A Recursive Algorithm Based on Fuzzy Neural Network for Target Number

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1 A Rcursv Algorth Basd on Fuzzy Nural Ntwork for argt Nubr Zhang Png School of raffc & ransport Engnrng, Cntral South Unvrsty,Changsha , Hunan, Chna Abstract o ovrco ths dfcncy, a copound fuzzy nural ntwork nad IMJ was ntroducd whch cobns th procss s rul-basd rasonng and functon approxaton functons. hs papr prsnts th concpt of targt nubrs, and ntroducs a nubr of rcursv algorth targt ssus. Suppos gvn postv ntgr n s A (1), A (),..., A (n-1), A (n), th nd to fnd th corrspondng nubr of targtng, guarantd to ak th su of th dffrnt lnts and th cobnaton of valus n th vcnty of th targt nubr (ay b qual or th absolut valu of th dffrnc btwn th nu cas) for th ost nubr of cobnatons. Such copound fuzzy nural ntwork phaszs th us of pror knowldg and studyng that aks t possbl to ffctvly prov th approxaton and gnralzaton of coplx nonlnar systs n larg sgnts rang. W ad a study of th ost nubr of cobnatons that can b consdrd a targt for loadng probls n th bst hght lvlng. h coon fuzzy nural ntwork dos not ntroduc th objct-orntd pror knowldg fro a practcal sns, such as ntroducng knowldg of opratng and controllng xprnc as wll as procss fld, rsultng n poor usng of fuzzy advantags. Kywords - argt Nubrs, Rcursv Algorth, Fuzzy Nural Ntwork I. INRODUCION BllazzR.tal. Study on th odl of fuzzy nural syst usng th thod of qualtatv sulaton [1]. F. Akhtov Daourntal proposd a fuzzy nural ntwork wth gntc factors to odl th nonlnar t srs syst []. F. Akhtov Daourntal proposd a fuzzy nural ntwork wth gntc factors to odl th nonlnar t srs syst [3]. h nubr of bnchark probls ar dscrbd as follows: gvn a postv ntgr n A (I) (1 n), {A (1), A (),..., A (n-1), A (n)} n th prsnc of such a bnchark nubr, so that th su of dffrnt lnts n th collcton and th absolut valu of th dffrnc btwn th valu and th valu of M, th cobnaton of dffrnt lnts s th ost [4]. h bnchark probl s NP coplt probl.h cobnaton of fuzzy logc syst tchnology wth nural ntwork s a nw da. hs nrv-fuzznss or fuzznss-nrv collaboratv ntgraton taks advantags fro nural ntwork and fuzzy syst n th followng two aspcts: th nural ntwork provds th fuzzy logc syst wth a connctv structur (wth natur of fault-tolranc and dstrbutd prsntaton) and studyng ablts; and fuzzy logc syst provds th nural ntwork wth a frawork wth advancd fuzzy ruls of f-thn thnkng and rasonng [5]. hr coplntary cobnaton aks th fuzzy nural ntwork bco a ajor dvlopnt trnd of ntllgnt control. Copard wth tradtonal thod, th fuzzy nural ntwork has ovrwhlng advantags n nonlnar syst odlng and dntfcaton aspcts. II. BENCHMARK NUMBER AND BENCHMARK NUMBER RECURSION ALGORIHM St S={A (1), A (),... (A), (n-1), A (n)}, th pol nubr s M, M s A (1), A (),..., A (n-1), A (n) ar postv ntgrs. If th gvn probl s solvd, th M should b A (1), A (),..., A (n-1), A (n), and th nubr of cobnatons that can b coposd of a cobnaton of a nubr of lnts n th ddl part of th lnt and (or th nu valu of th dffrnc btwn th valu of th absolut valu of th absolut valu). Rcursv algorth s a drct or ndrct way to call th procss of ts own algorth. In coputr prograng, rcursv algorth s vry ffctv for solvng a larg class of probls, whch oftn aks th dscrpton of th algorth s spl and asy to undrstand. Rcursv algorth to solv th probl: (1) th rcurson s calld n th procss or functon. () n th us of a rcursv stratgy, thr ust b a clar nd to th nd condton, known as rcursv xports. (3) th rcursv algorth s vry spl, but th opraton ffcncy of th rcursv algorth s vry low. So w do not advocat usng rcursv algorth dsgn procdurs. (4) n th procss of