application of rough set theory for clinical data ... - Science Direct

Msthl. Comput. Modelling Vol. 15, No. Printed in Great Britain

10, pp. 19-37, 1991

0895-7177/91 $3.00 + 0.00 Pergamon Press plc

APPLICATION OF ROUGH SET THEORY FOR CLINICAL DATA ANALYSIS: A CASE STUDY ABDALLA

S. A.

MOHAMED

Department of Computer Science, University of Regina, Regina, Sask. S4S 0A2, (Received

July 1989 and in revised form

Canada

June 1990)

Abstract-This paper is interested in the question of clinical data reduction and decision making using the basic rough set approach. In particular, we introduce another methodology based on the entropy measure instead of the quality of classification. This methodology is implemented on a microcomputer by which the physician (decision maker) can modulate his decision by some measures for diagnosis. An example of clinical data from patients with valvular heart diseases is considered to validate the efficiency of the proposed system.

1.

INTRODUCTION

Knowledge representation and knowledge processing are two important characteristics of the solution to the clinical data processing problem. Knowledge representation refers to the technique for representing data information in a computer. Knowledge processing refers to the technique for controlling the manipulation of knowledge in the system. There are four criteria to evaluate a knowledge representation in the system [KRS] [1,2]: flexibility, user-friendliness, expressiveness and efficiency of processing. The first three criteria are required to simplify the tasks of programming and comprehension. The efficiency dictates the tractability of a KRS for finding an efficient solution to the application. For knowledge processing, there are different reasoning methods associated with different KRS’s and they require different architectural supports [3]. In the use of a knowledge representation, the knowledge processing technique must be tailored to cope with the application requirements. The greatest need, in clinical data application, is the ability to deal with uncertain and incomplete information. With the proliferation of powerful personal computers among the general public and particularly due to the increased emphasis on the design of application oriented intelligent systems, a number of expert systems have been built in the decade [4]. Most of these systems were built by manually encoding the knowledge of human experts, which of course is a tedious process. On the other hand, much research efforts have been devoted to inductive learning, i.e., learning from examples [5]. To capture the essence of the decision making process (i.e., decision rules) of human experts (physicians, in our problem), is indeed a rather difficult if not impossible task. For this reason, many methods and theories of capturing information uncertainity have been examined. They include : 1) probability and Bayesian statistics [6]; 2) fuzzy logic [7]; 3) confidence factors [8]; and plausible inference [9]. On the other hand, when knowledge is incomplete, heuristic to provide a systematic solutions are utilized [lo]. The notion of rough sets [ll] was introduced framework for the study of problems arising from imprecise and insufficient knowledge [12] and We have been unable to communicate with the author with respect to galley proof corrections. Hence, this work is published without the benefit of such corrections. (Ed.) The author would like to acknowledge the meticulous corrections and helpful suggestions, including an improvement on results, provided by the referee. This work was supported by NSERC (Natural Sciences and Engineering Research Council) of Canada, which the author acknowledges with thanks. The author wishes to express his gratitude to Dr. Ziarko (University of Regina) especially with respect to the implemented computer package. Mailing Address: P.O. Box 1076, Heliopolis North 11737, Heliopolis, Cairo, Egypt.

19

20

A.S.A.

MOHAMED

for clinical data analysis [13-161. The original rough set method was proposed to handle deterministic KRS. An extension of rough sets to deal with non-deterministic KRS has been made in [17] and is known with the probabilistic approximate classification [PAC] 1181. Currently, there is both an increased amount of information to be used in the treatment of diseases, and an increased diffusion of the sources of this information. In order to explore the role of information-processing in clinical medicine, it is worthwhile to examine the nature of the process of patient care itself. The interaction between doctor and patient is an iterative information-gathering, decision making process. Although evaluation of the patient’s health and delineation of the disease process are prerequisites to a reliable selection of therapy, in clinical medicine, one is often forced to make decisions and take actions based upon incomplete information. The diagnostic treatment-planning process is initiated by the patient. The physician tries to discover all he can about the patient’s problem with the least amount of redundancy, minimal cast, and absence of harm to the patient. During the physical examination, the physician pays particular attention to things based on what he has learned thus far. Having identified the signs and symptoms, the physician evaluates this information and selects the appropriate clinical laboratory tests. These tests and procedures (non-invasive or invasive) monitor some physiological parameters related to heart function, blood contents,..., etc. During the process of refining his differential diagnosis, the physician selects particular actions based on their promise to provide patient care. These actions are decisions drawn from the The most significiant contribution to available uncertain clinical knowledge and experience. the physician’s decision, judging from the amount of information, is the reduction of the data size by eliminating the nonsignificant attributes. There needs to be a measure for describing the attributes dependency. This measure will be used for data reduction and decision rules deduction. For example, the clinical state of cardiac valves can be evaluated by echocardiography as a noninvasive diagnostic procedure. The measured and calculated attributes, here, for each patient will be arranged in a table with columns for attributes and rows for patients (objects). The rationale behind rough set analysis is to design a KRS which is able to help the physician find the dependency among the significant attributes and deduce the decision rules from the available clinical information. Although the decision rules seem to be certain and complete in the sense of univocal values given for each pair (object, attribute), there will be a problem of uncertainity and imprecision if some values corresponding to pairs (objects, attributes) are missing or imprecise

