.........
-
" .....
1 2 3
Iney, NSW 2006, Australia
Paul E. BEAUMONT* The Eye and Vision Research Institute, Sydney, NSW 2001, Australia Received June 1979 Revised July 1980 This paper examines the possibility of handling treatment effects as a fuzzy relational problem with particular reference to ophthalmology. An ophthalmological evaluation is considered to consist Of two majOr Stages. ~Firstly, an e~lanafion a patient's eye-health state in terms of known disease labels is formulated through diagnosis. Secondly, the effectiveness of a theral~euticstrategy de~igr,ed to restore a patient's normal eye-health, is then predicted via prognosis. The paper cxmsiders the latter aspect in the sense of assessing the effects of medical intervention by comparing pre- and post-therapeutic fuzzy subsets.
Keywords: Ophthalmological vector, Ophthalmological serts, Therapeutic strategy, Alternative fuzzy subsets, Predictably good, Effective, Goodness of a fuzzy choice.
The problem of medical diagnosis has received considerable attention in both medical and technical literatu~s. Croft [4] has asked if computerized diagnosis was possible. Perhaps,j the question should have been asked concerning the relevance, rather than the possibility, of computerized diagnosis. The papers by KulikowsM [8] on the pattern recognition approach, Wortman [24] on the information-processing approach, Hockstra and Miller [6] on the sequential g ~ e s method, Wechsler [20] on the fuzzy set-theoretic approach, Bouckaert and Thiry [2] on the physiopathological inference method, and Woodbury et al. [23] on the membership grade technique indicate the possibility of computerizing the process of diagnosis. The problems associated with computerized diagnosis are similar to those associated with the evaluation of treatment. Among researchers concerned with evaluating the worth of a treatment are Ledley [9, 10], Scheines [17], Bleich [1], Honigfeld etal. [7], Chodotf and ~ e w s [3], Vallbona et alo [19], Menn et al. [14], ~ d @oodbury and [22]~.Ledtey, for example, discusses treatment evaluation in terms of transition probabilities, conditional upon the treatment used and the type of disease. He then suggests the use of dynamic programming for selecting a treatment, Mille Woodbury and Clive discuss the applicability of the 0165-0114[82/0000-0000/$02.75 © 1982 North-Holand
:
O:O. O g u n ~ P . E . : B ~
i:
: : ~:, :
:: : :
ideas sets and membership grades quantification of c l i n i ~ judgement. As will be of m e d i ~ prognmis with or without treatment suffer ~nsiderab!e defi~encies. a
1: ri
approach that lacks crude or unrealistic statistical aspects in the form of descriptions or assumptions. ~ e method should knowledge. I t should not assume that all measuremen~ for apmient are available and des~aSbable precisely. Also. it should not t~sume that the patient has exactly one out of a set of diseases. : ~ :.
An ophthalmological evaluation consists of comparing a patient's ophthalmological vector (a set of observations of clinical variables, i.e. all:information a v ~ a b l e to the ophthalmolo~t concerning the patient's condition: signs, symptoms, laboratory data, medical history, etc.) with ideal ophthalmological sorts (representations of established medical knowledge in ophthalmological practice). At each stage of evaluation, manifestations of the concept of fuzzy subsets become prominent. Fuzziness in an ophthalmological procedure woutd naturally emanate from the definitions of ideal ophthalmological sorts and in the stipulation of a patient's ophthalmological vector. Due to the vagaries of ophthalmological vector measurements coupled with the diversity of disease manifestation among different patients with similar or identical diseases, further descriptional problems are bound to arise. To complicate matters even further, there is the possibility that within a single vatient several diseases may be vresent, as a result of which a he overlappproach to ~ d Woodb ~ [21] ihave testifiedto the inappropriateness of Bayes' theorem mainly independence assumption, i us~ of statiSti~ been tO place a mathematicallybased value o n the si~ifi~ahce Or t ~ statement or p r o ~ s i fion, Medlc~l sei~n~ has always been and will continue to b e thwarted by the rge a human science s ~extricab,y linked with imprecision. Thus, it is often ~ o s s i b l e or imr~racticable t o seek the
