semianr 2. (2)
TRANSCRIPT
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AN APPLICATION OF INTERVAL-
VALUED FUZZY SOFT SETS IN
MEDICAL DIAGNOSIS
Guide:Dr. Sunil Jacob John Jobish VD
M090054MA
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Contents.
1. Preliminaries.
2. Application of interval valued fuzzy
soft set in medical diagnosis.
3. Algorithm.
4. Case Study.
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1. Preliminaries.
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Definition 1.1[3]:
Let U - initial universe set
E - set of parameters.
P (U) - power set of U. and,
A - non-empty subset of E.
A pair (F, A) is called a soft set over U,
where F is a mapping given by F: A P (U).
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Example 1.1;
Let U={c1,c2,c3} - set of three cars.
E ={costly(e1), metallic color (e2), cheap (e3)}
- set of parameters. A={e1,e2} ⊂ E. Then;
(F,A)={F(e1)={c1,c2,c3},F(e2)={c1,c3}}
“ attractiveness of the cars” which Mr. X is going to buy .
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Definition 1.2[3]:
Let U - universal set,
E - set of parameters and A ⊂ E.
Let F (U) - set of all fuzzy subsets of U.
Then a pair (F,A) is called fuzzy soft set over
U, where F :A F (U).
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Example 1.2;
Let U = {c1,c2,c3} - set of three cars.
E ={costly(e1),metallic color(e2) , getup (e3)}
- set of parameters.
A={e1,e2 } ⊂ E.
Then;
(G,A) = { G(e1)={c1/.6, c2/.4, c3/.3},
G(e2)={c1/.5, c2/.7, c3/.8} }.
- fuzzy soft set over U.
Describes the “ attractiveness of the cars” which
Mr. S want.
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.
Definition 1.3[3]: An interval-valued fuzzy
sets X on the universe U is a mapping
such that;
X : U → Int ([0,1]).
where; Int ([0,1]) - all closed sub-intervals
of [0,1].
The set of all interval-valued fuzzy sets on U is
denoted by F (U).
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.1)()(0
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-membership of degree T he)](),([)(
),(~ˆ
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If,
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-bygiven is ˆ ˆ
by, denoted ,ˆ and ˆ of Union
Then, .) ( ~
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-bygiven is ˆ ˆ
by, denoted ,ˆ and ˆ ofon Intersecti
Then, .) ( ~
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)] ( ˆ
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-bygiven is nd
,cˆby denoted ˆ of comlement
Then, .) ( ~
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- 1 [
. )( ˆ
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Definition 1.7 [4]:
Let U universal set.
E set of parameters.
and A ⊂E.
set of all interval-valued fuzzy sets on
U.
Then a pair (F, A) is called interval-valued fuzzy
soft set over U.
where F : A
)(~
UF
).(~
UF
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Definition 1.8[5]: The complement of a
interval valued fuzzy soft set (F,A) is,
(F,A)C = (FC,¬A),
where ∀α ∈ A ,¬α = not α .
FC: ¬A F ( U ).
FC(β ) = (F ( ¬β ))C , ∀β ∈ ¬A
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Example2.3:
Let U={c1,c2,c3} set of three cars.
E ={costly(e1), grey color(e2),mileage (e3)},
set of parameters.
A={e1,e2} ⊂ E. Then,
(G,A) = {
G(e1)=⟨c1,[.6,.9]⟩,⟨c2,[.4,.6]⟩,⟨c3,[.3,.5]⟩,
G(e2)= ⟨c1,[.5,.7]⟩, ⟨c2,[.7,.9]⟩ ⟨c3,[.6,.9]⟩
}
“ attractiveness of the cars” which Mr. X want.
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Example 2.4:
In example 2.3,
(G,A)C = {
G(¬e1)=⟨c1,[0.1,0.4]⟩, ⟨c2,[0.4,0.6]⟩,⟨c3,[0.5,0.7]⟩,
G(¬e2)=⟨c1,[0.3,0.5]⟩, ⟨c2,[0.1,0.3]⟩⟨c3,[0.1,0.4]⟩
}.
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2. Application –
in
medical diagnosis.
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S - Symptoms, D – Diseases, and P - Patients.
Construct an I-V fuzzy soft set (F,D) over S
F:D→
A relation matrix say, R1 - symptom-disease
matrix- constructed from (F,D).
Its complement (F,D)c gives R2 - non
symptom-disease matrix.
We construct another I-V fuzzy soft set (F1,S)
over P, F1:S→
).(~
SF
).(~
PF
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We construct another I-V fuzzy soft set (F1,S)
over P, F1:S→
Relation matrix Q - patient-symptom matrix-
from (F1,S).
Then matrices,
T1=Q R1 - symptom-patient matrix, and
T2= Q R2 - non symptom-patient matrix.
).(~
PF
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The membership values are calculated by,
)},,(),({ b
)},,,(),({
],[),(
1
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deep
deepx
yxdp
U
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The membership values are calculated by,
)},(),({ q
)},(),({
),(
11
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j
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jiTU
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)},(),({ t
)},(),({
),(
22
22
2
j
j
ijTU
jiTU
ijTL
jiTL
jiT
dpdp
dpdps
tsdpS
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3. Algorithm.
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1. Input the interval valued fuzzy soft sets (F,D)
and (F,D)c over the sets S of symptoms, where
D -set of diseases.
