduction Algorithms

Algorithm 1: GMDIS ReductAlgorithm 1: GMDIS Reduct Algorithm 2: GMDIS PA AlgorithmAlgorithm 2: GMDIS PA Algorithm


A GMDIS = })})::{{,,},},::{{,,,,(( AtAtBBffAtAtaaDDAtAtUU BBaaaa

(2)(2) Do Do

{}{}RR(1)(1)

RRGMDISGMDIS (3)(3)

(4)(4) ))(( RRAtAtaaLoopLoop

}}{{ aaRRGMDISGMDIS (6)(6)

(7)(7) GMDISGMDISRR

(8) Until(8) Until DDRR (9) Return(9) Return RR

Where Where RR ReductReductR : A set of minimum attribute subset; AtAtRR

(5) If(5) If DaR }{

GMDIS Reduction Algorithms: GMDIS Reduct


}):{,},:{,,( AtBfAtaDAtU Baa A GMDIS =

(1)(1) {}{}PAPA

(2)(2) Do

(3)(3) Loop AtAtaa

(4)(4) Loop AtAtbb (5)(5) bbaa

If

(6)(6) }}{{aaPAPAPAPA

(7)(7) End LoopEnd Loop

(8)(8) End LoopEnd Loop

(9)(9) Return Return PAPA

PA: A set of principal attribute subset, AtAtPAPA

GMDIS Reduction Algorithms: GMDIS PA Algorithm


Rheumatic Fever GMDIS Reduction:

Worked ExampleApplying the new approach on MDIS Rheumatic Fever data to be a GMDIS by using the relation

}},,,,))(())((::)),,{({( AtAtBBBBaayyffxxffyyxx aacc

aaBB

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.

}{

}},,{{}}{{))(( KKSSAtAtREDRED


ФФ{F,A,P,H}{A,R,H}ФФФx7

Ф{F,A,E,P}{R,E}ФФФx6

Ф{F,A,R,P}{S,F,A,P}{F,A,K,E,P,H}

{S,F,A,K,P}x5

Ф{S,A,R}{R,K,E,H}{S,R,K}x4

ФФФx3

ФФx2

Фx1

x7x6x5x4x3x2x1

Rheumatic Fever Data Discernibility Matrix

Discernibility Matrix versus GMDIS


The discernibility functionThe discernibility function

}}{{}}{{}}{{

}}{{}}{{}}{{}}{{

}}{{}}{{}}{{}}{{

HHPPAAFFHHRRAAPPEEAAFF

EERRPPKKRRAAFFPPAAFFSSHHPPEEKKAAFF

PPKKAAFFSSRRAASSHHEEKKRRKKRRSSffAtAt

}}}}{{},},{{},},{{,,

}}{{},},{{},},{{},},{{},},{{{{))((ReRe

HHPPAAFFHHRRAAPPEEAAFF

EERRPPKKRRAAFFPPAAFFSSRRAASSKKRRSSAtAtdd

}},,{{}}{{))(( KKSSAtAtREDRED


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.

Final Note


ConclusionConclusion New approach for data reduction in GMDIS is considered as a New approach for data reduction in GMDIS is considered as a

generalization in the case of MDIS.generalization in the case of MDIS. This approach extended to Pawlak approach if the system is This approach extended to Pawlak approach if the system is

single-valued and the relations are equivalence.single-valued and the relations are equivalence. Opens the way for other approaches of data reduction Opens the way for other approaches of data reduction

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

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

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


AcknowledgmentAcknowledgment The authors greatly appreciate and thanks many The authors greatly appreciate and thanks many

peoples for their valuable comments and advices:peoples for their valuable comments and advices:

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

Prof. Aboul Ella HassanienProf. Aboul Ella Hassanien, Cairo University, Cairo University Prof. E. Rady,,Prof. E. Rady,, I.S.S.R., Cairo University. I.S.S.R., Cairo University. Dr. A. S. Salama.Dr. A. S. Salama. Pure Mathematics Dept., Pure Mathematics Dept.,

Faculty of Science, Tanta University.Faculty of Science, Tanta University.


شكراشكرالحسن لحسن استمااستماعكمعكم

by abd el-monem m. kozea, mohamed m. e. abd el-monsef, soaad abd el-badie attia el-afify

Documents

data reduction

information systemstablesis

decision attributes

missing data

incomplete data

information systems

decision rule generation

measure of decision