Data-Driven Nonlinear Hebbian Learning Method for Fuzzy Cognitive Maps

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1 Data-Drive Noliear Hebbia Learig Method for Fuzzy Cogitive Maps Wociech Stach, Lukasz Kurga, ad Witold Pedrycz Abstract Fuzzy Cogitive Maps (FCMs) are a coveiet tool for modelig of dyamic systems by meas of cocepts coected by cause-effect relatioships. The FCM models ca be developed either maually (by the experts) or usig a automated learig method (from data). Some of the methods from the latter group, icludig recetly proposed Noliear Hebbia Learig () algorithm, use Hebbia law ad a set of coditios imposed o output cocepts. I this paper, we propose a ovel approach amed data-drive () that exteds method by usig historical data of the iput cocepts to provide improved quality of the leared FCMs. is tested o both sythetic ad real-life data, ad the experimets show that if historical data are available, the the proposed method produces better FCM models whe compared with those formed by the geeric method. F I. INTRODUCTION uzzy Cogitive Maps (FCMs), itroduced by Kosko [] i 986, are a covieiet coceptual ad computig machiery for modelig ad simulatio of dyamic systems. They represet kowledge i a symbolic maer ad relate states, variables, evets, outputs ad iputs usig a cause ad effect approach. Whe compared to other techiques, FCMs exhibit a umber of highly appealig properties. I particular, kowledge represetatio becomes easy ad ituitive. Oe ca easily model feedback relatioships ad capture hidde depedecies betwee the cocepts. [2]. Applicatios of FCMs are i various areas icludig egieerig [3], [4], medicie [5], political sciece [6], ecoomics [7], earth ad evirometal scieces [8], etc. There are two mai groups of approaches to develop Fuzzy Cogitive Maps: () maual methods carried out by expert(s) who have kowledge of both FCMs ad the domai of applicatio, ad (2) automated or semi-automated methods, which use learig algorithms to establish models from historical data (simulatios of cocept values). The methods from the latter group exhibit umerous advatages over the maual methods, such as idepedece of the domai of applicatio which may lead to the developmet of ubiased models [9]. Oe of the paradigms used to automate developmet of FCMs stems from the Hebbia law. The first attempt to lear FCMs usig this approach was proposed by Dickerso ad Kosko i 994, ad was referred to as Differetial Hebbia Learig (DHL) [0]. This method was further exteded ito Noliear Hebbia Learig () []. The algorithm lears FCMs from iitial expert-derived FCM model ad a set of coditios imposed o output cocepts. The algorithm does ot use historical data ad requires a expert to develop a iitial map. To this ed, we propose a ovel extesio to method, called data-drive () which uses historical data to improve the quality of leared FCM models whe compared with geeric method. It is also worth emphasizig that the proposed method does ot rely o some iitial, expert-derived FCM model. The study is orgaized as follows. Sectio II presets backgroud iformatio o Fuzzy Cogitive Maps ad motivatio of this research. I Sectio III our algorithm,, is itroduced. Sectio IV ad Sectio V describe experimets that have bee performed ad elaborate o the results, whereas Sectio VI summarizes this paper. II. BACKGROUND AND MOTIVATION A. Fuzzy Cogitive Maps FCMs defie a give dyamic system by meas of cocepts associated by mutual cause-effect relatios. Each relatio is described by a umber from iterval [-, ], which correspods to its stregth. Positive values reflect promotig effect, whereas egative values correspod to ihibitig effect. The value of represets full egative, + full positive ad 0 deotes eutral relatio. Other values correspod to differet itermediate levels of causal effect. FCMs are coveietly expressed i the form of graphs. I a graph, the odes correspod to states, ad arrows associated with umbers correspod to relatios. The graph represetatio is equivalet to a square matrix, called coectio matrix, which stores all weight values of edges betwee correspodig cocepts. Figure shows a example a process cotrol problem [], which is modeled by the FCM show i Figure 2. This work was supported i part by the Alberta Igeuity, ad by the Natural Scieces & Egieerig Research Coucil of Caada (NSERC) W. Stach, L. Kurga, ad W. Pedrycz are with the Departmet of Electrical ad Computer Egieerig, Uiversity of Alberta, Edmoto, Caada ( {wstach, lkurga, pedrycz}@ece.ualberta.ca)

