Bek, Roman

Pokorný, Zdeněk

Růžička, Milan

English

Institute of Mathematics AS CR, v. v. i.

KybernetikaRoman Bek; Zdeněk Pokorný; Milan RůžičkaLarge variable systems of higher degrees in fuzzification process. Fuzzification ofsystems for technical and medical practice. IIKybernetika, Vol. 18 (1982), No. 5, 447--454Persistent URL: http://dml.cz/dmlcz/125867Terms of use:© Institute of Information Theory and Automation AS CR, 1982Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access to digitizeddocuments strictly for personal use. Each copy of any part of this document must contain theseTerms of use.This paper has been digitized, optimized for electronic delivery and stamped withdigital signature within the project DML-CZ: The Czech Digital Mathematics Libraryhttp://project.dml.czK Y B E R N E T I K A - V O L U M E 18 (1982), N U M B E R 5 LARGE VARIABLE SYSTEMS OF HIGHER DEGREES IN FUZZIFICATION PROCESS Fuzzification of Systems for Technical and Medical Practice II ROMAN BEK, ZDENĚK POKORNÝ, MILAN RŮŽIČKA The paper deals with fuzzification of large variable systems known from technical and medical sciences. Precise definitions of these systems ard description of their functioning can be formulated by means of languages having time terms ard formed on the base of predicate logics of higher degrees. The fuzzification of these systems is accompanied by their models modifications enabling calculation of certainty grades associated with corresponding entities of systems. fn the article methods of calculation of certainty grades for exactly defined notions are intro­duced, among others those of time variable system, element of system, input and output of system. Special care is taken of their applications in medical science and practice. This discourse follows the paper [ l ] and deals with precise definition and modelling of large variable systems of higher degrees. We first define on a selected ontological sphere a "sharp system" which later is modelled by a precise language system and fuzzified. At the beginning, let us consider a case of simple systems (known from medical practice) having a class of relations given by only one relation and the universe consists of only those objects which occur in the field of this relation at a given moment. That is the case, where we classify a set or signs wanting to form a clinical consequence which is at a given moment in a given relation with this collection. Theoretically the situation can be described as follows: there is given (n + I)-member relation R with the first (n — 1 (-tuple of elements from its field of the type <a,, ..., «„_,> at a moment /,• and since the relation R can be taken for mapping of the type R : <</, «„_,, /,) -> <«„, (,-> (this mapping may not be unique). we can consider the element an at a moment /,. We may not be sure about parameters of the type u{ «„_, at the moment /;. There are three cases possible: 447 1. Relation R is sharp, but relevance of given objects into ;;-tuples of the type <a, . . . . , a„„,, /,-> i.e. definition domain of the mapping R, is fuzzy. Hence /.>.(_,, ..., a„_,, a„, /,) = 1 for all (n + I)-tuples from the field of sharp relation R , /<II»IUI»I,«(«J> ',) < 1 for some aj (1 ^ j | n - l ) and for some contexts of the relation R . From here fuzzification of the n-th domain of R is inducted and its value can be calculated: /«(Rg)oniK(am, /,) = max (min {/;(cont)ot R(aj, t,) for all aj(\ £ j <? n — 1), forming the fc-th context <a,, ..., a„_,, a„, /,> e R} j for all contexts of the type <x,, ..., x„_,, a„, /,-> e R in which a„ occurs on the n-th place}. Example 1. Let R = {</;,, a2, a3, /,-> , <a,, a,, a3, /,-> , <a2, a3, a3, /,->} (<«,, a2, a3, /,-> is the first context of the relation R , <a,, a,, e;3, /,-> is the second context of the relation R , <a2, a3, a3, /,•> is the third context of the relation R) . Let / < ( C „ „ 0 " 1 . K ( « 1 > ti) = 0 ' 7 > /<,conO«l._(«2, ti) = 0 ' 6 5 /'<conO'>2,R(«l, ti) = 0 - 6 7 , / ' , conO»2 .« (« l ' ti) = ( > 6 2 / ' ( con , )"3 . K («2 . ti) = 0-8 , /',con.)«3.K(«3> ti) = 0 9 then /'(RC)"3.R(«3- t,) = max {min {0-7, 0-65}}, min {0-67, 062}. min {0-8, 0-9}} = 0-8 2. ;;-tupleofthe type<aJ ( ..., _„_,, /,> are sharp (definition domain of the mapping R is sharp), but relation R is fuzzy. Thus / ' (con.)0*._(Oj, ti) = 1 for all Oj{\ Sj^n — 1) forming the fc-th context <aj, ..., a„_,, a„, /,> E R and for all contexts of the type <x,, ..., x„_,, a„, /,-> G R in which a„ is on the last place, HRo(au ..., a„_,, a„, /,) < 1 for some <„., a2 , . . . , a„_,, a„, /,-> belonging before fuzzification into K. From here the fuzzification of the «-th domain of R is inducted and its value can be calculated: /'(RE,V_(fln> '.') = m a x {/'_°(xi> • • •> -vn-1> ',) | for all contexts of the type<x, , . . . , *._.,__, ( , • ) ._} . 