One-Dimensional Linear Local Prototypes for Effective Selection of Neuro-Fuzzy Sugeno Model Initial Structure

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1 One-Dmensonal Lnea Local Pototypes fo Effectve Selecton of euo-fuzzy Sugeno Model Intal Stuctue Jace Kabzńs Insttute of Automatc Contol, Techncal Unvesty of Lodz, Stefanowsego 8/22, Lodz, Poland, Abstact. We consde a Taag-Sugeno-Kang (TSK) fuzzy ule based system used to model a memoy-less nonlneaty fom numecal data. We develop a smple and effectve technque allowng to emove elevant nputs, choose a numbe of membeshp functons fo each nput, popose well estmated statng values of membeshp functons and consequent paametes. All ths wll mae the fuzzy model moe concse and tanspaent. The fnal tanng pocedue wll be shote and moe effectve. Keywods: fuzzy modelng, neuo-fuzzy systems. Intoducton We consde a Taag-Sugeno-Kang (TSK) fuzzy ule based system used to model a memoy-less nonlneaty fom numecal data. It s typcal fo such applcaton to pefom a tanng of the system to mpove modelng accuacy. One of the fst (and vey successful) appoaches to the poblem was Adaptve euo-fuzzy Infeence System (AFIS) poposed by Jang []. The dea was to ealze a fuzzy nfeence system as a neual netwo and to apply neual netwo leanng technques to tan the fuzzy model. The AFIS technque (whee least squaes optmzaton of consequent paametes s used and gadent decent eo bac popagaton s appled to tune membeshp functons paametes) s offeed as a standad tool of Matlab and s stll vey popula and employed to solve vaous modelng poblems - fo example [2,3]. Seveal othe leanng algothms wee developed to tan fuzzy models ncludng genetc, bacteal and PSO algothms [4,5,6]. Befoe the tanng of paametes s stated we have to popose ntal stuctue of the fuzzy model. Fo a eal-wold poblems we may consde a geat numbe of potental nputs, but t s essental to select only the eally mpotant ones fo system descpton. Also the numbe of membeshp functons fo each nput should be caefully chosen to compomse between model accuacy and complexty. The necessty of AFIS nput selecton was notced aleady by Jang [7] and ths aspect of fuzzy model complexty educton was nvestgated by many eseaches [8], but the poblem emans open. The most popula appoaches to ntal system achtectue selecton ae based on seveal clusteng technques[8].

2 The pupose of ths wo s to develop a smple and effectve technque allowng to emove elevant nputs, choose a numbe of membeshp functons fo each nput, popose well estmated statng values of membeshp functons and consequent paametes. All ths wll mae the fuzzy model moe concse and tanspaent. The esultng fnal tanng pocedue wll be shote and moe effectve. The poposed appoach s based on a concept of fuzzy pojecton of the data on each nput space [9,0], combned wth pece-wse lnea appoxmaton allowng to choose automatcally a numbe of membeshp functons accodng to the equed accuacy and to estmate statng values of all fuzzy system paametes. We demonstate the effectveness of the poposed technque nvestgatng seveal examples and compang esults wth popula clusteng methods. 2 euo-fuzzy nfeence system We consde a neual netwo ealsaton of TSK fuzzy model []. The netwo possesses nputs x,...,x, one output y, m membeshp functons, j,..., m assocated wth the -th nput,...,, j of the fom:, m IF ( x IS ) AD... AD ( x IS ) THE whee j,...,, j, j y p x... p x q f ( x,..., x ),, R ules () m and s the ule numbe. We can choose any smooth membeshp functon, fo example genealzed bell-shape functon: j, ( x) The ule fng stength s calculated as: x c a 2b (2) w, j ( x ) fo the -th ule and the nomalsed fng stength s: The fnal output s calculated as: w w R w. (3) (4)

3 R y w f ( x,..., x ). (5) It s well nown that ths Sugeno fuzzy model may be ealsed as a fve-laye neual netwo: the fst laye calculates the membeshp functons values, the second laye - fng stengths, the thd - nomalsed fng stengths, the fouth and the ffth calculates the fnal output []. The standad tanng pocedue poposed fo AFIS combnes gadent descent bac-popagaton method to tune membeshp functons paametes and least squaes method to fnd optmal consequent paametes. The system s supposed to model a memoy-less nonlneaty gven by nput-output data wth possble extaneous nputs. We assume that that the nput-output data ae gven by ~ x,, x ~ y,,..., m. (6),, 3 Intal stuctue selecton pocedue ~ To test the sgnfcance of the -th nput,,..., we consde a sngle-nput fuzzy model (SIFM) descbed below: nput - x, output - c, nput lngustc categoes: x IS x,,..., m, membeshp functons: ules:, ( x) x x a,, 2 b (7) IF p, x IS x y, q x, p, 0, q,..., m.,,, y THE 0 f x f x,, c p 0 0, x q,. (8) The acton cuve gven by the output c of ths system fo the nput data - x, genealses nfomaton coded by x, y. The degee of ths genealsaton depends on paamete a. As we want to cove the whole ange of the -th nput by a few (say 3 o 5) membeshp functons, t s easonable to tae such a that the set n whch a membeshp functon s actve, say { x: ( x) 03. } coves 0-30% of the,

