New mechanisms for reasoning and impacts accumulation for Rule Based Fuzzy Cognitive Maps

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1 New mechansms for reasonng and mpacts accumulaton for Rule Based Fuzzy Cogntve Maps Zdanowcz, P & Petrovc, D Author post-prnt (accepted) deposted by Coventry nversty s Repostory Orgnal ctaton & hyperlnk: Zdanowcz, P & Petrovc, D 2017, 'New mechansms for reasonng and mpacts accumulaton for Rule Based Fuzzy Cogntve Maps' IEEE, vol (n press), pp. (n press). DOI /TFZZ ISSN ESSN Publsher: IEEE 2017 IEEE. Personal use of ths materal s permtted. Permsson from IEEE must be obtaned for all other uses, n any current or future meda, ncludng reprntng/republshng ths materal for advertsng or promotonal purposes, creatng new collectve works, for resale or redstrbuton to servers or lsts, or reuse of any copyrghted component of ths work n other works. Copyrght and Moral Rghts are retaned by the author(s) and/ or other copyrght owners. A copy can be downloaded for personal non-commercal research or study, wthout pror permsson or charge. Ths tem cannot be reproduced or quoted extensvely from wthout frst obtanng permsson n wrtng from the copyrght holder(s). The content must not be changed n any way or sold commercally n any format or medum wthout the formal permsson of the copyrght holders. Ths document s the author s post-prnt verson, ncorporatng any revsons agreed durng the peer-revew process. Some dfferences between the publshed verson and ths verson may reman and you are advsed to consult the publshed verson f you wsh to cte from t.

2 Ths artcle has been accepted for publcaton n a future ssue of ths ournal, but has not been fully edted. Content may change pror to fnal publcaton. Ctaton nformaton: DOI /TFZZ , IEEE > REPLACE THIS LINE WITH YOR PAPER IDENTIFICATION NMBER (DOBLE-CLICK HERE TO EDIT) < 1 New mechansms for reasonng and mpacts accumulaton for Rule Based Fuzzy Cogntve Maps Pawel Zdanowcz, Dobrla Petrovc Coventry nversty, Faculty of Engneerng, Envronment and Computng,.K. Abstract Rule Based Fuzzy Cogntve Maps (RBFCMs) have been developed for modellng non-monotonc, uncertan, cause-effect systems. However, the standard reasonng and mpact accumulaton mechansms developed for RBFCMs assume that the level of varaton that a fuzzy set represents s drectly lnked wth the shape of the fuzzy set. It poses a bg restrcton on how the correspondng fuzzy sets have to be constructed. In ths paper we propose a new reasonng and mpact accumulaton mechansms whch take nto consderaton standard semantcs of fuzzy sets, where ther uncertanty s measured by fuzzness. New type of complex fuzzy relatonshps and reasonng on them s ntroduced to model a ont mpact of several causal nodes on one effect node. Wth these new mechansms, RBFCMs become much more flexble, provde more means to capture complexty of real world systems and are less computatonal demandng than standard mechansms. The advantages of the new RBFCMs are demonstrated usng dfferent examples and compared wth standard mechansms. Index Terms Fuzzy Cogntve Maps, Rule Based Cogntve Maps, Fuzzy Logc, Fuzzy Causal Relatonshp, Reasonng Mechansm

3 Ths artcle has been accepted for publcaton n a future ssue of ths ournal, but has not been fully edted. Content may change pror to fnal publcaton. Ctaton nformaton: DOI /TFZZ , IEEE > REPLACE THIS LINE WITH YOR PAPER IDENTIFICATION NMBER (DOBLE-CLICK HERE TO EDIT) < 2 I. INTRODCTION Fuzzy Cogntve Maps (FCMs) were ntroduced n [1] to model complex relatonshps between concepts. A FCM s a graph where nodes represent concepts of a system, whereas the behavour of the system s modelled by relatonshps among the nodes. There are two types of nodes; a node that has a causal nfluence on another node s the causal node, whereas a node that s mpacted by ths nfluence s an effect node. Relatonshps between the nodes n the FCM are represented usng adacency matrx [a], where a=0 means that there s no causal relatonshp, a (0,1] means that there s a postve relatonshp between nodes and,.e., f node s ncreased then node wll ncrease and f a [ 1,0), there s a negatve relatonshp,.e. f node s ncreased then node wll decrease. Each node n the FCM has a value n the range [0, 1] or [-1, 1]. The value of the effect node s calculated by addng multpled values of the correspondng causal nodes and degrees of causalty - weghts of the correspondng relatonshps. The drawback of FCMs s lnearty and monotony of a relatonshp between a causal node and an effect node. Regardless of how the causal node changes, ts mpact s multpled by a sngle weght defnng ts relatonshp wth the effect node. Therefore, the mpact receved by the effect node s lnearly dependent on the value of the causal node and weght of the relatonshp. A relatonshp n an FCM, s monotonc and symmetrcal about mdpont of the range of values that a causal node can take when the threshold functon s used to calculate the value of the effect node,.e. for range equal to [-1, 1], the relatonshp s symmetrcal n pont 0. Despte the lmtatons and smplcty of relatonshps FCMs could represent, they have become a wdely used modellng tool. An applcaton specfc modellng tool s Fuzzy Control System (FCS). In FCS, values of causal and effect nodes are expressed usng fuzzy sets and relatonshps between them usng IF-

4 Ths artcle has been accepted for publcaton n a future ssue of ths ournal, but has not been fully edted. Content may change pror to fnal publcaton. Ctaton nformaton: DOI /TFZZ , IEEE > REPLACE THIS LINE WITH YOR PAPER IDENTIFICATION NMBER (DOBLE-CLICK HERE TO EDIT) < 3 THEN rules. Ths method s wdely used n the engneerng doman to model fuzzy controllers. In contrast to FCMs, fuzzy IF-THEN rules enable modellng of non-monotonc and nonsymmetrcal relatonshps among nodes. A crsp nput nto the system has to be fuzzfed frst and then approprate rules, defnng a relatonshp between two nodes, are fred. Impact receved by the effect node s also fuzzy and has to be defuzzfed. Most often appled method of nference n FCS has been Mamdan nference method [2]. As a result, the mpact receved by the effect node from two causal nodes s an average of two mpacts, weghted wth ther frng strengths and the areas of the fuzzy sets. Another step n evoluton of fuzzy causal systems has been made by ntroducton of Rule Based Fuzzy Cogntve Maps (RBFCMs) by Tome and Carvalho [3]. As n FCS, values of nodes are represented usng fuzzy varables and relatonshps between enttes are expressed usng fuzzy IF-THEN rules. The rules represent mpact that varatons or changes n causal nodes have on effect nodes. Varatons and/or levels of nodes are modelled usng lngustc terms, such as Decreased, Increased, Hgh, Low, and so on. The man dfference between FCS and RBFCMs s n ther reasonng mechansms and the way mpacts receved by the effect node are accumulated. For example, f the frng levels of two rules are the same and the mpact s represented by two lngustc terms Increased Lttle and Increased, by applyng Mamdan nference, the defuzzfed output falls n between the two fuzzy sets. However, f these two rules are fred n an RBFCM, the result of accumulaton of two mpacts, Increased Lttle and Increased, s a fuzzy set that represents mpact More than Increased. RBFCMs have been successfully used n a few applcatons, such as forest fre modellng [4], soco economcal systems [5], fsherman behavour [6] [7], defence [8], and student-centred educaton [9].