448 Example 2. Let R be the same as in Example 1, but HR0(au a2, a3, /,) = 0-95 , HRo(au au a,. /,) = 0-82 fiRo(a2, a3, a3, /,) = 0-78 then / W 3 . „(a3, t,) = max {0-95, 0-82, 0-78} = 095 3. The relation R is fuzzy and so are its individual //-tuples of the type \au «2, ... a„-»0. Hence HRo(au . . . ,«„_„ a„, /,) < 1 for some <a., ..., „„_., a„, /,•> belonging before fuzzification to R , i«(cont)0*.R(a;. ' , ) < 1 for some a /1 _= j __ « — l), for some contexts of the type <xt, .... x„_i, a,„ /,•> e R . From lierc the fuzzification of the «-th domain of R is inducted and its value can be calculated: /'(Rg)"„,RK. '.•) = max {min {/%«(«,, ..., £.„_,,_•-, / , ) , V{c0at><>kAaJ>',) for all a /1 _ j _ n - l ) , forming the fc-th context <a,, ..., a„_1; a„, /,•> e R] I j for all contexts of the type <a, , . . . , a„_ ,, a„, /,•> e R} . Example 3. Let R be the same as in Examples 1 and 2, let contexts be fuzzified as in Example ! and so relation as in Example 2. Then /<<R,.)»3.R(«3. t/) = max {min {0-7, 0-65, 0-95}, min (0-67, 0-62, 0-82} , min {0-8, 0-9, 0-78} = 0-78 Let us return to the definition of term "system of the n-th degree". D 1. An ordered pair <"?/, :R) = ff forms a system of the rc-th degree at a time interval At (symbolically: {.9', At) e 9yii(n)\ iff: •'// is a class of objects (individuals = entities of zero degree), y% is a collection of objects of the type (s-R(J) where holds: for s = 1 : (S)R(J) c_ Ua> for s > ! : there is at least one element from the collection {su •.., s — 1} where s - J exists so that (s)R{J) c Ua°> x {<-">Rp> x ... x {<i_ l ' /l}«-«>. where .]<•••>;?}.<"> is the jrth power of the class of all relations of the s,-th degree from the set 0t, L JS 1 __ s - 1,/ = j 0 + j i + ... + j s - \ -Class °U is called the system universe. 449 Successive fuzzification of a system can be motivated by following reasons and can be formulated by precise language model modification of formerly "sharp" system. 1. Time interval At can be fuzzy, because we cannot say with full reliability, if given moment tt belongs to At. A reason may be relative accuracy regarding time items about events or our own uncertainty connected with unity structure of physical time (chronometer time) and biological one (in biological systems fuzzification). In many instances we may not be sure about relevancy of time interval, in which given event occurred. We doubt in this case about relevancy of this moment to given n-tuple of objects forming an element of this fuzzy relation. Fuzzy time interval At0 is a mapping from the class of all time moments into interval 2oi — (0> !>• Membership function of the type /*J,O(I() can have its values found at basic cases by empirical methods. 2. The system universe % can be fuzzy, since we are not able to decide with full certainty whether given objects from selected ontological sphere is from that. The cause of that may be limited accuracy or empirical data and variability of distin-quishing characteristics classifying relevancy of the object into the defines system universe and by that coincidentally into the fields of system relations. When fuzzi-fying terms of systems theory, we may further meet the system universe timely relativitated (given object from the language universe belongs to the system universe at a given moment). Such a type of universe we further denote by <%T. Fuzzy, timely relativitated universe °ll\. is a mapping from a class of pairs consisting of objects from selected ontological sphere and moments from interval At into numerical interval Q01. The membership function JXVTO (X, rf) can have its values determined in empirical way. 3. Fuzzification of relations 0t collection of the system. We cannot say surely, if a given relation belongs to 0t. The first fuzzification of the class 3& is a mapping from all "sharp" relations collection over selected ontological sphere (these relations can be of distinct degrees) into the interval Q0l. The membership function can have its values determined empirically in basic cases. We fuzzjfy further successively all objects from the field of every relation of the type (Sj)Rj- and so