P31* In this paper, the theory of computation is given first (Section-2) including basic concepts of rough sets. One important application of the rough sets approach to clinical data of valvular heart disease is outlined in Section-3 with the data acqusition procedure. Following the description of the experiment protocol, a computerized procedure for practical implementation is given. The results of application are arranged in Section-4. Discussions are given in Section-5. 2. THEORY

OF COMPUTATION

2.1 Basic Concepts Let U denotes a finite set of objects, and let R C U be an equivalance relation on U. R is space in referrred to as an indiscernibility relation. The pair A = (U, R) is called an approximate the deterministic rough set model. Let P be a probabilistic measure defined on the Sigma-algebra of subsets of U, then one can define a probabilistic approximation space [PAS] A,, as a triple A, = (U, R, P). In this context, each subset of U,corresponds to a random event representing a certain concept of interest. Let R’ = {X1,X2,..., X,} denote the partition induced by equivalance relation R , where xi,i= 1,2 ,...) n, is an equivalance class of R (an elementary set of A ). Our primary objective is to characterize an expert concept Y C U in A, by the known concepts Xi. To deal with random events, the probability measures need to be available especially probabilities of occurence of events Y and Xi. Let P(Y]Xi) = P(Y n Xi)/P(Xi) denote the probability of occurence of event Y conditioned on event Xi. In other words, P(Y]Xi) is the probability that a randomly selected object with

Application

of rough set theory

21

the description of concept Xi belongs to Y. But there will be a serious problem to apply this approach due to the need for verifying the statistical significance of the sampled data considered during analysis. The main advantage of basic rough set theory is that it does not need any such additional information. By the way, it is an advantage over stochastic approaches such as discrimant analysis. Having no such requirements, basic rough set theory can be used to analyze statistically non-significant samples of information in order to find out rules deduced {Des from any experience (large or small). I n order to measure how well the set of descriptions (Xi)lXi E V} can categorize the objects in Y, the notion of PAC with a level of certainity p, 0.5 < p 5 1, is introduced as follows: - The lower approximation APL(Y)

of Y, in A, = (U, R, P) is defined = {u E U~U E Xi and P(YlXi)

as

1 p}

(I)

It is in fact the union of all those classes Xi with conditional probabilities It is also referred to as the P-positive region POSp(Y) of the concept Y. - The upper approximation of Y is defined as A,,(Y)

= {ZJ E Ulu E Xi and P(YlXi)

P(Y IXi) 2 P.

> 0.5)

(2)

It is the union of all those classes Xi with conditional probabilities P(Y(Xi) > 0.5. region or the /3- The set BND A,L(Y) = A,,(Y) - A,L(Y) is called the ,k-doubtful boundary region of Y. It consists of all classes Xi with conditional probabilities 0.5 5

-

p(yIxi)

.(V

Figure 1. Approximate

classification of set Y in PAS.

The level /3 is a minimum certainity requirement. Then, any object matching the description of any one of the classes Xi in the POSp(Y) may b e p ossibly a member of concept Y with a probability greater than or equal to ,i3. If an object satisfies the description class Xi c BND,L(Y), no decision can be made in this case. However, we may conclude that all objects in the negative region NEG(Y) possibly do not belong to concept Y. If A,L(Y) = A,,(Y) (i.e., BND we may say that the concept Y is definable in the PAS A,; otherwise, Y is APL(Y) = 0), nondefinable or a rough set.