Opht~n'~v~a£ Wo~osis via fuzzy subsets
125
readily accept a low probabifity of redictable and an inexorable loss of 1.~robabJli~~ of success despite a would be satisfied with the know,our. Odds such as 0.1, which could [ered as 'unacceptable', may in the eptable. If, however, his disease had a as remission and retention of vision then the odds m~ Thus, given a disease and a treatment, a realistic method of evaluating the treatment effect and a treatment, a realistic method of evaluating the treatment effect ought to provide a means of balancing the disease prognosis against the treatment morbidity and mortality in such a manner that the imprecision arising from the use of 'reasonable' and 'acceptable' is not purposely suppressed for the sake of mathematical tractability and convenience. The fear of imprecision which besets medical science often compels the medical scientist to shroud himself in a rigid and fashionable employment of statistical methods as a means of self-protection. The use of 'in vogue' statistic~ methods usually displaces an appraisal of these methods. A case in point is the double masked randomized clinical trial. It is considered the most powerful tool for determir~,ing a treatmeut effect. See, for example, Masden et al. [13]. its power, precision and position of respect would tend to suggest that other methods are barely worth considering. The design of the randomized trial is, bewcver, often associated with unpredictable factors such as the differential losses of patients to follow up, inadequate recruitment, the lack of treatment flexibility, the impossibility of masking patient bias, and so on. Such clinical trials are expensive, arduous, and time consuming. The personnel, finance, and time required in the examination of all treatment effects via the double masked clinical trial method are riot realistically available. The progress of medic~ science, consistent with maximum speed and economy, would therefore seem to rely on alternative approaches with respect to the assessment of the reasonability and acceptability of propositional statements concerning medical evaluation. For example, in the evaluation of the treatment effects on patients with haemorrhag;c disciform macular degeneration it might be advantageous to examine treatment and control groups with respect to the following imprecise stipulations: 'fairly h i ~ ' incidence of disease in a given location, 'very accurate' diagnostic categorization, 'favourable' geographical influence, 'good' seasonal influence, 'more or less predictable' genetic constitution of population, 'fairly reliable' visual assessment technique, 'considerably low' incidence of extraneous factors, etc. This is exactly the trend of thottghts pursued in this paper.
3° Comple~es of o p h ~ o l o ~ c ~
~ea~en~ e v ~ o n
We take particular cognizance of the fact that the personnel, finance, and t~-ae required in the examination of all treatment effects by using traditional methods such as the double masked cfinical trial could be hard to find. Consequent upon
<
:
L
•
:/
~
/
•
51!~
)ii!, ¸
i d
~:k
.
•
of: S:i:
vector refers to: thes i ~ s and ical vecto~ deoend n
these
~d such
vector: . . . . . . . . . . . . . . .
:
medie "knowonand so :forth. we: uid
:. . . . .
then we w o ~ d h a v e the following lingu~tie ~seriptions representing components of
:
& i d e a l . o p h @ ~ o , ~ g ~ c ~ ~orts with respect to dlsease k, -~pre-therapeutic ophthalmological vector for patient ] with respect to disease k for all i = l , 2 , . . . , m , ]=1,2,...,n, and k = 1,2....,p, A
......... ::
&~t~era~utic o~hth todisease~k ~ for ~1 i, fi ~
g i ~ vector for patient j with respect .~::
For any ideNs6~,: y~; let {w(,~,},' r = L 2 , 2 : . ,sz, d e n o t e : a c N l e ~ i 0 n o f a t N b u t e s
~
127
~ m ~ z z y subsets
(e.g., good, low, high, accurate, etc,) ~ a t are applicable to Yu, such that
For example, ff we set I = k = 1, and
:3)~-A average. Also, let us set l = 2, k = 3, and sz = 2 so that Yz3~- diagnostic categorization; w(uzs~accurate,
and
w(2~sA--precise,
and so on and so forth. Then we have that z(1)aa~ high disease incidence, zcs)~ ~ average disease hacidence, z(z)2s&predse diagnostic categorization, etc. Next, we shall let x(l) i~k.(,) =
&ik ] zo.)l~.
