Decision Support Systems 11 (1994) 37-43 37 North-Holland

A weighted fuzzy reasoning algorithm for medical diagnosis *

S h y i - M i n g C h e n 1. Introduction

National Chino Tung University, Hsinchu, Taiwan, ROC It is obvious that many physicians' knowledge

This paper presents a weighted fuzzy reasoning algorithm for in m e d i c a l d i agnos i s a n d m a n y p a t i e n t s ' s y m p t o m handling medical diagnostic problems, where fuzzy set theory m a n i f e s t a t i o n s invo lve fuzzy c o n c e p t s [8]. I t is and fuzzy production rules are used for knowledge represen- often t he case tha t w h i l e m e d i c a l d i a g n o s t i c p r o b - tation. The algorithm can perform fuzzy matching between

lems can be handled easily by physicians, they are the patient's symptom manifestations and the antecedent por- tions of fuzzy production rules to determine the presence of often too difficult to be handled by computers. diseases, where the result is interpreted as a certainty level Therefore, there is an increasing demand to de- indicating the degree of certainty of the presence of the sign a medical diagnostic system to handle medi- disease. Because the algorithm allows each symptom in medi- cal diagnostic problems. In recent years, the ap- cal diagnosis to have a different degree of importance, it is plication of fuzzy set theory [14] in medical diag- more flexible than the ones we presented in [3] and [4]. The algorithm can be executed very efficiently. If the knowledge nosis has been investigate [1], [3], [4], [9], [11], base contains n fuzzy production rules and there are p [12]. symptoms, then the time complexity of the algorithm is O(np). The theory of fuzzy sets was proposed by Zadeh

[14] in 1965. Let U be the universe of discourse, Keywords: Fuzzy production rules; Fuzzy set theory; Knowl-

U = {u 1, u 2 . . . . . Up}. A fuzzy set A of U is a set edge base; Knowledge representation; Similarity function; Similarity measures, of ordered pairs {(u i, fA(Ui))l U i ~ U}, where f n

is the membership function, f n : U ~ [0, 1], and fA(u i ) is the grade of membership of u i in A. Let A and B be two fuzzy sets in the universe of discourse U, i.e.,

u = {u~, u2 . . . . . u~} ,

......... Shyi-Ming Chen was born in Taipei, A = {(u i , f A ( U i ) ) [ . i E U}, Taiwan, Republic of China, on Jan- uary 16, 1960. He received the B.S. B = {(Ui, fB(Ui))]Ui E U}, degree in electronic engineering from National Taiwan Institute of Technol- where f a and fB are the membership functions

~ ogy~ Taipei, Taiwan, in 1982, and the M.S. and Ph.D. degrees in electrical of the fuzzy sets A and B, respectively. Then, the engineering from National Taiwan intersection operation between the fuzzy sets A University, Taipei, Taiwan, in 1986 and 1991, respectively. From October and B is d e f i n e d by: 1982 to August 1984, he served in the Chinese Navy as an Electronics Offi- A (~B = {(u i, f A n B ( U i ) ) l f A n B ( U i ) cer. Since August 1987, he has been

,n the faculty of the Department of Electronic Engineering, = Min( fA(Ui) , fB (u i ) ) , U i ~ U~. ?u-Jen University, Taipei, Taiwan. He is currently an Associ- / ~te Professor in the Department of Computer and Informa- :ion Science, National Chiao Tung University, Hsinchu, Tai- In [1], Adlassnig defined inexact medical enti- van. His research interests include knowledge-based systems, ties as fuzzy sets and used fuzzy logic [13] to tatabase systems, artificial intelligence, and fuzzy systems. Dr. 2hen is a member of the IEEE Computer Society and the Phi p e r f o r m e a p p r o x i m a t e r e a s o n i n g [10, p.30] fo r Fau Phi Scholastic Honor Society. medical diagnosis. In [9], Leung et al. developed Correspondence to: Dr. Shyi-Ming Chen, Department of Com- a novel expert system shell SYSTEM Z-II based puter and Information Science, National Chino Tung Univer- sity, Hsinchu, Taiwan, R.O.C. o n fuzzy sets, fuzzy logic, and fuzzy numbers [8]. * This work was supported by the National Science Council, In [11] a n d [12], S a n c h e z p r e s e n t e d a t r e a t m e n t