A.S.A.

22

MOHAMED

2.2 Attributes Dependency in a KRS The KRS can be formally

defined

as 5-tuple

Q = (U,C,D,V,f),

(3)

where U C D V f:

denotes a set of objects. is a set of condition attributes. is a set of decision (action attributes C n D # 8). UV,,where V, is the domain attribute a E (C U D). U x (CUD) -+ V in an information function such that f(e, u) E V, for e E U and u E CUD (i.e., f assigns attribute values to objects in U).

induced by two arbitrary LetA*={Ai,Az ,..., A,}andB*={Bi,Bz ,..., B,} be partitions subsets of attributes in the KRS a. The degree of information dependency of set A and set B can be measured either by the quality of classification introduced by Pawlak [19] or using the entropy measure. The entropy of a set A having n supporting elements is defined by

H(91,Q2,...,@,)

=

C

QA(Ai)

* ln{Q’A(A)}l

(4

i

where

‘A(Ai) = Ci

@A(h) @A(Ai)

i=

’

1,2 ,...,

(5)

n,

@A(Ai) denotes the degree to which an event Ai may be a member of A or belong to A . This characteristic function, in fact, can be viewed as a weighting coefficient which reflects the ambiguity in a set, and as it approaches unity, the membership values of an event in A becomes higher. For example, QA(Ai) = 1 indicates strictly the containment of the event Ai in A. If, on the other hand, Ai does not belong to A, @A(Ai) = 0. Any intermediate value would represent the degree to which Ai could be member of A . Thus, OA(Ai) associates with each element in A a real number in the interval [O,l] with each element in A a real number in the interval [O,l] with the value of @A at Ai, i = 1,. . . , n. The entropy is seen to lie between 0 and 1 in a way Hmjn = 0

for

Jrj

=

1,

Qi = 0, H mar --1

Q1=Q2

for

j E {1,2,.

. . , n}

(6)

i # j, =...=

a,+.

(7)

The entropy measure does not depend on the absolute values of Q but their relative values. In other words, a set with a single nonzero entropy and a set having constant @value for all elements would have H = 1. The entropy H(A*jB*) may be defined by [20]:

H(B*lA*)

=

2 [P’Ai/n~~*lAi)],

(8)

i=l

where H(B*IAi)

= -e[P(BjIAi)h(P(Bj[Ai))]. j=l

Thus,

it provides

a plausible

measure

of dependency

of A on B.

(9)

Application

of rough set theory

23

B’ is functionally dependent on A* iff H(B*lA*) = 0. Othervise, 8” is functionally independent of A’ iff H(B*IA’) = 1, where 0 5 H(B*lA*) 5 1. A subset of condition attributes A C C is said to be a dependent set in R with respect to D if there exists a proper subset BcA Otherwise,

such that

A is said to be an independent

An attribute

c is said to be dispensible H(D*I(C

Otherwise

c is an indispensible

attribute

H(D*]B*)

= H(D*]A*). to D.

set with respect

to D if

in C with respect - {c})*)H(D*IC*). in C with respect

(11)

(12)

to D.

3. APPLICATION In cardiology and more specifically the valvular heart diseases [VHD], echocardiography becomes the most prominant noninvasive diagnostic procedure. It depends on the interaction of ultrasound beam with the tissues of myocardium. The produced ultrasound beam from a piezoelectric crystal traverses tissues and substances at different velocities depending on their acoustic impedence without any possible morbidity involved in radiography [21]. Part of the ultrasound energy is reflected at the interface between two structures of different acoustic impedences. The reflected echo is converted into electrical pulses by the piezoelectric crystal and it can be displayed and recorded. The technical procedure of clinical data acquisition will be described in the next part (3.1). An example of VHD is considered with the special focus on the mitral valve regurgitation [MVR].

