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Fuzzy model of inexact reasoning in medicine
Computer Methodr and Programs in Biomedicine, 30 (1989) l-8 Elsevier
COMMET
01020
Section
I. Methodology
Fuzzy model of inexact reasoning in medicine Pave1 Pi5 and Radko
Mesiar
Slovak Technical University, Bratislava, Czechoslovakia
In this paper a fuzzy model of inexact reasoning in medicine is developed. Both diagnosis and choice of therapy are considered. In the context of diagnosis a model used as the mathematical basis of MYCIN is given in a fuzzy interpretation. Single combining functions are identified with basic fuzzy operations and principles. In choosing therapy, the approach adopted in MYCIN formalised in fuzzy set terminology. A modified model of therapy selection is developed based upon the certainty factors of possible significant organisms’ identities. Medical
decision
making;
Fuzzy set theory;
Inexact
reasoning
1. Introduction
tainty factors identities.
The medical decision making modelling in consulting systems consists of two main parts. The first is the determination of diagnosis, the second is the therapy recommendation. For the first one we recall a model used as the mathematical basis of the consulting program MYCIN [12] and we give its fuzzy interpretation. Following the principles developed for a general rule-based consulting system working with certainty factors [5], we present a general model of inexact reasoning. We identify single combining functions with some of basic fuzzy operations and principles. As far as the therapy part is concerned, we formalise MYCIN’s system of choosing the recommending therapy in the terminology of fuzzy set theory as well. In MYCIN’s therapy selection all the significant organism’s identities are equivalent, i.e. they are not distinguished on the basis of their certainty factors. We propose a modified model of therapy selection based upon the cer-
Correspondence: Radko Mesiar, Katedra Matematiky, Stavebna Fakulta, SVST, Radlinskeho 11, 81368 Bratislava, Czechoslovakia. 0169-2607/89/$03.50
0 1989 Elsevier Science Publishers
of possible
2. Fuzzy interpretation inexact reasoning
significant
organisms’
of Shortliffe’s
model
of
Shortliffe [11,12] works with the measure of belief MB, the measure of disbelief MD and the certainty factor CF = MB - MD. All these measures of evidential strength are supposed to be given for any single hypothesis h and any single piece of information e, i.e. MB(h,e), MD(h,e). and CF(h,e) are known. For a general hypothesis H and a general piece of information E these measures are computed from the known values by means of some rules of combination. The inexactness is hidden just in the approximation technique of the combination rules. We present them in what follows. For more details, especially on the limiting cases, see [12]. Note that as CF = MB - MD, there is no need to describe any combining rule for CF. I. Incrementally acquired evidence: (a) MB(H,E, and E2)=MB(H,E1)+ MB(H,E,)-MB(H,E,).MB(H,E,)
B.V. (Biomedical
Division)
2
(b) MD(H,E,
and E,) = MD(H,E,)
+
MD(H,E,)-MD(H,E,).MD(H,E,).
II. Conjunction of hypotheses: (a) MB(H,hH,,E) = min{MB(H,,E), MB(H,,E))
(b) MD(H,hH,,E)=max{MD(H,,E), MD(H,,E)}.
III. Disjunction of hypotheses: (a) MB( H,VH,, E) = max{ MB(H,,
TABLE
1
Abbreviation
Organism
MBMDCF
EC PM KP SF ES ss
Echerichia coli Proteus mirabilis Klebsiella pneumonine Streptococcus faecalis Enterobacter species Staphylococcus sapr.
0.8 0.5 0.5 0.2 0.1 0.15
0.1 0.2 0.35 0.05 0.05 0.2
CF+
0.7 0.3 0.15 0.15 0.05 -0.05
0.7 0.3 0.15 0.15 0.05 0.00
E),
MB(H,,E)l (b) MD(H,VH,,E)=min{MD(H,,E), MD(H,,E)}.
IV. Strength of evidence: (a) MB(H,S)=E(H,S).CF+(S,E)
shown in Table 1. For this example the universal space is U = {EC,
PM, KP, SF, ES, SS}.
(b) MD(H,S)=@H,S)+CF+(S,E). Here H, Hi are general hypotheses; E, E, and S are general pieces of evidence. In rule IV, the evidence S is not known with certainty, but only with certainty factor CF(S, E) based upon prior information E. z and z are the MB and MD for H, when S is known to be true, CF+ = max{O,CF}. Combining rules I-IV is equivalent to some basic fuzzy connective operations and principles. On the other hand, in several rule-based consulting programs the quantification technique is introduced by means of fuzzy set theory, see for instance [6,8]. All these facts lead us to give a fuzzy interpretation of Shortliffe’s model, see also [lo]. Let U be a non-empty system of some observed objects. We will call it a universal set. So U may be, for instance, a system of all possible elementary diagnoses in the current case. A fuzzy set A is identified with its membership function mA? mA: U+ [OJ]. For more details see for instance [4]. We will use the following notation of Zadeh: A =
c
m,(x)/x.
xsu
Here a fuzzy set A is described in a pointwise manner for each element x E U by the degree of membership mA( x) of x into A. An example is
Shortliffe in MYCIN’s therapy recommendation takes into account only non-negative certainty factors, i.e. he works with CF+. So our opinion about the current organism is hidden in the CF+ quantities and may be expressed by the fuzzy set A,
A =
c
CF,+,,/ORG
ORGEU
= 0.7/EC + 0.3/PM +O.l5/SF
+ O.l5/KP
+ O.OS/ES + O.OO/SS.