Republic of China, under Grant NSC 81-0408-E-009-520. o f l inguis t ic en t i t i e s in m e d i c i n e b a s e d on fuzzy

0167-9236/94/$07.00 © 1994 - Elsevier Science Publishers B.V. All rights reserved

38 Shyi-Ming Chen / Fuzzy reasoning for medical diagnosis

sets, allowing the assignment of graded diagnosis Table 1 to patients. In [3] and [4], we have presented the Fuzzy quantifiers and their corresponding numerical intervals.

techniques for handling medical diagnostic prob- Fuzzy quantifiers Numerical intervals lems based on fuzzy set theory. However, the always [1.00, 1.00] techniques presented in [3] and [4] assume that very strong [0.95, 0.99] all symptoms in medical diagnosis are of equal strong [0.80, 0.94] importance (i.e., each symptom in medical diag- more or less strong [0.65, 0.79] nosis has the same weight). If we can allow each medium [0.45, 0.64]

more or less weak [0.30, 0.44] symptom in medical diagnosis to have a different weak [0.10, 0.29] degree of importance, then there is room for very weak [0.01, 0.09] more flexibility, no [0.00, 0.00]

In this paper, we propose a weighted fuzzy reasoning algorithm to handle medical diagnostic problems, where fuzzy set theory and fuzzy pro- In order to make the real-world knowledge duction rules [3], [4], [5], [10] are used for knowl- suitable for being processed by computers, fuzzy edge representation. The algorithm allows each production rules have been used for knowledge symptom in medical diagnosis to have a different representation. The fuzzy production rule allows degree of importance. It can perform fuzzy the rule to contain some fuzzy quantifiers [3] (i.e., matching between the patient's symptom mani- strong, weak, very strong, more or less weak, festations and the antecedent portions of fuzzy etc.). production rules, where the result is interpreted Let R be a set of fuzzy production rules, as a certainty level indicating the degree of cer- R = {R 1, R 2 . . . . . Rn}. tainty of the presence of the disease. The algo- rithm is more flexible than the ones we presented The general formulation of the rule Ri, 1 ~< i ~< n, in [3] and [4] due to the fact that it allows each is as follows:

symptom in medical diagnosis to have a different R i: IF Di T H E N d i (CF = ] . i , i ) , where

degree of importance. (1) D~ represents the antecedent portion of R i

This paper is organized as follows. In section which may contain some fuzzy quantifiers. 2, the concepts of fuzzy production rules are

The definitions of the fuzzy quantifiers and introduced. In section 3, the definition of the

their corresponding numerical intervals are similarity function F is presented to measure the given in Table 1 [3]. degree of similarity between fuzzy sets. In section

(2) d~ represents the consequence portion of R i 4, a weighted fuzzy reasoning technique is pre- (3) /z i is the certainty factor which indicates the sented. In section 5, we present a weighted fuzzy certainty that R~ is believed in. The certainty reasoning algorithm for handling medical diag- levels which we used and their corresponding nostic problems. The conclusions are given in

numerical intervals are given in Table 2. section 6.

Table 2 Certainty levels and their corresponding numerical intervals.

2. Knowledge representation Certainty levels Numerical intervals

absolutely certain [1.00 1.00] It is obvious that much knowledge in the real- extremely certain [0.96 0.99]

world is fuzzy rather than precise. The fuzziness very certain [0.86 0.95] [8] occurs when the boundary of a piece of infor- pretty certain [0.76 0.85] mation is not clear cut. For example, the follow- quite certain [0.66 0.75]

fairly certain [0.56 0.65] ing is a piece of fuzzy knowledge: " I f you have a more or less certain [0.46 0.55] strong headache, then you might have caught a little certain [0.30 0.45] cold" where "strong" is a fuzzy concept because very little certain [0.16 0.29] the meaning of the term "strong" is not clear cut hardly certain [0.01 0.15] and is also context dependent, absolutely uncertain [0.00 0.00]

Shyi-Ming Chen/ Fuzzy reasoning for medical diagnosis 39

In medical diagnostic problems, a physician's where a i ~ [0, 1], b i ~ [0, 1], and 1 ~< i ~<p. By us- knowledge can be represented as ing the vector representation method [3], A and

