A Framework for Selection of Membership Function Using Fuzzy Rule Base System for the Diagnosis of Heart Disease

Published Online October 2013 in MECS

Heart disease, sometime define as coronary artery disease (CAD), is a well-known term that can be referred to any condition that affects the heart. Most of the people with heart disease have symptoms such as chest pain, blockage and fatigue, as many as 50% have no symptoms until a heart attack occurs.

R.Das,Ibrahim Turkoglu b, Abdulkadir Sengur[1] have used a neural network based model for the diagnosis of heart disease from the available data. Initially the data have been partitioned for the purpose of the usage of the neural network and for the validation of data sets. Three types of neural networks have been used. The authors have made a comment that a multilayer feed forward neural network has shown the excellent performance among the three types of neural networks.

Vanisree K,Jyothi Singaraju[2] have used a Decision Support System for the diagnosis of Congenital heart disease from the available database .They have used back propagation neural network which contains one input layer, one output layer and one or more hidden layers. As the name implies the input layer receives signals from the external nodes and transmits these signals to other layers without performing any computation at that layer. The output layer receives the signals from an input layer through a weighted connection links, performs computations at that layer and produces output of the network

It has been trained using a supervised delta learning rule. The dataset used in this study are the signs, symptoms and the results of physical evaluation of a patient. The proposed system has achieved an accuracy of 90%.

Priti Srinivas Sajja, Dipti M Shah[3] have used a knowledge-oriented decision support system for the diagnosis of abdomen pain .They have used a modified Prolog rule format, for their decision making process and have created awareness in the area especially where trained manpower is in scarce.

Ali.Adeli, Mehdi.Neshat[4] have used a fuzzy expert system for the diagnosis of heart disease. The authors have used several variables viz as chest pain type, blood pressure, cholesterol, resting blood sugar, resting maximum heart rate, sex, electrocardiography (ECG), exercise, old peak (ST depression induced by exercise relative to rest), thallium scan and age as inputs. The status of the patients as healthy or sick has been used as output. Four types of sickness have been used as output. These are Sick s1,Sick s2, Sick s3, Sick s4.

Narendra S. Chaudhari and Avishek Ghosh [5] have used a projection algorithm with an objective to map p-dimensional patterns to q-dimensional space such that the structure of the data is preserved. They have implemented an extension of Sammon’s algorithm using fuzzy logic approach. Finally they have shown that their result was excellent.

Ranjana Raut, S. V. Dudul[6] have used a neural network based approach for intelligent system in the diagnosis of heart disease. They have used classifiers based on various neural networks namely multilayer perceptron, Jordan recurrent neural network, generalized feed forward neural network, Modular Radial Basic Function, Self Organizing Feature Map, other techniques like Support Vector Machine and conventional statistical techniques such as Data Analysis and CART. They have proposed Muitilayer Perceptron Neural Network based classifier has shown the excellent result as compared to others.

K. Usha Rani[12] have used a classification approach based on neural network. They have used single and multilayer neural network mode for classification of heart disease dataset. For training the network, they have used back propagation algorithm and parallelism technique at the same time at each neuron in all hidden and output layers to speed up the learning process. They have proved that neural network based technique has given satisfactory result.

K.Rajeswari, V.vaithiyanatham, P.Amirtharaj[13] have proposed a Decision support system for reliable heart disease risk prediction of Indian patients using machine learning technique. They have used genetic algorithm to determine high impact pattern and their optimal value. They have used theoretical approaches to implement the machine learning algorithm.

Ersin Kaya, Bulent Oran and Ahmet Arslahn[14] have used a fuzzy rule based classifier for diagnosis of congential heart disease which defines structural and functional disease of heart. They have used weighted vote method and single winner method . The result has shown that the weighted vote method generally has increased the classification accuracy of congenital heart disease.

E.P.Ephzibah1, V. Sundarapandian[15]have proposed a neuro fuzzy expert system that finds a solution to diagnose the disease using some of the evolutionary computing techniques like genetic algorithm, fuzzy rule based learning and neural networks. Their proposed neuro-fuzzy method refers to the combination of artificial neural network and fuzzy logic based system in which the 13 attributes have been applied taken from UCI Machine learning repository. The result has helped the doctors to arrive at a conclusion about the presence or absence of heart disease in patients.