Note that here ‘ + ’ is not the ordinary addition, but Zadeh’s fuzzy notation. The fuzzy set A which has been introduced may be used as a basis for therapy selection. Atanasov and Stoeva [l] introduced the concept of the intuitive fuzzy set A*,
A* =
c (m,*bb,*(x))/x,
XECJ
where m,*(x) E [O,l] defines the degree of membership and u,,(x) E [O,l] defines the degree of non-membership of a given element x E U into A *. Let us return to the previous example. Our knowledge about the identity of the current
3
organism set A*.
may be expressed
by the intuitive
fuzzy
Similarly H*/E
c
A* =
(MB,,, , MD,,, )/ORG
ORGEU = (0.8 ; O.l)/EC
+ (0.5 ; 0.2)/PM
+ (0.5 ; 0.35)/KP
+ (0.2; O.O5)/SF
+ (0.1; O.OS)/ES + (0.15 ; 0.2)/SS. This intuitive fuzzy set A* includes our actual knowledge about the identity of the current organism. Throughout MYCIN’s diagnostic determination A* develops according to the incrementally acquired knowledge and to the application of the individual rules of MYCIN. The modelling presented is confirmed by the subjective logic ideas (see e.g. [12]) and by the fact that in general we have MD( H, E) = MB( - H, E ), where - H is the negation of a hypothesis H (see the complement operator COMP* in what follows). Let .%’ be a set of all possible hypotheses in Shortliffe’s model and Let d be a set of all possible pieces of evidence. We can suppose 6’~ 3. For a piece of evidence E E E denote by MBE the fuzzy set MBE =
c
MB( H,E)/H,
we denote
MDE =
c
MD(H,E)/E
and
H E2
EE8
q&E) uH*(E)
Here A, B and C are fuzzy sets. INT corresponds to some triangular norm, UN to some triangular conorm and COMP to some decreasing automorphism of the unit interval [O,l]. For more details see for instance [4]. We give some examples:
k>
=MB(H,E) = MD(H,E).
UN,(a,b)
= max{ u,b}
INT,(u,b)
= min{ u,b}
-1
=u.b
UN,(u,b)=h-‘(h(u)+h(b)) denote
(MB(H,E),MD(H,E))/E,
H* = C
fuzzy union: AUB = C iff m&x) = UN(m,(x),m,(x)), x E U; fuzzy intersection: AnB = C iff m&x) = INT(m,(x),m,(x)), x E U; fuzzy complement: A’ = C iff mJx) = COMP( mA( x)), x E U.
INT,(a,b)
c CF+(S,E)/S. SE8
For a hypothesis tive fuzzy set
In the last case the evidence S is not known to be true but we know only the certainty factor CF( S, E) based upon prior evidence E. In the crisp case, we have for the sets some basic operations: the union, the intersection and the complement. They are defined in a pointwise manner by means of corresponding connectives. A similar situation applies in the case of the fuzzy sets. Let us denote single fuzzy connectives by UN, INT and COMP. Then:
UN,(u,b)=u+b-u-b
HEX
CFE=
H,S))/S.
COMP,(a)=(l-a)/(l+ka),
= MB(H,E).
Similarly
= C ( MB(H,S),MD( SE8
COMP(a)=l-a
i.e.
HE310 y,m(H)
we denote
and
by H * the intui-
i.e.
INT,hb) =f’tfb)
+ftb))-
Here a, b are the reals of the unit interval, h is an increasing bijection from [O,l[ to the [O,co[, h-’ is its inverse, f is a decreasing bijection from ]O,l] to [O,co[ and f ’ its inverse. For more detail see for instance [2]. Note that if f(u) = h(COMP(u)), then UN, and INT, satisfy the De Morgan rules;
4
they are commutative and associative. Certainly the same properties are satified by UN, and INT, and by UN, and INT,. It is possible to define also some non-associative fuzzy connectives, see for instance [lo]. These last may perhaps be useful in the modelling of decision making in non-associative domains. In the case of intuitive fuzzy sets we are in a similar situation. Here the basic operations are defined in a pointwise manner by means of corresponding connectives UN *, INT * and COMP *: UN*((a,b),(c,d))
= (UN(a,c),INT(b,d));
INT*((a,b),(c,d))
= (INT(a,c),UN(b,d));
COMP*(a,b)
= @,a).
Here a, b, c, d are the reals of the unit interval, and UN and INT are the basic fuzzy connective described above of the same type (i.e. with the same index). Note that if A* is a Zadeh fuzzy set, then u,, = COMP(m,,) = mA+. The Shortliffe rules of combination can be presented in the following form: I. (4 MB( E, and E2) = MBE, UMBE,, i.e. we (2) use the UN, connective; (b) MD( E, and E2) = MDE, UMDE,; (2) = H,*nH,*, i.e. we (1) use the INT,* connective (INT, = min for the membership degree MB, UN, = max for the non-membership degree MD); =H,*UH,*, i.e. we III. (a) and (b) (H,VH,)* (1) use the UN: connective. Rule IV corresponds to the modus ponens principle. If we know with certainty b that the membership degree is a, then as a real membership degree we take c = MPP(a, b) (here MPP is the modus ponens principle operator). For more details see for instance [7]. Some examples of MPP: II.
(a) and
(b) (HiAH,)*
MPP,(u,b)=max{O,u+b-1) MPP,(u,b)
= a. b
MPP,(u,b)
= min{ u,b}.
In the case of the intuitive fuzzy sets we are in a similar situation. If we know with certainty c that the degrees of membership and non-membership are (a, b), then as real membership and non-membership degrees we take (u, o) = MPP *((a, b), c) = (MPP(u,c),MPP(b,c)). Now, we are able to give a fuzzy interpretation of rule IV. IV. (a) and (b) H*/E = MPP,*(H*,CFE), i.e. MB(H,S) and
= MPP$%(H,S),CF+(S,E))
MD(H,S)
= MPP@(H,S),CF+(S,E)).