4 nteval mn( x, ), max( x, ). It may be poved that the shape of the acton cuve of SIFM s obust to sngle outles n the measued data and to the measuement nose. If the -th nput s nessental the cuve geneated by coespondng SIFM wll be flat, f t s meanngful the cuve wll cove sgnfcant pat of mn( y ), max( y ). Buldng SIFM fo evey nput, plottng and testng t s acton cuve A x, c ( x )), m we ae able to classfy the mpotance of the nput (,,, and to select sgnfcant nputs wth coespondng cuves. We must emembe that sometmes data symmety may lead to false dagnoss that the nput s elevant one. In ths case othe technques fo nput selecton must be used [8], o the modellng doman should be naowed, o nonlnea data tansfomaton may be appled. Selecton of membeshp functons fo each nput s based on pece-wse lnea appoxmaton of acton cuves deved above. Unfom o mean-squae appoach ae both applcable. The nput data have to be soted n ascendng ode and epeated values must be emoved t s mpotant especally fo gd-type data. It s not necessay to obtan contnuous pece-wse lnea appoxmaton. Mean squae appoxmaton means that statng fom the left we ae to appoxmate (by a lnea functon) maxmal numbe of subsequent ponts x, c ( x )), unde the condton (,, that the mean-squae eo s lmted by a gven paamete. Unfom appoxmaton conssts n placng the longest possble lne segments ( stcs ) nsde a -wde A x, c ( x )), m on the plane. Both tube suoundng the cuve (,,, appoaches ae easy to mplement numecally and poduces equvalent esults. Of couse desgn paamete nfluences the numbe of ntevals and so the numbe of membeshp functons. As the esult of pece-wse lnea appoxmaton fo the -th sgnfcant nput we obtan m ntevals and a lnea polynomal x x j,2, m I,. (9), j mn, j, max, j x, x j,2, m P,. (0), j ( x ) p, j x p0, j, x mn, j max, j fo each nteval. The next step s to buld a neuo-fuzzy (AFIS) model to mtate cuve A x, c ( x )), m wth small numbe of membeshp functons. The (,,, poposed membeshp functons fo the -th sgnfcant nput wll be m genealzed bell-shape functons ( ) wth paametes, j x x x, c x x, j,2, m a, j max, j mn, j, j max, j mn, j,. () 2 2 So the functon, j( x) s spanned ove I, j and cented at the mddle pont of I, j. The choce of the thd paamete b,j s abtay t s easonable to stat wth

5 b, j.5 fo all membeshp functons. The ules fo the poposed SISO neuo-fuzzy model wll be: IF x IS THE c p x p, j,2,..., m, (2), j, j 0, j whee statng values of paametes ae taen fom pecewse lnea appoxmaton esults and ae gven by (0) and (). Each ule descbes a lnea, local fuzzy pototype of fuzzy-flteed one-dmensonal data. Sngle nput sngle output neuo-fuzzy models poposed fo each sgnfcant nput ae x, c ( x )),, m. As the statng values of taned usng the data (,,, paametes (0, ) ae caefully chosen and as we tan one-dmensonal model, the tanng s fast and the esults ae accuate. Fnally, afte the tanng we get onedmensonal neuo-fuzzy models, the -th one s equpped wth m membeshp functons ( ) wth optmsed paametes a,j, b,j, c,j,, j m ules IF x IS, j THE c pˆ, jx pˆ 0, j wth optmsed paametes ˆ, p. p, j ˆ0, j The mult-nput model, whch wll be a statng pont fo the fnal tanng s equpped wth R IF ( x IS, j m q ules: ) AD... AD ( x pˆ... pˆ, j,2,, m 0, j IS, j 0, j ) THE y pˆ, j x... pˆ, j x (3) and membeshp functons paametes ae a,j, b,j, c,j. The above neuo-fuzzy nfeence system s taned by a selected pocedue to obtan the fnal model. 4 Examples The esults of the poposed appoach wee compaed wth thee standad methods of geneaton ntal stuctue of fuzzy nfeence system fom numecal data appled n Matlab []: genfs - geneates a FIS stuctue fom a tanng data set, usng a gd patton on the data (no clusteng), numbe and type of membeshp functons s gven by the use; genfs2 - geneates Fuzzy Infeence System stuctue fom data usng subtactve clusteng, cluste ad s gven by the use; genfs3 - geneates Fuzzy Infeence System stuctue fom data usng FCM clusteng, numbe of clustes s gven by the use. Use-defned paametes wee chosen to obtan the same numbe of membeshp functons as geneated automatcally by the poposed method.