2 Fig.. Simple process cotrol problem Valve ad valve 2 provide two differet liquids ito the tak. The liquids are mixed ad a chemical reactio takes place. The cotrol obective is to maitai the desired level of liquid ad its specific gravity. Valve 3 is used to drai liquid from the tak. Three experts have developed the iitial FCM model for this system []. It cosists of five cocepts that are defied as follows: C the amout of the liquid i the tak C2 the state of Valve C3 the state of Valve 2 C4 the state of Valve 3 C5 the specific gravity of the liquid ito the tak C C2 C3 C4 C5 C C C C C Fig. 2. Example of a FCM graph alog with its coectio matrix I FCMs, each cocept has a value that reflects the degree to which the cocept is active i the system at a particular iteratio (discrete time momet). This value, called activatio level, is a floatig-poit umber betwee 0 (iactive) ad (active). For the above example, activatio level of each valve determies degree to which it is ope. The value of 0 meas that a give valve is closed, value of meas that it is fully opeed, ad other values represet partially opeed valve. Similarly, values of the two remaiig cocepts correspod to differet amout of the liquid ad its gravity. Oce a FCM has bee formed ad iitial values of all cocepts were determied as iitial state of the etire system, the model ca be simulated. Simulatio boils dow to calculatig future values of cocepts at discrete time poits based o equatio (), which takes ito accout the activatio levels at the previous iteratio ad the coectio matrix N {,..., N}, C ( t + ) = f C + eici i= where: C (t) activatio level of cocept th at iteratio t e i stregth of relatio from cocept C i to cocept C f trasformatio fuctio The trasformatio fuctio is used to maitai the values of the weighted sum withi a certai rage. The ormalizatio hiders quatitative aalysis, but, at the same time, it allows to compare activatio levels of differet cocepts. A sapshot of activatio levels of all odes at a particular iteratio defies the system state. It ca be coveietly represeted by a state vector, which cosists of the odes activatio values. Iitial state vector refers to the system state at the first iteratio. Successive states are calculated by iterative applicatio of the formula (). Figure 3 shows a sample simulatio result of the model from Figure 2 started from iitial state vector suggested i [], i.e. C ( 0) = [0.4, 0.8, 0.62, 0.77, 0.3] Value of ode C C2 C3 C4 C Iteratio umber Fig. 3. Sample FCM simulatio result The values of all cocepts stabilize after a few iteratio steps, ad they correspod to a stable state of the system. Particularly iterestig are values of cocepts C ad C5 sice they represet target variables i this system. B. Developmet of Fuzzy Cogitive Maps from Data Differetial Hebbia Learig (DHL) [0] is a automated method for learig FCMs, i particular learig the coectio matrices, which is based o the law which correlates chages of causal cocepts. de i = e i + dc i dc (2) where de i is the chage of weight betwee cocept i th ad th e i is the curret value of this weight, ad dc i, dc are chages i cocepts i th ad th values, respectively. The learig process iteratively updates values of all weights from the FCM graph util the desired structure (coectio matrix) is foud. Nevertheless, the results of experimets performed with this learig method were icoclusive. As a result, several other learig approaches based o the Hebbia priciple have bee proposed. Oe of them, the Noliear Hebbia Learig algorithm () that was recetly itroduced by Papageorgiou ad colleagues [], [2] is used i this paper. This method adusts the weights based o iitial experts kowledge, i.e. iitial sketch of the map, ad additioal iformatio o the ()