3. General fuzzy model of inexact reasoning The work of a rule-based consulting system consists roughly of the propagation of uncertain knowledge throughout the net of rules according to some combining functions. Hajek [5] introduced the next four combining functions: NEG (the weight of a negation of a proposition); CONJ (the weight of a conjuction of some propositions); GLOB (global weight from the particular contributions); CTR (the contribution yielded by a rule where the antecedent is not known to be true). Due to the De Morgan rules we may add the fifth combining function: DISJ (the weight of a disjunction of some propositions). The combining functions introduced correspond to the next basic fuzzy (and logical) connectives or principles: COMP negation NEG INT CONJ conjunction UN disjunction DISJ disjunction GLOB UN MPP CTR modus ponens Here the INT and UN of CONJ and DISJ are of the same type and together with COMP they satisfy the De Morgan rules. We propose a general model based on the modification of the fuzzy connectives used in the Shortliffe model. It seems to be natural to use INT, (INT:) and UN, (UN:) in the case of CONJ and DISJ. It is for instance the method of MYCIN, EMYCIN, PROSPECTOR, etc. Our main interest is focused on the GLOB and CTR combining functions. For the GLOB function, i.e. the incrementally acquired evidence, it is natural to preserve the strong monotonicity (the global weight growth after each posi-
5
tive contribution). Then the corresponding UN needs to be of type 3, i.e. we use UN, (see for instance [2,4]). Note that UN, is a special case of UN,, where h(x) = - ln(1 - x). The same results were obtained by Hajek [5] without the tools of fuzzy set theory. For the MPP of the CTR function we have the next limits (see for instance [lo]): the correMPP, I MPP I MPP,. In MYCIN sponding MPP is MPP,. In AL/X [9] it is MPP,. We propose to use our model with convenient fuzzy connectives as a mathematical basis of a consulting system for diagnosis. The adequacy of the single fuzzy connectives used in concrete domain of determining diagnosis is beyond the scope of this paper and needs to be justified separately for every special case.
Here COT is co-trimoxazole, AMOX is amoxicillin, NF is nitrofurantoin and SF is sulfadimidine. We can now suppose that certainty factors of possible identities of offending organisms are known. Let E be total information obtained before choosing the therapy. We express all possible identities of the organism i (with their CF) as a fuzzy set CE;E, CTE
=
c
Ce+(ORG,
E)/ORG.
ORG
We deal with all confirmed hypotheses about the identity of organism i (see for instance [12], p. 188). In our example we take into account only one organism and we have: A = CF,E = 0.7/EC + 0.3/PM
4. Therapy selection
+O.l5/SF
MYCIN selects drugs on the basis of the identity of significant organisms only. Shortliffe’s model of inexact reasoning is used to decide, for each current organism deemed to be significant, which hypotheses regarding the organism’s identity are sufficiently likely so that they must be taken into account in choosing a therapy. In the following text, we describe MYCIN’s therapy selection (for more details see [12]) using the results of fuzzy set theory which permits its modification. As an illustration the following example is used.
+ O.l5/KP
+ O.OS/ES + O.OO/SS.
Therapy will be chosen to cover all significant identities (on the basis of their CFs). Denote all such significant identities (with their CF) of the organism i as a fuzzy set SC4 E. Then clearly SCTE C C&E (note that if A, B are two fuzzy sets, then A c B iff mA I mg). We use the usual notation for fuzzy sets for support and cardinality (see for instance [4]): F-card A =
c m,(x) x=lJ
4.1. Example We have examined an acute case of cystitis. Using a model of inexact reasoning, we have found out information about the possible identities of offending organisms described in the example given in Table 1. The sensitivity list, expressed in percent, is as follows: ORGANISM
EC PM KP SF ES ss
DRUG COT
AMOX
NF
SF
78 90 84 00 71 92
38 78 04 88 62 a9
86 28 72 95 91 95
23 47 04 00 24 49
Here C is the usual summation symbol. The fuzzy set SCFjE is uniquely determined by the following three conditions: F-card SC4 E > 0.9 . F-card CE;]E card supp SC4 E = min F-card SCE;; E = max. This is accomplished by ordering the CF+s from the highest to the lowest and selecting all those on the list until their sum exceeds firstly 0.9. F-card CE]E. All possible identities of the organisms that seem to be significant are expressed as a fuzzy set CFE, CFE = USCqE. Here the union is the
6
classical maximum one, i.e. we use UN,. Return our example. Then:
to
F-card CF, E = 0.7 + 0.3 + 0.15 + 0.15 + 0.05 + 0 = 1.35 supp CF,E = {EC, PM, KP, SF, ES}, card supp CF,E = 5. We obtain SCFtE = 0.7/EC + 0.3/PM + O.l5/KP + O.l5/SF, as F-card SCF,E = 1.30 > 1.215 = 0.9. 1.35 = 0.9. F-card CF, E, card supp SCF, E = card {EC, PM, KP, SF} = 4 = min, F-card SCF;E = 1.3 = max (with regard to 4 elements obtained in the support). CFE = SCF,E.
The therapy lists Se (DRUG)
lists D(ORG)
induce
the sensitivity
= C Se (DRuG/ORG)/ORG. SI T be given.
Let a therapy is
Then
its sensitivity
Se(T) Se(T)
=
U
Se(DRUG),
DRUGE
T
where the fuzzy union is the UN,, i.e. maximum. ‘maximal therapy sensitivity’ The so-called Se( MT) is given by the maximum sensitivities for each identity ORG E SI, i.e. it is equivalent to the sensitivity of a therapy consisting of all drugs being taken into account. In our case we have Se( MT)
= 0.86/EC
+ 0.90/PM
+ 0.84/KP
Up until now MYCIN omits the acutal values of CFs in choosing the recommended therapy. It works on the ordinary set of significant identities SI only, SI = supp CFE. However, let us suppose that it is more convenient to work with the essential information hidden in CFs. From now on, we shall deal with the identities of SI only. In our case it is SI = {EC, PM, KP, SF}. We have a list of therapy indications (sensitivities) D(ORG) for each possible identity ORG E SI,
Similarly, the recommended therapy sensitivity Se( RT) is given by the maximum sensitivities of the drugs of the recommended therapy RT, i.e.