~-pre-therapeutic liguistic process V i, ], k, 1, r.
descriptions
of
the
evaluation
For example, for all ], k let x~k denote blood sugar level, x2ik denote visual acuity. Then set l = k = 1. qhus
X(1) 2i 1(3) =
X2i 1 [ Z(3)l 1
~ t h e visual acuity of patient ] conditional upon average disease incidence before (or without) treatment, X(2) lj1(2) _-- XI}I ] Z(2)23
~ t h e blood sugar level of patient ] conditional upon precise diagnostic categorization before treatment, and so on and so forth. Similarly define: ~k(r)
iik
as the post-therapeutic linguistic descriptions of the process Vi~ k, l, r. From the above considerations we find that
Z(r)li ~- w(r)~kQ Y~k
tions in turn would give rise to various impredse:statements about:a patient's ophthalmological conditions. Let X denote the space of the possible ophthalmolOgiCal ~nditions and Z the space of the possible contexts for prognosiS. For any ~ let F(A) denote the infinite set of the membership functions of A w h i l e P(A)denotes the infinite set of all hnprecise statements about the elements o f A, Then an appropfi~.te linguistic transformation, @ say, would be a fuzzy relation betwven X and Z. As is wet,' kno W n, d~ is a fuzzy subset of the Cartesian product, X x Z. More precisely d~aF(XxZ) and ~k:XxZ-->[0,1]. Moreover,
. . . .
~P(x×g), since IF(-) and P(.) are isomorphic [30]. In this way, we find that both forms of the ~kt,), can be considered to linguistic descriptions for prognosis, namely, x~k)~,)and ~(t) be elements of X x Z, for all i, ], k, l, and r. Now, let
Also, 'by considering another f ~ possibly), i,e., ~:XxZ~[0,1] let
A
^
relation, ~, between X and Z (~ = d~,
and ~ P ( X × Z ) , ~
~,
~
-
~(x~)) and 4~(x~,~), both in [0, 1], are subjectively determined for all i, L k, ~, and r.. One way of determirfing theseis by using .the following fuzzy restrictions :
....
"
~,~
~
^
- ~
(|)
,
Or
(~)
" "
129
ophthaladitioned attribute. visual acuity), n
"" 2
p=3 q=3
¸
(k = ocular photocoagulafion, macular degeneration, capillary retinopathy), (l = disease incidence, diagnostic categorization, population genetics)
such that the s~'s are given by s a i ~ i.~id~.~ = 2
(r = acceptable, high),
sai~g.o~ti. ~atcg,,.~tio. = 2
(r = u n r e a s o n a b l e , g o o d ) ,
and Spopulafion genetics ---~2
(r = stable, unreliable).
Then the following list shall be obtained: y~l = (disease) incidence of ) Y2~= (diagnostic) categorization for ~ ocular photocoagulation, Y31= (population) genetics with respect t o ) Y12 = incidence of ) Y22= categorization for ~ macular degeneration, Y32= genetics with respect to Y~3= incidence of ) Y23=categorization for ~ capillary retinopathy, Y33= genetics with respect to ) Let b.t. and a.t. denote before and after treatment, respectively x111 (~111)= JAN's sugar level b.t. (a.t.) ) xull (~2~) = JAN's visual acuity b.t. (a.t.)~ under ocular X|21 (X121) = J O N ' s sugar level b.t. (a.t.) | photocoagulation, x221 (~222~)=JON's visual acuity b.t. (a.t.) x2 (~12) = JAN's sugar level b.t. (a.t.) "] x2~2 02212)= JAN's visual acui~5, b.t. (a.t.) ! under macular x122 (J2122)= JON's sugar ievel b.t, (a.t.) degeneration, x222 (x222)= JON's visual acuity b.t. (a.t.) xa13 (J2tt3) = JAN's sugar level b.t. (a.t.) "~ x2~3 (322~3)=3AN's visual acuity b.t. (a.t.) I under capillary X123 (X123) = JON's sugar level ~.t. (a.t.) retinopathy, x22s (~z23)= .~ON's visual acuity b.t. ~a.t.)
•
Z(1)22 = unreasonablel Z(1)22 , " g ~
j
'~' ..... categon~ :i t m n f o r
[: i
• degeneratmn, ....
| Z0)32 = s t a b l e
....
Z(2)3 2 = u n r e l l a O l e
..