9.1 Data Acquisition An experiment was conducted on twenty patients with MVR. They were studied by ultrasound echocardiography and all echocardiograms were measured in the supine position. A standard IREX echocardiographer was employed using 2.25 MHz, 1.5 cm unfocused transducer with a repetition rate 1000 set-‘. Simultaneous caroid pulse tracings were obtained with a HewlettPackard APT-16 pulse transducer. An ECG(lead-11) was also recorded simultaneously. The ultrasound transducer was placed in the third, fourth, and fifth left intercostal to record the echocardiograms. The subjects were selected to have age (13-54 years), and body surface area [BSA] 1.45-2.22 m2. The considered attributes (listed in Table 1) contain: (1) Physical measurements which contain systolic blood pressure [SBP], diastolic blood pressure [DBP], and pulse pressure [PP]; (2) M-mode echocardiographic measurements including left atria1 [LA], Aortic root [AO], end diastolic diameter [EDD], end systolic diameter [ESDI, systolic posterior wall thickness [SPW], and diastolic posterior wall thickness [DPW] ; attributes such as end diastolic volume index [EDVi], end (3) C a 1cu 1a t e d ec h ocardiographic systolic volume index [ESVi], stroke volume index [SVi], ejection fraction [EF], fractional shortening [FS], circumfrential velocity of shortening [Vcfl, and left ventricular mass index [LVmi]; and (4) The ejection time [ET] determined from caroid pulse tracing. According to the normal ranges of these attributes [22], some norms for these attributes are established as depicted in Table 2. The question of norms and their influence on robustness of decision rules has been recently investigated in [23]. These norms are normal (n), abnormal high (UH), and abnormal low (UL). For sex, different normas are considered with male (M), and female (F). Following the attributes categorization, the clinical data is converted to those classes (M, F, N, UL, and UH), and the attributes having only one class are removed from the table.

24

A.S.A.

MOHAMED

Table 1. An example of KRS

U

Attributes {A}

(Patients)

Condition Attributes {C}

Sex

AiF

1

Physical Measurements

Echocardiographic measurements

1

l---L

A0

DPW

SPW

ET

EDD

ESD

male

30

3.6 3.5

1.2 1.1

1.4 1.3

0.28 0.36

5.5 8.0

3.2 5.6

u3

female

19

4.3

1.

1.3

0.32

6.2

4.2

u20

male

22

3.1

1.2

1.3

0.3

6.5

5.2

Ul

u2

1.75 1 130 1

50

1 80 1 4.5

U

Attributes {A}

(Patients)

Decision Attributes {D}

Ul u2 u3

u20

EDVi

ESVi

SVi

89.6 305.5 141.6

17.6 111.8 45.8

71.9 193.7 95.9

157.6

84.4

73.2

EF

FS

LVmi

0.8 0.63 0.68

0.42 0.3 0.32

1.5 0.88 1.08

0.46

0.2

0.67

Table 2. Domain of attributes of CKRS

No. Attribute Name

l-

classes N

1 2 3 4

UL

UH

Sex Age (13-54) BSA (1.45-2.22) SBP

115-125

< 115

> 125

5 6 7

DBP

75-85 70-80 1.9-4

< 75 < 70 < 1.9

>80

2-3.7

PP LA

>80 >4 > 3.7

8

A0

9

DPW

0.6-1.1

1.4

11 12 13 14 15 16 17 18 19

ET

< < < <
> > > >

EDD ESD EDV ESV sv EF FS V cf

0.275-0.313 3.65.2 2.3-3.9 76.1-114.9 29.1-48.1 47-66.8

0.275 3.6b 2.3 76.1 29.1

< 47 0.538-0.662 < 0.538 0.18-0.42 < 0.18 1.02-1.94

< 1.02

> 1.1 0.313 5.2 3.9 114.9 48.1

> 66.8 > 0.662 > 0.42 > 1.94

25

Application of rough set theory Table 3. Decision and condition attributes Decision Attributes

Condition Attributes

EDVi

SEX, SBP, DBP, LA, AO, ET, EDD, ESD, ESVi, SVi, EF, FS, LVmi

ESVi SVi

SEX, SEX, SEX, SEX, SEX,

EF FS LVmi

3.2 Procedure

SBP, SBP, SBP, SBP, SBP,

DBP, DBP, DBP, DBP, DBP,

LA, LA, LA, LA, LA,

AO, AO, AO, AO, AO,

ET, ET, ET, ET, ET,

EDD, EDD, EDD, EDD, EDD,

ESD, EDVi, SVi, EF, FS, LVmi ESD, ESD, ESD, ESD,

EDVi, EDVi, EDVi, EDVi,

ESVi, ESVi, ESVi, ESVi,

EF, SVi, SVi, SVi,

FS, LVmi FS, LVm.i EF, LVmi

EF, FS

For Computation

At the beginning, the KRS will be built to define the decision, and condition attributes. Only one decision attribute will be considered at a time from the available set of calculated attributes using echocardiogram measurements. All other attributes are considered to be condition attributes as shown in Table 3. Thus, the KRS S may be defined as follows: u= {w,w,...,~20} C = { SEX, SBP, DBP, PP, LA, AO, . . . } D = {EDVi, ESVi, SVi, EF, FS, Vcf, LVmi } with range for information function f: VO = VI = . . - ={ M,F,N,UL,UH In this particular application, for all u E U, and c E C f(e, c) = N