D(oRG)
Let a therapy consisting of co-trimoxazole amoxicillin be given, T = {COT, AMOX}. its sensitivity Se(T) is
=
C
S~(DRUG/ORG)/DRUG.
DRUG
Each drug is associated here with its sensitivity (for more details see [12]). We have
+0.95/SF.
Se(RT)
=
U
Se(DRUG).
DRUGERT
Se(T)
= 0.78/EC
+ 0.90/PM
and Then
+ 0.84/KP
+0.88/SF. D(EC)
= 0.78/COT
+ 0.38/AMOX
+ 0.86/NF
+0.78/AMOX
+ 0.28/NF
+ O.O4/AMOX
+ 0.72/NF
0.88/AMOX
+ 0.95/NF
+0.23/SF D(PM)
=0.90/COT + 0.47/SF
D(KP)
= 0.84/COT
We define a strong similarity relation in the space U of all possible therapies (U = 2MT) as a fuzzy set R on the Cartesian product U x U. Let T,, T2 be two therapies against the identities of SZ. Then we define
m,(T,,T,) =l-
“,“;‘{
]Se(T,/ORG)-Se(T,/ORG)]}.
+ O.O4/SF D(SF)
=
Here Se(T,/ORG) = m,,(?,(ORG) ity of a therapy T for the identity
is the sensitivORG.
The two therapies Ti and T2 are considered to be almost identical if m,(T,, T2) 2 0.95. In our example we have m,(T,MT) = 1 - max{ (0.78 - 0.86 (, IO.90 0.90 ], IO.84 - 0.84 1, IO.88 - 0.95 I} = 1 - 0.08 = 0.92, i.e. our T and MT are not almost identical. MYCIN’s choice of recommended therapy RT is based on the following principles: (1) RT is almost identical to MT; (2) card RT is minimal; (3) relative efficiency of RT is maximal; (4) the contra-indications of RT are minimal. For more detail see [12]. In our example the first two principles are satisfied only by RT = { COT,NF}. We propose a modification of the first principle. We propose to use another strong similarity relation on U, namely S, to explore the CFs of possible identities ORG of SI,
RT, = { COT,NF}. For general use it is possible to use another strong similarity relation on U to get an appropriate criterion for the first principle.
5. Conclusion We have examined the essential principles of the well-known and successful consulting program MYCIN in terms of fuzzy set theory. Such a formalisation covers all known rule-based knowledge systems. Moreover, it allows the creation of a variety of models of decision making which can serve as a mathematical basis for the formulation of more efficient consulting systems. We have presented some fuzzy connectives allowing the modifications of the model of inexact reasoning in diagnostic determination, as well as a strong similarity relation S modifying the recommended therapy choice. The adequacy of the use of single fuzzy connectives and relations in concrete medical domains is beyond the scope of this paper and needs to be justified for every event separately.
= 1 -X(&(T,)AS~(T,))/X(S~(MT)), where A is a measure the space SI, A(Se(T))
=
c
defined
on the fuzzy sets of
Se(T/ORG)
. CF(ORG,E).
ORG E SI
We use A as a symbol for strong symmetric difference (see for instance [4]), mAIB = 1mA -,mB I. Now, the two therapies Tl and T, are considered to be almost identical if m,(T,,T,) 2 0.95. Then our modified first principle of choosing the recommended therapy RT is equivalent to the condition h(Se(RT))
>0.95.h(Se(MT)).
In our example we have: X(MT)=0.86~0.7+0.90~0.3+0.84~0.15+0.95 .0.15 = 1.1405 X(COT) = 0.942; X(NF) = 0.9365; x(COT,AMOX) = 1.074; h(AMOX,NF) = 1.0865; x(COT,NF) = 1.1405. As far as 0.95 . A( MT) = 1.0835, now the first two principles are satisfied by RT, = {AMOX,NF} and
References fuzzy sets, in: HI K. Atanasov and S. Stoeva, Intuitionistic Proceedings Polish Symposium Interval and Fuzzy Mathematics, pp. 23-26 (Poznaii, 1983). 121J. Dombi, A general class of fuzzy operations, the De Morgan class of fuzzy operators and fuzziness measures induced by fuzzy operators, Fuzzy Sets Syst. (1982) 1499163. ]31 D. Dubois and H. Prade, Fuzzy Sets and Systems, Theory and Applications (Academic Press, New York, 1980). connec141 D. Dubois and H. Prade, Fuzzy set aggregation tives, Inf. Sci. 36 (1985) 85-121. functions for certainty degrees in 151 P. Hajek, Combining consulting systems, Int. J. Man-Machine Stud. 22 (1985) 59-16. and I. Stanev, Fuzzy rule-based systems, 161 K. Kratchanov fuzzy algorithms, fuzzy reasoning, in: Abstracts of 1st Joint IFSA-EC and EURO-MG Workshop on Progress in Fuzzy Sets in Europe, p. 40 (Warsaw, 1986). [in ]71 V. Novak, Fuzzy Sets Theory and its Applications Czech] (SNTL, Prague, 1986). S. Zadroiny and J. Kacprzyk, A fuzzy ]81 J.W. Owsihski, mathematical programming model for the analysis of resource use and needs in agriculture, in: Abstracts of 1st Joint IFSA-EC and EURO-MG Workshop on Progress in Fuzzy Sets in Europe, p. 59 (Warsaw, 1986).
8 [9] A. Paterson, AL/X User Manual (Intelligent Terminals, 1981). [lo] P. PIS and R. Mesiar, Fuzzy model of inexact reasoning in medical consulting systems, in: Proceedings 1st Joint IFSA-EC and EURO-MG Workshop on Progress in Fuzzy Sets in Europe (Warsaw, 1986).
[ll]
E.H. Shortliffe and B.G. Buchanan, A model of inexact reasoning in medicine, Math. Biosci. 23 (1975) 351-377. [12] E.H. Shortliffe, Computer Based Medical Consultation MYCIN (Elsevier, New York, 1976).
