Compuer Applcaons n Elecrcal Engneerng Accuracy of he nellgen dynamc models of relaonal fuzzy cognve maps Aleksander Jasrebow, Grzegorz Słoń Kelce Unversy of Technology 25-314 Kelce, Al. Tysącleca P. P. 7, e-mal: enegs@u.kelce.pl Applyng fuzzy relaonal cognve maps n dynamc modellng work of he sysems nvolves resrcons dervng from he assumed model parameers. The selecon of hese parameers depends mosly on ables of calculang equpmen used for he smulaon and on he modellng purposes. In mos cases s necessary o balance beween ncreasng he mappng accuracy (whch s conneced wh he calculaon me lenghenng) and shorenng he calculaon me (whch, n consequence, worsens he accuracy). Addonally, amng a he accuracy maxmzaon no always can really mprove, bu s always conneced wh he growng of he compuaonal load. In hs chaper he analyss of he nellgen cognve maps work accuracy n he realzaon dynamc models s elaboraed. As a resul of he numercal analyss here was shown he exsence of ceran opmal parameers of analyzed sgnals fuzzyfcaon and conneced wh hem samplng parameers n fuzzy arhmecal operaons performed durng he modellng processes. 150 1. Inroducon In works [1-5] here were nroduced and analyzed applyng sac and dynamc models of fuzzy relaonal cognve maps n decsonal monorng low-srucural objecs. The research resuls showed he exsence of ceran dependences beween accuracy of such models work, seleced parameers of fuzzy cognve maps and he lengh of he samplng sep chosen for numercal calculaon. I s specally demonsraed n dynamc models, where he sablzaon of he sysem afer smulang by exernal sgnals needs ceran number of he cycles of he sgnals flow hrough feedbacks. In he chaper, he resuls of he smulaon analyss of he relaonshp beween acons accuracy, membershp funcons parameers and fuzzness measure of he appled sgnals, wll be presened on he example of chosen fuzzy relaonal cognve map. The resuls wll be shown n he form of approprae dagrams, derve from whch he exsence of ceran opmal parameers of he models fuzzyfcaon. 2. Model of he analyzed cognve map Generally speakng, a cognve map can be presened as followng par of ses: <X, R> (1) where: X = [X 1,..., X N ] T he se of values of he map conceps (sae vecor), R = {R j } marx of relaons beween varables X and X j (,j = 1,..., N).
Fgure 1 presens graphcal model fuzzy relaonal cognve map chosen for numercal analyss. Fg. 1. The explored cognve map graphcal model (N = 5) Dynamc model of fuzzy relaonal cognve maps for he objec from Fg. 1 can be presened n he form (2) [4]: X ( 1) X ( ) [ ( ) k k 5 1 X X ( 1)] R, k (2) where: X k he k-h concep value (k = 1,..., 5), dscree me, operaon of fuzzy addon, operaon of fuzzy subracon, R,k ndvdual fuzzy relaon beween fuzzy conceps numbered and k, operaon of max-mn fuzzy composon. Machng fuzzy parameers, should be consdered: he ype of membershp funcon, accordng o whch ndvdual conceps wll be fuzzyfed (Fg. 2), unversum doman (whch depends on he expeced concep values) and he number of he unversum samplng pons. The essenal queson s also deermnng he mehod of normalzaon of conceps, whch s necessary owng o her dsnc physcal characerscs. For he needs of hs analyss here was chosen non-dmensonal normalzaon o he doman [-1, 1]. For fuzzyfcaon, accordng o he algorhm presened n [3, 4], Gauss ype funcon (3) whose graphcal represenaon (for seleced parameers) s presened n Fg. 3, was chosen. 2 ( xc) ( x ) e (3) where: µ(x) membershp funcon, x argumen, c µ(x) funcon cenre, σ µ(x) funcon wdh coeffcen. 151
Gauss ype membershp funcon µ(u ) -2,0-1,5 - -0,5 0,5 1,5 2,0 u Trangular membershp funcon µ(u ) -2,0-1,5 - -0,5 u 0,5 1,5 2,0 Trapezodal membershp funcon µ(u ) -2,0-1,5 - -0,5 u 0,5 1,5 2,0 Fg. 2. Examples of membershp funcons, whch can be used for fuzzyfcaon of cognve map conceps, u unversum, µ(u) membershp funcon 152