for normal

cardiac

}, and Vd ={N,UL,UH}.

valve.

= UL

for abnormal

less than the normal

= UH

for abnormal

higher than

range.

the normal

range.

To derive the decision rules from the available information (KRS), we have to identify the SUperfluous attributes and remove them before the decision rules will be generated. The process of identifying and removing the superfluous attributes will be as follows: Let D’ = {Yi, .. . , Ym} and C’= {Xi,... ,X,,}, then P-dependency of set C on D is defined by ,‘+(,,I(-)

=

(13)

where POSp(D*) and

= jiiPOSp(Yj)

(14)

(15)

lpo%(~‘)I = ~ww3w)l)* j

The &significance,

pp of an attribute fqr(c) =

c E C is given by

‘%PIC) - WW

If /q(c) = 0, then attribute C is superfluous. To find the most significant attributes, the following (1) Choose R, R , and & of size ICI. (2) Let R = 0, R = 0 (3) FOR cl and c,

BEGIN Find pp (c); IF &c)

> 0 add c to 52

- W>

algorithm

can be used:

(16)

A.S.A.MOHAMED

26

ELSE add c to R END (4) WHILE R # 0 BEGIN Q=0 FOR each c E R BEGIN Find pp(c); IF pp(c) > 0 add c to R ELSE add c to Q END IF Q # 0 remove Q[O] from R and C END (5) END At the termination of this algoritm R will contain the most P-significant attributes which are the reduct of C. It is important to note that, a reduct is just one hypothesis about what is significant, in general, many different reducts can be computed. The sequence in which attributes are eliminated depends mainly on: 1) the order in which attributes are acquired by the system from original input; and 2) the order of attributes in list R after Step-3 in the algorithm. To discriminate among the attributes available in the resulting reducts (due to order of attribute removal), a suitable preference function [24] will be used to select the best reduct, assuming a level of satisfaction with alternative is supposed to be known. Let S = {Sl,S:!,... , SL} be our set of alternatives, the possible reducts of decision under consideration. Assume we have associated with each alternative a number between 0 and 1 indicative of the satisfaction with that alternative. We can define a preference function which then selects our performance between two alternatives. DEFINITION. Assume 2 = O.O,O.l, 0.2, . . . ,1.0. Let &, Z2 E Z. Assume Z1 is the satisfaction associated with alternative S1, and Zs is the satisfaction ated with S2. Consider the function F F:ZxZ-+W

W =

associ-

{O,f3l,S2),

defined as follows : F(Zl,

22)

F(Zl,Zz)

=f%

if Zl > Z2,

(17a)

=S2

if Z2 > Zl,

(17b)

if Z1 > Z2,

(17c)

F(Z1,472) =O

which indicates if the satisfaction of S1 is greater than S2, we choose S1 and if satisfaction of S2 is greater than 4, we choose S2 and if satisfaction is even, we are indifferent. The function is then basically a preference function [251. That is if SlRS2 signifies “Sl is prefered or indifferent to S2,” then our function F generates R which satisfies (1) For all Si and Sj, either Sj R Si or Si R Si (2) For allSi,Sj,Sk p.t. SiRSj andSjRSk, thenSiRSk Therefore, using this function we can associate a unique ordering, up to an equivalence class, of equal Z’S among our preferences for the alternatives. Let F(Zl,Zz) = T, where T is a subset of W. Then since W is of the form

where h, q, r are numbers between [O,l] indicating ulated by Z.

to what degree S satisfies the conditions stip-

Application of rough set theory

27

Table 4. Accuracy of approximation

Class

Decision

No. of

ESVi

SVi

EF

LVmi

Accuracy

1 .oo

7

7

7

7

13

13

13

13

1.00

N UH UL

7

5

10

7

0.71

8

5

10

8

0.62

5

5

5

5

1.00

N UH UL

3 13

4

3

3

1 .oo

13

13

13

1.00

4

4

4

4

1.00

N UH

6

4

9

6

0.66

13

10

15

13

0.76

UL

1

1

1

1

1.00

19

19

19

1

1 .oo

UH

1

1

1

19

1.00

N

1

1

1

1

1 .oo

19

19

19

19

1 .oo

N

FS

Boundary

Upper Approx.