rcursv calls to th syst for ach layr of th rturn pont, th aount of local volu, such as opnng up th stack to stor. oo uch rcurson can asly lad to stack ovrflow, tc.. So w do not advocat usng rcursv algorth dsgn procdurs. Bnchark nubr rcurson algorth basd on th actual stuaton can b dscrbd as: n a gvn st, S={A (1), A (),... (n-1), A (), A (n)}, bcaus t s possbl to fnd a plan, ths papr consdrs that th hght and hght lt of dffrnt hght of th goods ar pld up, and th hght lt s st to prov th vhcl hght. In th rsarch procss, th nubr of bnchark usng xhaustv thod, naly fro a lowr bound valu to th uppr lt valu ach t n ordr to pass hghr dgr dstanc of 1 n ascndng and allows th loadng of th goods on th hght and nubr of bnchark tolranc valu of 1, th data for DOI /IJSSS.a ISSN: x onln, prnt

2 coputng and fnd th bnchark nubrs and satsfy dffrnt cobnatons. th bnchark nubr to t all th cobnaton of tolranc. Fgur. argt nubr algorth chart Fgur 1. Sarch targt nubr chart III. ALGORIHM ANALYSIS A. Algorth dscrpton Accordng to fgur 1, th analyss of th rcursv algorth for solvng th probl of pol nubr s gvn. h whol algorth conssts of two stags. h frst stag s th frst plntaton of th prprocssng, and thn prfor th followng stps. h stps ar dscrbd as follows: Prprocssng stp: frst, th lnts n A ar stord n th st B, th lnts n th orgnal A ar not arrangd n ordr. Stp 1: f th A s pty, th nd of ths phas.. Stp : P, P3, P1,... Pn untl th rgnc of a crtan valu and valu n th spcfd bnchark wthn th tolranc rang, rcord th nubr of cobnatons of dffrnt bncharks and th cobnaton of lnts. Accordng to th calculaton rsults, th followng cass ar dvdd nto th followng: (1): th valu of th tolranc rang of th nubr of bncharks not found to t th condtons of th bnchark s dtrnd to b th nd of th algorth; (): fnd a Ps qual to th rang of tolranc wthn th bnchark, rcord th cobnaton, and fro th nxt lnt bgan to contnu to travrs th calculaton, to fnd B. Algorth t coplxty In th algorth, th hghst frquncy of xcuton s th addton of stp. In th frst phas of th algorth, th ntal lnts of th st A ar assud to b n, thn th frst stp s, and th axu s n. Scond plntaton stps, du to th collcton of A has bn rducd by th 1 lnts, so th axu n-1 ts. hrd plntaton stps, du to th collcton of A has bn rducd by th lnts, so th axu n- ts. Su up, th total nubr of frst stags of th algorth s not or than: n+ (n-1) = (1+n) n/. So, a total of M of th of th wth th uppr bound and lowr lt valu s not or than (1+n) n/. hrfor, th t coplxty of th algorth s O (N). C. Algorth spac coplxty Spac coplxty s a asur of th sz of th storag spac n th runnng procss of an algorth, an algorth for th storag spac occupd by th coputr ory, ncludng th storag spac occupd by th storag algorth tslf [6], th storag spac and th algorth of th storag spac occupd by th algorth n th runnng procss. h ory spac occupd by th nput and output data of th algorth s dtrnd by th probl of th probl. h algorth uss a rcursv algorth, th spac coplxty of th stack spac usd by rcurson, t s qual to DOI /IJSSS.a ISSN: x onln, prnt

3 th nubr of tporary storag spac allocatd by a call ts th nubr of calls (that s, th nubr of rcursv calls to add 1, th 1 tabl s not th bgnnng of a non rcursv call) [7-10]. h spac coplxty of th algorth s also gvn n th for of ordrs of agntud. If th spac coplxty of th algorth s changd, t can b xprssd as O (1); f th spac coplxty of th algorth s proportonal to th logarth of th n of, O (1ogn) s O (n) [11]. h rcursv algorth for paratr s an array, you only nd to assgn t a stord by argunt to transt a pontr addrss spac, naly a achn word spac. For ths rcursv algorth, th t coplxty and spac coplxty s th utual nflunc [1]. Whn th pursut of a bttr t coplxty, th prforanc of spac coplxty can b changd, whch can lad to or storag spac, and whn th pursut of a bttr spac coplxty, th prforanc of t coplxty can b changd, can lad to longr run t [13]. hrfor, ths t, th nubr of bnchark dsgn of th rcursv algorth, to consdr th prforanc of th algorth, th algorth usng frquncy, algorth procssng of th sz of th data, th algorth dscrbs th charactrstcs of th languag, th algorth runnng th achn syst nvronnt and othr factors. D. h basc prncpls of dffrnt quvalnt rcurson algorth For th paralll structur systs coposd by n falur ods, whn syst falur probablty s Pfs, thr s Pfs P ak Z k k1 1 1 k k P a Z a Z a Z k3 P a Z ak Z k k3 P a Z a3 Z 3 ak Z k k4 Pa Z In th forula, (1) s a on-dnsonal standard 0 noral cuulatv dstrbuton functon. a Z s an quvalnt falur od. s th rlabl ndx of quvalnt falur od, on whch th solvng quatons by usng JC thod on pag of ths papr. P s th probablty of vnt. Forula (-1) ndcats th basc da of dffrnt quvalnt rcurson algorth, splfyng th calculatng probls of coplx hgh-dnsonal noral dstrbuton cuulatv dstrbuton functon to b on of ondnsonal standard noral cuulatv dstrbuton functon. Whn AB, C ln C 1 ln 1 AB P f K P f, () E. Modfr forula of falur probablty on dffrntal quvalnt rcurson algorth calculatng syst hr ar sgnfcant calculatng xapls ndcat that whn th quvalnt falur od s drctly usd, th obtan falur probablty n paralll syst on th on hand s too uch on th othr hand n tand syst t s rlatvly sall. On ths background, th papr convrss and rducs th falur probablty obtand fro th quvalnt falur od AB and falur od C.Suppos PA B P AB A P AB B K βab and βc ar th rlabl ndxs for quvalnt falur od AB and falur od C rspctvly. Manwhl, Pf P AB C (4) Pf PABC h calculatng thod of Pf s convrsd and rducd [53] as pr th followng forula: Whn P 1K P (5) AB C f IV. RECURSIVE COMPOUND FUZZY NEURAL NEWORK h coon fuzzy nural ntwork dos not ntroduc th objct-orntd pror knowldg fro a practcal sns, such as ntroducng knowldg of opratng and controllng xprnc as wll as procss fld, rsultng n poor usng of fuzzy advantags. o ovrco ths dfcncy, a copound fuzzy nural ntwork nad IMJ was ntroducd whch cobns th procss s rul-basd rasonng and functon approxaton functons. Aong ths, th rul ntwork appls fuzzy nural ntwork structur that basd on rul s rasonng and studyng, through th us of pror knowldg, to achv dvdng of opraton aras. Manwhl, th functon ntwork s usng th fuzzy nural ntwork to achv th nonlnar functon approxaton wthn th opraton rgon of corrspondng procss. Such copound fuzzy nural ntwork phaszs th us of pror knowldg and studyng that aks t possbl to ffctvly prov th approxaton and gnralzaton of coplx nonlnar systs n larg sgnts rang. Structur of th copound fuzzy nural ntwork s as follows: th functon approxaton ntwork s coposd of fv parts, ncludng th nput layr L1, fuzzy layr L, rul layr L3, ds-fuzzy layr L4 and output layr L5. h rul f (3) DOI /IJSSS.a ISSN: x onln, prnt

4 rasonng ntwork coposd of four parts, ncludng th nput layr L1, fuzzy layr L, rul layr L3 and output layr L5. h frst layr s th nput layr L1. In ths layr, ach nod s drctly connctd wth th nput vctor: X and pass ach nput valu X x1xx n to th nxt lvl. n s th quantty of nput varaton, =1 s functon ntwork and = s rul ntwork ,,, n ; 1, (6) O u 3 3 j j ( 1,,, 1 O u (8) s ; j 1,,, n 1 ) h fourth layr s th ds-fuzzy layr L4: achvng th ds-fuzzy calculaton of th fuzzy concluson of functon approxaton ntwork. Nodal pont nubr of ths layr s th sa as that of th rul ntwork n th thrd layr. W provds th conncton wght btwn rul layr and dsfuzzy layr j ( j 1,,, s ) O w u (9) h ffth layr s th output