Fg. 3. Hypohecal course of one of npu sgnals, marked as X 1 (afer Gauss ype fuzzyfcaon wh σ = 0.2), wh momenary value (cenre) equals o 0.35 (afer non-dmensonal normalzaon o doman [-1, 1]) Smlarly o fuzzyfcaon of conceps, also a he choosng of fuzzy relaon characerscs one can use dfferen membershp funcons, whch can be he base for creang such he relaons (Fg. 4). For he needs of presen smulaon here were chosen he Gauss ype relaons, srenghs of whch were correspondng wh values of he crsp relaons presened n (2). Durng he process of deermnng he unversum doman, wo conceps should be aken no consderaon: normalzaon doman boundares and wdh coeffcen of he membershp funcon σ. Normalzaon doman boundares should be chosen n he way whch secures maxmal symmery of fuzzy conceps shapes n full range of normalzed values. I means e.g. ha f σ < 0.8 for normalzaon range [0, 1], unversum doman can amoun from -1 o 2, bu for normalzaon range [-1, 1], unversum doman should be wder from -2 o 2. Generally, for hgher values of σ he unversum should be wder owng o necessy of keepng he above menoned symmery. Fnally he unversum wh doman [-2, 2] was chosen. In he furher par of he chaper here wll be presened resuls of he research on dependency of he accuracy of fuzzy relaonal cognve map acves on membershp funcon parameer σ (fuzzness degree FUZ) of he model presened n (4)-(5) [6] and on samplng sep Δx of he fuzzy ses unversum X = [-2, 2] (x k = -2 + Δx k, k = 0,..., K). 1 FUZ( X ) 1 D2( μ ( x)) 1 2 X K (4) D K 2 μx ( xk ) 2 2 μx ( x)) 1 k0 ( (5) where: X (x) membershp funcon of fuzzy se X ype (3), = 1,..., 5. μ 153
Gaussodal relaon µ(r,u 1,u2) -2,0-1,5 - -0,5 0,5 1,5 2,0 u1 2,0-2,0 - u2 Trangular relaon µ(r,u 1,u2) -2,0-1,5 - -0,5 0,5 1,5 2,0 u1 2,0-2,0 - u2 Trapezodal relaon µ(r,u 1,u2) -2,0-1,5 - -0,5 0,5 1,5 2,0 u1 2,0-2,0 - Fg. 4. Examples of fuzzy relaons bul on he bass of dfferen knds of membershp funcons; u 1, u 2 unversum varables, µ(r,u 1,u 2 ) membershp funcon of he relaon u2 154
3. Seleced resuls of smulaon analyss Marx of relaons r = {r,k } (, k = 1,..., 5) was deermned as follow (accordng o Fg. 1): 0 0 0 0 0 0 0,3 r 0 0 0 0,1 0,1 (6) 0,1 0 0 0 0,3 0,5 0 0 Elemens of marx r are values of reference for he consrucng ndvdual fuzzy relaons [3, 4], whch can be desgned by expers or durng he learnng process. These relaons are elemens of he fuzzy relaons marx R, whch s he bass of he operang of he relaonal fuzzy cognve map used n he esed model. I was assumed ha he sysem wll be acng under nfluence of one-sho forcng seleced conceps o ceran values. These values are shown n Table 1. Table 1. Values of smulang sgnals Concep number 1 2 3 4 5 Smulang value (normalzed) 0.5 0.4 0 0 0 In consecuve calculaon seps he sysem obans a ceran sae of equlbrum, whch s he bass for he concluson. The smulaon was carred ou for 200 seps of dscree me. A. Comparave resuls of analyss for crsp and defuzzyfed courses of conceps presened n Fg. 1, accordng o marx R from (6) for σ = 0.2 and dfferen values of K In Fg. 4 here s presened comparson of me courses of esed sysem conceps n dynamc crsp model and fuzzy model for dfferen number of samplng pons of he unversum (afer defuzzyfcaon wh weghed average mehod). From Fg. 4 resuls ha he lower number of he unversum samplng pons he larger dfferences beween me courses of he fuzzy model conceps. Furher research also shows dependency of hs dfference on he value of σ coeffcen (Fg. 5). Therefore can be saed ha accuracy of fuzzy models depends on he number of he unversum samplng pons K (whch s dencal wh he number of lngusc funcons seleced for fuzzyfcaon) and he coeffcen of he membershp funcon wdh σ. I s que nuve assessmen, based on vsual comparson of me courses, bu he conclusons objecfcaon bears he presenaon numercal creron, whch would allow o apprase he accuracy of he mappng more accuraely and o pon he suffcen level of hs accuracy. 155
- a) - b) - c) d) - e) - Fg. 4. of conceps for dfferen models. a) crsp sysem, b) e) fuzzy sysems wh dfferen numbers of lngusc funcons (K) on he unversum doman (wh consan value of he coeffcen σ = 0.2): b) K = 101, c) K = 65, d) K = 41, e) K = 33 156
- a) b) - c) - d) - e) - Fg. 5. of conceps for dfferen models. a) crsp sysem, b) e) fuzzy sysems wh dfferen values of σ coeffcen (wh consan number of he lngusc funcons on he unversum doman K = 41): b) σ = 0.3, c) σ = 0.4, d) σ = 0.5, e) σ = 0.7 157
