Zjavka: Recognition of Generalized Patterns by a Differential Polynomial eural etwork

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1 ETASR - Egeerg Techolog & Appled Scece Reserch Vol. o. 6-6 Recogto o Geerlzed Ptters Deretl Polol Neurl Netor Ldslv Zjv cult o Mgeet Scece d Iortcs Uverst o Žl Žl Slov lzjv@gl.co Astrct A lot o proles volve uo dt reltos detcto o hch c serve s geerlzto o ther qultes. Reltve vlues o vrles re ppled ths cse d ot the solute vlues hch c etter e use o dt propertes de rge o the vldt. Ths reseles ore to the uctolt o the r hch sees to geerlze reltos o vrles too th coo ptter clsscto. Deretl polol eurl etor s e tpe o eurl etor desged the uthor hch costructs d pprotes uo deretl equto o depedet vrles usg specl tpe o root ult-pretrc polols. It cretes rctol prtl deretl ters descrg utul dervtve chges o soe vrles lese the deretl equto does. Prtculr polols ctch reltos o gve cotos o put vrles. Ths tpe o detcto s ot sed o hole-ptter slrt ut ol to the lered hdde geerlzed reltos o vrles. Keords - polol eurl etor; depedece o vrles detcto; deretl equto pproto; rtol tegrl ucto I. INTRODUCTION The prcpl dsdvtge o the rtcl eurl etor (ANN detcto geerl s the dslt o put ptter geerlzto. ANNs c ler to clss put ptters ut utlze ol the solute vlues o vrles. Hoever the ltter der sgctl hle ther reltos e the se. Tht s h ANNs re le to correctl recogze ol slr or coplete ptters copred to the tr set. I the put cosdered s e.g. shpe oved or szed the put tr the eurl etor detcto ll l. A pproch to loo t the put vector o vrles ot s ptter ut s depedet oud pot set o N-desol spce could e ttepted. A eurl etor hch ould e le to ler d det uo dt reltos s to cot ultpretrc polol uctos to ctch prtl depedece o gve puts. Its respose ould e the se to ll ptters (sets hch vrles re perored th the tred depedece regrdless o the ctul vlues [9]. Bologcl eurl cell sees to ppl slr prcple. Its dedrtes collect electrcl sgls cog ro other euros. But ule the rtcl euro soe o the sgls lred terct sgle rches (dedrtes o eurl cell (see gure lese the ultpled vrles o ult-pretrc polol do. Preters o polol ters c represet the sopss o the cell dedrtes. These eghted cotos re sued the od cell d trsored to reltve vlues usg te-deled dc perodc ctvto uctos (the ctvted eurl cell geertes seres o te-deled output pulses respose to ts put sgls. Ao psses electrcl pulse sgls o to dedrtes o other eurl or eector cells []. The perod o ths ucto depeds o soe put vrles d sees to represet the dervtve prt o prtl ter o deretl equto coposto. Deretl polol eurl etor (D-PNN costructs d tres to pprote uo deretl equto descrg reltos o put vrles tht re ot etrel ptters. It ors ts output s geerlzto o put ptters slr to the oes utlzed the hu r. It cretes structurl odel o uo reltoshps o put vrles descrpto. D-PNN s sed o GMDH (Group Method o Dt Hdlg polol eurl etor hch s creted the Ur scetst Alese Ivheo 968 he the c-propgto techque s ot o et. He ttepted to decopose the coplet o process to spler reltoshps ech descred lo order -vrle polol processg ucto o sgle euro []. g.. A ologcl eurl cell

2 ETASR - Egeerg Techolog & Appled Scece Reserch Vol. o II. DIERENTIAL POLYNOMIAL NEURAL NETWORK The sc de o the D-PNN s to crete d pprote deretl equto (DE ( hch s ot o dvce [] th specl tpe o root (poer rctol ultpretrc polols (5. Coto degree (5 / / euros u c j j j u u u( - serched ucto o ll put vrles B( C(c c - preters ( Π polol Bloc output Eteded euros ourer s ethod o prtl DE soluto serches the soluto or o the product o uctos o hch t lest depeds ol o vrle. A prtl dervto o ucto z( o put vrles c e epressed ( []. z( ( ( z ( Eleetr ethods o deretl equto soluto epress the soluto specl eleetr uctos polols (e.g. Bessel s uctos ourer s poer seres. Nuercl tegrto o deretl equtos s sed o ther pproto through: rtol tegrl uctos trgooetrc seres The st d ore sple hs ee selected usg the ethod o tegrl logues hch replces thetcl opertors d sols DE the rto o correspodg vrles. Dervtves re replced the tegrl