5 Ths artcle has been accepted for publcaton n a future ssue of ths ournal, but has not been fully edted. Content may change pror to fnal publcaton. Ctaton nformaton: DOI /TFZZ , IEEE > REPLACE THIS LINE WITH YOR PAPER IDENTIFICATION NMBER (DOBLE-CLICK HERE TO EDIT) < 4 The man characterstc of the method proposed by Carvalho and Tome s that there s a lnk between a varaton represented by the fuzzy set and the shape of the fuzzy set [3]. The greater the area and support of the fuzzy set s the greater the varaton t represents. Carvalho and Tome proposed a set of predefned membershp functons that can be used n an RBFCM and that ensures that the results obtaned are correct. Consequently, experts are requred to use the set of proposed membershp functons and ths reduces flexblty and applcablty of the method. As one of the man aspects of modellng usng fuzzy logc s possblty to represent experts knowledge n a flexble way, n ths paper we propose a new reasonng mechansm and a new mechansm for accumulaton of mpacts n RBFCMs. New approaches use standard semantcs of fuzzy sets and mprove RBFCM s flexblty by removng the constrant prevalent n the standard mechansms that lmt the range of membershp functons that can be used by experts. One of the man dffcultes n constructng fuzzy cogntve models s elctaton of knowledge. It can be ether bult by experts, resultng n subectvty of the model and facng dffcultes n extractng knowledge for complex maps, or nduced from hstorcal data usng learnng algorthms. The latter became an mportant topc of research of FCMs n depth revewed n [17]. Learnng of FCMs can be grouped nto two categores Hebban learnng and evolutonary algorthms (EA). Hebban learnng based methods [18] requre nput, output and an ntal set of weghts to be defned pror to learnng. The algorthm teratvely changes weghts to mnmze the dfference between the output of the smulaton and the desred values of nodes. An example of Hebban learnng based reasonng was proposed n [19], where a learnng approach s used to change weghts between nodes to mprove stablty of the system. The second category EA, has receved n the recent years much more attenton than Hebban learnng. The EA encapsulate a broad range of populaton based, meta heurstc methods ncludng memetc algorthms, whch

6 Ths artcle has been accepted for publcaton n a future ssue of ths ournal, but has not been fully edted. Content may change pror to fnal publcaton. Ctaton nformaton: DOI /TFZZ , IEEE > REPLACE THIS LINE WITH YOR PAPER IDENTIFICATION NMBER (DOBLE-CLICK HERE TO EDIT) < 5 combne multple methods to mprove local search capabltes of the EA [20]. The EA explore the search space and evaluate canddate solutons usng a ftness functon, whch consst of desred set of output nodes values and optonally other parameters. The advantage of the EA s that they do not requre specfyng the ntal set of weghts between concepts as those are generated randomly at the begnnng of the learnng process. An example of applcaton of the EA s Dynamc Multagent Genetc Algorthm, where elements of populaton are represented by agents that can nteract wth ther neghbours to mprove the populaton. The algorthm s used to reconstruct Gene Regulatory Network by buldng a FCM whch represents relatonshps between 200 genes [21]. FCS learnng methods can be dvded nto two categores: neural nets based and EAs. There are three aspects of the FCS that can be optmsed: the shape of the membershp functons, the rule base that defnes a relatonshp between nodes and parameters of the nference used [22] [23]. The paper s organsed as follows. In Secton II, a case study s ntroduced and n the subsequent secton semantcs of fuzzy sets are dscussed. In secton IV, a standard RBFCM reasonng mechansm and ts characterstcs are presented followed by a descrpton of a new reasonng mechansm proposed and benefts of employng t. Secton V presents the standard RBFCM accumulaton mechansm, ts lmtatons and a soluton to the problems by ntroducng a new method for accumulaton of mpacts. In Secton VI, new types of relatonshps that can be defned and used n RBFCMs and the correspondng reasonng mechansm are explaned. Conclusons are gven n Secton VII.

7 Ths artcle has been accepted for publcaton n a future ssue of ths ournal, but has not been fully edted. Content may change pror to fnal publcaton. Ctaton nformaton: DOI /TFZZ , IEEE > REPLACE THIS LINE WITH YOR PAPER IDENTIFICATION NMBER (DOBLE-CLICK HERE TO EDIT) < 6 II. CASE STDY One of the strengths of modellng usng fuzzy logc s the ablty to model systems on the hgher level of abstracton and descrbng concepts that are dffcult to represent usng precse mathematcal relatonshps. An example of such system s a cyber defence system of an enterprse. One of ts subsystems s detecton of compromses to Communcaton and Informaton Systems wthn organsaton. Detecton of compromses s one of the key aspects of provdng securty of nformaton that s crucal for most organsatons. Ths subsystem can be modelled usng RBFCM as presented n Fgure 1. CIS Communcaton and Informaton Systems Fg. 1 An RBFCM for sybsystem of cyber defence - compromses detecton Some of the elements of the detecton system are easly measurable,.e. Number of CIS compromses detected. On the other hand, some elements, such as: nderstandng Cyber tools and tradecrafts are concepts representng human capabltes that are dffcult to quantfy, and, therefore, relatonshps between them are dffcult to defne. In the presented model relatonshps between nodes are defned usng fuzzy IF-THEN rules,.e. IF Ablty to detect compromses to CIS s Increased Much THEN Number of CIS compromses detected s Increased. Fuzzy terms used to defne these relatonshps are modelled usng membershp functons. As n [3], we use seven values of varatons n causal nodes and consequent varatons n effect nodes, Decreased Much (DM), Decreased (D), Decreased Lttle (DL), Mantaned (M), Increased Lttle (IL),