3 modeled system expressed by restrictios imposed o some cocepts, to derive the coectio matrix. Therefore, the mai applicatio of method is to fie-tue the iitial model. The algorithm is based o the oliear Hebbiatype learig rule that was itroduced for Artificial Neural Networks [2], [3]. More precisely, it uses Oa learig rule origially itroduced to lear weights of euros. Adapted to Fuzzy Cogitive Maps, the rule is expressed as follows ei ( k) = ei ( k ) + η C ( Ci sg( ei ) C ei ( k ) ) (3) where: C i, C are the curret activatio values of the cocept i th ad th calculated for each iteratio accordig to the formula (), e i (k) is the weight value of the relatio betwee cocept i th ad th at the iteratio k, ad η is the learig coefficiet. Hebbia learig priciple assumes that the update of the weight e i is proportioal to the product of the C i ad C cocepts activatios. However, this may lead to ifiite growth of the weight value. The key idea behid the Oa learig rule is to avoid this effect by usig forgettig term, which is subtracted from the right had side. I this particular method, the forgettig term is proportioal ot oly to the value of the weight, but also to the square of the value of the target cocept (for e i C is the target cocept). method assumes that all the cocepts are sychroously triggered at each iteratio ad that they sychroously chage their values. The learig algorithm takes the iitial FCM ad iitial values of all cocepts ad iteratively updates the model util the desired map is foud. This method has two termiatio coditios. The first oe utilizes iformatio o desired values of some cocepts, which come from expert kowledge or problem specificatio. These cocepts are called Desired Output Cocepts (DOCs) ad they usually have predefied rage of desired values. Depedig o the particular problem, learig may termiate whe all the desired cocepts reach the desired activatio levels or whe they are close eough to these levels. For the example show i Figure 2, cocepts C ad C5 have bee determied as DOCs. The secod coditio takes ito cosideratio the variatio of the subsequet values of the DOCs ad is held if all of them chages less tha predefied very small costat e. Whe the variatio of the DOCs is smaller tha e it is poitless for the algorithm to cotiue the learig process. C. Motivatio The algorithm lears FCMs from iitial expert kowledge ad a set of coditios imposed o Desired Output Cocepts. It does ot exploit ay additioal iformatio that could improve learig ad geerate more accurate models. We ote that historical data that describe give system (a simulatio of cocept values) are ofte available. Sice the algorithm does ot take advatage of these data to improve the learig, i this paper we propose a ovel extesio to method that utilizes the historical data to develop models of better quality tha the models leared usig the geeric. The two mai obectives of the study ca be outlied as follows: to propose a ovel approach to lear FCMs from data based o algorithm, which is called data-drive oliear Hebbia Learig method () to test the proposed method by comparig the quality of leared maps obtaied from to the maps obtaied from the geeric. III. DATA-DRIVEN NONLINEAR HEBBIAN LEARNING METHOD The geeric Noliear Hebbia Algorithm () for learig Fuzzy Cogitive Maps cosists of seve steps [2], [] Algorithm STEP. Give: values of cocepts C(0) ad iitial coectio matrix E(0), ad restrictios imposed o desired values of DOCs i the form of iequalities MIN MAX C C C STEP 2. For each iteratio step k STEP 3. Update the weights accordig to equatio (3) STEP 4. Calculate C(k) for each cocept accordig to () STEP 5. Evaluate termiatio coditios usig C(k) from STEP 4, E(k), ad E(k-). STEP 6. Util both termiatio coditios are met, go to STEP 2 STEP 7. Retur the fial coectio matrix W FINAL The two coditios from STEP 6 ca be expressed as follows CONDITION (miimizig the cost fuctio F) STEP. Calculate cost fuctio for each F = DOC 2 C ( k) where T is the mea target value T of the cocept C, i.e. T C = MAX C MIN 2 The obective of the traiig process is to determie the set of weights that miimize fuctio F. CONDITION 2 (termiate algorithm after a limited umber of steps) STEP. Calculate the maximum differece e max betwee e i (k) ad e i (k-) STEP 2. If the absolute value of e max is less tha ε retur TRUE, otherwise retur FALSE Now, let us assume that historical data are available for the give system. They form a matrix D, where d i correspods to the value of i th cocept at the th time poit. I