logues.e. dervtve opertors re reoved d sulteousl ll opertors re replced slrl or proporto rs equtos ll vectors re replced ther solute vlues. Desol ters re dvded soe others hch results serched o-desol leess crteros [5]. ( ( (5 ( coto degree o -put vrle polol o uertor coto degree o deotor (< g.. A loc o deretl euros Ech ler o the D-PNN cossts o locs hch cot dervtve euros oe or ech rctol polol (5 deg the prtl dervtve depedet chge o soe put vrles. A loc lso cots ddtol eteded euros (EN hch or copoud uctos (5 pplg prevous ler loc outputs. Ech loc cots sgle polol (thout dervtve prt hch ors ts output etrce to the et hdde ler (gure.. Neuros do t ect the loc output ut re ppled ol or the totl output clculto (DE coposto. Ech euro hs vectors o djustle preters d ech loc cots vector o djustle preters o the output polol. The root uctos o deotors (5 re loer th ccordg to the coto degree hch te the polols o euros to copetet poer degree. The c e replced poer uctos o deotors. Iputs o costt coto degree ( org prtculr cotos o vrles eter ech loc here the re susttuted to polols (gure.. It s ecessr to djust ot ol the polol preters ut lso the D-PNN s structure. Ths es soe euros ters o role o the DE re to e let out. p j q j Y.. t r u v t eghts o ters The rctol polols (5 hch c descre prtl depedece o -put vrles o ech euro re ppled s ters o the DE (6 coposto. The prtl crete uo ult-pretrc o-ler ucto hch codes reltos o put vrles. The uertor o (5 s polol o coplete -put coto degree o sgle euro d relzes e ucto z o orul (. The deotor o (5 s dervtve prt hch gves prtl utul chge o soe euro put vrles d ts polol coto degree s less the. It rose ro the prtl dervto o the coplete -vrle polol copetet vrle(s. (6 g.. Deretl polol eurl etor

3 ETASR - Egeerg Techolog & Appled Scece Reserch Vol. o III. IDENTIICATION O SIMPLE LINEAR DEPENDENCIES Cosder ver sple depedece o -put vrles hch ultplct s costt (e.g.. D-PNN ll cot ol loc o polol euros ((8 s ters o DE (gure.. As the put vrles do t chge costtl t s ecessr to dd oth ters (rctol polol o dervtve vrle d the DE (loc. D-PNN ll ler ths relto esl ccordg to sples o the trg dt set es o geetc d evoluto lgorth (GA []. ( (8 Cosder ore coplcted ler depedece here vrles deped o rd. or eple su o the rst vrles equls the rd vrle (. The coplete DE (or dervtves d -cotos o loc cossts o 6 ters (euros ut ol o the ll e eough or dervtve ters (9 ( (. I other ters (euros re dded the D-PNN ll or ss (see gure 5. To-vrle coto polols o uertors ((8 c e lso ppled hch could prove the D-PNN uctolt d crese the uer o the DE ters. Ths - vrle depedece s descred ore coplcted epoetl uctos. The D-PNN s ell s chrged the possle -sded depedet chge o put vrles. or eple 9 s the se su s 9. The prcpl phse o ts djustet resdes eltg o soe euros ( ters o the DE. g.. Idetcto o costt quotet o vrles ( g. 5. Idetcto o the su depedece (9 ( ( Mult-lered D-PNN cretes copoud polol uctos. M epoetl uctos o hgher lers crr soe secodr uctos o prevous lers descrg the prtl reltos o ts vrles. ro thetcl pot o ve the st hdde ler ors the er uctos hch susttute the put vrles o d hdde ler euro d loc polols - the outer uctos. Provded ths ssupto e re le to clculte the prtl dervtves o copoud uctos vrles o prevous lers s DE ters ( ro the er uctos ( o outer ucto (. These copoud DE ters re ored s products o prtl dervtves o d er uctos (5 [6]. ( ( X ( ( ( (φ (X φ (X φ (X (. ( X ( ( φ (5 g. 6. Idetcto o the -vrle depedece th -coto locs / Bloc output Neuro Neuro ( (8 Bloc output Neuro Neuro Neuro (9 ( (