8 Ths artcle has been accepted for publcaton n a future ssue of ths ournal, but has not been fully edted. Content may change pror to fnal publcaton. Ctaton nformaton: DOI /TFZZ , IEEE > REPLACE THIS LINE WITH YOR PAPER IDENTIFICATION NMBER (DOBLE-CLICK HERE TO EDIT) < 7 Increased (I), Increased Much (IM). However, the correspondng fuzzy sets are defned by cyber defence experts, n a dfferent way compared to [3]. They are gven n Fgure 2, where x axs represents percentage of varatons. Examples from ths case study are used to explan mprovements to the RBFCM mechansms proposed n ths paper. Fg. 2 Lngustc terms that represent varatons n causal and effect nodes III. SEMANTICS OF FZZY SETS Two man types of fuzzy sets used n RBFCMs represent varaton of an attrbute whch descrbes a node (small ncrease, bg ncrease, etc.) and level of attrbute (low level, hgh level, etc.). The standard nterpretaton of these fuzzy sets s one proposed by Bellman and Zadeh [10], where membershp functon s defnng a degree of preference or belef, ( x), of obect x belongng to fuzzy set A over other obects, where ( x) 0 means that the obect does not belong to fuzzy set A, whereas ( x) 1 defnes the full membershp. In ths paper, trapezodal membershp functons are consdered but any type of membershp functons can be used provded they fulfl the followng two requrements [11]: 1) Two consecutve membershp functons defnng varatons overlap and cross n the pont x, where ( x) 0.5 and 2) The sum of all membershp degrees for a gven pont x s equal to 1. (1)

9 Ths artcle has been accepted for publcaton n a future ssue of ths ournal, but has not been fully edted. Content may change pror to fnal publcaton. Ctaton nformaton: DOI /TFZZ , IEEE > REPLACE THIS LINE WITH YOR PAPER IDENTIFICATION NMBER (DOBLE-CLICK HERE TO EDIT) < 8 A reasonng mechansm and accumulaton of mpacts n RBFCMs proposed n [3], have been bult assumng specfc relatonshps between degrees of varatons of fuzzy sets and ther areas and postons on the unverse of dscourse [11]. Ths condton greatly affects the way fuzzy sets of degrees of varaton (or levels) have to be constructed and lmts expert s flexblty when defnng these fuzzy sets. The fuzzy set A represents a greater varaton than fuzzy set B when followng condton s fulflled: A B X A B support support Area Area,, A B A B where X represents fuzzy sets defned n the unverse of dscourse. In Fgure 3 (a) and (b), fuzzy set A, Increased, represents a greater varaton than fuzzy set B, Increased Lttle, as t lays further from the begnnng of the scale (pont 0). In the case presented n Fgure 3 (b), fuzzy set B has a bgger area than fuzzy set A, and, therefore, these fuzzy sets cannot be used to model knowledge on degrees of varatons n an RBFCM. Fg. 3 An example of fuzzy sets whch represent varatons (or levels) Fuzzy sets presented n Fgure 2, defned by the cyber defence experts, do not fulfl the condton (2) as fuzzy set Mantaned has greater area than fuzzy set representng greater varaton Increased Lttle, therefore cannot be used n a standard RBFCM.

10 Ths artcle has been accepted for publcaton n a future ssue of ths ournal, but has not been fully edted. Content may change pror to fnal publcaton. Ctaton nformaton: DOI /TFZZ , IEEE > REPLACE THIS LINE WITH YOR PAPER IDENTIFICATION NMBER (DOBLE-CLICK HERE TO EDIT) < 9 In the new RBFCM reasonng mechansm, ths condton has been removed to allow usng standard understandng of the fuzzy sets semantcs, where the area, support, core and nner and outer bases are used to represent uncertanty modelled by the fuzzy set, presented n Fgure 4. Metrc that we use to measure uncertanty n RBFCMs s degree of fuzzness. Degree of fuzzness defnes how dfferent from ts complement the fuzzy set s. The more dfferent from ts complement the fuzzy set s, the less fuzzy t s; the fuzzness of a crsp fuzzy set s equal to 0. Degree of fuzzness can be used to assess uncertanty of experts. Dfferent functons can be used to measure fuzzness of a fuzzy set. In ths paper, we use Index of fuzzness [12]. Index of fuzzness of fuzzy sets A for Mnkowsk class of dstances [13] s defned as: 1 w d 1 w w fc,w A d a c, A xdx, 1, a w (3) w where, ca, s the dstance between fuzzy set A and ts complement ca ( ) rased to the power of w, a and d are the mnmum and maxmum of support of fuzzy set A, respectvely. In ths paper, Hammng dstance s used, where w 1: 1 c,1 c, A a d f A d a x dx 1 where, x x x s the Hammng dstance between degree of belef of fuzzy c, A A c( A) set A and ts complement ca, ( ) for a gven pont x, and x c( A) 1 A x. It can be observed that for any trapezodal membershp functon: d 1 ba boa ca, x dx d a 2 2 a

11 Ths artcle has been accepted for publcaton n a future ssue of ths ournal, but has not been fully edted. Content may change pror to fnal publcaton. Ctaton nformaton: DOI /TFZZ , IEEE > REPLACE THIS LINE WITH YOR PAPER IDENTIFICATION NMBER (DOBLE-CLICK HERE TO EDIT) < 10 represented as the shaded area n Fgure 4. Therefore, Index of fuzzness s 1 c,1 c, A a d f A d a x dx d a d a and normalzed fuzzness s: fˆ c,1 A b bo b bo A A A A ba boa 2 ba boa ba boa d a 2 support 2 ( b bo core ) A A A A Fg. 4 Hammng dstance between fuzzy set A and ts complement Factors that mpact the normalsed fuzzness are the length of the core, nner and outer bases of the fuzzy set. The greater the nner and outer bases are, for a gven support, the fuzzer the fuzzy set s. When nner and outer bases are equal to 0,.e. the fuzzy set s crsp, there s no fuzzness. The same relaton, an ncrease of fuzzness, can be observed when the sze of the nner and outer bases remans unchanged, but the sze of core and support become smaller. If the length of core s equal to 0,.e. t s 1 pont, the fuzzness s the hghest, equal to 0.5. The excepton s a sngleton fuzzy set, where fuzzness s the lowest. The reasonng mechansm proposed n ths paper s defned n such a way as to preserve ths metrc and use t n reasonng nstead of the area of the fuzzy set.