4 other words, values of the cocepts are expressed as timeseries. The size of D is KxN where K is the umber of available data poits ad N is the umber of cocepts i the modeled system. For istace, the data show i Figure 3 would form a matrix D of size 0x5. Data-drive Noliear Hebbia Algorithm () utilizes available historical data i STEP 4, which is modified to the followig form: STEP 4. Assig C(k) with the ext row of matrix D Thus, the matrix update is carried out based o the available data, which are used at each iteratio step of the algorithm. I case all data are poits exploited ad the termiatio coditios are ot satisfied, we start usig the same data poits agai (first row of matrix D). Basically, still uses the Oa learig rule, but istead geeratig data used for learig oly from the curret model, it takes advatage of data available for a give system. still eeds the iitial coectio matrix but istead of expert-geerated map, it ca use a radomly geerated iitial map. Sice the problem of learig i this case is to obtai a FCM model, which if started from a give iitial state vector coverges to a state that fulfills certai coditios we also updated the first termiatio coditio. After each iteratio of weight update (learig), curret FCM is simulated from the iitial state vector util a stable state is reached. The, the values of DOCs from this state are compared with the desired values of DOCs. This procedure guaratees that if the solutio is foud, it would meet all the learig requiremets. Therefore, the Coditio is ow expressed i the followig way: CONDITION (checkig coditios imposed o DOCs) STEP. Simulate the curret FCM defied by E(k) startig from the iitial coditio C(0) util a fixed state is reached STEP 2. For each C that has bee defied as DOC check whether the fixed value C () meet the restrictio MIN MAX C C ( ) C STEP 3. If there is at least oe C that does ot meet the restrictio from STEP 2, retur FALSE STEP 4. Otherwise, retur TRUE IV. EVALUATION I this paper, we used both sythetic ad real-life data to evaluate the algorithm. The goal of the experimets is to compare quality of the solutios foud by algorithm with method. A. Data Sets ) Sythetic data The sythetic data used i experimets were obtaied by geeratig radom FCMs alog with radom iitial vectors. Three groups of data have bee prepared with maps that cosist of 5, 0, ad 20 cocepts, respectively. As oted i [4], i practice FCMs are usually relatively small, ad typically ivolve 5 0 odes. Additioally, for each group we geerated two subsets with differet map desities (defied as the ratio of the o-zero weights to the total umber of weights) equal to 20% ad, respectively. Agai, our choice is motivated by the results of aalysis preseted i [4], which reveals that the typical desity of FCMs is i the rage of 20 30%. Cosequetly, for the experimets we formed six differet setups. I additio, te idepedet maps were geerated for each setup to assure statistical validity of the results. 