IF symptoms THEN concluded disease (CF = B can be represented by the vectors X and B, /zi). For example, let U be a set of symptoms, respectively, where U = {vomiting, fever, knee pain, right lower ab-

A = (a l , a 2 . . . . . ap), dominal pain, leukocyte increased}, and appen- dicitis be a concluded disease, then a physician's B = ( b l , b 2 . . . . . bp). knowledge may be represented by the rule R 1 as follows: Assume that each u i in U has a different degree R I : IF {always vomiting A always fever A no of importance and the importance of u i is wi,

knee pain A always right lower ab- where w i ~ [0, 1] and 1 ~< i ~<p, then the degree of dominal pain A always leukocyte in- importance of each u i in U can be described by a creased} weighted vector W, where

THEN appendicitis (CF = very certain) According to Table 1 and Table 2, the rule R 1 W= (wl, w2,. . . ,Wp). may subjectively be expressed as follows: RI: IF {(vomiting, 1.00)), (fever, 1.00), (knee In this case, the degree of similarity between the

pain, 0.00), (right lower abdominal fuzzy sets A and B can be measured by the pain, 1.00), (leukocyte increased, similarity function F, F(A, B, W) ~ [0, 1], where 100 [

THEN appendicitis (CF = 0.90) F ( A , B, W) = ~ T(aj, bj)* ~ - - . (2) where D 1 --{(vomiting, 1.00), (fever, 1.00), (knee

J=' Ewk pain, 0.00), (right lower abdominal pain, 1.00),

k = l (leukocyte increased, 1.00)}. It is obvious that D~ is a fuzzy set of the universe of discourse U, The larger the value of F(A, B, W), the higher where U = {vomiting, fever, knee pain, right lower the similarity between the fuzzy sets A and B. abdominal pain, leukocyte increased}. Example 3-1: Let A and B be two fuzzy sets of

the universe of discourse U, where

U = {Ul, U2, U3, U4} , 3. Similarity measures

A = {(u,, 0.9), (u2, 0.5), (u 3, 0.1), (b/n, 0.2)}, In this section, the similarity function F is

introduced to measure the degree of similarity B = {(u l, 0.3), (u2, 0.8), (u 3, 0.6), (u 4, 0.2)}.

between fuzzy sets. Then, by using the vector representation method, Let x and y be two real values between zero

the fuzzy sets A and B can be represented by the and one. The degree of similarity between x and

vectors X and B, respectively, where y can be measured by the function T [6],

T ( x , y ) = l - l x - y l , (1) A = (0.9, 0.5, 0.1, 0,2),

where T(x, y) ~ [0, 1]. The larger the values of B = (0.3, 0.8, 0.6, 0.2). T(x, y), the higher the similarity between x and

Assume that the weighted vector W is as follows: y. For example, if x = 0 . 9 and y = 0 . 7 , then W = (0.5, 1.0, 0.8, 1.0), then T(x, y ) = 0.8. It indicates that the degrees of

0.5 similarity between the real numbers 0.9 and 0.7 is F ( A , B, W) = (1 - 10.9 - 0.31) * 3.3 0.8. It is obvious that if x = y , then T(x, y )= 1.

Let U be the universe of discourse and let A + ( 1 - 10.5- 0.8 I)* L0 .Z5.3 and B be two fuzzy sets of U, i.e.,

+ (1 - 10.1 - 0.6 I) * (,~33 u = { u l , u2 . . . . . u , } ,

• ~ -- 0.6969. A = {(u 1, a , ) , (u 2, a2) . . . . , (up, ap)}, +(1 - 1 0 . 2 - 0 2 1 ) l.O

B = {(u 1, bl), (u 2, b2), (Up, bp)}. It indicates that the degree of similarity between " ' " the fuzzy sets A and B is about 0.6969.

40 Shyi-Ming Chen / Fuzzy reasoning for medical diagnosis

4. A weighted fuzzy reasoning technique nostic problems. Let A be a threshold value, U be a set of symptoms, and V be a set of diseases,

In this section, we present a weighted fuzzy where U = {ml, m 2 , . . . , mp} and V = reasoning technique based on the similarity func- {dl, d 2 . . . . ,dn}. Assume that the knowledge base tion F. contains the following fuzzy production rules:

Let U be a set of symptoms and V be a set of R i : IF D i THEN d i ( C F = ~Li) , diseases, where U = {m 1, m2 , . mR} where D i = {(m 1, t i l ) , (m2, t i2) . . . . . (mp, tip)} , ]£i

' " ' ~ [0, 1], and 1 ~< i ~< n, then, based on the vector V= {dl, d 2 . . . . ,dn}. representation_ method, D i can be represented by

the vector Oi, where Assume that the knowledge base contains the following fuzzy production rule: O i = ( t i l , t i 2 , . . . , tip) and 1 ~< i ~< n.