12 Ths artcle has been accepted for publcaton n a future ssue of ths ournal, but has not been fully edted. Content may change pror to fnal publcaton. Ctaton nformaton: DOI /TFZZ , IEEE > REPLACE THIS LINE WITH YOR PAPER IDENTIFICATION NMBER (DOBLE-CLICK HERE TO EDIT) < 11 IV. RBFCM REASONING MECHANISMS A. Standard RBFCM reasonng mechansm A smple fuzzy causal relatonshp, FCR, between two nodes, causal Node X and effect Node Z s defned usng IF-THEN rules where the antecedent of a rule defnes the value of Node X and the consequent of the rule specfes the value of Node Z. The FCR can be modelled by up to N rules, where N s the number of fuzzy terms that the causal node can have; n ths case, seven fuzzy sets are used, N = 7. The FCR s defned as follows: R: IF Node X s A THEN Node Z s C, =1 N The reasonng algorthm developed n [14] assumes that the condtons (1) are satsfed. Consequently, ether one or two rules R, = 1,,N can be fred. To explan the standard reasonng mechansm, let us consder a relatonshp between two nodes Number of CIS compromses detected and Ablty to assess and learn from system compromses represented n Fgure 1. Fuzzy sets modellng varatons are defned n Fgure 5, where membershp functons are defned respectng condton (2). Let us assume that the nput nto the node Number of CIS compromses detected s decreased and falls n between supports of two fuzzy sets Decreased Lttle and Mantaned. Ths trggers two rules and, wth degrees of belef m and A m, respectvely, where m m 1. R : IF Number of CIS compromses detected s Decreased Lttle ( m ) THEN Ablty to assess and learn from system compromses s Mantaned ( m ) R : IF Number of CIS compromses detected s Mantane d ( m ) THEN Ablty to assess and learn from system compromses s Increased Lttle ( m ) where degrees of belef of the antecedent and the consequent are gven n the brackets. A A A A A C C

13 Ths artcle has been accepted for publcaton n a future ssue of ths ournal, but has not been fully edted. Content may change pror to fnal publcaton. Ctaton nformaton: DOI /TFZZ , IEEE > REPLACE THIS LINE WITH YOR PAPER IDENTIFICATION NMBER (DOBLE-CLICK HERE TO EDIT) < 12 Fg. 5 Defnton of varatons fuzzy sets respectng condton (2) To handle the case when two rules are fred, an algorthm for combnng the mpact of two fred rules was proposed [14]. The process of reasonng on these two rules s presented n Fgure 6, where nodes X and Y represent nodes Number of CIS compromses detected and Ablty to assess and learn from system compromses, fuzzy set A and A are fuzzy set Decreased Lttle and Mantaned, C and C are fuzzy set Mantaned and Increased Lttle, respectvely. Frst, fuzzy sets of the consequent of fred rules, n ths example, Mantaned, C, and Increased Lttle, C, are cut nto C ' and C ', respectvely, correspondng to the frng levels m and of these two cut fuzzy sets s determned usng Max-product method as follows: m A C ' C ( z) m m m m A C ' C ( z) m ( z) max[ C ' ( z), C ' (z)] m. non where m s frng level of fuzzy set A, m s frng level of fuzzy set A A A. Centrod xc of unon represents the crsp mpact receved by Node Z - Ablty to assess and learn from system compromses. The next step of the process s to determne a new fuzzy set

14 Ths artcle has been accepted for publcaton n a future ssue of ths ournal, but has not been fully edted. Content may change pror to fnal publcaton. Ctaton nformaton: DOI /TFZZ , IEEE > REPLACE THIS LINE WITH YOR PAPER IDENTIFICATION NMBER (DOBLE-CLICK HERE TO EDIT) < 13 Causal Output Set ( ), whch represents combned varatons and preserves the trapezodal shape of the two consequent fuzzy sets. followng condton: s determned n such a way as to satsfy the Area Area (8) Fg. 6 Reasonng mechansm when two rules are fred The values of core and b are calculated based on the dstance between defuzzfed consequent fuzzy sets C and cut fuzzy sets, C, xc, as follows: xc C and xc C xc xcc, core mn core core core core C C C C xcc xc C, respectvely, and defuzzfed unon of the two xc xcc, b mn b b b b C C C C xcc xc C where defuzzfcaton of fuzzy set s carred out usng the Centrod method [16]. b and core are used to calculate the remanng characterstcs of : support and bo, as follows: bo 2 Area (2 core b ) support b core bo

15 Ths artcle has been accepted for publcaton n a future ssue of ths ournal, but has not been fully edted. Content may change pror to fnal publcaton. Ctaton nformaton: DOI /TFZZ , IEEE > REPLACE THIS LINE WITH YOR PAPER IDENTIFICATION NMBER (DOBLE-CLICK HERE TO EDIT) < 14 The centrod of new fuzzy set and, then, the fuzzy set s calculated usng the Centrod defuzzfcaton method s shfted so that xc xc as n Fgure 7. Fuzzy set represents the mpact of the causal node Number of CIS compromses detected on the effect node Ablty to assess and learn from system compromses. Fg. 7 Transformaton of unon nto the fuzzy set If the nput from the user falls wthn a core of one of the fuzzy sets, then only one correspondng rule s fred wth frng level equal to 1. As a result, fuzzy set consequent fuzzy set of the fred rule. becomes the B. A new reasonng mechansm In ths paper, a new reasonng mechansm for calculaton, when more than one rule s fred, s proposed. The algorthm s focused on mantanng uncertanty of the fred fuzzy sets rather than the relatonshp between Area and Area. Therefore, condton (2) does not have to be respected when membershp functons of fuzzy varatons are defned. The new method s based on the dea that to whch centrod the centrod of should have a shape smlar to the consequent fuzzy set s closer to. In ths way, the fuzzness of the be smlar to the fuzzness of the nearer consequent fuzzy set. wll Let us consder the example of relatonshp between two nodes Number of CIS compromses detected and Ablty to assess and learn from system compromses, as n the prevous secton. The two rules are fred as follows:

16 Ths artcle has been accepted for publcaton n a future ssue of ths ournal, but has not been fully edted. Content may change pror to fnal publcaton. Ctaton nformaton: DOI /TFZZ , IEEE > REPLACE THIS LINE WITH YOR PAPER IDENTIFICATION NMBER (DOBLE-CLICK HERE TO EDIT) < 15 R : IF Number of CIS compromses detected s Decreased Lttle ( m THEN Ablty to assess and learn from system compromses s Mantaned ( m R : IF Number of CIS compromses detected s Mantaned ( m where THEN Ablty to assess and learn from system compromses s Increased Lttle ( m m A s frng level of rule and m C m, A A ) A ) C ) C ) m A s frng level of rule and m C m, and m m 1. A A A It can be observed that the centrod of unon of two cut fuzzy sets followng value (Fgure 8 (a) and (b)): xc xc, when m 1 and m 0, C C C xc xc, when m 0 and m 1, C C C 0,1 and xc xc xc when m m 0,1, C C C C where m m 1 C C C ' and C ', takes the For example, assume that m C m C (Fgure 8 (a)). Then: dst C dst C where dstc xc xcc s the dstance between centrod of C, xc, and centrod of, xc, C dst C xc xc s the dstance between centrod of C, xc, and centrod of, xc. C C The mportance of mpact of a cut consequent fuzzy set s represented as ts nverted dstance from unon of the cut consequent fuzzy sets; the nearer the centrod of the cut consequent fuzzy set to centrod of s, the hgher ts mpact s. In Fgure 8 (a), centrod C s nearer to centrod Fgure 8 (b), centrod xc of unon and, therefore, ts mpact on xc C s nearer to centrod xc and, therefore, mpact of xc C of s hgher, whle n C s hgher.

17 Ths artcle has been accepted for publcaton n a future ssue of ths ournal, but has not been fully edted. Content may change pror to fnal publcaton. Ctaton nformaton: DOI /TFZZ , IEEE > REPLACE THIS LINE WITH YOR PAPER IDENTIFICATION NMBER (DOBLE-CLICK HERE TO EDIT) < 16 Fg. 8 Relaton between frng levels m, C m and centrod of unon of the cut fuzzy sets C ' and C ' C The mpacts are normalsed usng normalsng factor c as follows: c dst dst C C s determned usng the followng formulas: core core C C core c dst C dst C b b C C b c dst C dst C bo bo C C bo c dst C dst C support b core bo Once the characterstcs of are obtaned, t s shfted so that xc xc.

18 Ths artcle has been accepted for publcaton n a future ssue of ths ournal, but has not been fully edted. Content may change pror to fnal publcaton. Ctaton nformaton: DOI /TFZZ , IEEE > REPLACE THIS LINE WITH YOR PAPER IDENTIFICATION NMBER (DOBLE-CLICK HERE TO EDIT) < 17 C. Characterstcs of the reasonng mechansms (1) Preservaton of fuzzness In some applcatons, the expert can defne fuzzy sets volatng condton (2). An example of such two fuzzy sets, Mantaned and Increased Lttle, s gven n Fgure 9 (a) and (b), where the support and area of fuzzy set Mantaned are greater than the respectve values of fuzzy set Increased Lttle. Let us assume that both fuzzy sets Mantaned and Increased Lttle are fred wth degrees m and m 1 m C C C, respectvely. For frng levels mc approachng 1, the centrod of the unon of cut fuzzy sets Mantaned and Increased Lttle s approachng the centrod of fuzzy set Mantaned; therefore, the sze of core fuzzy set Mantaned. On the other hand, when frng level s also approachng the sze of the core of mc s approachng 0, the centrod of unon s approachng the centrod of fuzzy set Increased Lttle and, n consequence, the sze of core s approachng the sze of the core of fuzzy set Increased Lttle (Fgure 9 (b)). Fg. 9 Comparson of the new and standard reasonng mechansms

19 Ths artcle has been accepted for publcaton n a future ssue of ths ournal, but has not been fully edted. Content may change pror to fnal publcaton. Ctaton nformaton: DOI /TFZZ , IEEE > REPLACE THIS LINE WITH YOR PAPER IDENTIFICATION NMBER (DOBLE-CLICK HERE TO EDIT) < 18 In each Fgure 9 (a) and (b), two fuzzy sets are shown; one, resultng from frng fuzzy set Mantaned wth degree of belef mc 0.9 and fuzzy set Increased Lttle wth degree of belef m 0.1, the second, resultng from frng fuzzy set Mantaned wth degree of belef m 0.1 C C and fuzzy set Increased Lttle wth degree of belef mc 0.9. It can be observed that when the new reasonng mechansm s used, the shape of fuzzy set resembles the shape of the consequent fuzzy set t s closer to, Mantaned or Increased Lttle, and mantans the fuzzness of the consequent fuzzy set t s closer to. When the standard reasonng mechansm s used, the shape and fuzzness of (2) Normalty and convexty of the are dfferent from fuzzy sets Mantaned and Increased Lttle. The standard reasonng mechansm s based on the dea that the area and varaton represented by the fuzzy set are related,.e. larger varatons are represented by fuzzy sets wth larger supports and areas. However, n some cases, ths approach can result n beng a trapezod wth a core longer than ts support. Therefore, the obtaned result s no longer a fuzzy number. To llustrate ths problem, let us consder a relatonshp between Node X and Node Z wth membershp functons defned as n Fgure 5. Let us assume that for any nput between 10 and 20, the followng two rules wll be fred: R: IF Node X s Increased Lttle ( m A ) THEN Node Z s Increased Lttle ( m C ) R: IF Node X s Increased (1 m A ) THEN Node Z s Increased Much (1 m C ) where mc m. A The sze of the core of the fuzzy set Increased Lttle s core 5 and ts area s Area 11.25, whereas the core of fuzzy set Increased Much s core 50 and ts area s IL Area IM 55. For example, for the nput 13 the frng level m of the frst rule s equal to A IL IM

20 Ths artcle has been accepted for publcaton n a future ssue of ths ournal, but has not been fully edted. Content may change pror to fnal publcaton. Ctaton nformaton: DOI /TFZZ , IEEE > REPLACE THIS LINE WITH YOR PAPER IDENTIFICATION NMBER (DOBLE-CLICK HERE TO EDIT) < 19 m 0.3 and, therefore, of the second rule s 1 m 0.7. sng formulas (8) to (12), s A calculated as Area 41.8, core 46.3, b 2.6, bo 11.6 and support As t can be seen, the support s shorter than the core of A. In Fgure 10 (a) the relatonshp between the area, core and support of dfferent frng levels wth respect to ma of the two rules gven n the Example s presented. For frng levels m A n the nterval [0.15, 0.95], the sze of the core s greater than the sze of the support, and, therefore, these trapezodal functons cannot be used as membershp functons for. To avod ths problem, n the reasonng mechansm proposed, the relaton between the area of the unon of the cut consequent fuzzy sets and the area of s removed. Fg. 10 Relatonshp between area, core and support of, for dfferent frng levels of the two rules The results of usng the new mechansm, for the dscussed example are presented n Fgure 10 (b). When compared wth the results of the standard RBFCM reasonng mechansm, t can be observed that the sze of core s smaller than the sze of support for all frng levels. The new reasonng mechansm calculates the same sze of the core, as the standard reasonng, but as t does not preserve the areas of and, t determnes dfferent area and support of than the standard reasonng mechansm. It calculates core and support n such a way that