Shradhanjali Rout[16] have used a Fuzzy Petri net based system instead of fuzzy expert system and have analyzed the system through fuzzy rule based reasoning algorithm. They have used 11 attributes for input and 1 attribute as output for diagnosis of the of heart disease. They have shown that their result was excellent by using their proposed petri net based fuzzy system.

Various authors ([1][2][6][12]) have used several neural network models like back propagation neural network, multilayer perceptron neural network, radial basis neural network etc. Their effort was to detect the heart disease using fuzzy expert system, but they have not shown how much accuracy they have achieved in their result. That is the reason for taking this work as described in this paper.

In this paper an effort has been made to diagnose the heart disease by using fuzzy rule base. Initially 7 attributes of patients have been used as input values from the Clevand database[11]. The data values used have been portioned into several intervals based on certain intermediate values of the available data values. Thereafter fuzzy rule base has been applied on the partitioned input data values. Based on the fuzzy value of the output variable, the heart disease can be ascertained.In this context various membership functions have been used for each input variable and output variable. Finally absolute residual and mean absolute residual have been calculated using the output value produced by fuzzy rule base in comparison to the actual output value as available in the database [11].

In the first section the necessity of this research work and related literature review have been furnished. The methodology used including various fuzzy membership functions (triangular, trapezoidal, gaussian, Z shaped, bell shaped ,sigmoid based ,gausssian combination membership have been discussed in the second section. The detailed procedures of research work have been elaborately stated in the third section. This includes fuzzification of input data, formation of rule base, estimation of heart status using fuzzy rule base and various membership functions. Based on the minimum value of mean absolute residual, the particular membership function has been selected for fuzzification in the rule base. In the fourth section the estimated heart status using fuzzy rule base as against the original heart status has been furnished. The justification of this research work has been discussed in the fifth section.

• II.    Methodology
  
  2.1 Expert System

2.2 Fuzzy Membership Function

A knowledge base can be formed by combining the knowledge of multiple human experts. A rule is a conditional statement that links given conditions to actions or outcomes. A frame is another approach used to capture and store knowledge in a knowledge base. It relates an object or item to various facts or values. A frame-based representation is ideally suited for object-oriented programming techniques. Expert systems making use of frames to store knowledge are also called frame-based expert systems.

expression s=2).The gaussian membership function µ A of vector x has been represented by

P a ( x : p , q , 5 ) = exp

1 x - p

Fuzzy membership functions determine the membership functions of objects to fuzzy set of all variables. A membership function provides a measure of the degree of similarity of an element to a fuzzy set. There are different shapes of membership functions, triangular, trapezoidal, gaussian, bell-shape,Z shape, sigmoid base, gaussian combination.

2.2.1 Triangular membership function

Let p, q and r represent the three vertices of the X coordinates and µ A (x) represents the Y coordinate in a fuzzy set A,where A is the membership value .In this equation p: lower boundary and r: upper boundary where membership degree is zero, q: the centre where membership degree is 1.

0      if x <  p

x - p

P a ( x ) = ■	q - p
	r - x
	r - q
	0

if P < x < q if q < x < r if x > r

2.2.2 Trapezoidal Membership function

The trapezoidal curve is a function of µ A of vector x, and depends on four scalar parameters p, q, r and s where p and s allocate the "feet" of the trapezoid and the parameters q and r allocate the "shoulders."

2 q

2.2.4 Z shape base Membership function

p _A ( x : p , q )

^if x <  p ²

1 - 2

x - p

v q- p > 2

x - q

if p <  x <  p + q /2

if ( p + q ) /2 < ^x < q

2.2.5 Bell shape base Membership function

The Bell shape base membership function µ A of vector x, depends on three scalar parameters p,q and r, where the parameter q is usually positive. The parameter r locates the center of the curve.