UH

N

EDVi

UH

EXAMPLE.

Lower Approx.

Patients

Attributes

Let 21 be satisfaction 1 and assume

2~ is any satisfaction

other

than

1

1

21 =

0

i'

{ --PO Ql

z2=

42

QlO

>’

0 ' 0.1’0.2’““1

where some qi is not zero. Using the extension

F(Z1,Z2)=

fyZ1,

z2>=

bdqil,

principle

1g,$,$ i E

M

[26]

,..., fg,y

qlo

>,

M = {O,O.l,. . . ,0.9}.

Sl

Since we assumed at least one is not zero, then our choice is to select Si over SZ. We select as our preference either Sr or S2, depending upon whether q or r is larger. The procedure for obtaining W is as follows: If Zl = { z2 =

{

a,

Qz ‘3

Cl

c2

r1

r2 r3

rn

cz'c3

Cfl>

-I-

Cl

g

’ q ’ * * . ’ c, > -,...,-

ci E [O,11, a, pi E [O,11,

then,

4. RESULTS It was found that {age, BSA, PP, SPW, and V,j } have only one class. Therefore, they were not considered during computation. The decision attributes and the corresponding condition attributes are listed in Table 3. The accuracy of approximation for the individual decision attribute is given in Table 4. The resulting core attributes are depicted in Table 5. Actually, the dependence between the decision attribute and the condition attributes will be influenced if one of the significant condition attributes will be removed during the process of attribute elimination.

28

A.S.A.

MOHAMED

Table 5. Core of attributes

Decision

Reducts of Condition Attributes

Attributes EDVi

ESVi, SVi, FS

ESVi

1 1 1 1 (

SVi EF FS LVmi

The percentage of decrease of is eliminated, can be followed in ESVi, SVi, EF, FS, and LVmi}, computation during the process relate the decision attribute with

SEX. SBP. EDVi. SVi. FS ET, EDD, EF, LVmi SEX. DBP. ESVi. SVi. LVmi EDVI, ESVi SVi, EF

i

such dependency, if one of the significant condition attribute Figures 2, 3, 4, 5, 6, and 7 for the decision attributes {EDVi, respectively. These figures contain also the precision of core of attribute elimination. The expected decision rules which the condition attributes are listed in Tables 6-11.

Table 6. Decision rules for EDVi

Condition Attributes

Rules

SVi

ESVi

FS

Decision

UL

-_.

UH

N

_____

N

#2 #3

UH

UL

N

N

#4

UH

UL

UH

UH

#5

UH

UH

UH

#6

N

N

#7

N

UH

#8

N

UL

_-___ _____

Rule #l

UH

UH N N

Table 7. Decision rules for ESVi

Condition Attributes

Rules ET Rule#l #2

SVi

EF

Decision FS

N

N

---

---

UH

N

UL

---

---

UL

#3

N

UH

---

---

N

#4

UH

---

N

---

N

#5

UH

UH

UL

UL

UL

#6

UH

UH

UL

N

UH

#7

UH

N

UL

---

N

#8

UH

UL

UL

---

UH

#9

UH

N

UH

---

N

0.

None

0.6

0.8

l*qf#

$

SEX SBP

@

.: 4: -I” #

SEX

-

SEX SBP DBP LA

SEX SBP DBP LA A0

B

E

E

SEX SBP DBP

#

#

#

SEX SBP DBP LA A0 ET EDD

6

$ t@

SEX SBP DBP LA A0 ET

#.

#.