2. Write the soft medical knowledge R1 and R2
representing the relation matrices of the
IVFSS (F,D) and (F,D)c respectively.
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3. Input the IVFSS (F1,S) over the set P of
patients and write its relation matrix Q.
4. Compute the relation matrices T1=Q R1 and
T2=Q R2.
5. Compute the diagnosis scores ST1 and ST2
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6. Find Sk= maxj { ST1 (pi , dj) ─ ST2 (pi,┐dj)}.
Then we conclude that the patient pi is
suffering from the disease dk.
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4. Case Study.
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Patients - p1, p2 and p3.
Symptoms (S) - Temperature, Headache, Cough
and Stomach problem
S={ e1,e2,e3,e4} as universal set.
D ={d1,d2}.
d1 - viral fever, and
d2 - malaria.
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Suppose that,
F(d1) ={ ⟨e1, [0.7,1]⟩, ⟨e2, [0.1,0.4]⟩,
⟨e3, [0.5,0.6]⟩, ⟨e4,[0.2,0.4]⟩) }.
F(d2) ={ ⟨e1,[0.6,0.9] ⟩, ⟨e2,[0.4,0.6] ⟩,
⟨e3,[0.3,0.6] ⟩, ⟨e4,[0.8, 1] ⟩ }.
IVFSS - (F,D) is a parameterized family
={ F(d1), F(d2) }.
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IVFSS - (F,D) can be represented by a relation
matrix R1 - symptom-disease matrix- given by,
R1 d1 d2
e1 [0.7, 1.0 ] [ 0.6, 0.9 ]
e2 [0.1, 0.4 ] [0.4, 0.6 ]
e3 [0.5, 0.6 ] [0.3, 0.6 ]
e4 [0.2, 0.4 ] [0.8, 1.0 ]
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The IVFSS - (F, D)c also can be represented by
a relation matrix R2, - non symptom-disease
matrix, given by-
R2 d1 d2
e1 [0 , 0.3 ] [ 0.1, 0.4 ]
e2 [0.6, 0.9 ] [0.4, 0.6 ]
e3 [0.4, 0.5 ] [0.4, 0.7 ]
e4 [0.6, 0.8 ] [0 , 0.2 ]
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We take P = { p1, p2, p3} - universal set .
S = { e1, e2, e3, e4} - parameters.
Suppose that,
F1(e1)={⟨p1, [.6, .9]⟩, ⟨p2, [.3,.5]⟩,⟨p3, [.6,.8]⟩},
F1(e2)={ ⟨p1, [.3,.5] ⟩, ⟨p2, [.3,.7] ⟩, ⟨p3, [.2,.6] ⟩},
F1(e3)={⟨p1, [.8, 1]⟩, ⟨p2, [.2,.4]⟩,⟨p3, [.5,.7]⟩} and
F1(e4)={⟨p1, [.6,.9] ⟩,⟨p2, [.3,.5] ⟩, ⟨p3, [.2,.5] ⟩},
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IVFSS - (F1,S) is a parameterized family
={ F1(e1), F1(e2), F1(e3), F1(e4) }.
gives a collection of approximate
description of the patient-symptoms in the
hospital.
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Q e1 e2 e3 e4
p1 [0.6, 0.9] [0.3, 0.5] [0.8, 1] [0.6, 0.9]
p2 [0.3, 0.5] [0.3, 0.7] [0.2, 0.4] [0.3, 0.5]
p3 [0.6, 0.8] [0.2, 0.6] [0.5, 0.7] [0.2, 0.5]
(F1,S) - represents a relation a relation matrix
Q - patient-symptom matrix - given by;
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Combining the relation matrices R1 and R2
separately with Q. we get,
T1=Q o R1 - patient-disease matrix.
T2=Q o R2 - patient-non disease -
matrix.
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T1 d1 d2
p1 [0.1 ,0.9] [0.3 ,0.9]
p2 [0.1 ,0.5] [0.2 ,0.6]
p3 [0.1 ,0.8] [0.2 ,0.8]
T2 d1 d2
p1 [0 , 0.8] [0 , 0.7]
p2 [0 , 0.7] [0 , 0.6]
p3 [0 , 0.6] [0 , 0.7]
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ST1-ST2 d1 d2
p1 0.2 0.6
p2 -0.7 -0.4
p3 0.5 -0.1
Now we calculate,
The patient p3 is suffering from the disease d1.
Patients p1 and p2 are both suffering from
disease d2.
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References1. Chetia.B, Das.P.K, An Application of Interval-
Valued Fuzzy Soft Sets in Medical
Diagnosis, Int. J. Contemp. Math. Sciences, Vol.
5, 2010, no. 38, 1887 - 1894
2. De S.K, Biswas R, and Roy A.R, An application
of intuitionistic fuzzy sets in medical
diagnosis, Fuzzy Sets and
Systems,117(2001), 209-213.
3. Maji PK, Biswas R and Roy A.R, Fuzzy Soft
Sets, The Journal of Fuzzy Mathematics
9(3)(2001), 677-692.
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4. Molodtsov D, Soft Set Theory-First
Results, Computers and Mathematics with
Application, 37(1999), 19-31.
5. Roy MK, Biswas R, I-V fuzzy relations and
Sanchez’s approach for medical
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