COMMET
01020
Section
I. Methodology
Fuzzy model of inexact reasoning in medicine Pave1 Pi5 and Radko
Mesiar
Slovak Technical University, Bratislava, Czechoslovakia
In this paper a fuzzy model of inexact reasoning in medicine is developed. Both diagnosis and choice of therapy are considered. In the context of diagnosis a model used as the mathematical basis of MYCIN is given in a fuzzy interpretation. Single combining functions are identified with basic fuzzy operations and principles. In choosing therapy, the approach adopted in MYCIN formalised in fuzzy set terminology. A modified model of therapy selection is developed based upon the certainty factors of possible significant organisms’ identities. Medical
decision
making;
Fuzzy set theory;
Inexact
reasoning
1. Introduction
tainty factors identities.
The medical decision making modelling in consulting systems consists of two main parts. The first is the determination of diagnosis, the second is the therapy recommendation. For the first one we recall a model used as the mathematical basis of the consulting program MYCIN [12] and we give its fuzzy interpretation. Following the principles developed for a general rule-based consulting system working with certainty factors [5], we present a general model of inexact reasoning. We identify single combining functions with some of basic fuzzy operations and principles. As far as the therapy part is concerned, we formalise MYCIN’s system of choosing the recommending therapy in the terminology of fuzzy set theory as well. In MYCIN’s therapy selection all the significant organism’s identities are equivalent, i.e. they are not distinguished on the basis of their certainty factors. We propose a modified model of therapy selection based upon the cer-
Correspondence: Radko Mesiar, Katedra Matematiky, Stavebna Fakulta, SVST, Radlinskeho 11, 81368 Bratislava, Czechoslovakia. 0169-2607/89/$03.50
0 1989 Elsevier Science Publishers
of possible
2. Fuzzy interpretation inexact reasoning
significant
organisms’
of Shortliffe’s
model
of
Shortliffe [11,12] works with the measure of belief MB, the measure of disbelief MD and the certainty factor CF = MB - MD. All these measures of evidential strength are supposed to be given for any single hypothesis h and any single piece of information e, i.e. MB(h,e), MD(h,e). and CF(h,e) are known. For a general hypothesis H and a general piece of information E these measures are computed from the known values by means of some rules of combination. The inexactness is hidden just in the approximation technique of the combination rules. We present them in what follows. For more details, especially on the limiting cases, see [12]. Note that as CF = MB - MD, there is no need to describe any combining rule for CF. I. Incrementally acquired evidence: (a) MB(H,E, and E2)=MB(H,E1)+ MB(H,E,)-MB(H,E,).MB(H,E,)
B.V. (Biomedical
Division)
2
(b) MD(H,E,
and E,) = MD(H,E,)
+
MD(H,E,)-MD(H,E,).MD(H,E,).
II. Conjunction of hypotheses: (a) MB(H,hH,,E) = min{MB(H,,E), MB(H,,E))
(b) MD(H,hH,,E)=max{MD(H,,E), MD(H,,E)}.
III. Disjunction of hypotheses: (a) MB( H,VH,, E) = max{ MB(H,,
TABLE
1
Abbreviation
Organism
MBMDCF
EC PM KP SF ES ss
Echerichia coli Proteus mirabilis Klebsiella pneumonine Streptococcus faecalis Enterobacter species Staphylococcus sapr.
0.8 0.5 0.5 0.2 0.1 0.15
0.1 0.2 0.35 0.05 0.05 0.2
CF+
0.7 0.3 0.15 0.15 0.05 -0.05
0.7 0.3 0.15 0.15 0.05 0.00
E),
MB(H,,E)l (b) MD(H,VH,,E)=min{MD(H,,E), MD(H,,E)}.
IV. Strength of evidence: (a) MB(H,S)=E(H,S).CF+(S,E)
shown in Table 1. For this example the universal space is U = {EC,
PM, KP, SF, ES, SS}.
(b) MD(H,S)=@H,S)+CF+(S,E). Here H, Hi are general hypotheses; E, E, and S are general pieces of evidence. In rule IV, the evidence S is not known with certainty, but only with certainty factor CF(S, E) based upon prior information E. z and z are the MB and MD for H, when S is known to be true, CF+ = max{O,CF}. Combining rules I-IV is equivalent to some basic fuzzy connective operations and principles. On the other hand, in several rule-based consulting programs the quantification technique is introduced by means of fuzzy set theory, see for instance [6,8]. All these facts lead us to give a fuzzy interpretation of Shortliffe’s model, see also [lo]. Let U be a non-empty system of some observed objects. We will call it a universal set. So U may be, for instance, a system of all possible elementary diagnoses in the current case. A fuzzy set A is identified with its membership function mA? mA: U+ [OJ]. For more details see for instance [4]. We will use the following notation of Zadeh: A =
c
m,(x)/x.
xsu
Here a fuzzy set A is described in a pointwise manner for each element x E U by the degree of membership mA( x) of x into A. An example is
Shortliffe in MYCIN’s therapy recommendation takes into account only non-negative certainty factors, i.e. he works with CF+. So our opinion about the current organism is hidden in the CF+ quantities and may be expressed by the fuzzy set A,
A =
c
CF,+,,/ORG
ORGEU
= 0.7/EC + 0.3/PM +O.l5/SF
+ O.l5/KP
+ O.OS/ES + O.OO/SS.