SEX SBP DBP LA A0 ET EDD ESD

#

@

0

SEX SBP DBP LA A0 FT E”3 ESD EF

Attribute

e

CI

SEX SBP DBP LA A0 ET EDD ESD EF LVmi

Removal

@tFS

e

c:

2

u

.

g B @ f

& c 5. g F

(I

D'BP

0.c N>ne

Core precision

0

u 0.4 % -z m c e. Ei e. 0.2 B

1

Fpa ii I? 2

0,

DkP LA D$P LA A0

0

DB'P LA A0 ET

a

DkP LA A0 ET EDD

0

DkP LA A0 ET EDD ESD

DtiP LA A0 ET EDD ESD EF

Attribute removal

B

DkP LA A0 ET EDD ESD EF LVmi

+-SBP

,rrEDVi

?-ESD

t+=S

A

z

H t;

,,

F 6 5

i

; z 3 5

B pj

2:: 0 -z p e.

ifs _ ‘;

f

s

t

a

iid

8

P

-

if

0.0.

0.2.

0.4.

06..

0

N>ne

.

X

LO.{;

8, ,,,,,,

SI!X

@

i

Y

iX X

sI?x SBP

1;

B

,,

I

T

sl?x SBP DBP LA

X Y

{

0 X Y

SIIX SBP DBP

0 X Y

sE!x SBP DBP LA A0

{ ;

0

1;:

X Y

SEX SBP DBP LA A0 ESD

0

,,

,,,

sE!x SBP DBP LA A0 ESD EDVi

Attribute

X

Y

SEX SBP DBP LA A0 ESD EDVi ESVi

removal

X

Y

T

V

l

LVmi

SEX SBP DBP LA A0 ESD EDVi ESVi FS

%fEF

YWEDD

T-ET

v-

b

Y

g s

i

$ B

s E:

b 8 E 6 f.

0.6.

0. a.

z 1. E" l7

g g e

B 1 ; 9 B i

s 1.

l

,,

,,

,,,I#

0.0. NLbne

0.2.

fk 'd g c !k ff f 7 a:: g 4 om4.

? ,a 3 i &

az! 3 1 tc.

1.0. d m -I s*

SdP

?

Core

ln

-G

I

precision

!?BP LA

@

pl

SBP LA A0

G

s

SkP LA A0 ET

*

*

SE!P LA A0 ET EDD

,,

,,,

SbP LA A0 ET EDD ESD

Attribute

EDSi

SbP LA A0 ET EDD ESD

removal

EDVi FS

, SBP LA A0 ET EDD ESD

h-SVi

*-ESVi

k-DBP

4-LVmi

se--Sex

0.c

0.2

0

SEX

I

bne

c

SEX SBP

SEX SBP DBP

SEX SBP DBP LA

Core precision

‘\

0

*

slzx SBP DBP LA A0

sex SBP DBP LA A0 ET

sE!x SBP DBP LA A0 ET EDD

SEk SBP DBP LA A0 ET EDD ESD

0

removal \ SEX SEX SBP SBP DBP DBP LA LA A0 A0 ET ET EDD EDD ESD ESD SVi SVi EF

Attribute

8

I SEX SBP DBP LA A0 ET EDD ESD SVi EF LVmi

o_EDVi

0

one

b

skx

‘\

stx SBP

Core

sex SBP DBP

precision

stix SBP DBP LA

stix SBP DBP LA A0

SEk SBP DBP LA A0 ET

SEk SBP DBP LA A0 ET EDD

SEX SBP DBP LA A0 ET EDD ESD

1

Attribute

SEX SBP DBP LA A0 ET EDD ESD EDVi

removal SEX SBP DBP LA A0 ET EDD ESD EDVi ESVi

zvi

SEX SBP DBP LA A0 ET EDD ESD EDVi

#-cSVi

&-EF

x

#

0

l

Application

of rough set theory

35

Table 8. Decision rules for SVi

Condition Attributes

Rules

Decision

.

ET

EF

EDD

LVmi

Rule#l

UH

UH

UH

- --

UH

#2

UH

UH

N

N

UL

#3

UH

UH

N

UH

N

#4

UH

N

N

---

UL

#5

UH

N

UH

---

UH

#6

UH

UL

---

---

N

#7

UL

- -_

_--

___

UL

#6

N

___

___

___

N

Table 9. Decision rules for EF Condition Attributes

Rules

LVmi

DBP Rule#l

ESVi

SEX _-_ -__

UL

UH

UL

___

UL

UH

UL

UH

UH

UH

UH

#3

UL

#4

UL

UH UH UH

#5

UL

UH

UH

UH

#6

UL

UH

N

#7

UL

UH UH

UH

N

#6

UL

N

UL

N

UH

#lO

UL

#11

UL

#12

UH

#13

UH

UH UH UH N -___-

N

#9

#14

N

__-

#2

Decision

SVi

F F M F F ___ m-m _-_ m-v

N

UL

-__

___

--_

N

___

UH

m-m _-_

___

_-_

_-_

N UH N UH UH N UH N UL UH UH N UH UH

Table 10. Decision rules for FS

Rules

.