~
'
genetics with respect t o ]
j
~'
z(x)t3 = acceptable) z(2)~3 = gn j
:-
~. . . . . . ...,
ztx)23 = unreasonable ~2~...~,,_ c^~ .... i ~ c a t e ~ v r ~ , u n zo. Z(2)23 = g o o o
J
za)33 = stable z unreliable ~ (2)33 =
:
J
i
capillary retinopathy,
. . genet,cs w,th respect t o j
....
xO) re(t) 111(2) ~.-~' 111(2)/~ = p r e - (post-)therapeutic sugar level for J A N conditioned on high incidence of ocular photocoagulation, X(3)
~(3) h 213(2) 1V~Z13(zv=P re- (post-)therapeufic visual acuity for J A N conditioned on unreliable genetics with respect to capillary retinopathy,
xC2~2(I)(x~2cx))=pre - (post-)therapeutic visual acuity for J O N conditioned on
There are 72 possibilities for each of the pre- and post-therapeutic linguistic descriptions in the case patients with respect to 3 disc a has 2 attributes. It is helpful t o note that the denofions ACZ) = { . . . , Ix(Z) zt,~(t) ~ ik(v) ~, iik(r)~ ~,~iik(r)]}~
I ,,(o
. . . . [ "~'ffk(r)
~ X x Z}
and
can be respectively shnpfified thus: A=
x xz}
A={(D,.g&(D))]DeX.~:,-Z}
-- ~. .....:
(3.2)
(3.3) duct of the spaces of the :ontexts for prognosis. In strategy, in wh/ch case a represents a 'therapeutic'
~ .
Aand::~ :are alternative ~ z z y subsets.
(A 'do-nothing' stategy could, in fact, cover the case of administering an alternative treatment to the so'called 'therapeutic' strategy.) g g~ples.
(a)
(b)
A = {(sug','x level with high incidence of ocular photocoagulation, fairlylow), (visual acuity ~ven unreliable genetics with respect to capillary retinopathy, reasonably- keen)}, .~,={(sugar level with high incidence of ocular photocoagulation, normal), (visual acuity ~ven unreliable genetics with respect to capillary retinopathy, very-keen)}. A = {(visual acuity given reasonable categorization for macular degeneration, good), (sugar level with rare incidence of macular degeneration, acceptable)}, .~ = {(visual acuity given reasonable categorization for macular degeneration, not-so-good), (sugar level with rare ir~cidence of macular degeneration~ too-high)}.
In both examples of the nature of 'do-nothing' and 'therapeutic' strateg/es, three points should be noted. (0 ~ e fuzzy s u b ~ t s A and A represent 1/nguistic descriptions of the evaluations of the ophthalmolog/cal states or conditions of a particular patient. (ii) The membership functions P.A(') and #&(.)are themselves fuzzy subsets so that the strategies are essentially represented by higher level fuzzy subsets. (i/i) I t is being supposed that the contexts under which the evaluations are performed require the following correspondences, in the event that A stands for a 'do-nothing' strategy while ]k represents a strategy of satisfacxory consequence, or vice versa:
normal ~ not-fairly-low very-keen <-, nm-reasonably-keen not-so-good <-~not-good too-high <-~not-acceptable Different contexts would possibly generate different correspondences. The contexts would also determine whether A a n d A are alternative (possible) treatments
132
Now, let g and F be fm
h(E) > hff(E, ~)), where h ~ ) denotes the h e i s t of E, ~ denotes an alternative to E, and t ( ~ , ) denotes a choice of E a n d ~ (which ought to be a fuzzy Subset).
Lemma. The goodness associated with f(g, E) is given by
max(E, E) for any E, where m"~ is a fuzzified maximum operator defined by w ~ - ~ ~:~(d)= amax = ~ {~E(x)^ ~(~)}, with d, x, and Y~belonging to the Cartesian product described above. (See Baldwin and Guild [29]). l~lm,~ Let A and A be alternative fuzzy subsets of the Cartesian product of X and Z such that A = {(D,/~A(D)) ] D e X x Z}, h = { ( f ) , ~ ( D )- ) I D e x x z},
' and the iernma on 'goodness' above together Ex2mpl~. (I) ~ t £={(/5,, ~. ), (D2,0.3), (D3,0.2), (D4,0.7)} and A " D ~, 0.0), (D2, 0.1), (D3, 0.8), (D4, 0.7)} so that f( fn~x(g, A)~,
/
----
D*=]DvD
~ ~(D*)= max{0.0, 0.1, 0.5, 0,5~=0.5, a ~f D~¢:2)=m~{o.o, o:L: 0.3, 0.3}= o,3~
. . . . .0,7}=0.7, ' ~ (D * 4)= rn~{O.0, 0.l, 0.7,
~ h t h a t m o l o ~ l pvognos/sv/a [uzzy subsas
133
A ) = {D~, 0,5)~ (D*, 0i3), (D~, 0.2), (D*, 0.7)} :~ :~ is not effective. t.0)} and B = {(Da, 0.6), (D2, 0.8), (D3, 1.0)}, :~:~ : ~ ~ . m ( D ~ •
=max{0.6, 0;6, 0.6}= 0.6.