Ri: IF D i THEN d i ( C F =/xi) , Assume that the degree of importance of each

where D i = {(mj, t i ) l tij ~ [0, 1], 1 ~<j ~<p}, /tz i ~ symptom rnj in D i is wij, where 1 <~j~<p, and [0, 1], and 1 ~< i ~< n, and assume that the patient assume that the degrees of importance of all has a set M of symptom manifestations, where symptoms_ in D i can be described by a weighted M = {(m j, x ) l xj ~ [0, 1], 1 ~< j ~< p}. It is obvious vector IV//, where

that D i and M are fuzzy sets of U, where U = ~//= (wil ' wi2, . . . ,w ip) . {ml, m 2 . . . . . mp}. By using the vector representa- tion method_ D i and_ M can be represented by Assume that the patient has a set M of symp- the vectors D i and M, respectively, where tom manifestations,

Di = ( til, ti2 . . . . . t ip), M = {(mj, xj ) i xj ~ [0, 1], 1 ~<j ~<p},

= (x l , x2 , . . . , xp). then M_can also be represented by the vector M, where M = ( x 1, x 2 . . . . , Xp). The algorithm is now

Let W/ be the weighted vector_ of the symp- presented as follows: toms appearing in D i, where W i = ( W i l , w i2 . . . . .

Wip). By applying (2), we can get Weighted fuzzy reasoning algorithm

[ ] P Wq F ( M , Di, W i ) = 2 T ( x j , t i j )* p , (3) begin

j=l ~ W~ k let the fuzzy set Q be the result of the intersection between the fuzzy sets M and

k=l Oi, i.e.,

where F ( M , D__~, W i) ~ [0, 1]. The larger the value of F ( M , D i, Wi), the higher the similarity be- Q = M n D i

tween M and D i. = {(my, sy) ] (m i, xy) ~ M , (my, tiy ) ~ O i , Let A be a threshold value. If F ( M , Di, W i) >~

A, then the rule R i can be fired; it indicates that Sy = Min(xy, tiy ) and 1 ~<j ~<p}.

the patient might have the disease d i with the let T be a set of symptoms which have degree of certainty of about c i, where c i = nonzero membership value in the fuzzy set F ( M , Di, Wi)* I~ i and c i ~ [0, 1]. The larger the T, i.e., value c i, the higher the possibility that the patient might have the disease d i. If F ( M , Di, W/) < A, T = {my I(my, sy) ~ Q, 0 <sy ~ 1 then the rule R i can not be fired, and 1 ~<j ~<p}.

if T ~ O then 5. A weighted fuzzy reasoning algorithm 2y i ~ F ( M , Di, Wi);

begin Yi >1 A then In this section, we present a weighted fuzzy begin

reasoning algorithm for handling medical diag- ci 4-Yi * ~i;

Shyi-Ming Chen / Fuzzy reasoning for medical diagnosis 41

(* the patient might have the dis- R3: IF{(ml, 0.00), (m2, 0.00), (m3, 0.90), ease d i with the degree of cer- tainty of about c i . ) (m4, 0.00), (m 5, 0.30), (m6, 0.00)}

if there exists a tuple (P, [a, b]) in THEN d 3 (CF = 0.92) Table 2 which is shown as fol- R4: IF{(ml, 0.00), (me, 0.00), (m3, 0.00), lows:

(m4, 0.90), (ms, 0.00), (m6, 0.60)}

Certainty Numerical THEN d 4 (CF = 0.95)

levels intervals Rs: IF{(ml, 0.00), (m2, 0.00), (m3, 0.00),

(m4, 0.00), (m5, 0.90), (m6, 0.40)} P [a, b] THEN d s (CF = 0.96). Then

/z I = 0.98,/x 2 = 0.90, ~3 = 0.92, ]A4 = 0.95,

/x s = 0.96,

where c i ~ [ a , b ] and 0 < a ~ < b D, = {(m,, 0.90), (m 2, 0.00), (m 3, 0.00), ~< 1, then the corresponding cer-

tainty level of c i is P (m 4, 0.00), (m s, 0.30), (m 6, 0.00)}, end

end D 2 = {(m,, 0.00), (m 2, 0.95), (m 3, 0.00),

end. (m4, 0.50), (ms, 0.00), (m6, 0.10)},

In the following, we use an example to illus- 03 = {(ml' 0.00), (m2, 0.00), (m3, 0.90), trate the medical diagnostic process, where the (m4, 0.00), (m 5, 0.30), (m6, 0.00)}, result of any arithmetic operation is represented D 4 = {(m l, 0.00), (m2, 0.00), (m 3, 0.00), by 2 digits of significant numbers.

(m4, 0.90), (m5, 0.00), (m6, 0.60)}

Example 5-1: Let U be a set of symptoms, V D 5 = {(m 1, 0.00), (m2, 0.00), (m 3, 0.00), be a set of concluded diseases, and M be a set of the patient's symptom manifestations, where ( m4, 0.00), ( m s, 0.90), ( m6, 0.40) }. U = {m 1, m 2, m 3, m4, m s, rn~), Based on the vector representation method,

M, DI, D2, 03, D4, and D 5 can be represented V= {dl, d2, d3, d4, ds}, by the vectors M, D1, D2, O3, O4, and Ds, M = {(ml, 0.80), (m 2, 0.00), (m3, 0.50), respectively, where

(m4, 0.00), (ms, 0.20), (m6, 0.00)}. M = (0.80, 0.00, 0.50, 0.00, 0.20, 0.00)

Assume that the threshold value A is 0.20 (i.e., D1 = (0.90, 0.00, 0.00, 0.00, 0.30, 0.00)

A = 0.20), and the knowledge base of a medical D2 -- (0.00, 0.95, 0.00, 0.50, 0.00, 0.10) diagnostic system contains the following fuzzy production rules: D3 = (0.00, 0.00, 0.90, 0.00, 0.30, 0.00)

RI: IF{(m,, 0.90), (m2, 0.00), (m 3, 0.00), On = (0.00, 0.00, 0.00, 0.90, 0.00, 0.60)

(m4, 0.00), (m s, 0.30), (m6, 0.00)} D5 = (0.00, 0.00, 0.00, 0.00, 0.90, 0.40).

THEN dl (CF = 0.98) Assume that the weighted vectors of D 1, D_D_2, D3, D4, and D s are W1, W 2, W 3, W 4, and W5,

R2: IF{(ml, 0.00), (m 2, 0.95), (m3, 0.00), respectively, where

(m4, 0.50), (ms, 0.00), (m6, 0.10)} WI = (1.00, 0.00, 0.00, 0.00, 0.20, 0.00)

THEN d2 (CF = 0.90) W2 = (0.00, 1.00, 0.00, 0.50, 0.00, 0.10)

42 Shyi-Ming Chen / Fuzzy reasoning for medical diagnosis

W 3 = (0.00, 0.00, 1.00, 0.00, 0.20, 0.00) certainty level of c 5 is "little certain".

W4 = (0.00, 0.00, 0.00, 1.00, 0.00, 0.50) Thus, we can obtain the following results:

The patient might have the disease d 1 with W 5 = (0.00, 0.00, 0.00, 0.00, 1.00, 0.20), then the degree of certainty of about 0.88 (very when i = 1, certain)

Q = M n D 1 The patient might have the disease d 3 with = {(m 1, 0.80), (m2, 0.00), (m3, 0.00), the degree of certainty of about 0.60 (fairly

certain) (m4, 0.00), (ms , 0.20), (m6, 0.00)};

T = {m~, ms} . The patient might have the disease d 5 with the degree of certainty of about 0.34 (little

Because T4: 0 , we get Yl = F ( M , D1, W 1) = 0.90. certain). Since 0.90 > A, we obtain c I = 0.90*0.98 --- 0.88. From Table 2, we can see that the corresponding

certainty level of c I is "very certain". 6. Conclusions when i = 2,

Q = M A D 2 In this paper, we present a weighted fuzzy

-~ {(ml, 0.00), (m2, 0.00), (m3, 0.00), reasoning algorithm for handling medical diag- nostic problems. The algorithm is more flexible