6 We ae to model two 2-dmmensonal functons: a) y = x 2 + x 2 2 cos 8x cos 8x 2 (Rastagn functon), b) 2 2 sn(6 2)) y ( x x ove [0,][0,]. We geneate 400 ponts of gd data n [0,][0,] and calculate coespondng values of y. We mae a false assumpton that the thd nput nfluences the functon we consde the tples (x, x 2, x 3 ) whee value of x 3 s geneated on andom fom [0,]. As we see n both cases the cuve A 3. s flat - we can elmnate nput x 3 as an nessental one. Pece-wse lnea mean squae appoxmaton was appled and fo numbes of membeshp functons was: fo poblem a) m, m 2 = 6,6, fo poblem b) m, m 2 = 2,3. Because of paametes geneated by the lnea, local pototype appoach, the ntal poston of one-dmensonal FIS s close to the desed one and the tanng s fast and effectve. Fnally the ntal FIS was geneated accodng to (3) and compaed wth those geneated by genfs-3. Fg.. Sngle-nput fuzzy model cuves left: functon a), ght: functon b) Fg.2. Appoxmaton of sngle-nput fuzzy model cuves left: functon a), ght: functon b). Fo both functons the statng eo as well as the whole leanng pefomance was supeo n case of the poposed method. The desgn pocedue s llustated n fg. - 5.

7 Fg.3. Tanng of a sngle-nput fuzzy model to the acton cuve, functon b). Fg.4. Acton suface of an ntal FIS (befoe the fnal tanng) and numecal data +, functon b). Fg.5. Fnal model tanng: left functon a), ght functon b). Fg.6. Modelng a multdmensonal ball: left - sngle-nput fuzzy cuves, ght - fnal model tanng The next tas was modelng a secton of a multdmensonal ball wth one dumb nput. 000 ponts wee geneated on andom n the unt mult-dmensonal cube. The esults ae pesented n fg. 6. Agan the poposed method s bette than the standad clusteng pocedues.

8 5 Conclusons We popose smple and effectve pocedue fo ceatng neuo-fuzzy TSK models of complex systems fom the numecal nput-output data. Sgnfcant nputs, numbe of membeshp functons fo each nput and ntal values of all system paametes ae set automatcally based on accuacy equements. Because the ntal stuctue and paametes ae set popely, we need a few tanng teatons fo the neual netwo epesentaton of ou model to convege. The geneated FIS may be used as a statng pont to any tanng algothms also those ncopoatng educton of fuzzy ules what s hghly ecommended, although was not dscussed hee. The poposed technque was compaed wth clusteng methods mplemented n Matlab. Results obtaned fom the poposed method povded seveal tmes smalle statng tanng eo, leadng to smple models, smalle eos and educton of necessay tanng tme. Seveal mpotant poblems wee not dscussed hee, fo example elmnaton of elevant but dependent nputs. Refeences. Jang JR.: AFIS: Adaptve-netwo-based fuzzy nfeence system. IEEE Tans. Syst. Man Cyben. 23, (993) 2. Buyubngola E., Ssmanb A., Ayldzc M., Alpaslanb.F and Adejaed A.: Adaptve neuo-fuzzy nfeence system (AFIS): A new appoach to pedctve modelng n QSAR applcatons: A study of neuo-fuzzy modelng of PCP-based MDA ecepto antagonsts, Booganc & Medcnal Chemsty, 5, , (2007) 3. Kovac., Bau S.: The AFIS-based oute pefeence estmaton n sea navgaton, Jounal of Matme Reseach, III, (2006) 4. Alcalá R., Alcalá-Fdez J., Casllas J., Codón O., F. Heea F., Local Identfcaton of Pototypes fo Genetc Leanng of Accuate TSK Fuzzy Rule-Based Systems, Intenatonal Jounal Of Intellgent Systems, 22, (2007) 5. Cheng-Jan Ln, Shang-Jn Hong, The desgn of neuo-fuzzy netwos usng patcle swam optmzaton and ecusve sngula value decomposton, euocomputng 7, (2007) 6. Gal L., Botzhem J., Koczy L.T., Modfed bacteal memetc algothm used fo fuzzy ule base extacton, Poceedngs of the 5th ntenatonal confeence on Soft computng as tansdscplnay scence and technology, (2008) 7. Jang JR.: Input selecton fo AFIS leanng, IEEE Int. Conf. Fuzzy Systems, 2, (996) 8. Chenguln Hu, Feng Wan, Input selecton n leanng systems: a bef evew of some mpotant ssues and ecent developments, Poceedngs of the 8th ntenatonal confeence on Fuzzy Systems, (2009) 9. Ln Y., Cunnngham III G.A., Coggeshal S.V., Usng fuzzy pattons to ceate fuzzy systems fom nput-output data and set the ntal weghts n a fuzzy neual netwo, IEEE Tans. on Fuzzy Systems, 5, (997) 0. Kabzńs J. Woźna P., Kuźmńs K., Effectve selecton of neuo-fuzzy Sugeno model achtectue, Poc. of 7-th IASTED Intenatonal Conf. Modellng, Identfcaton & Contol (998). The MathWos, Inc. genfs.html, genfs2.html, genfs3.html,