New Approaches for Data Reduction in Generalized Multivalued Decision Information System (GMDIS): Case Study of Rheumatic Fever Patients. By Abd ElMonem M. Kozea, Mohamed M. E. Abd ElMonsef, Soaad Abd ElBadie Attia ElAfify Mathematics Department, Faculty of Science,
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WRSTA2006, 13 August 2006
By Multivalued Decision Information System (GMDIS):
Abd ElMonem M. Kozea,
Mohamed M. E. Abd ElMonsef,Soaad Abd ElBadie Attia ElAfify
Mathematics Department, Faculty of Science,
Tanta University, Egypt
Email:[email protected]
WRSTA2006, 13 August 2006
WRSTA2006, 13 August 2006
WRSTA2006, 13 August 2006
WRSTA2006, 13 August 2006
WRSTA2006, 13 August 2006
Information Systems/Tables Multivalued Decision Information System (GMDIS):
WRSTA2006, 13 August 2006
Decision Systems/Tables Multivalued Decision Information System (GMDIS):
WRSTA2006, 13 August 2006
= Multivalued 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 TypesWRSTA2006, 13 August 2006
WRSTA2006, 13 August 2006
Rheumatic Fever Data: Characteristics Multivalued 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
WRSTA2006, 13 August 2006
by ( Collecting of Attributes).
WRSTA2006, 13 August 2006
S Multivalued 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 ( Multivalued 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 Multivalued 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 MultiValued Decision Information System (GMDIS) Multivalued Decision Information System (GMDIS):
WRSTA2006, 13 August 2006
WRSTA2006, 13 August 2006
= Multivalued 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 Multivalued Decision Information SystemA Generalized Multivalued Information System can be defined as follows.
A Generalized Multivalued Decision Information System can be defined as follows.
WRSTA2006, 13 August 2006
(1) Multivalued 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 Multivalued 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
2. Multivalued 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
Y Multivalued 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)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 Multivalued Decision Information System (GMDIS):
WRSTA2006, 13 August 2006
WRSTA2006, 13 August 2006
Í Multivalued 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 Multivalued 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 Multivalued 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
WRSTA2006, 13 August 2006
x Multivalued 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
= Multivalued 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 Multivalued 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
WRSTA2006, 13 August 2006
WRSTA2006, 13 August 2006