g~,~.m)(Dz*) = max{0.6, 0,8, 0.8} = 0.8, W~-~c~.B)(D3*)= max{0.6, 0.8, 1:0} = 1.0.
Thus, A
max(B, B) = B = B ::> h(B) = h(max(B, ~'~ B))= 1.0. "
Accordingly, B is not effective. This example shows that if the result of administering a treatment leaves the patient's state unaltered, then the effect of the treatment is obviousl~ nil, (3) Suppose ~ = {(D1, 0.7), (D2, 0.3), (D3, 0.5)} and C = {(D1, 0.4), (D2, 0.2), (D3, 0.3)}. Then m"~(C, C)= {(D~*, 0.4), (Dz*, 0.3), (D3*, 0.4)}. Since h(~) = 0.7
and
h(ma"~(C, C)) = 0.4,
it follows that C is effective. Remarks. A close examination of these examples reveals that h(max(E,E)) equals h(E) or h(E), for any pair o f alternatives (E, E), so that E is effective or not effectwe according as h(max(E, E)) < or I> h(E). Thus ~effective is [not effective
~less than if h(E) is ~ greater than or equal to h(E).
so that E is effective or not In any event, h(max(E,E))=min(h(E),h(E)), ~ ^ effective according as min(h(E), h(E))equals h(E) or h(E). Merely comparing the heights of the alternative fuzzy subsets may at first seem unrealistic an approach for determining the effectiveness of a treatment. But by remembering that the components o f E and E are linguistic descriptions, it becomes clear that the heights represent some sort of levels of an overall evaluation because an evaluation of one aspect or component is invariably performed in relation to the other aspects. The use of (3.4) requires a method for generating explicitly the alternative fuzzy subsets A and A vis-a-vis the relations (3.1), (3.2), and (3.3). Oguntade and Gero E32] have developed a simple procedure for obtaining evaluative values oe;~uzzy variables. Such a procedure, with a slight modification, can be used to generate alternative fuzzy subsets explicitly. The procedure is given compactly as i 1 [/zj(Yik)JJ
k = 1, 2 , . . . , n,
(3.5)
4. A ~ ~
case
An ophthalmological treatment was administered to two patients J A N and .ION. Both were suffering from diabetic eye disease admLxed with senile heredomacular degenerative diseases. JAN was diabetic but JON was not. The ophthalmological vector (visual acuity, blood sugar level, diabetic resistance) was linguistically transformed, subject to the ideal ophthalmological sort 'reliable diagnostic categorization', to the vector of linguistic variables
(4.1)
(Xl, x2, X3, x4, Xs, x6) = X
where x l denotes good visual acuity "~ xe denotes nodal b l ~ d sugar level ], x3 denotes high diabetic resistance J
for (diabetic) JAN,
~35
x~
x2
x3
x4
x5
x6
xl 1 x 2 0.6
0.2 1
0.4 0.2
0.4 0.2
0.2 0.1
x~
0;7
1
::
0.1 1
0.1
0.4
x6 0.3 0 . 5
0.2
0.3
0.7
0.8 0.9 0.7 0.5 0.3 1
0.6
Table lb. ~A(Yk~)(after treatment)
xl x2 x3 x4 xs x6
gl
X2
X3
X4
X5
X6
1 0.2 0.2 1'6 0,I ~;.1
0.7 1 0.7 1 0.2 0.1
0.8 0.3 1 0.9 0.3 0.2
0.5 0.1 0.1 1 0.I 0.1
0.5 0.3 0.3 0.6 I 0.1
0.9 0.6 0.6 0.8 C.7 1
of such contexts are 'state of medical practice', 'accuracy of diagnosis', etc. The subjective observations are shown in Tables l a and lb. In general, the matrices which represent the subjective observations are not symmetric. Whereas g-(Yk~) uses xk as a reference frame for determining the skrdlarity of effect betwee£ xk and x~, ~(Y~k) employs x~ as a reference frame. These effects are not generally identical. For example, the fact of a nondiabetic possessing good visual acuity has little in common with the knowledge that a diabetic possesses good visual acuity. Yet, the fact cff a diabetic possessing good visual acuity could well lead one to presume that a nondiabetic counterpart would also possess good visual acuity, depending on the circumstances of the disease that is admixed with diabetic eye disease. Computational forms for T and _k must now be stipulated in order to use (3.5). As previously :remarked, we shall use the following forms: A T B = {(x; min(~A(x), ~B(x))},
(4.2)
A £ B = {(x; ~'~(A.m(x))}.