Condition Attributes ESVi

Rule#l

EDVi

Decision

UL

UH

UH

#2

UL

N

N

#3

UH

- - -

N

#4

N

---

N

Table 11. Decision rules for LVmi

Rules .

Condition Attributes SVi

EF

#2 #3

UL UL UH

N UH ---

#4

N

---

Rule#l

Decision UH N UH UH

36

A.S.A.

MOHAMED

5. DISCUSSION Many decision classification systems have been applied to medical diagnosis problems. Such systems can generally be interpreted as comprising a knowledge base and a method of reasoning from that knowledge base. It is useful to identify two essentially basic approaches according to how the knowledge base is compiled: (1) The data is compiled as tables of conditional probabilities and decisions method of Bayesian probability inference. Other methods of reasoning algorithms and discriminant function analysis. The common attribute is a numerical tabular knowledge base. (2) The second alternative approach is the rule based expert system in base of facts and rules are compiled by questioning expert clinicians establish the significant relationships among the available attributes.

are taken using the include clustering to all these systems which a knowledge and attempting to

In this paper, the problem of clinical knowledge base representation has been formulated with a solution within the framework of the rough set approach using the entropy measure instead of the quality of classification introduced by Pawlak [ll]. A procedure for computation is explained to be easily implemented on microcomputers. A preference function is defined to select the best reduct of condition attributes for the selected decision attribute. The relationship between the decision attribute and condition attributes (in medical literatures) depends on an important assumption. This assumption deals with the geometry of the left ventricle which is described by a prolate ellipse with equal short axes in normal persons. Comparing the attribute’s dependency obtained from this analysis with the deduced relationships from the prescribed model, satisfactory and acceptable results are obtained. If we investigate the obtained results for EF, for example, you will find: - In normal subjects with the the model). - In the computed results from contains SEX, DSP, ESVi, proposed model, there is no

available

relationship

EF is related

to SVi, and EDVi (from

diseased subjects with MVR, the core of condition attributes SVi, and LVmi. Comparing the obtained results with the significant difference due to the following facts:

(1) During the MVR, the left ventricle [LV] pumps the blood to both left atrium and aortic root. This causes a remarkable decrease in the cardiac output and makes no difference between ESVi and EDVi. (2) DBP is influenced with the decrease of cardiac output. (3) The LV chamber b ecomes continuously filled with blood which increases its load and consequently including the LVmi does not contradict with the present situations. The selection of preference function overcomes the multi-reducts resulting from the attribute elimination. In this way, the size of the clinical table is reduced and dependency among the decision attributes and the condition attributes is determined. Moreover, a set of decision rules are deducted to describe this dependency. A unique advantage of this approach is the flexibility to update the knowledge base with new data and the constraint imposed on the attributes to be in a numerical form is avoided. REFERENCES 1. W.A. Wood, Important issues in knowledge representation, Proc. IEEE 79, 1322-1334(1986). 2. M. King and M. Rosner, Special issue on knowledge representation, Proc. IEEE 79, 1299-1303(1986). 3. W.D. Hillis, The Conneefion Machine, MIT Press, Cambridge, MA, (1985). 4. D. Micbie, Knowledge-based systems, Report #lOOl, Dept. of Computer Sci., Univ. Of Illinois, Urbana, Illinois, (1960). 5. J.R. Quinlan, Learning efficient classification procedures and their application to chess endgames, In Machine Learning: The Artificial Intelligence Approach, Edited by R.S. Michalski, J.G. Carbonel and T. M. Mitchell, Tiogu Press, Palo Alto, (1963). 6. R.O. Duda, P.E. Hart and N.J. Nilson, Subjective Bayesian methods for rule-based inference systems, In Proc. National Computer Conference, pp. 1075-1082, (1986). 7. L.A. Zadeh, Approximate reasoning based on fuzzy logic, In Proc. 6ih. Int. Conj. Artijicial Inielligence, pp.

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Application of rough set theory

37

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