2) Real-life data For real-life experimets we chose a FCM model for a process cotrol problem that was itroduced i [] ad already show i Figure ad 2. The model has 5 cocepts ad the desity of the map is 32%. We decided to use a real FCM model rather tha raw data sice the method requires a iitial coectio matrix as a iput. This way we could ivestigate the impact of providig the true or a radomized matrix as the iput. B. Experimets ) Sythetic data For each setup, we took the geerated models ad iitial vectors ad performed simulatio util covergece was reached. The, we arbitrary chose of cocepts to be DOCs (this value was selected to be cosistet with the reallife model reported i []). It correspods to 2, 4, ad 8 DOCs for maps that iclude of 5, 0, ad 20 odes, respectively. For the selected cocepts the desired rage of values were established as ±0. of the stable value. It meas that if a give cocept C i stabilized after the simulatio at the activatio level of a, mic i ad maxc i has bee calculated as a-0., ad a+0., respectively. These values were used as the first termiatio criterio i both ad methods. The secod criterio e was set to 0.02 as suggested i []. The learig procedure for both ad was carried out as follows. Firstly, we radomly geerated iitial map. Next, we performed learig usig both ad DD-. Additioally, if the termiatio criteria could ot be met after 00 iteratios, the procedure was restarted with a ew iitial map. 2) Real-life data The model of a process cotrol system from [] was used to perform two groups of experimets:. I the first oe, we took the iitial model proposed by experts (Figure 2) ad performed learig with method. The map that has bee established met all the restrictios defied i [], i.e N 0. ad 0.74 N 5 0., ad, therefore, was cosidered as the desired model of this system for the remaiig experimets. Next, we geerated data by simulatig this

5 model, ad performed learig usig method. 2. I the secod group of experimets, we radomly chose iitial matrix ad carried out learig usig both ad usig the same procedure as for sigle experimet with sythetic data. C. Evaluatio Criteria We cosidered three evaluatio measures to assess the performace of the proposed method. They are cosistet with the FCM evaluatio criteria proposed i [4] ad they have bee used with experimets o both sythetic ad reallife data: i-sample error differece betwee the available data, ad data geerated by simulatig the leared model from the same iitial vector. The criterio is defied as a ormalized average error betwee correspodig cocept values at each iteratio betwee the two state vector sequeces. K N error _ iitial = C ˆ C (4) ( K ) N t= = where C (t) is the value of a ode at iteratio t i the iput data, Cˆ is the value of a ode at iteratio t from simulatio of the leared model, K is the iput data legth, ad N is the umber of cocepts. out-of-sample error evaluatio of the geeralizatio capabilities of the leared FCM. To compute this criterio, both the desired ad leared models are simulated from te radomly chose iitial state vectors. Subsequetly, the value of the measure error_iitial is computed for each of the simulatios to compare state vector sequeces geerated by these models, ad a average of these values is computed. P K N p p error _ behavior = C Cˆ (5) P ( K ) N p= t = = where C p (t) is the value of a ode at iteratio t for data geerated by desired model started from p th iitial state vector, Cˆ p is the value of a ode at iteratio t for data geerated by leared model started from p th iitial state vector, K is the iput data legth, N is the umber of cocepts, ad P is the umber of radom iitial state vector. fial-state accuracy evaluatio of meetig the restrictios o DOCs. The model is simulated from the iitial coditio util fixed state is reached, ad the the followig formula is used DOCi = % (6) DOCi where DOC is the umber of DOCs that meet the i restrictios after simulatig correspodig model from the iitial vector, ad DOC i is the total umber of DOCs. This measure is calculated for both i-sample ad out-ofsample experimets. TABLE I IN- ERROR RESULTS FOR SYNTHETIC DATA IN- IN- 5 20% 0 20% 20 20% 5 20% 0.53 (0.05) 0.49 (0.04) 0.82 (0.07) 0.93 (0.009) 0.23 (0.05) 0.20 (0.05) 0.29 (0.03) 0.29 (0.008) 0.76 (0.05) 0. (0.06) 0. (0.08) (0.04) TABLE II ERROR RESULTS FOR SYNTHETIC DATA 0 20% 20 20% 0.52 (0.007) 0.49 (0.0) 0.82 (0.04) 0.93 (0.06) 0.23 (0.04) 0.20 (0.07) V. RESULTS 0.29 (0.0) 0.29 (0.05) 0.75 (0.00) 0. (0.0) 0. (0.04) (0.06) A. Sythetic Data Table I ad Table II summarize the experimetal results for the sythetic data. Reported values have bee calculated as averages obtaied from 0 idepedet experimets (with differet models) for each setup. The rows correspod to differet experimetal setups i terms of maps sizes (5, 0, ad 20) ad desities (20% ad ), ad the value i each cell expresses the average value of a correspodig criterio. For i-sample ad out-of-sample errors the average values are followed by stadard deviatios (i brackets) across all te rus. Experimetal results show that the data-drive approach is o average better whe it comes to the i-sample errors, i.e., average errors equal 0.83 for vs for datadrive. This is because we use historical data for learig data-drive, whereas the geeric algorithm takes ito cosideratio oly the fial state. However, more importat are results icluded i Table II, as they determie how well a give model captures (geeralizes) the kowledge of the target domai. This is because out-of-sample experimets are performed o

6 previously usee data. They show that cosistetly, over differet map sizes ad desities, produces better FCM models whe compared with. The quality of learig decreases slightly with the icrease the map size for both methods. The out-of-sample errors are worse for 20 odes maps whe compared with errors for 5 odes i case of, ad 49% worse i case of. Also, we ote that the map desity does ot have sigificat ifluece o the learig process. Takig ito cosideratio the last criterio, i.e. fial-state accuracy, the advatage of usig method becomes evidet. The solutios foud by method do ot meet the learig obective i 20- of experimets. Our method is guarateed to meet the coditios for DOCs for the i-sample experimets. However, it turs out that the DOC coditios were also satisfied i all out-of-sample experimets performed with method. Table III covers the results of statistical aalysis of the differeces betwee the results produced by the ad We performed paired t-test at 95% cofidece betwee the correspodig pairs of 0 experimets performed with ad methods. TABLE III STATISTICAL RESULTS COMPARISON THROUGH PAIR T- TESTS vs. T-VALUE 5 20% % % Sice the critical t-value at 95% cofidece equals 2.26, the results show that the differeces are statistically sigificat, i.e., the method provides statistically sigificatly lower error rate, for 5 out of 6 differet setups. The differece is ot statistically sigificat oly for the largest map with the high desitiy. B. Real-life Data Tables IV ad V show the results for the first experimet. We ote that the method was used to geerate the desired model, ad therefore all the error measures are equal 0 i this case. TABLE IV IN- ERROR RESULTS FOR THE FIRST EXPERIMENT WITH REAL MODEL IN- IN % TABLE V OUR-OF- ERROR RESULTS FOR THE FIRST EXPERIMENT WITH REAL MODEL % C C2 C3 C4 C5 C C C C C C2 C3 C4 C5 C C C C C C Fig. 4. vs. models for the first experimet The learig performed by startig from the iitial matrix geerated by experts (Figure 2) resulted i better learig quality whe compared to results o sythetic data. This is because the iitial map was similar to the desired map. Figure 4 illustrates coectio matrices of the models foud by ad. Bolded are values that differ by more that 0.25 betwee the two solutios. The two models are very similar ad differ by more tha 0.25 ust for oe weight. O average, the differece is 7%. Tables V ad VI report experimetal results for the secod experimet. Similarly to the experimets for sythetic data, 0 idepedet experimets were carried out ad the average values are show i the tables. TABLE VI IN- ERROR RESULTS FOR THE SECOND EXPERIMENT WITH REAL MODEL IN- IN % 0.58 (0.009) 0.35 (0.0) TABLE VII OUR-OF- ERROR RESULTS FOR THE SECOND EXPERIMENT WITH REAL MODEL % 0. (0.00) 0.37 (0.02) The tests show that produces FCMs that are 4% better (they reduce the correspodig error rates by ( )/0.58=4.6% for i-sample ad ( )/0.=4.4%) whe compared with maps geerated by algorithm for both i-sample ad out-of-sample measures. The criterio show that the method is capable to fid models that fulfill the learig obectives (restrictios imposed o DOCs), whereas the approach satisfies the obectives i -% of