Note that here ‘ + ’ is not the ordinary addition, but Zadeh’s fuzzy notation. The fuzzy set A which has been introduced may be used as a basis for therapy selection. Atanasov and Stoeva [l] introduced the concept of the intuitive fuzzy set A*,
A* =
c (m,*bb,*(x))/x,
XECJ
where m,*(x) E [O,l] defines the degree of membership and u,,(x) E [O,l] defines the degree of non-membership of a given element x E U into A *. Let us return to the previous example. Our knowledge about the identity of the current
3
organism set A*.
may be expressed
by the intuitive
fuzzy
Similarly H*/E
c
A* =
(MB,,, , MD,,, )/ORG
ORGEU = (0.8 ; O.l)/EC
+ (0.5 ; 0.2)/PM
+ (0.5 ; 0.35)/KP
+ (0.2; O.O5)/SF
+ (0.1; O.OS)/ES + (0.15 ; 0.2)/SS. This intuitive fuzzy set A* includes our actual knowledge about the identity of the current organism. Throughout MYCIN’s diagnostic determination A* develops according to the incrementally acquired knowledge and to the application of the individual rules of MYCIN. The modelling presented is confirmed by the subjective logic ideas (see e.g. [12]) and by the fact that in general we have MD( H, E) = MB( - H, E ), where - H is the negation of a hypothesis H (see the complement operator COMP* in what follows). Let .%’ be a set of all possible hypotheses in Shortliffe’s model and Let d be a set of all possible pieces of evidence. We can suppose 6’~ 3. For a piece of evidence E E E denote by MBE the fuzzy set MBE =
c
MB( H,E)/H,
we denote
MDE =
c
MD(H,E)/E
and
H E2
EE8
q&E) uH*(E)
Here A, B and C are fuzzy sets. INT corresponds to some triangular norm, UN to some triangular conorm and COMP to some decreasing automorphism of the unit interval [O,l]. For more details see for instance [4]. We give some examples:
k>
=MB(H,E) = MD(H,E).
UN,(a,b)
= max{ u,b}
INT,(u,b)
= min{ u,b}
-1
=u.b
UN,(u,b)=h-‘(h(u)+h(b)) denote
(MB(H,E),MD(H,E))/E,
H* = C
fuzzy union: AUB = C iff m&x) = UN(m,(x),m,(x)), x E U; fuzzy intersection: AnB = C iff m&x) = INT(m,(x),m,(x)), x E U; fuzzy complement: A’ = C iff mJx) = COMP( mA( x)), x E U.
INT,(a,b)
c CF+(S,E)/S. SE8
For a hypothesis tive fuzzy set
In the last case the evidence S is not known to be true but we know only the certainty factor CF( S, E) based upon prior evidence E. In the crisp case, we have for the sets some basic operations: the union, the intersection and the complement. They are defined in a pointwise manner by means of corresponding connectives. A similar situation applies in the case of the fuzzy sets. Let us denote single fuzzy connectives by UN, INT and COMP. Then:
UN,(u,b)=u+b-u-b
HEX
CFE=
H,S))/S.
COMP,(a)=(l-a)/(l+ka),
= MB(H,E).
Similarly
= C ( MB(H,S),MD( SE8
COMP(a)=l-a
i.e.
HE310 y,m(H)
we denote
and
by H * the intui-
i.e.
INT,hb) =f’tfb)
+ftb))-
Here a, b are the reals of the unit interval, h is an increasing bijection from [O,l[ to the [O,co[, h-’ is its inverse, f is a decreasing bijection from ]O,l] to [O,co[ and f ’ its inverse. For more detail see for instance [2]. Note that if f(u) = h(COMP(u)), then UN, and INT, satisfy the De Morgan rules;
4
they are commutative and associative. Certainly the same properties are satified by UN, and INT, and by UN, and INT,. It is possible to define also some non-associative fuzzy connectives, see for instance [lo]. These last may perhaps be useful in the modelling of decision making in non-associative domains. In the case of intuitive fuzzy sets we are in a similar situation. Here the basic operations are defined in a pointwise manner by means of corresponding connectives UN *, INT * and COMP *: UN*((a,b),(c,d))
= (UN(a,c),INT(b,d));
INT*((a,b),(c,d))
= (INT(a,c),UN(b,d));
COMP*(a,b)
= @,a).
Here a, b, c, d are the reals of the unit interval, and UN and INT are the basic fuzzy connective described above of the same type (i.e. with the same index). Note that if A* is a Zadeh fuzzy set, then u,, = COMP(m,,) = mA+. The Shortliffe rules of combination can be presented in the following form: I. (4 MB( E, and E2) = MBE, UMBE,, i.e. we (2) use the UN, connective; (b) MD( E, and E2) = MDE, UMDE,; (2) = H,*nH,*, i.e. we (1) use the INT,* connective (INT, = min for the membership degree MB, UN, = max for the non-membership degree MD); =H,*UH,*, i.e. we III. (a) and (b) (H,VH,)* (1) use the UN: connective. Rule IV corresponds to the modus ponens principle. If we know with certainty b that the membership degree is a, then as a real membership degree we take c = MPP(a, b) (here MPP is the modus ponens principle operator). For more details see for instance [7]. Some examples of MPP: II.
(a) and
(b) (HiAH,)*
MPP,(u,b)=max{O,u+b-1) MPP,(u,b)
= a. b
MPP,(u,b)
= min{ u,b}.
In the case of the intuitive fuzzy sets we are in a similar situation. If we know with certainty c that the degrees of membership and non-membership are (a, b), then as real membership and non-membership degrees we take (u, o) = MPP *((a, b), c) = (MPP(u,c),MPP(b,c)). Now, we are able to give a fuzzy interpretation of rule IV. IV. (a) and (b) H*/E = MPP,*(H*,CFE), i.e. MB(H,S) and
= MPP$%(H,S),CF+(S,E))
MD(H,S)
= MPP@(H,S),CF+(S,E)).
