DIAGNOSING CARDIOVASCULAR DISEASE USING AN ENHANCED ROUGH SETS APPROACH

Figures & data

FIGURE 1 Partitioning process of minimize entropy principle approach.

FIGURE 2 Research procedure.

Algorithm 1 The proposed_approach procedure

1: begin

2: Discretization (D);

3: LEM2 (ℜ);

4: Filter (ℜ, U, supp);

5: end;

TABLE 1 Thresholds of All Attributes in the Breast Cancer Dataset

Attribute	min	sec 1	pri	sec 2	max
Clump_Thickness	1	4.5	6.5	8.5	10
Cell_Size	1	1.5	2.5	4.5	10
Cell_Shape	1	1.5	2.5	4.5	10
Marginal_Adhesion	1	1.5	3.5	6.5	10
Single_Epithelial_Cell_Size	1	1.5	2.5	3.5	10
Bare_Nuclei	1	1.5	2.5	5.5	10
Bland_Chromatin	1	2.5	3.5	7.5	10
Normal_Nucleoli	1	1.5	2.5	8.5	10
Mitoses	1	1	1.5	2.5	10

FIGURE 3 Membership function of Clump_Thickness.

TABLE 2 Breast Cancer Dataset Result Rule Set Example Using LEM2

Rules	Support
(Mitoses=L_1)&(Bare_Nuclei=L_1)&(Normal_Nucleoli=L_1)	152
&(Marginal_Adhesion=L_1)&(Single_Epithelial_Cell_Size=L_3)= > (Class=FALSE)
(Mitoses=L_1)&(Normal_Nucleoli=L_1)&(Cell_Size=L_1)&(Bare_Nuclei=L_1)	122
&(Marginal_Adhesion=L_1)&(Cell_Shape=L_1)
&(Single_Epithelial_Cell_Size=L_3)= > (Class=FALSE)
(Mitoses=L_1)&(Normal_Nucleoli=L_1)&(Cell_Size=L_1)& (Bare_Nuclei=L_1)	115
&(Bland_Chromatin=L_1)&(Single_Epithelial_Cell_Size=L_3)= > (Class=FALSE)
(Mitoses=L_1)&(Normal_Nucleoli=L_1)&(Cell_Shape=L_1)&(Cell_Size=L_1)	100
&(Marginal_Adhesion=L_1)&(Clump_Thickness=L_1)
&(Single_ Epithelial_Cell_Size=L_3)= > (Class=FALSE)
(Normal_Nucleoli=L_1)&(Bland_Chromatin=L_1)&(Clump_Thickness= L_1)	69
&(Cell_Size=L_1)&(Cell_Shape=L_1)&(Marginal_Adhesion=L_1)
&(Single_Epithelial_Cell_Size=L_3)= > (Class=FALSE)

Algorithm 3 The LEM2 procedure

Procedure LEM2 ℜ: ℜ T

begin

ℜ i : = ℜ;

ℜ T : = Ø

while (ℜ i  ≠ Ø)

begin

T: = Ø;

T(ℜ i ): = {t | [t] ∩ ℜ i  ≠ Ø};

while (T = Ø or [T] ⊂ /ℜ)

begin

select a pair t ∊ T(ℜ i );

T: = T ∪ {t};

ℜ i : = R i ∩ t;

T(ℜ i ): = T(ℜ i ) − T;

end;

for ∀ t ∊ T do begin

if [T − {t}] ⊆ ℜ then T: = T − {t};

end;

ℜ T : = ℜ T  ∪ {T};

ℜ i : = ℜ −∪ T ∊ ℜ T [T];

end;

for ∀ T ∊ ℜ T do begin

if ∪ S ∊ ℜ T −{T} [S] = ℜ then ℜ T : = ℜ T  − {T};

end;

Algorithm 4 The FILTER procedure

Procedure filter (ℜ T , U, supp): ℜ f

begin

ℜ s : = sort rules in ℜ T descending according to their support;

ℜ u : = Mark ℜ T as unused;

for ∀ u ∊ U do begin

c: = 0;

for ∀ r ∊ ℜ s do begin

if r matches u then

mark r as used;

u: = u + 1;

if c =supp then

break;

end;

end;

end;

end;

ℜ f : = ℜ T − ℜ u ;

end;

TABLE 3 Result of Proposed Approach and Rough Set Approach

Method	Train accuracy	Test accuracy	Number of rules
SVM (Platt et al., 1999)	97.4 ± 0.4%	96.9 ± 0.2%	NA
Rough set (Bazan and Szczuka, 2001)	100.0 ± 0%	98.3 ± 0.3%	31.7 ± 2.3
without discretization
Rough set (Bazan and Szczuka, 2001)	100.0 ± 0%	97.6 ± 1.1%	20.7 ± 0.6
with local discretization
Proposed approach	99.9 ± 0.1%	98.3 ± 0.3%	27.7 ± 3.5

Algorithm 2 The discretization procedure

Procedure Discretization (D): ℜ

begin

for ∀ a i  ∊ A do begin

sort x j ;

for ∀ x j do begin

x m  = (x j  + x j+1)/2;

Calculate_entropy (x m );

if S < S min then S min: = S;

end;

end;

use {S} build the membership function: f i (x) for a i ;

end;

use f i (x) convert D to ℜ;

end;

Procedure Calculate_entropy (x): S

begin

calculate with Eq. (1);

return entropy_value S;

end;

TABLE 4 UCI Arrhythmia Dataset Experiment Result

Method	Train accuracy	Test accuracy	Number of rules
Fuzzy-weighted AIRS	NA	80.7%	NA
(Polat, Sahan, and Günes, 2006)	91.2 ± 0.6%	70.4 ± 4.1%	NA
SVM (Platt et al., 1999)
Rough set (Bazan and Szczuka, 2001)	100.0 ± 0.0%	67.3 ± 6.9%	49 ± 5.3
without discretization
Rough set (Bazan and Szczuka, 2001)	100.0 ± 0.0%	77.0 ± 3.6%	55.3 ± 2.3
with local discretization
Proposed approach	100.0 ± 0.0%	87.6 ± 4.8%	49.7 ± 3.1

TABLE 5 Result of Cardiovascular Disease Dataset

Method	Train accuracy	Test accuracy	Number of rules
SVM (Platt et al., 1999)	86.7 ± 1.1%	85.3 ± 1.5%	NA
Rough set (Bazan and Szczuka, 2001)	100.0 ± 0%	78.2 ± 4.3%	22.3 ± 4.6
without discretization
Rough set (Bazan and Szczuka, 2001)	100.0 ± 0%	91.5 ± 0.4%	98 ± 7.5
with local discretization
Proposed approach	100.0 ± 0%	92.2 ± 0.5%	94 ± 7.9

Platt , J. C. , B. Scholkopf , C. Burges , and A. J. Smola . 1999 . Fast training of support vector machines using sequential minimal optimization . In: Advances in Kernel Methods – Support Vector Learning . Cambridge , MA : MIT Press .

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Bazan , J. and M. Szczuka . 2001 . RSES and RSESlib – A collection of tools for rough set . Lecture Notes in Computer Science 2005 : 106 – 113 .

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Polat , K. , S. Sahan , and S. Günes . 2006 . A new method to medical diagnosis: Artificial immune recognition system (AIRS) with fuzzy weighted pre-processing and application to ECG arrhythmia . Expert Systems with Applications 31 : 264 – 269 .

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