7 experimets. Similarly to results from Table III, the statistical aalysis of the results was performed. The paired t-test value betwee ad was equal to 5.72, which meas that the differece is statistically sigificat at 95% cofidece. VI. CONCLUSIONS I this paper, a ovel method for automated learig of FCMs from data, amed, is itroduced ad experimeted with. The proposed method applies a oliear Hebbia priciple ad available data to geerate FCM models. The mai idea behid the method was to use historical data to improve learig quality. Whe compared to method for learig FCMs, i the stoppig criterio was modified to achieve models that fulfill the iitial requiremets o certai, predefied cocepts called Desired Output Cocepts (DOCs). Experimetal results for both sythetic ad real-life data show that is capable of formig better quality FCMs whe compared with the costructs resultig from the algorithm. Results for sythetic data are o average 9% better for for the out-of-sample tests. Both methods produce slightly worse solutios for larger maps with both errors growig liearly. Experimets with real-life data show that method provides satisfactory solutios whe started from a matrix that is predefied by the expert ad close to the desired solutio. Comparative aalysis shows the advatages of usig over method, i terms of both lower i-sample ad out-of-sample errors as well as better ability to satisfy the coditios set o DOCs; the results are cosistet with these obtaied for sythetic data. To sum up, the proposed algorithm lears better models tha the geeric method. However, this improved quality requires the availability of historical data. I cotrast, algorithm does ot require historical data, but it relies o expert-derived iitial map ad a set of coditios o DOCs. [7] D. Kardaras, ad G. Metzas, Usig fuzzy cogitive maps to model ad aalyse busiess performace assessmet, i Advaces i Idustrial Egieerig Applicatios ad Practice II, J. Che, ad A. Mital, (Eds), pp , 997 [8] R. Giordao, G. Passarella, V. F. Uricchio, ad M. Vurro, Fuzzy cogitive maps for issue idetificatio i a water resources coflict resolutio system, Physics ad Chemistry of the Earth, vol. 30, o. 6 7 (Special Issue), pp , 2005 [9] W. Stach, L. A. Kurga, ad W. Pedrycz, A survey of fuzzy cogitive map learig methods, I: P. Grzegorzewski, M. Krawczak, ad S. Zadrozy, (Eds.), Issues i Soft Computig: Theory ad Applicatios, Exit, pp. 7 84, 2005 [0] J. A., Dickerso, ad B. Kosko, Virtual worlds as fuzzy cogitive maps, Presece, vol. 3, o. 2, pp , 994 [] E. I. Papageorgiou, C. D. Stylios, ad P. P. Groumpos, Fuzzy cogitive map learig based o oliear Hebbia rule, I: T. D. Gedeo, ad L. C. C. Fug, (Eds.), Lecture Notes i Artificial Itelligece, Spriger Verlag, vol. 2903, pp , [2] E. Oa, Neural etworks, pricipal compoets ad subspaces, Iteratioal Joural of Neural Systems, vol., pp. 6-68, 989 [3] E. Oa, H. Ogawa, J. Wagviwattaa, Learig i oliear costraied Hebbia etworks, I: Kohoe T., et al. (Eds.), Artificial Neural Networks, North-Hollad, pp , 99 [4] W. Stach, L. Kurga, W. Pedrycz, ad M. Reformat, Geetic learig of fuzzy cogitive maps, Fuzzy Sets ad Systems, vol. 53, o. 3, pp , 2005 REFERENCES [] B. Kosko, Fuzzy cogitive maps, Iteratioal Joural of Ma- Machie Studies, vol. 24, pp , 986 [2] E. I. Papageorgiou, C. Stylios, P. P. Groumpos, Usupervised learig techiques for fie-tuig fuzzy cogitive map causal liks, Iteratioal Joural of Huma-Computer Studies, vol. 64, pp , 2006 [3] W. Stach, L. Kurga, W. Pedrycz, ad M. Reformat, Parallel fuzzy cogitive maps as a tool for modelig software developmet proect, North America Fuzzy Iformatio Processig Society Coferece (NAFIPS 04), pp , 2004 [4] M. A. Stybliski, ad B. D. Meyer, Sigal flow graphs versus fuzzy cogitive maps i applicatio to qualitative circuit aalysis, Iteratioal Joural of Ma Machie Studies, vol. 35, pp , 99 [5] P. R. Iocet, ad R. I. Joh, Computer aided fuzzy medical diagosis, Iformatio Scieces, vol. 62, o. 2, pp. 8 04, 2004 [6] M. Kha, ad M. Quaddus, Group decisio support usig fuzzy cogitive maps for causal reasoig, Group Decisio ad Negotiatio Joural, vol. 3, o. 5, pp , 2004