far these objects are also relations we fuzzify even them. So we continue in further fuzzification process till the universe elements in fields of successi-vely fuzzified relations. Successive fuzzification can be thus formed until the 2s-th degree. As a whole, the "sharp" collection if = {(s,)R!,..., (Sn)R„} can be fuzzified till the (2sj + l)-th degree, where Sj is the highest degree among degrees of relations from the collection R : s = max {slf ..., s„}. In such a case we shall simply write gi° = ^ ° ( 2 S J + 1 ) a n c j this fuzzy collection is mapping from that of mostly fuzzified relations over selected region into the interval Q01-The membership function values can be calculated if functional values are given 450 (obtained empirically or as results of some other calculations): Hmoi((s,)Ri) for each (Si)R ; e 9t (in "sharp" system), further functional value H(sj)Rjoi((Sjl)Rji, •••, (SJn)Rj„, t,) for each (SJ)RJ e P$(in "sharp" system), where <(Sj'l)RJ-1, •••, {Sjn)Rj„, ti> 6 (Si'Rj- (The method of calculation has been shown in the previous paper). 4. Fuzzification of notion "system": we cannot decide with full reliability, if °UT is a class of selected elements from ontological region and if 9t is a collection of selected relations over that. S"° is thus a mapping from a class of triples consis-ting of a class of objects with zero type level from the selected region, a class of moments andaclass of relations of various degrees defined over the region into the interval Q0l. The membership function values can be calculated: \iyo(^lT, 9t) = min {nmoT(xt, (,), fi^o((Si)Rj) I for each xt and tt e At for every relation of the type (SJ)RJ belonging to collection 9t\ . Since existentional interval At of the system is fuzzy (for above mentioned reasons), fuzzification of the class Sf^ot is motivated by uncertainty, if the pair Sf = <^CT, 9t} is really a system at a time interval At to which belong moments appearing in relations from the class R. The membership function values can be calculated: V-(<r9<>t)°(&'> At) = min {^0(<?/T, 9t), nMo(t^) [ for all t{ e At occurring on the last places in relations of the type (SJ)RJ e 9t and in couples of the type (xt, f,> e e °UT for Sf = (<WT, 9t}, (Sf, At}e Sfyot} . E. 3.1. Example. Let a "sharp" system be given: Sf = <^T , ®}, %T = {(x, tt}, (y, tj}} , 9t = {(1)R, ( 2 )R , (3 )R} (Sf, At}eSfyot, thtjeAt, let relations ( 1 )R , ( 2 )R , (3)R be the same as in Example E. 2.1. Let their fuzzification be the same as in E. 2.L Let values of further membership functions be p^tt) = 0-95 (iAt0(tj) = 0-87 completely for tt and tj in At is then 0-87. Let HvoT(x, tt) = 0-75 , iivoT(y, tj) = 0-82 and totally for x and y in <SUT is 0-75 ^o , ( ( I ) R ) = 0-82 ^ 0 1 (( 2 ) R ) = 0-68 /%<H((3)R) = 0-57 . 451 Since relations from the collection 31 can be fuzzified only five times (for example relation (3 )R), the whole collection can be fuzzified utmost six times. ^o.( ( 1 )R ) = 0-82 M»o,((2)R) = 0-68 ^ 0 2 (( 3 ) R ) = 0-6 We can calculate for all relations: m i n ^ r R ) , ^ 0 2 (( 2 ) R ) , ^0 2(( 3 )R)} =0-6 Further calculations only in brief: /iao2((1)R) = A%o2((2)R) = 0-82 ^ 0 3 (( 2 ) R ) = 0-68 fi^((3)R) = 0-6 total calculation: min {0-82, 0-68, 0-6} = 0-6 further ^„ 5 (( ! ) R ) = ^o2(( 1 )R) = 0-82 ^o , ( ( 2 ) R ) = ^o , ( ( 2 ) R ) = 0-68 /%o5((3)R) = 0-6 then min {0-82, 0-68, 0-6} = 0-6 and ^oc((1>R) = / ^ ( ( 1 ) R ) = 0-82 = ^ 0 (( 1 ) R ) ^o , ( ( 2 ) R ) = ^o . ( ( 2 ) R ) = 0-68 = ^ 0 (( 2 ) R ) ^o*( (3 )R) = 0-6 = M ( 3 ) R ) totally min {0-6, 0-68, 0-6} = 0-6 and M * r , St) = 0-6 HirraAsr, At) = 0-6 Remark. We have a right to ask a question, if a couple of the type <<.VT, 3$) charac-terized by its membership function value relatively very small (for example in interval (0, 0-5)) with respect to relevancy to class S^^o/(n), is to be still considered as a system and further operated with as that. However we cannot lift up our attention in clinical practice from those couples which are doubtful as to the system formation. Majority of diseases begins namely with very indefinite predroms. To recognize a kind of illness beyond these symptoms is very important. Relation of these signs often resolves in a system with a low grade of membership. With increasing time formerly unidenti-fied symptoms are developing to characteristic ones whose physiological significance is well known. (Symptoms of a disease can be understood as input values of system defined on a related