21 Ths artcle has been accepted for publcaton n a future ssue of ths ournal, but has not been fully edted. Content may change pror to fnal publcaton. Ctaton nformaton: DOI /TFZZ , IEEE > REPLACE THIS LINE WITH YOR PAPER IDENTIFICATION NMBER (DOBLE-CLICK HERE TO EDIT) < 20 core support. Therefore, the resultng can be consdered as a convex and normal trapezodal fuzzy set. V. ACCMLATION OF IMPACTS A. Standard RBFCM accumulaton mechansm Accumulaton process s carred out when more than one causal nodes are affectng an effect node. Let us consder a case where two causal nodes: Ablty to detect compromses to CIS and Number of compromses of CIS on whch nformaton resdes are n a causal relatonshp wth one effect node, Number of CIS compromses detected, as shown n Fgure 1. If, for example, Ablty to detect compromses to CIS s Increased and causes Number of CIS compromses detected to Increase and Number of compromses of CIS on whch nformaton resdes s also Increased and causes Number of CIS compromses detected to Increase Lttle, t seems approprate that Number of CIS compromses detected should change to More than Increase. However, the standard nference mechansms, such as Mamdan [2] or Sugeno [15], wll result n Number of CIS compromses detected changng somewhere between value Increased Lttle and Increased dependng on the strength of each mpact. Therefore, a dfferent mechansm for accumulaton of mpacts n RBFCMs s requred, as proposed n [14]. The accumulaton of mpacts mechansm requres shftng the fuzzy set representng a lower varaton towards the fuzzy set representng a greater varaton. Before shftng and accumulaton are conducted, t s necessary to determne whch fuzzy set represents lower varaton. Fuzzy set A represents lower varaton then fuzzy set B when: mn( core ) mn core A B Fuzzy set A s shfted towards fuzzy set B untl the followng condton s met (Fgure 11 (a)):

22 Ths artcle has been accepted for publcaton n a future ssue of ths ournal, but has not been fully edted. Content may change pror to fnal publcaton. Ctaton nformaton: DOI /TFZZ , IEEE > REPLACE THIS LINE WITH YOR PAPER IDENTIFICATION NMBER (DOBLE-CLICK HERE TO EDIT) < 21 A shfted B mn support mn core Therefore, fuzzy set A s shfted towards fuzzy set B by dstance: shft mn core mn support A B A The standard RBFCM accumulaton process s a dscrete, recursve algorthm. It accumulates degrees of belef of every pont x as the sum of the correspondng degrees of belef of two fuzzy sets, shfted A and B ; f t s greater than 1, then the surplus n degrees s carred forward towards ponts where the sum of degrees of belef s lower than 1. The resultng Varaton Output Set, VOS VOS, representng an accumulated value of postve mpacts, s calculated as follows: x mn1, B x A x shft A carry x 1 B A A 1 carry x max 0, x x shft carry x 1 carry x (19) 1 0 The result of accumulatng two mpacts A and B, presented n Fgure 11 (a), s shown n Fgure 11 (b). The centrod of xc s calculated usng the Centrod defuzzfcaton method [16]. VOS Descrbed algorthm performs accumulaton of postve mpacts. All accumulated negatve mpacts form negatve Varaton Output Set, VOS In order to accumulate negatve mpacts, when fuzzy sets A and B represent negatve varatons, descrbed algorthm needs to be altered accordngly,.e. fuzzy set A s shfted to fuzzy set B untl the followng condton s fullfled: max support max core A shfted B. After accumulaton of both postve and negatve mpacts, the total varaton receved by the effect node s determned as the sum of the centrods xcvos xc and VOS of the two varaton output sets, postve VOS and negatve VOS respectvely, weghted by ther respectve areas:

23 Ths artcle has been accepted for publcaton n a future ssue of ths ournal, but has not been fully edted. Content may change pror to fnal publcaton. Ctaton nformaton: DOI /TFZZ , IEEE > REPLACE THIS LINE WITH YOR PAPER IDENTIFICATION NMBER (DOBLE-CLICK HERE TO EDIT) < 22 xc Area xc Area varaton Area Area VOS VOS VOS VOS VOS VOS Fg. 11 Accumulaton of mpacts usng the standard accumulaton mechansm The only characterstc of the nvolved fuzzy sets that s preserved durng ths shftng approach s the nner base of the fuzzy set representng the greater varaton. In the gven example, the nner base of about poston of the shfted fuzzy set s lost. VOS s the same as the nner base of fuzzy set B. The nformaton B. A new accumulaton mechansm The standard accumulaton mechansm reles on the relatonshp between lngustc varaton and the shape of the fuzzy set expressed n condton (2). We are proposng a new method of accumulatng mpacts when varatons of a causal and effect nodes are defned n a flexble way,.e. the lngustc varaton values are not lnked wth the shapes and areas of the correspondng fuzzy sets. The new method accumulates mpacts consderng fuzzness of fuzzy sets.

24 Ths artcle has been accepted for publcaton n a future ssue of ths ournal, but has not been fully edted. Content may change pror to fnal publcaton. Ctaton nformaton: DOI /TFZZ , IEEE > REPLACE THIS LINE WITH YOR PAPER IDENTIFICATION NMBER (DOBLE-CLICK HERE TO EDIT) < 23 Centrod xcvos of fuzzy set VOS, s calculated consderng centrods of fuzzy sets A and B, xc A and xc B, as follows: xc ( vos xc ) B xca p ˆ ˆ p 1 f A f B 2s Parameter p depends on the fuzzness of fuzzy sets nvolved n the accumulaton. The hgher the fuzzness ˆf A and/or ˆf B s, the smaller the parameter p and xcvos are. Parameter p takes the maxmum value 1 when sets A and B are crsp sets or sngletons.e., when fˆ A fˆ B 0. As a result, xcvos s equal to the arthmetc sum of centrods of the respectve sets and mpacts are fully accumulated. Parameter p takes the mnmum value 0.5 when both ˆ ˆ.5 and s 1. fuzzy sets A and B are trangular fuzzy sets, f A f B 0 Parameter s s ntroduced to ncrease the mpact of accumulaton. The hgher the value of parameter s s, the stronger the accumulaton of mpacts. Table I summarzes the mpact of the parameter s on the parameter p. Fuzzy set VOS s calculated usng the standard summaton of fuzzy sets A and B, VOS A B (Fgure 12). Characterstcs of VOS are calculated as follows: core core core VOS VOS B A b b b VOS A A B bo bo bo B support b core bo VOS VOS VOS VOS After they are calculated the fuzzy set s shfted so that the centrod of calculated usng (21). VOS s equal to centrod