P a ^{( x :} p , q, r ) =

1 +

x - r

2 q

.^(5)

p

	0      if x <  p
	x—p if p <  x <  q
p ( x : p , q , r , 5 ) = ■	q - p	^   (2)
	r - x   „
	---- if q <  x <  r
	r - q
	0       if x <  r

• 2.2.3    Gaussian Membership function

• 2.2.6    Sigmoid base Membership function

• 2.2.7    Gaussian combination Membership function

The gaussian curve is a function of µ A of vector x, and depends on three scalar parameters p, q and s where p: center and q: width and s: fuzzification factor(in

The sigmoid function, µ A (x,[p,r]), as given in the following equation by µ A ( x,p,r) is a mapping on a vector x, and depends on two parameters p and r . Depending on the sign of the parameter p , the sigmoidal membership function is inherently open to the right or to the left, and thus is appropriate for representing concepts such as "very large" or "very negative." More conventional-looking membership functions can be built by taking either the product or difference of two different sigmoidal membership functions.

^P A ⁽ x ^{: p} , r ) = 1 + e p (x - r )

,      \      — ( x — c )

M a ( x o , c ) = e       2   ...... ⁽7)

The function gauss2mf is a combination of two parameters sig and c. The first function, specified by sig1 and c1, determines the shape of the left-most curve. The second function specified by sig2 and c2 determines the shape of the right-most curve. Whenever c1 < c2, the gauss2mf function reaches a maximum value of 1. Otherwise, the maximum value is less than one.

Absolute residual

The absolute residual is represented by

| (estimated status - actual status) |

En absolute residual i=1

Mean absolute residual is

(sum of absolute residual) _n

• III.    Implementation
  
  Step 1:

From Clevand database [11],out of 76 attributes, 7 attributes have been taken as input and 1 attribute has been used as output. The input attributes are chest pain type, resting blood pressure in mm(Trestbps),serum cholesterol in mg(Chol),number of years the person behaves as a smoker(years), fasting of blood sugar (fbs), maximum heart rate achieved(thalrest), resting blood rate(tpeakbps).The output attribute is the angiographic disease status of heart of patients.

Step 2:

The first input field “Chest pain type” comprises of four types as denoted by numerical value. The value 1 indicates” typical angina”, 2 indicates “atypical angina, 3 indicates “non-angina” and 4 indicates “asymptomatic angina” .The second and third input fields are resting blood pressure measured in mm and serum cholesterol measured in mg respectively. The fourth input field is the number of years the person behaves as a smoker that denoted by a numeric value. The fifth input field indicates the contents of fasting blood sugar. When fasting blood sugar is greater than or equal to 120 mg, the level of fasting blood sugar is 1 otherwise 0.The sixth and seventh input fields are maximum heart rate achieved and resting blood rate respectively. The output field refers to the angiographic disease status of heart of patients. It is to state that diagnosis of heart disease is made by the angiographic disease status which is a number between 0 to 1. Where 0 indicates the diameter of coronary arteries is less than 50% of original diameter and 1 indicates the diameter of coronary arteries is more than 50% of its original diameter.

Step 3:

The data items(7 attributes per patient) have been used as input. The input data items have been furnished in table 1.

Table 1: Input data values

Patients No	Chest pain type	Resting blood pressure (t restbps)	Serum cholesterol (chol)	Numbers of Years as a smoker (years)	Fasting of blood sugar (fbs)	Maximum heart rate achieve (thalrest)	Resting blood rate (tpeakbps)
1	1	145	233	20	1	60	90
2	4	120	229	35	0	78	140
3	3	130	250	0	0	84	195
4	2	130	204	0	0	71	160
5	2	120	236	20	0	73	165
6	4	120	354	0	0	84	165
7	4	140	203	25	1	86	185
8	4	140	192	25	0	86	180
9	2	140	294	0	0	85	204
10	2	120	263	30	0	70	165
11	3	172	199	35	1	91	220
12	3	150	168	40	0	60	192
13	2	110	229	0	0	75	175
14	4	140	239	30	0	86	180
15	3	130	275	20	0	62	165

Step 4:

Thus input and output variables have been classified based on their interim values, which have been furnished in table 2.