layr L5: usng gravty thod to calculat th jont output of functon approxaton ntwork and rul s rasonng ntwork. s s / y O O O O (10) Fgur 3. Structur opology of CFNN s Ntwork. h scond layr s fuzzy & rcursv layr L: ach nod rprsnts a lngustc varabl valu, for xapl NB, ZE and PS, tc. It calculats ach nput coponnts of subordnatng dgr functon that blong to th brshp functon of ach fuzzy collcton of lngustc varabl valu. Hr th Gaussan functon s adoptd. u c O xp (7) ( 1,,, n ; j 1,,, s ; 1. ) Aong ths, c and rspctvly rprsnt th cntr and wdth of brshp functon, s s th fuzzy dvson valu of X. For th rul ntwork, th fuzzy dvson valu of prrqust of rasonng s dcdd by pror knowldg whch nabls th provnt of ntwork s rat of convrgnc. h thrd layr s th rul layr L3: ach nod rprsnts a fuzzy rul wth whch to atch th fuzzy rul s prcondton and achv fuzznss and applcablty of ach calculatd rul. Functon ntwork ruls can b adjustd as pr th actual stuatons, whras th fuzzy ruls and fuzzy ruls valus of rul s rasonng ntwork hav to b dcdd by th pr-dtrnd syst rasonng ruls. Blow provds th calculatng forula of functon ntwork, n whch th rul ntwork calculaton s dcdd by th actual rul conncton. h calculaton thod of copound fuzzy nural ntwork lstd as follow: whn th fuzzy dvson valu s for crtan, th paratr that ndd studyng ar th wght w of functon ntwork, th cntr c & wdth of scond layr brshp functon as wll as th rcurson conncton wght. Hr usng th rror back propagaton thod to adjust paratrs, n whch th rror cost functon as follow: E 1 y y (11) d Basd on th advantags of dynac fuzzy nural ntwork and copound fuzzy nural ntwork, ths papr put forwards th rcursv copound fuzzy nural ntwork. On th bass of copound fuzzy nural ntwork t ntroducs rcursv nods nsd th scond layr of functon approxaton ntwork, whch not only takng advantags of objctv pror knowldg but also postvly rspondng to th dynac syst. h structur of rcursv copound fuzzy nural ntwork as th followng dagra 4. Fgur 4. Structur opology of RCFNN s Ntwork DOI /IJSSS.a ISSN: x onln, prnt

5 Its dffrnc wth copound fuzzy nural ntwork ls n th scond layr of functon approxaton ntwork. h nput of abov functon ntwork layr s, 1 1 u f k O k O k O k O k 1, ans th conncton wght of rcursv unt. In ths layr, 1 O k rcords th ntwork nforaton n th prvous t, so th dynac appng can b ralzd. Whn th larnng calculaton of rcursv copound fuzzy nural ntwork was appld to calculat th brshp functon cntr c and wdth and, thr ar th followng forulas: E E E c O 1 1 O c 1 1 x O k 1 c E O 1 1 O x1 O k 1 c (1) (13) E E O 1 O (14) x O k 1 c O k1 Nvrthlss, th fuzzy nural ntwork as wll as th rsarch concntratd anly on th statc ssus hav ltd accss to th dynac syst,of whch ost ar n th stag of thortcal rsarchs and lab sulatons. In consdraton of ost actual systs ar dynac nonlnar systs, ths papr focuss on how to dsgn a rcursv fuzzy nural ntwork to ak t sutabl for dynac syst, and ract to th rqurnts of practcal ngnrng applcaton to sarch for thods of takng full advantags of pror knowldg as wll as ralzng th control of th dynac syst wth ffctv, rlabl, practcal and splst odls. V. ES OF POLE NUMBER ALGORIHM h bnchark nubr algorth s dvdd nto 4 groups accordng to th nubr of A lnts n n. h uppr lt, lowr lt and tolranc of pol nubr ar th sa as th pact of th sz of th nput probl. Each group gnrats 50 rando sapl (A, ). Each probl s solvd ndpndntly by th abov bnchark nubr algorth, and th corrspondng optal bnchark nubr s found. A. st nvronnt PC for tstng th wndows confguraton for th ordnary ho coputr, XP opratng syst. Accordng to th probl of th sz of th n=50 of a rando lnt, th lvl of coputr opratons, can b drawn fro th rando data, th thr tst data s tabl. Exprnt 1: ABLE RANDOMLY GENERAED SAMPLES FROM HE EXPERIMEN ABLE. RESULS OF EXPERIMEN 1 targt nubr targt nubr h bnchark olranc copost array Bst bnchark nubr 110 Exprnt : ABLE RANDOMLY GENERAED SAMPLES FROM HE EXPERIMEN B. st rsults and analyss Fro th abov xprntal data, w can s that n th dtrnaton of th bnchark nubr, n all th sapl spac, t s dtrnd that contans th nubr of th axu nubr of bnchark, whch shows that th algorth has practcal applcaton to dtrn th rsults. DOI /IJSSS.a ISSN: x onln, prnt