25 Ths artcle has been accepted for publcaton n a future ssue of ths ournal, but has not been fully edted. Content may change pror to fnal publcaton. Ctaton nformaton: DOI /TFZZ , IEEE > REPLACE THIS LINE WITH YOR PAPER IDENTIFICATION NMBER (DOBLE-CLICK HERE TO EDIT) < 24 TABLE I IMPACT OF PARAMETER S ON THE STRENGTH OF ACCMLATION s p mn p max Fg. 12 Accumulaton of mpacts usng the new accumulaton method Negatve mpacts are accumulated n the smlar way to the accumulaton of postve mpacts. We are proposng a new approach to varaton calculatons whch consders fuzzness of postve VOS and negatve VOS mpacts receved: xcvos xcvos varaton xcvos f VOS xc f VOS 1 ˆ 1 ˆ VOS ( ) 2 fˆ VOS fˆ ( VOS ) f VOS f VOS otherwse

26 Ths artcle has been accepted for publcaton n a future ssue of ths ournal, but has not been fully edted. Content may change pror to fnal publcaton. Ctaton nformaton: DOI /TFZZ , IEEE > REPLACE THIS LINE WITH YOR PAPER IDENTIFICATION NMBER (DOBLE-CLICK HERE TO EDIT) < 25 Impacts set VOS and VOS are weghted wth ther levels of fuzzness. The less fuzzness n fuzzy VOS or VOS s, the more mportant n accumulaton of mpacts t s. C. Characterstcs of the accumulaton mechansms (1) Commutatvty and assocatvty In an RBFCM, an effect node can be mpacted by more than two causal nodes, therefore the possblty of recevng, smultaneously, more than two fuzzy mpacts exst. To assure that the order of recevng mpacts s not predomnant, accumulaton of mpacts mechansm should be commutatve and assocatve. The standard accumulaton of mpacts mechansm s commutatve; regardless whether mpact A or B s receved frst, the result of accumulaton of mpacts A and B s always the same as the fuzzy set representng the smaller varaton s shfted towards the other one representng the greater varaton. The fact that the nner slope of the fuzzy set representng the greatest varaton becomes the nner slope of the VOS causes the standard accumulaton mechansm to be nonassocatve; the order of the mpacts receved does matter when there are three or more mpacts receved smultaneously. On the other hand, the accumulaton of mpacts mechansm proposed, s both commutatve and assocatve, as t consders all the mpacts receved by an effect node n a sngle operaton that s both commutatve and assocatve (see formulas (21) (26)). (2) Accumulaton senstvty Let us consder three examples to demonstrate dfferences between the standard and the new mechansms for accumulaton of mpacts. In Example 1, defntons of fuzzy sets A Increased and B Increased Lttle volates condton (2), as presented n Fgure 13 (a). Let us assume that an effect node receves both

27 Ths artcle has been accepted for publcaton n a future ssue of ths ournal, but has not been fully edted. Content may change pror to fnal publcaton. Ctaton nformaton: DOI /TFZZ , IEEE > REPLACE THIS LINE WITH YOR PAPER IDENTIFICATION NMBER (DOBLE-CLICK HERE TO EDIT) < 26 mpacts A and B, and xc 15, Area 15, xc 27.5 and Area 10, respectvely. In A A Example 2, presented n Fgure 14 (a), the effect node receves two dentcal mpacts Increased and xc xc 27.5 and Area Area 10. A B A B B B Fg. 13 Comparson of results of the two mechansms for accumulaton of mpacts n Example 1 Fg. 14 Comparson of results of the two mechansms for accumulaton of mpacts n Example 2 Table II shows the results of the two accumulaton mechansms. Parameter s n the new mechansm s arbtrarly set to s 4 to ncrease the mpact of accumulaton. As t can be seen, f the standard RBFCM accumulaton s used, the combned effect of two mpacts Increased n Example 2 s equal to However, t s smaller than the combned effect of two mpacts Increased and Increased Lttle, n Example 1, whch s equal to Ths means that recevng two mpacts representng a larger varaton, such as Increased, results n a smaller accumulated value than f one smaller, Increased Lttle, and one larger mpact, Increased, are receved and accumulated usng the standard accumulaton algorthm. It s due to the larger area of the fuzzy set Increased Lttle than the area of the fuzzy set Increased. On the other hand, the new mechansm accumulates mpacts based on ther fuzzness levels, rather than areas. Therefore, n

28 Ths artcle has been accepted for publcaton n a future ssue of ths ournal, but has not been fully edted. Content may change pror to fnal publcaton. Ctaton nformaton: DOI /TFZZ , IEEE > REPLACE THIS LINE WITH YOR PAPER IDENTIFICATION NMBER (DOBLE-CLICK HERE TO EDIT) < 27 Example 1, the combned effect of Increased and Increased Lttle s equal to 36.3, and the combned effect of two mpacts Increased s hgher, equal to 45.83, when s 4. TABLE II COMPARISON OF RESLTS OBTAINED SING THE TWO ACCMLATION MECHANISMS xc VOS Example 1 Example 2 Example 3 Example 4 xc A xc B standard mechansm xc VOS new mechansm Example 3 and 4 demonstrate the results of the two accumulaton mechansms when dentcal mpacts, n terms of ther centrod values, are receved, but the mpacts have ndces of fuzzness. In Example 3, fuzzy sets A and B have centrods xc 16.5 and xc 37.5 and B s a crsp set wth fuzzness 0 (Fgure 15 (a)). In Example 4, fuzzy set B changes to a trangular fuzzy set wth the maxmum fuzzness 0.5 (Fgure 15 (b)). In both examples, fuzzness of fuzzy set A remans the same. Both mechansms accumulate dfferent mpact when the degree of fuzzness of fuzzy set B s ncreased (Table II); 48.7 and 49.1 usng the standard mechansm and 51.5 and 48.2 usng the proposed mechansm when s 4, n Example 3 and 4, respectvely. However, the change n the accumulated value obtaned by the standard mechansm s neglgble. A B Fg. 15 Defntons of fuzzy sets A and B The new mechansm s much more senstve to changes n fuzzness of fuzzy set B. Fgure 16 presents the accumulaton of mpacts when fuzzness of fuzzy sets B changes from 0 to