Table 2: Input data values with respect to Fuzzy Set

Chest pain type	Numeric value
Typical angina	1
Atypical angina	2
Non typical-angina	3
Asymptomatic angina	4
Fasting blood sugar (fbs)	Fuzzy set value
fbs=0,(fbs<=120mg)	no
fbs=1,(fbs>120mg)	yes
years as (years_input4)	Fuzzy set value
years_input4<=30	Low possibility
years_input4>30	High possibility
Resting blood rate achieved (tpeakbps)	Fuzzy set value
tpeakbps < =140	low
tpeakbps >140	high

Resting blood pressure(t restbps)	Fuzzy set value
t restbps<=145( from 110 -145)	min
t restbps>145( from 145 -above)	max
Serum cholesterol (chol)	Fuzzy set value
chol<=240	min
chol>240	max
Maximum heart rate achieved (thalrest)	Fuzzy set value
Thalrest< =70	Min
thalrest,>70	Max
Angiographic disease	Fuzzy
0(diameter of coronary arteries <=50%	mild
1(diameter of coronary arteries >50%	massive

Step 5:

The fuzzy rules have been applied on the data to estimate the angiographic status of heart of the patients .The fuzzy rules have been furnished in table 3.

Table 3: Rule base

Step 6:

Initially triangular membership function has been applied to all input and output variables. The fuzzy rule base toolbox of Matlab 7 have been applied and that has produced the estimated output as furnished in table 4.

Step 7:

Thereafter trapezoidal, Z shaped, Bell shaped, sigmoid and Gaussian combination membership functions have been applied to all input and output variables. The fuzzy rule base toolbox of Matlab 7 has been applied to produce the estimated output as furnished in table 4.

Table 4: Original heart status and estimated heart status

A	B	C	D	E	F	G	H	I
1	0	0.415	0.433	0.276	0.460	0.401	0.499	0.500
2	1	0.575	0.510	0.845	0.500	0.608	0.510	0.566
3	0	0.515	0.415	0.276	0.425	0.508	0.499	0.499
4	0	0.515	0.400	0.276	0.425	0.434	0.499	0.499
5	0	0.415	0.478	0.233	0.425	0.434	0.459	0.499
6	0	0.515	0.478	0.278	0.460	0.434	0.498	0.400
7	1	0.575	0.628	0.833	0.500	0.575	0.500	0.566
8	0	0.515	0.500	0.313	0.425	0.401	0.498	0.499
9	0	0.515	0.500	0.313	0.425	0.401	0.498	0.499
10	0	0.515	0.500	0.278	0.460	0.508	0.499	0.400
11	0	0.415	0.478	0.276	0.460	0.434	0.499	0.499
12	0	0.515	0.500	0.276	0.425	0.434	0.498	0.499
13	1	0.575	0.535	0.833	0.500	0.508	0.510	0.566
14	0	0.515	0.478	0.276	0.425	0.401	0.499	0.499
15	0	0.515	0.500	0.276	0.425	0.401	0.498	0.499

Step 8:

The absolute residual value of the estimated output with respect to the original status have been calculated based on certain commonly used membership function(triangular, Gaussian, trapezoidal, z shaped),which have been furnished in table 5.The absolute residual value of the estimated output with respect to the original status have been calculated based on other membership function (Bell shaped, sigmoid and Gaussian combination)with have been furnished Table 6

Table 5: Absolute residual based on membership function

A	J	K	L	M
1	0.415	0.433	0.276	0.46
2	0.425	0.490	0.155	0.5
3	0.515	0.415	0.276	0.425
4	0.515	0.400	0.276	0.425
5	0.415	0.478	0.233	0.425
6	0.515	0.478	0.278	0.46
7	0.425	0.372	0.167	0.5
8	0.515	0.500	0.313	0.425
9	0.515	0.500	0.313	0.425
10	0.515	0.500	0.278	0.46
11	0.415	0.478	0.276	0.46
12	0.515	0.500	0.276	0.425
13	0.425	0.465	0.167	0.425
14	0.515	0.478	0.276	0.425
15	0.515	0.500	0.276	0.425