ontological sphere). Definition of system with vague characte-ristic terms can be therefore very important for prevention and prophylacs of deve-loping sickness. D 2. An object x is said to be element of system Sf at a moment ff e At (abbreviated: 452 <x, y, f;> e gt) iff: <y,At}eSf¥oi Sf = ^tiT,@y, <x, fj> e <8rr, fie/if. Fuzzification of class of system elements can be motivated by several grounds. First we do not have certainty regarding relevancy of object x to the system universe °tiT and moreover we may not be quite sure about membership of formerly stated class °tiT as the system Sf universe at all moments of interval At. Further we may not insist in membership of object x to a field of some relation from collection 3k at a moment t{ and then the collection 01 is really a collection of system Sf relations. For already known reason we neednot be sure whether moment f; belongs to the interval At. Fuzzy set (Sef is a mapping of the class consisting of triples of the type <x, Sf, f(> into the interval Q01. Calculation of membership function values for the triple <x, Sf, /,•> can be done by the following algorithm: Algorithm. Step 1: Determine max value from membership functions values for all contexts of the type contkR in which element x occurs at a moment tt e At for all relations of the type l i e f . Step 2: Determine min from the step 1 and from membership function value ^o(R). Step 3: Determine min from the step 2 and from the values figq(R) for all relations of the type R in which x occurs at t,. Step 4: Determine max of the result from the step 3 for all relations of the type R e 0t in which x appears at tt e At, <Sf, At) e Sfyoi. Step 5: Find min from the result of the step 4 and from the value y.voT(x, ff), Sf — = <e,., my. This min is the value of ji^)o for the triple <x, Sf, f;>. Development diagram: lB| 453 Remark. When formulating the function /v<0° w e ta^e c a r e °f t n e degree of x relevancy to the system universe, of that of /,• to At and coincidentally that x is at t{ in the field of a relation belonging to 01 at a certain degree. The element x may occur in the first relation in various contexts which we consider all and we choose max value of membership grade of x at tt to these contexts. There may be more of these relations envolving an element x at a moment tt. Hence we have to select the max value of memberships grades. E. 3.2. Example, (continuation of Example E. 3.1.) hs()Jx, Sf>, tt) = 0-75 hse)0(y, 9>, tf) = 0-82 Let us notice that our conception of fuzzy system represents certain generalization in a comparison with common notion of fuzzy abstract automata, where "sharp" knowledge of input events and given state are assumed, but only fuzzy knowledge of proceeding occurrence. Despite the fact that we do not define in this paper notions of "fuzzy input event" and "fuzzy state" our conception does admit definitions of this kind. It is obvious that in discussed technical and medical practice with large and variable systems such a conception would be convenient. (Received June 26, 1981.) R E F E R E N C E S  [1] R. Bek, M. Ruzicka: On modelling of large variable systems of higher degree by means of language systems. Kybernetika 15 (1979), 6, 408 — 429. [2] R. Bek, Z. Pokorny, M. Ruzicka: Principle concepts of systems fuzzification — Fuzzification of systems for technical and medical practice I. Kybernetika 18 (1982), 3, 234—246. Doc. Dr. Roman Bek, DrSc, katedra aplikovane matematiky strojni fakulty CVUT (Department of Applied Mathematics, Faculty of Mechanical Engineering — Czech Technical University), Suchbdtarova 4, 160 00 Praha 6. Czechoslovakia. Doc. MUDr. ZdenSk Pokorny, CSc, tjstav pathologicke fyziologie fakulty vseobecneho lekafstvi UK (Institute of pathological physiology, Medical Faculty — Charles University), U nemocnice 5, 128 53 Praha 2. Czechoslovakia. Dr. Milan Ruzicka, CSc, katedra matematiky strojni fakulty CVUT (Department of Mathe-matics, Faculty of Mechanical Engineering— Czech Technical University), Karlovo nam. 13, 121 35 Praha 2. Czechoslovakia. 454 

Large variable systems of higher degrees in fuzzification process. Fuzzification of systems for technical and medical practice. II

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Large variable systems of higher degrees in fuzzification process. Fuzzification of systems for technical and medical practice. II

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Institute of Mathematics AS CR, v. v. i.