(m n, 0.00), ( m 5, 0.00), (m6, 0.00)); than the ones we presented in [3] and [4] due to T = Q. the fact that it allows each symptom in medical when i = 3, diagnosis to have a different degree of impor-

tance. The algorithm can be executed very effi- Q = M n D 3 ciently. If the knowledge base contains n fuzzy

----- {(ml, 0.00), (m2, 0.00), (m3, 0.50), production rules and there are p symptoms, then

(m4, 0.00), (m 5, 0.20) (m6, 0.00)}; the time complexity of the algorithm is O(np). In ' the above we assumed that the same set of symp-

T = {m 3, ms} . toms is used to describe all disease features.

Because T4; Q, we get Y3 = F ( M , D3, W 3) = However, if we can divide the diseases into sub- 0.65. categories, such that different categories of dis-

Since 0.65 > A, we obtain c 3 = 0.65 * 0.92 = 0.60. eases are described by different vectors, then the From Table 2, we can see that the corresponding number of symptoms included in each vector certainty level of c 3 is "fairly certain", description will be reduced and the execution when i = 4, time of the systems will be improved.

Based on the proposed algorithm, we have Q = M n D 4 implemented a medical diagnostic expert system

= {(ml, 0.00), (m 2, 0.00), (m 3, 0.00), on a P C / A T by using Turbo Pascal version 5.5 for diagnosing the diseases of the gastro-intesti-

(m4, 0.00), (m 5, 0.00), ( m6, 0.00) }, nal system including esophagus cancer, esophag-

T = 0 . ism, gastric ulcer, stomach cancer, gastritis acute, when i = 5, appendicitis, duodenum ulcer, peritonitis acute, Q = M A D 5 gallbladder acute, pancreatitis acute, diarrhea,

hemorrhoid, large intestine cancer, and spleno- = {(ml, 0.00), (m2, 0.00), (m 3, 0.00), hepatomegalia.

(m4, 0.00), ( m 5, 0.20), (m6, 0.00)};

T = { ms} . References Because T ~ O, we get Y5 = F ( M , Ds, W 5) = 0.35. Since 0.35 > ,~, we obtain c 5 = 0.35 * 0.96 -- 0.34. [1] K.P. Adlassnig, Fuzzy Set Theory in Medical Diagnosis, From Table 2, we can see that the corresponding IEEE Trans. Syst. Man Cybern., 16, 2 (1986) 260-265.

Shyi-Ming Chen / Fuzzy reasoning for medical diagnosis 43

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[3] S.M. Chen, A New Approach to Handling Fuzzy Deci- 1 (1989) 2-10. sionmaking Problems, IEEE Trans. Syst. Man Cybern., [10] C.V. Negoita, Expert Systems and Fuzzy Systems, Ben- 18, 6 (1988) 1012-1016. jamin/Cummings, California, 1985.

[4] S.M. Chen, J.S. Ke, and J.F. Chang, An Efficient Algo- [11] E. Sanchez, Medical Application with Fuzzy Sets, in A. rithm to Handle Medical Diagnostic Problems, Cybernet- Jones, A. Kaufmann, and H.-J. Zimmermann (eds.) Fuzzy ics and Systems: An International Journal, 21, 4 (1990) Sets Theory and Applications, D. Reidel, Holland, 1986, 377-387. 331-347.

[5] S.M. Chen, J.S. Ke, and J.F. Chang, Knowledge Repre- [12] E. Sanchez, Medical Diagnosis and Composite Fuzzy sentation Using Fuzzy Petri Nets, IEEE Trans. on Relations, in M.M. Gupta, R.K. Ragade, and R.R. Yager Knowledge and Data Engineering, 2, 3 (1990) 311-319. (eds.) Advances in Fuzzy Set Theory and Applications,

[6] S.M. Chen, J.S. Ke, and J.F. Chang, Techniques for North-Holland, 1979, 437-444. Handling Multicriteria Fuzzy Decision-Making Problems, [13] L.A. Zadeh, The Role of Fuzzy Logic in the Manage- Proc. of the 4th International Symposium on Computer ment of Uncertainty in Expert Systems, Fuzzy Sets and and Information Sciences (1989) 919-925. Systems, 11 (1983) 199-227.

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