3. General fuzzy model of inexact reasoning The work of a rule-based consulting system consists roughly of the propagation of uncertain knowledge throughout the net of rules according to some combining functions. Hajek [5] introduced the next four combining functions: NEG (the weight of a negation of a proposition); CONJ (the weight of a conjuction of some propositions); GLOB (global weight from the particular contributions); CTR (the contribution yielded by a rule where the antecedent is not known to be true). Due to the De Morgan rules we may add the fifth combining function: DISJ (the weight of a disjunction of some propositions). The combining functions introduced correspond to the next basic fuzzy (and logical) connectives or principles: COMP negation NEG INT CONJ conjunction UN disjunction DISJ disjunction GLOB UN MPP CTR modus ponens Here the INT and UN of CONJ and DISJ are of the same type and together with COMP they satisfy the De Morgan rules. We propose a general model based on the modification of the fuzzy connectives used in the Shortliffe model. It seems to be natural to use INT, (INT:) and UN, (UN:) in the case of CONJ and DISJ. It is for instance the method of MYCIN, EMYCIN, PROSPECTOR, etc. Our main interest is focused on the GLOB and CTR combining functions. For the GLOB function, i.e. the incrementally acquired evidence, it is natural to preserve the strong monotonicity (the global weight growth after each posi-
5
tive contribution). Then the corresponding UN needs to be of type 3, i.e. we use UN, (see for instance [2,4]). Note that UN, is a special case of UN,, where h(x) = - ln(1 - x). The same results were obtained by Hajek [5] without the tools of fuzzy set theory. For the MPP of the CTR function we have the next limits (see for instance [lo]): the correMPP, I MPP I MPP,. In MYCIN sponding MPP is MPP,. In AL/X [9] it is MPP,. We propose to use our model with convenient fuzzy connectives as a mathematical basis of a consulting system for diagnosis. The adequacy of the single fuzzy connectives used in concrete domain of determining diagnosis is beyond the scope of this paper and needs to be justified separately for every special case.
Here COT is co-trimoxazole, AMOX is amoxicillin, NF is nitrofurantoin and SF is sulfadimidine. We can now suppose that certainty factors of possible identities of offending organisms are known. Let E be total information obtained before choosing the therapy. We express all possible identities of the organism i (with their CF) as a fuzzy set CE;E, CTE
=
c
Ce+(ORG,
E)/ORG.
ORG
We deal with all confirmed hypotheses about the identity of organism i (see for instance [12], p. 188). In our example we take into account only one organism and we have: A = CF,E = 0.7/EC + 0.3/PM
4. Therapy selection
+O.l5/SF
MYCIN selects drugs on the basis of the identity of significant organisms only. Shortliffe’s model of inexact reasoning is used to decide, for each current organism deemed to be significant, which hypotheses regarding the organism’s identity are sufficiently likely so that they must be taken into account in choosing a therapy. In the following text, we describe MYCIN’s therapy selection (for more details see [12]) using the results of fuzzy set theory which permits its modification. As an illustration the following example is used.
+ O.l5/KP
+ O.OS/ES + O.OO/SS.
Therapy will be chosen to cover all significant identities (on the basis of their CFs). Denote all such significant identities (with their CF) of the organism i as a fuzzy set SC4 E. Then clearly SCTE C C&E (note that if A, B are two fuzzy sets, then A c B iff mA I mg). We use the usual notation for fuzzy sets for support and cardinality (see for instance [4]): F-card A =
c m,(x) x=lJ
4.1. Example We have examined an acute case of cystitis. Using a model of inexact reasoning, we have found out information about the possible identities of offending organisms described in the example given in Table 1. The sensitivity list, expressed in percent, is as follows: ORGANISM
EC PM KP SF ES ss
DRUG COT
AMOX
NF
SF
78 90 84 00 71 92
38 78 04 88 62 a9
86 28 72 95 91 95
23 47 04 00 24 49
Here C is the usual summation symbol. The fuzzy set SCFjE is uniquely determined by the following three conditions: F-card SC4 E > 0.9 . F-card CE;]E card supp SC4 E = min F-card SCE;; E = max. This is accomplished by ordering the CF+s from the highest to the lowest and selecting all those on the list until their sum exceeds firstly 0.9. F-card CE]E. All possible identities of the organisms that seem to be significant are expressed as a fuzzy set CFE, CFE = USCqE. Here the union is the
6
classical maximum one, i.e. we use UN,. Return our example. Then:
to
F-card CF, E = 0.7 + 0.3 + 0.15 + 0.15 + 0.05 + 0 = 1.35 supp CF,E = {EC, PM, KP, SF, ES}, card supp CF,E = 5. We obtain SCFtE = 0.7/EC + 0.3/PM + O.l5/KP + O.l5/SF, as F-card SCF,E = 1.30 > 1.215 = 0.9. 1.35 = 0.9. F-card CF, E, card supp SCF, E = card {EC, PM, KP, SF} = 4 = min, F-card SCF;E = 1.3 = max (with regard to 4 elements obtained in the support). CFE = SCF,E.
The therapy lists Se (DRUG)
lists D(ORG)
induce
the sensitivity
= C Se (DRuG/ORG)/ORG. SI T be given.
Let a therapy is
Then
its sensitivity
Se(T) Se(T)
=
U
Se(DRUG),
DRUGE
T
where the fuzzy union is the UN,, i.e. maximum. ‘maximal therapy sensitivity’ The so-called Se( MT) is given by the maximum sensitivities for each identity ORG E SI, i.e. it is equivalent to the sensitivity of a therapy consisting of all drugs being taken into account. In our case we have Se( MT)
= 0.86/EC
+ 0.90/PM
+ 0.84/KP
Up until now MYCIN omits the acutal values of CFs in choosing the recommended therapy. It works on the ordinary set of significant identities SI only, SI = supp CFE. However, let us suppose that it is more convenient to work with the essential information hidden in CFs. From now on, we shall deal with the identities of SI only. In our case it is SI = {EC, PM, KP, SF}. We have a list of therapy indications (sensitivities) D(ORG) for each possible identity ORG E SI,
Similarly, the recommended therapy sensitivity Se( RT) is given by the maximum sensitivities of the drugs of the recommended therapy RT, i.e.