Table 6: Absolute residual based on membership function

A	N	O	P
1	0.415	0.433	0.276
2	0.425	0.490	0.155
3	0.515	0.415	0.276
4	0.515	0.400	0.276
5	0.415	0.478	0.233
6	0.515	0.478	0.278
7	0.425	0.372	0.167
8	0.515	0.500	0.313
9	0.515	0.500	0.313
10	0.515	0.500	0.278
11	0.415	0.478	0.276
12	0.515	0.500	0.276
13	0.425	0.465	0.167
14	0.515	0.478	0.276
15	0.515	0.500	0.276

A= Patients No, B= Original status (Available from the database), C= Estimated Status (using triangular membership function), D= Estimated Status (using trapezoidal membership function)E= Estimated Status (using Gaussian membership function )F= Estimated Status (using Z shaped membership function)G= Estimated Status (using bell shaped membership function)H= Estimated Status (using sigmoid membership function)I= Estimated Status (using Gaussian combination membership function)

A= Patients No, J= Absolute residual using Triangular membership function, K= Absolute residual using Trapezoidal membership function, L= Absolute residual using Gaussian membership function M= Absolute residual using Z shaped membership function N= Absolute residual using bell shaped membership function O= Absolute residual using sigmoid based membership function P= Absolute residual using Gaussian combination membership function .

Step 9:

The mean of absolute residual of the estimated output value for each membership function has been calculated and furnished in table 7.

Table 7: Mean absolute residual against membership function

Membership function	Mean absolute residual
Triangular	0.4770
Trapizoidal	0.4726
Gaussian	0.2567
Z shapes	0.2470
Bell shapes	0.4770
Sigmoid Shape	0.4726
Gaussian combination	0.2667

Step 10:

From table 7 it has been observed that mean of absolute residual based on Z shape membership function has achieved minimum value as compared to triangular and trapezoidal, Gaussian, bell shape, Gaussian combination membership functions. Therefore it is desirable that Z shaped membership function has to be used in the fuzzy rule base for the diagnosis of heart disease.

• IV.    Result

Since the Z shaped membership function is suitable for implementation of fuzzy rule base for heart disease diagnosis system. The fuzzy rule based system using Z shaped membership function can be used for the diagnosis of heart disease. The estimated status of heart disease based on fuzzy rule base using Z shaped membership function has been furnished in table 8.Snapshot of Z shaped membership function based Rule editor has been furnished in figure 1.

A= Patients No,

B=Original status Available from the database

F=Estimated Status using Z shape membership function

D=Absolute residual values using Z shape membership function

Table 8: Estimated status, actual status based on Z shape membership function

A	B	F	D
1	0	0.460	0.46
2	1	0.500	0.5
3	0	0.425	0.425
4	0	0.425	0.425
5	0	0.425	0.425
6	0	0.460	0.46
7	1	0.500	0.5
8	0	0.425	0.425
9	0	0.425	0.425
10	0	0.460	0.46
11	0	0.460	0.46
12	0	0.425	0.425
13	1	0.500	0.5
14	0	0.425	0.425
15	0	0.425	0.425

Fig. 1: Rule editor of Z shaped membership function

• V.    Conclusion

It has been shown that Fuzzy rule base system using Z shaped membership function has given the best result as compared to other membership functions. For patient 1 with the characteristics blood pressure during resting time as 145, serum cholesterol as 233 mg, maximum heart rate as 60,blood rate during rest as 90 , the person is having fasting blood sugar, the person can be treated as a smoker for 20 years with a typical angina(chest pain type angina value 1 ), the output ( angiographic disease status) has been observed as 0.460 using fuzzy rule base system. The original status for patient 1 is 0. Thus for patients 1 ,the absolute residual (derivation from original) is 0.460.For patient 2, the observed output status using fuzzy rule base system is 0.500 as against the original status as 1 with absolute residual (derivation from original) is 0.5.For patient 19, the observed status using fuzzy rule base is 0.500 as against the original status as 1 with absolute residual (derivation from original) is 0.5.Accordingly the output (angiographic disease status) for all the patients can be ascertained.