29 Ths artcle has been accepted for publcaton n a future ssue of ths ournal, but has not been fully edted. Content may change pror to fnal publcaton. Ctaton nformaton: DOI /TFZZ , IEEE > REPLACE THIS LINE WITH YOR PAPER IDENTIFICATION NMBER (DOBLE-CLICK HERE TO EDIT) < It can be observed that when the standard mechansm s used, the accumulated value abruptly ncreases when the degree of fuzzness of fuzzy set B reaches the value around It s because the sze of the area of the fuzzy set representng the greater varaton, fuzzy set B, s smaller than the sze of the area of fuzzy set A representng the smaller varaton, when the fuzzness of B s below 0.45; therefore, the standard mechansm does not accumulate mpacts correctly. Fg. 16 Comparson of results of the two mechansms for accumulaton of mpacts n Example 3 (3) Computatonal complexty The new accumulaton of mpacts mechansm s consderably less complex than the standard reasonng mechansm. It requres 6 smple arthmetc operatons to be performed (equatons (21) to (25)). On the other hand, the standard mechansm requres performng at least 20 operatons for every fuzzy set whch s accumulated, resultng n a hgh number of calculatons requred (equatons (18) to (20)). Ths s due to the recursve nature of the standard mechansm that need to be performed for all the ponts on the unverse of dscourse, where two accumulated fuzzy sets overlap. The dfferences between both mechansms are summarzed n the Table III. For example, the RBFCM developed for the cyber defence case study, conssts of 31 nodes and 44 relatonshps between them. Runnng the model for 20 teratons took 1.2 seconds on the core 5 powered laptop.

30 Ths artcle has been accepted for publcaton n a future ssue of ths ournal, but has not been fully edted. Content may change pror to fnal publcaton. Ctaton nformaton: DOI /TFZZ , IEEE > REPLACE THIS LINE WITH YOR PAPER IDENTIFICATION NMBER (DOBLE-CLICK HERE TO EDIT) < 29 TABLE III COMPARISON OF NMBER OF OPERATIONS REQIRED TO CALCLATE THE ACCMLATED IMPACT Calculaton of the shape of VOS Standard Mnmum of 20 operatons per accumulated fuzzy set Calculaton of the centrod 2 (centrod of VOS and VOS ) 2 Defuzzfcaton 1 1 New 3 VI. COMPLEX RELATIONSHIPS In some cases, to determne an mpact on the effect node, nformaton about the state of more than one causal node should be consdered smultaneously, rather than separately. Let us consder a complex relatonshp cfcr presented n Fgure 1, where Ablty to detect compromses to CIS depends on the nderstandng of Cyber tools and Cyber tradecraft. Whle the relatonshp between these three nodes could be modelled usng two FCR relatonshps, t would not capture the nuanced character of ths relatonshp. An ncrease n understandng of Cyber tools needs to be followed by an ncrease n understandng of Cyber tradecraft to have a postve mpact on the Ablty to detect compromses to CIS. If two separate FCR relatonshps are used, then any postve change n one of the nodes nderstandng of Cyber tools or nderstandng of Cyber tradecraft would have a postve mpact on the Ablty to detect compromses to CIS. However, accordng to cyber defence experts, t would not represent the realty where knowledge about the states of the two nodes smultaneously s requred to defne the mpact. Generally, a cfcr conssts of fuzzy IF-THEN rules whose antecedent part ncludes all the causal nodes nvolved n the relatonshp, oned wth AND operator. Rules are defned as follows: R : IF X s A AND Y s B THEN Z s C

31 Ths artcle has been accepted for publcaton n a future ssue of ths ournal, but has not been fully edted. Content may change pror to fnal publcaton. Ctaton nformaton: DOI /TFZZ , IEEE > REPLACE THIS LINE WITH YOR PAPER IDENTIFICATION NMBER (DOBLE-CLICK HERE TO EDIT) < 30 where R s the th rule, =1,,N N, X, Y, Z are nodes nvolved n the relatonshp and A, B, C, =1 N, are fuzzy sets defnng these nodes. Whle n the case of FCR, up to two rules can be fred, n the case of cfcr up to four rules can be fred. To calculate, when several, complex rules are fred t s necessary to calculate the unon of the cut consequent sets of the effect node: ( z) max[c' ( z),...,c' ( z),...,c' ( z)] 1 NN where, C' ( z) m C ( z), =1,..., N N s the cut consequent fuzzy set and m m m, =1,..., N N s the th rule frng level. A B sng an extenson of the reasonng mechansm proposed n Secton IV.B, s calculated n such a way that ts core, nner and outer bases depend on the shape of all fuzzy sets C, =1,,N N: core NN C c 1 core dst C b bo NN C c 1 1 b dst bo C NN C c dst C support b core bo where c NN dst C dst xc xc, s the dstance between defuzzfed unon of cut fuzzy sets and fuzzy C C set C, where only those fuzzy sets whose frng level s greater than 0 are consdered, m 0, 1,..., N N.

32 Ths artcle has been accepted for publcaton n a future ssue of ths ournal, but has not been fully edted. Content may change pror to fnal publcaton. Ctaton nformaton: DOI /TFZZ , IEEE > REPLACE THIS LINE WITH YOR PAPER IDENTIFICATION NMBER (DOBLE-CLICK HERE TO EDIT) < 31 Once the characterstcs of are obtaned, t s determned n such a way that xc xc. An example of determnng unon resultng from frng four rules R, R, R and k presented n Fgure 17 where: R : IFX s A ( m ) AND Y s B ( m ) THEN Z s C ( m ) A B C R : IF X s A ( m ) AND Y s B ( m ) THEN Z s C ( m ) A B C R :IFX s A ( m ) AND Y s B ( m ) THEN Z s C ( m ) k k Ak k Bk k Ck R : IFX s A ( m ) AND Y s B ( m ) THEN Z s C ( m ) l l Al l Bl l C l A A, A A, B B, B B and C, C, C and C are dfferent fuzzy sets. k l k l k l R l s Fg. 17 Fuzzy reasonng for a complex FCR As t can be seen, because of nputs x and y beng receved by Nodes X and Y, rules defnng the complex relatonshp between nodes X, Y and Z are fred wth the followng frng levels: m A A m m 1 m m Ak Al A Bk m m B m m 1 m Bl B B