D(oRG)
Let a therapy consisting of co-trimoxazole amoxicillin be given, T = {COT, AMOX}. its sensitivity Se(T) is
=
C
S~(DRUG/ORG)/DRUG.
DRUG
Each drug is associated here with its sensitivity (for more details see [12]). We have
+0.95/SF.
Se(RT)
=
U
Se(DRUG).
DRUGERT
Se(T)
= 0.78/EC
+ 0.90/PM
and Then
+ 0.84/KP
+0.88/SF. D(EC)
= 0.78/COT
+ 0.38/AMOX
+ 0.86/NF
+0.78/AMOX
+ 0.28/NF
+ O.O4/AMOX
+ 0.72/NF
0.88/AMOX
+ 0.95/NF
+0.23/SF D(PM)
=0.90/COT + 0.47/SF
D(KP)
= 0.84/COT
We define a strong similarity relation in the space U of all possible therapies (U = 2MT) as a fuzzy set R on the Cartesian product U x U. Let T,, T2 be two therapies against the identities of SZ. Then we define
m,(T,,T,) =l-
“,“;‘{
]Se(T,/ORG)-Se(T,/ORG)]}.
+ O.O4/SF D(SF)
=
Here Se(T,/ORG) = m,,(?,(ORG) ity of a therapy T for the identity
is the sensitivORG.
The two therapies Ti and T2 are considered to be almost identical if m,(T,, T2) 2 0.95. In our example we have m,(T,MT) = 1 - max{ (0.78 - 0.86 (, IO.90 0.90 ], IO.84 - 0.84 1, IO.88 - 0.95 I} = 1 - 0.08 = 0.92, i.e. our T and MT are not almost identical. MYCIN’s choice of recommended therapy RT is based on the following principles: (1) RT is almost identical to MT; (2) card RT is minimal; (3) relative efficiency of RT is maximal; (4) the contra-indications of RT are minimal. For more detail see [12]. In our example the first two principles are satisfied only by RT = { COT,NF}. We propose a modification of the first principle. We propose to use another strong similarity relation on U, namely S, to explore the CFs of possible identities ORG of SI,
RT, = { COT,NF}. For general use it is possible to use another strong similarity relation on U to get an appropriate criterion for the first principle.
5. Conclusion We have examined the essential principles of the well-known and successful consulting program MYCIN in terms of fuzzy set theory. Such a formalisation covers all known rule-based knowledge systems. Moreover, it allows the creation of a variety of models of decision making which can serve as a mathematical basis for the formulation of more efficient consulting systems. We have presented some fuzzy connectives allowing the modifications of the model of inexact reasoning in diagnostic determination, as well as a strong similarity relation S modifying the recommended therapy choice. The adequacy of the use of single fuzzy connectives and relations in concrete medical domains is beyond the scope of this paper and needs to be justified for every event separately.
= 1 -X(&(T,)AS~(T,))/X(S~(MT)), where A is a measure the space SI, A(Se(T))
=
c
defined
on the fuzzy sets of
Se(T/ORG)
. CF(ORG,E).
ORG E SI
We use A as a symbol for strong symmetric difference (see for instance [4]), mAIB = 1mA -,mB I. Now, the two therapies Tl and T, are considered to be almost identical if m,(T,,T,) 2 0.95. Then our modified first principle of choosing the recommended therapy RT is equivalent to the condition h(Se(RT))
>0.95.h(Se(MT)).
In our example we have: X(MT)=0.86~0.7+0.90~0.3+0.84~0.15+0.95 .0.15 = 1.1405 X(COT) = 0.942; X(NF) = 0.9365; x(COT,AMOX) = 1.074; h(AMOX,NF) = 1.0865; x(COT,NF) = 1.1405. As far as 0.95 . A( MT) = 1.0835, now the first two principles are satisfied by RT, = {AMOX,NF} and
References fuzzy sets, in: HI K. Atanasov and S. Stoeva, Intuitionistic Proceedings Polish Symposium Interval and Fuzzy Mathematics, pp. 23-26 (Poznaii, 1983). 121J. Dombi, A general class of fuzzy operations, the De Morgan class of fuzzy operators and fuzziness measures induced by fuzzy operators, Fuzzy Sets Syst. (1982) 1499163. ]31 D. Dubois and H. Prade, Fuzzy Sets and Systems, Theory and Applications (Academic Press, New York, 1980). connec141 D. Dubois and H. Prade, Fuzzy set aggregation tives, Inf. Sci. 36 (1985) 85-121. functions for certainty degrees in 151 P. Hajek, Combining consulting systems, Int. J. Man-Machine Stud. 22 (1985) 59-16. and I. Stanev, Fuzzy rule-based systems, 161 K. Kratchanov fuzzy algorithms, fuzzy reasoning, in: Abstracts of 1st Joint IFSA-EC and EURO-MG Workshop on Progress in Fuzzy Sets in Europe, p. 40 (Warsaw, 1986). [in ]71 V. Novak, Fuzzy Sets Theory and its Applications Czech] (SNTL, Prague, 1986). S. Zadroiny and J. Kacprzyk, A fuzzy ]81 J.W. Owsihski, mathematical programming model for the analysis of resource use and needs in agriculture, in: Abstracts of 1st Joint IFSA-EC and EURO-MG Workshop on Progress in Fuzzy Sets in Europe, p. 59 (Warsaw, 1986).
8 [9] A. Paterson, AL/X User Manual (Intelligent Terminals, 1981). [lo] P. PIS and R. Mesiar, Fuzzy model of inexact reasoning in medical consulting systems, in: Proceedings 1st Joint IFSA-EC and EURO-MG Workshop on Progress in Fuzzy Sets in Europe (Warsaw, 1986).
[ll]
E.H. Shortliffe and B.G. Buchanan, A model of inexact reasoning in medicine, Math. Biosci. 23 (1975) 351-377. [12] E.H. Shortliffe, Computer Based Medical Consultation MYCIN (Elsevier, New York, 1976).