6 ABLE 4. RESULS OF EXPERIMEN targt nubr h bnchark targt nubr olranc copost array Bst bnchark nubr 108 Exprnt 3: ABLE RANDOMLY GENERAED SAMPLES FROM HE EXPERIMEN ABLE 6. RESULS OF EXPERIMEN 3 targt nubr h bnchark targt nubr olranc copost array Bst bnchark nubr 109 VI. CONCLUSION In ths papr, th algorth s proposd to solv th probl of pol nubr, but th algorth s basd on th dffrnt pol nubr stat and dffrnt substs of th cobnaton of th nubr of probls. In accordanc wth our bnchark nubr algorth, th gratr th M, th largr th bnchark nubr, th or dffcult to solv. For a gvn nubr s not a postv ntgr, but a ral nubr can b transford nto th cas by odfyng th paratrs. Howvr, n th procss of wrtng cod, spcally n th prparaton of rcursv functons, bcaus n th procss of rcurson to produc a larg ntrdat varabl, although dffrnt cobnatons hav to rturn th valu of th portfolo, but w can not drctly to th slcton of th. hrfor, how to ralz th slcton of th rcursv functon and th choc, or to furthr rsarch and thnkng. REFERENCES [1] S. Chn, C. F. N. Cowan, S. A. Bllngs, t al., Paralll rcursv prdcton rror algorth for tranng layrd nural ntworks, Intrnatonal Journal of Control, vol. 51, no.6, p.p [] Bllngs S A, Jaaluddn H B, Chn S, A coparson of th backpropagaton and rcursv prdcton rror algorths for tranng nural ntworks, Mchancal Systs & Sgnal Procssng, vol. 5, no.91, p.p [3] Stono R, Basns B, Mus C, Rcursv nural ntwork rul xtracton for data wth xd attrbuts. IEEE ransactons on Nural Ntworks, vol. 19, no., p.p [4] Bouchard M, Nw rcursv-last-squars algorths for nonlnar actv control of sound and vbraton usng nural ntworks, IEEE ransactons on Nural Ntworks, vol. 1, no.1, p.p [5] Huang D S, Zhao W B, Dtrnng th cntrs of radal bass probablstc nural ntworks by rcursv orthogonal last squar algorths, Appld Mathatcs & Coputaton, vol. 16, no.1, p.p [6] Paras M, Srvastava A K, oonobu S, t al., A nw rcursv nural ntwork algorth to forcast lctrcty prc for PJM day ahad arkt, Intrnatonal Journal of Enrgy Rsarch, vol.34, no.6, p.p [7] Xaorong S, Fan Y, L Jng, t al. A Study on Robust BP Algorth Basd on Rcursv Fuzzy Nural Ntwork. Arospac Control, vol., no.3, p.p [8] Da-Jun D U, F M R, L-Xong L I, Radal-bass-functon nural ntwork basd on fast rcursv algorth and ts applcaton, Control hory & Applcatons, vol. 5, no.5, p.p [9] Zhang X L, Cao C X, M B, A Nw Larnng Algorth Basd on Radal Bass Functon Nural Ntwork --Rcursv Orthogonal Last Squars (ROLS) Algorth, Journal of Chongqng Unvrsty, vol.5, no.10, p.p [10] Mu L, h Dsgn of Fdward Nural Ntwork Basd on Rcursv Prdcton Error Algorth. Coputr Engnrng & Applcatons, vol. 39, no.19, p.p [11] D'Acrno A, Vaccaro R, On paralllzng rcursv nural ntworks on coars-grand paralll coputrs: A gnral algorth, Paralll Coputng, vol. 0, no.94, p.p [1] Laufr C W, Coghll G, A Rgularzd Invrs QR Dcoposton Basd Rcursv Last Squars Algorth for th CMAC Nural Ntwork, vol. 15, no.1, p.p , 01. [13] Daohang Sha, Vladr B. Bajc, An optzd rcursv larnng algorth for thr-layr fdforward nural ntworks for o nonlnar syst dntfcatons, Intllgnt Autoaton & Soft Coputng, vol. 17, no., p.p DOI /IJSSS.a ISSN: x onln, prnt