(4.3)
and Hence, the observations shown in Tables 2a and 2b. By letting E~(X)=
xk; T {g,i(Yk~)/g'i(Y~.)}
k = 1,..., 6
i---I
where j AB or A and T is given by (4.2), we shall have
EB(X) --~-.-{(x!, 0,4), (x2, 0.3), (x3~. 0.3)~, (x4, 0.3), (x 5, 0.3), (x6, 0.4)}
(4.4)
IN
0 . 5 . LO . . 4,0.
t., 0
Z0
1.7 :
:
. . . . i:
!.0 z o 3,o oi5
.... ~
:
O~
0~6
0'3
06
2'3
1.0
:' 7
...........
=.
' .=
~
•
Table 2b. V.A(Y~)/V~.(Y,,) (after treatment) X1
X2
X3
X4
X5
X6
1.0 0.3 0.3 1.2 0.2
3.5 1.0 2.3 10.0 0.7
4.0 0.4 i.0 9.0 1.0
0.8 0.1 0.1 1.0 0.2
5.0 1.5 1.O 6.0 1.0
9.0 6.0 3.0 8.0 7.0
0.1
0.2
0.3
0.!
0.1
LO
and
EA(X) = {(xl, 0.1), (x2, 0.2), (x3, 0.3), (x,, 0.1), (x5, 0.1), (x6, 1)}.
(4.5)
In terms of the old notations (see (3.1), (3.2) and (3.3)), F_~(X) and EA(X) are, respectively, given by respectively. given from (4.3) by Using ill,
t from (4:4)'and (4.5) that
m~(EA(X), E~(X))={(xl, 0.I), (x2, 0.2), (xa, 0.3), (x4, 01,), (xs, 0.1), (x6, 0.4)} whence we find that h(EA(X))=I.0
and
h(m"~(
,
))=0.4
which implies from (3.4) that the treatment was effective.
on ~ect
I37
If ~t
t~
~ea! ~.tates,whic-h are of course linguistically described,
th¢~-
hat the teehrfique would yield a reliable approach to r~rognosis
~ ~............. p l a ~ d o n ~ h t h a t m o l o g y , the technique seems to be :~ ;e :ii~o t h e r disciplines of the medical profession. In fact, it does n¢~ losS~6fW generali~ ould,," result if' the technique is applied to any h~ : t ~e "~ system, when the evaluation of the system's effectivenes~ is under ~s study. A general approach to such evaluation, based on context-dependency, re~mtiv~sm, and related concepts has been attempted by Oguntade [3!]. It s h o e d be indicated that this paper is addressed primarily to the philosophy of ~ sets as an instrument of enormous potential for prognosis, its possible use has been stressed, with a .special. emphasis on linguistic transformations of basic evaluative variables into manipulafible fuzzy subsets. More theoretical research as wei~ as empirical tests are being pursued in the hope that a precise technique would aomeday emerge to allow us to manipulate imprecision ~n the medical sci e~c~=~ as a whole. R~eren~ [!] H.L. Bleich, Computer evaluation of acid-based disorders, .~, Clinical laves,. ~8 (1969) I6~91696. [2] A. Bouchaert and S. Thiry, Physiopathological inference by computer, lr~. J. Bio-Medica! Computing 8 (1977) 85-94. [3] P. Chod~off and J.H. Drews, Decision ~mking in anesthesiology, 3. Am, Med. A'~soc. 298 (197i) 85-88. [4] D J . Croft, Is computerized diagnosis possible? Comput. B~omed. Re.~. 5 ~t972~ 35~-367 [5] D.G. Fryback, Baye,~~ ~heorem and conditiona| nonindependence of data in medical diagnosis, Comput. Biomed. Res. 11 (1978) 423-434. [6] D.J. Heckstra and S.D. Miller, Sequential game.:, and medical diagnosis, Comput. Biomed. Res. 9 (1976) 205-215. [5] G, Honi£feld, D.F. Klein and S. Feldman, Prediction of psychopharmacologic effects in man: development and vafidadon of a computerized dmgnostic decision tree, Comput. Biomed. Res~ 2 (1969) 350-361. [8] C.A. Kulikowski, Pattern reeogni6on approach to medical diagnosis, IEEE Trans. ~yst. Sci. Cybernet. 