4 ETASR - Egeerg Techolog & Appled Scece Reserch Vol. o. 6- Ech loc o the D-PNN ors prtl DE ters utlzg ts sc d eteded euros. Sgle djustle polol (P gure 6. thout dervtve prt cretes the loc output (pplg the et hdde ler ut the euros re ppled ol or the totl DE coposto. The locs o the d d the ollog hdde lers crete copoud ters (CT o the DE usg ther ddtol eteded euros outputs d puts o c coected locs o prevous lers. Cosder or stce the st loc o the lst hdde ler hch tes ts o euros s sc ters (6 o the DE (6. Susequetl t cretes eteded ters o the d (prevous hdde ler vrles usg reverse output polols d puts o oud locs. It cretes rctol copoud ters o the DE or dervtve put vrles o prevous hdde ler usg dervtos o copoud d er uctos (. As couples o vrles o the er uctos φ ( d φ ( der ro ech other ther prtl dervtos re d so the su (5 ll cosst ol o ter. ( ( ( ( ( c c ( ( ( d d ( ( (6 ( (8 ( The prevous ler loc reverse outputs re used to crete ecessr prtl dervtos o the outer d er uctos (o polols o deretl euros (. Lese copoud ters c e creted or the st hdde ler (8. The led locs org 8 ters o the DE ere ttched to the presetl djusted loc. Ths c e perored ell recursve lgorth. It s ot ever ter tht s used the coplete DE; soe o the ere ecessrl let out. Ths dctes or the euros o locs d s ese to use the s gees o GA. Preters o polols re represeted rel uers. A chroosoe s sequece o ther vlues hch c e es utted. The D-PNN s totl output Y s the su o ll ctve prtl DE ter vlues ccordg to (9 hch the preset ctve out c e ult. Y totl out o DE ters (9 It c e see tht the -vrle D-PNN (gure 6. susttll cossts o overlg edge etors (WN ech gog c out ro the locs o the lst hdde ler d grdull ttchg to the dervtve vrles o prevous lers. The D-PNN o the depedet put vrles usg - coto locs ll hve totll 6 locs o ll put coto couples the st hdde ler. The uer o cotos or ll vrles creses eorousl ech et hdde ler. Ths could e solved pplg WNs s ol soe o the locs re creted d used. The totl out o D-PNN s hdde lers could equl t lest to the uer o put vrles (.e. s t ust e le to crete ech coto o hch d to rech c ll dervtve vrles o the st ler. So WNs o the st hdde ler ll volve. rdo locs cosequetl the d ler ll cot. locs etc. Ths the uer o ll WN locs decreses ech et hdde ler utl s reched just loc. D-PNN ll hve severl overlg WNs prtl the lers g. Soe WN lers overl ech other d so the locs c e used severl tes deret WNs (gure.. The locs o the d d ollog hdde lers c e recoected d ths could copeste ssg coto locs. The coectos o the coplete st hdde ler locs re ed. Lese the prevous -vrle D-PNN tpe does t c costruct the prtl rctol ters o the DE ro c-coected locs o prevous lers. All WN locs ttch c grdull the dervtve vrles o prevous lers. The serchg spce cots gret out o locl error solutos hch GA c sh esl. Ths prole s cused lot o possle cotos o loc puts d coposed DE ters (ol soe o the e eploed hch selecto s crtcl phse o the D-PNN s costructo esdes the sulteous preter djustet []. g.. Wedge etors o the -vrle -coto D-PNN The D-PNN o the 6 depedet put vrles usg - coto locs ll hve totll 5 locs o ll put coto couples the st hdde ler d 6 hdde lers. Hoever eperet th rght trgles (gure 9 d gure. t could e sucet th hdde lers g ecuse there s the u o -vrle depedece to det. IV... DE EXPERIMENTS -vrle D-PNN s le to det ro/colu or dgol depedece o chess peces (gure 8.. Iput vector s ored ther postos (ro colu. I the hte roo checs the lc shop ther or postos equl d ths c D-PNN ler to det. Aother relto occurs the lc shop checs the hte roo the su or derece o ther d -postos re equl A A B B or A A B B. Tle d Tle sho etor resposes to depedet d

5 ETASR - Egeerg Techolog & Appled Scece Reserch Vol. o. 6- depedet vrles o put vectors. Trg dt set c cosst o ollog 5 dt sples d -postos o the chess peces (the chessord s elrged [9]: {} {65} {} {68} {558} {5} 5 6 posto the put tr (gure 9.. Iput vector o the D- PNN s ored (ro colu coordtes o the trgle pees A B C. The depedece o pots A C s dgol A B vertcl d B C horzotl. As there re sulteousl occurred tpes o pot reltos t s ecessr to crese the uer o locs o D-PNN s hdde lers. Trg dt set c cosst o ollog 5 dt sples rght equl-sed trgles : {69} {} {555} {} {555555} Testg rdo rght trgles ust eep the pees A B C depedet to e correctl recogzed D-PNN. Tle shos resposes o tred etor to depedet rght trgles. Tle. pples ol vertcl deortos ( d o rght trgles C pees (g.. do to e sho trspret seprtg ple detchg the reltve trgles. 