B. Resuls of numercal apprasal of he creron of nearness beween crsp and defuzzyfed values Apprasal of he accuracy level of he mappng of courses by fuzzy model was performed usng nearness creron (7) consderng devaon beween fuzzy course and crsp course ha was aken as he comparson base. The am of he sudyng of he above menoned creron s an aemp a fndng he number of he unversum K samplng pons and he membershp funcon wdh coeffcen σ ha secure he mnmal value of he creron (7) n specfc crcumsances. J ( FUZ) 1 200 200 w o 2 X ( n) X ( n) FUZ n0 where: X w (n) defuzzyfed course of -h concep of he cognve map (1), X o (n) crsp course of -h concep of he cognve map (n equaon (1) fuzzy operaors was replaced wh arhmecal operaors), = 1,..., 5. Deermnng he course of funcon J(FUZ) allows o dscover s mnmum for gven value of K. Ths mnmum can ake dfferen values for dfferen values of K, moreover also depends on earler assumed lmaons of he calculang sysem (e.g. on he unversum u doman boundares). I should be also consdered ha such research s carred ou ndependenly for each concep and s resuls can be dfferen for dfferen conceps. Fgs. 6 and 7 presen dagrams of funcon J(FUZ) for concep X 1 for wo dfferen values of K. I should be noced ha de faco hey presen dependency on he membershp funcon wh coeffcen σ because FUZ s funcon of σ. mn (7) Fg. 6. The course of funcon J(FUZ) values of concep X 1 for K = 33 158
Fg. 7. The course of funcon J(FUZ) values of concep X 1 for K = 41 The comparson of courses from fgs. 6 and 7 leads o he observaon ha mnmal value of he nearness creron J(FUZ) can occur for dfferen values of he fuzzness degree FUZ of gven concep and s locaon depends on he assumed echncal parameers of calculang sysem (number of he unversum K samplng seps and, ndrecly, he unversum u doman wdh). Therefore can be saed ha for a consan value of K (or Δx) here s opmal value FUZ * dependen on he parameer σ. Accordng o hs here can be formulaed he problem of fndng up he opmal value of σ. Generally, analyzng he resuls of A. and B. can be saed ha here s a problem of he opmzaon of he selecng parameers σ, Δx and R, whch can be solved by usng dfferen opmzaon algorhms (e.g. graden or genec) [7]. 4. Conclusons Resuls of he paral numercal analyss of he accuracy of nellgen dynamc models of fuzzy cognve maps presened n he work, lead o exsence of ceran opmal parameers of he fuzzyfcaon wh usng max-mn composon beween fuzzy conceps and approprae fuzzy relaons. The problem of seekng opmal parameers of he model s conneced no only wh he modelled sysem parameers hemselves. I also requres consderaon of he calculang equpmen echncal ables and expeced calculaon me. For fndng opmal parameers here s proposed usng ceran opmzaon mehods based on graden or genec algorhms ha wll be presened n furher works. References [1] Borsov V. V., Kruglov V. V., Fedulov A. C. Fuzzy models and neworks. Telekom, Moscow 2007 (n Russan). [2] Jasrebow A., Gad S., Słoń G., Analyss of he fuzzy cognve maps dynamcs n dagnosc monorng of sysems. Proc. of XIV Scenfc Conference Compuer Applcaons n Elecrcal Engneerng ZKwE 2009, Poznan 2009, pp. 287-288 (n Polsh). 159
[3] Jasrebow A., Słoń G., Fuzzy cognve maps n relaonal modellng of low-srucural sysems. In: Jasrebow A. (red.) Compuer Scence n XXI Cenury. Informaon Technologes n Scence, Technology and Educaon. Scence Publshng House of Insue of Exploaon Technology Naonal Research Insue, Radom 2009, pp. 35-38 (n Polsh). [4] Jasrebow A., Słoń G., Fuzzy cognve maps n relaonal modellng of monorng sysems. In: Kowalczuk Z. (red.) Sysems of fauls deecon, analyss and oleraon. PWNT, Gdansk 2009, sr. 217-224 (n Polsh). [5] Kosko B., Fuzzy cognve maps. In. Journal of Man-Machne Sudes, Vol. 24. pp. 65-75, 1986. [6] Osowsk S., Neural neworks for he nformaon processng. Prnng house of Warsaw Unversy of Technology, Warszawa 2000 (n Polsh). [7] Sach W., Kurgan L., Pedrycz W., Reforma M., Genec Learnng of Fuzzy Cognve Maps, Fuzzy Ses and Sysems, Vol. 153, Augus 2005, pp. 371-401. 160