6 (3) (I97~) 173-178. [9] R.$. Ledley~ Computer aids to medical diagnosis, J. Am. Med. Assoc. 196 (1966) 933-943. [10] R.S. Ledley, Practical problems in the use of computers in medical diagnosis, Proc. IEEE 57 (1969) 1900--1918. [11~1 R.S. Ledley, Automatic pattern recognitio~ for c~incial medicine, Proc. IEEE 57 (1969~ 20172035. [12] S.G. Loo, Fuzzy relations in the social and behavioural sciences, 3. Cybernetics 8 (1978) 1-16. [13] B.W. Madsen, T.L. W ~ d i n ~ , K.F. llett, G.M. Shenfield, 3.M. Potter and J.W. Paterson, Clinical trial experience by simulation: a workshop report, Br. IVied. J. 2 (t978) 1333-1335. [14] S.L Menn, G.O, l~arnett, D. Schmechel, W.D. Owens ~nd H. Pontoppidan, A computer program to assist in the care of acute respiratovv fa~u.re, j. Am. Medo Assoc. 223 (i973) 308-311. [15] E.A. Patrick, F.P. Stelmack and L.Y.L. $hen, Review of pa~ern rec~3grfition in medic~Adiagnosis and consulting relative to a new system model, IEEE Trans. Sy~t, Man and Cybernet. SMC-4 (1) (1974) 1-16.
O.O.
138
:.
[18] E.: T a ~ ' ~ and M~
~
.....
~
P,K
. %
,,:/.
:
-
t ..... ?
,Structural model'mg i n a c t a s s of systems using:fuzzy sets theory,
FJe,ctronics, Liege, Belgium (1963) ~ 8 6 8 .
[22] M.A. Woodbury and L Clive, Clinical pure ~ a s a
fuzzy partition. L Cybemet. 4(3):(1974)
111-121.
[23] M.A. Woodbury, J. Clive and A. Garson, Mathematical topology: a grade of membership technique for obtaining disease definition, Comput. Biomed. Res. 11 (1978) 277-298.
[24] P.M. Wortman; Medical diagnos/s: an information-processingapproach, Comput. Bioraed, Res. 5 (1972) 315-328.
[25] L.A. Zadeh, Fuzzy sets and systems, in: Proceedings of the Symposium on System Theory, [26] [27] [28] [29] [30] [31] [32]
Microwave Research Institute Symposia S e r ~ 15 (Polytechnic Inst. Press, Brooklyn, I ~ , 1965) 21-37. L.A. Zadeh, Fuzzy sets, Information and Control 8 (3) (1965) 338-353. L.A. Zadeh, Outline of a new approach to the analysis of complex systems and decision processes, IF~E Trans. Syst. Man and Cybemet. 3 (1) (1973) 28--44. L.A. Zadeh, A fuzzy-algorithmic approach to the definition of complex ef imprecise concepts, Intemat. L Man-Machine Studies 8 (1976) 249-291. LF. B~dwin and N.C.F. Guild, Comments on the 'fuzzy max' operator of Dabois and Prade, ~ntemat. J. Systems St. I0 (9) (1979) 1063-1064. C.V. Negoita and D.A. Ralescu, Applications of Fuzzy Sets to Systems Analysis (Birkhatw~r Vertag, Basel/Stuttgart, 1975). O.O. Oguntade, Semantics and pragmatics of fuzzy sets and systems, Fuzzy Sets and Systems, 6 (I981) 119-143. O.O. Oguntade and J.S. Gero, Evaluation of architectural design profiles using fuzzy sets, FtLzzy Sets and Systems 5 (1981) 221-234.