5 6 g. 8. Reltos o chess peces TABLE I. RESPONSES TO RANDOM INPUT VECTORS WITH DEPENDENT VARIABLES Iput vector Output TABLE II. RESPONSES TO RANDOM INDEPENDENT INPUT VECTORS O VARIABLES Iput vector Output Iput vector Output ( ( < > < 5. 5 > < > < 8.9 > < > < > < 8.65 > < >.9 5 < > 5.9 g. 9. Depedet rght equl-sded trgle shpes A seprtg ple could e otced detched ro the reltve clsses hch hve the se chrcterstc ( the su o the st couple s less the t should e the output s less the the desred roud d other hd roud. D-PNN c e tred ol th sll put-output dt sples (lese the GMDH polol eurl etor does to ler depedece [8]. 6-vrle D-PNN c ler to det (geerlze chgele shpe (e.g. trgle regrdless o ts sze or g.. Idepedet deored rght trgles pees A d C

6 ETASR - Egeerg Techolog & Appled Scece Reserch Vol. o. 6- TABLE III. RESPONSES TO RANDOM INPUT VECTORS WITH 6 DEPENDENT VARIABLES (RIGHT TRIANGLES Iput vector (A A B B C C Output TABLE IV. RESPONSES TO RANDOM INDEPENDENT INPUT VECTORS O 6 VARIABLES (DEORMED RIGHT TRIANGLES IN C APEXES Iput vector (A B C Output Iput vector (A B C Output V. DISCUSSION Ol ler depedeces o vrles hve ee ssued or splct the eples preseted. I there s occurrece o o ler depedece o the put dt the squre poer epoet vrles ould eted coto polols o euros d ppled lso s copetet dervtve ters (. The deotors o (( result ro the prtl dervtves o the coplete DE ter polols o uertors. The root squre (or poer uctos re lel ot volved to rctos ( the dereces o vlues o put vrles re ot too g ( cse rel dt odel s occurred preseted eples. ( ( ( j 5 5 ( ( ( Accordg to (-(5 t s possle to dee hgher degree prtl dervtos o -vrle copoud ucto ((uv ( [6]. As the vrles o the D-PNN s er uctos uφ( d ψ( re deret the d rd d 5 th ters o eq. ( re (s the prtl dervto o ψ s. Lese the ters o ( d (5 do [6]. ( ( u v [ ( ψ ( ] uφ( vψ( ( ψ ψ ψ ( v v v ψ ψ ψ ( v v ψ ψ u u v ψ ψ ψ v v v (5 A rel dt eple ght solve the ether orecst sed o soe tred dt reltos hch re used or clcultg the et stte o sste. Let s te severl tpes o vrles (e.g. pressure dp teperture prtl descrg sttes o ths ver cople sste. The put vector o the D-PNN s ored vlues o these vrles deed tr coordtes o eteorologcl p. The trg dt set cludes dete sttes o te tervl d desred etor outputs. The output could r the ether orecst s rlls cloud sushe. There c turll rse possle trset sttes (e.g... The output s coputed or loclt o the p d could lso predct the tospherc pressure or other qutt. VI. CONCLUSION Artcl eurl etors geerl respod to relted ptters th slr output. The det put ptters o the ses o ther reltoshp. Lese the detcto o uo depedeces o the dt vrles could lso e cosdered. Ths could e regrded s ptter o strcto slr to tht utlzed the hu r hch pples the pproto th te-deled perodc ctvto uctos o ologcl euros hgh dc sste o ehvor. D- PNN s e tpe o eurl etor hch perors detcto sed o uo geerlzed reltos o put vrles. D-PNN ors ts uctol output s coposto o deretl equto ters (hch descre sste o depedet vrles ro rtol tegrl uctos. The prole o the ult-lered D-PNN costructo resde cretes ever prtl coto ter or coplete DE utlzg soe ed lo coto degrees ( hle the out o vrles s s rule hgher. REERENCES [] Ľ. Beňušová Neuro d r. Cogtve sceces Cllgr Brtslv ( Slov. [] A.G. Ivheo Polol theor o cople sstes IEEE Trsctos o sstes Vol. SMC- No. pp [] J. Hroec Deretl equtos II. SAV Brtslv 958 ( Slov. [] R. Rchovsý J. Výorá Prtl deretl equtos d soe o ther solutos Pul. SNTL Prh 9 ( Czech. [5] J. Kueš O. Vvroch V. rt Prcples o odellg SNTL Prh 989 ( Czech. [6] I. Kluváe L. Mší M. Švec Mtetcs I. II. SNTL Brtslv 966 ( Slov. [] S. Ds A. Arh A. Kor Prtcle sr ptzto d deretl evoluto lgorths: Techcl slss pplctos d hrdzto perspectves Coputer d Iorto Scece Vol. 8 pp [8] B. B. Msr S. Dehur P.K. Dsh G. Pd A reduced d coprehesle polol eurl etor or clsscto Ptter recogto letters Vol. 9 No. pp [9] L. Zjv Geerlzto o ptters detcto th polol eurl etor Jourl o Electrcl Egeerg Vol. 6 No. pp. - [] L. Zjv Costructo d djustet o deretl polol eurl etor Jourl o Egeerg d Coputer Iovtos Vol. No. pp. -5