nonlinear pranali olakh

pranali olakh hii pranalichya input aanee outputchya mojamapatoon pranaliche ganiteey model olakhanyachi kinwa mojanyachi eka paddhat aahe. pranali olakhanyachya anuprayoganmadhye ashee konatihi pranali samavisht aahe jithe input aanee output mojale jaau shakatat aanee tyat audhyogik prakriya, niyantran pranali, aarthik deta, jivashastra aanee jeevan vidnyan, aushadh, samajik pranali aanee itar anekancha samavesh aahe.
nonlinear pranalichi vyakhya ashee kelee jaate qee konatihi pranali jee resheey nahee, mhanje ashee konatihi pranali jee superposition tattva purn karat nahee. hii nakaratmak vyakhya heey aspasht karate qee anek vegvegalya prakarchya nonlinear pranali aahet. aitihasikdrushtya, pranalichya vishisht varganvar laksh kendrit karoon araikhik pranalinsathi pranali olakh vikasit jhaali aahe aanee tyache vyapakpane paach moolbhut drushtikonanmadhye vargikaran kele jaau shakate, pratyek model vargadvare paribhashit kele jaateh
pranali olakhanyasathi chaar anek palavya lagatilah mahitee gola karane, namuna gruheet dharane, mapdand olakhane aanee namuna pramanikaran. mahitee gola karane haah olakh paribhashetil pahila aanee aavashyak bhag manla jaato, joou nantar tayar kelelya pratimasathi input mhanoon vaparla jaato. tyat yogya deta sanch nivadane, purv-prakriya aanee prakriya yancha samavesh asato. yaat uddan tepche pratilekhan, deta sanchayan aanee deta vyavasthapan, calibration, prakriya, vishleshan aanee sadarikaran yasah dnyat algoridamchi amalbajavani samavisht aahe. shivay, ekhadya vishisht pratimasamor vishvas midvanyasathi kinwa nakarnyasathi pratiman pramanikaran aavashyak aahe. visheshtah, mapdand andaj aanee pratiman pramanikaran heey pranali olakhanyache avibhajya bhag aahet. manyata mhanje sankalpanatmak pratimechi pushti karanyachi aanee pratimeche sanganakeey parinam aanee vastavik mahitee yanchyatil puresa sanvad darshavinyachi prakriya.
voltera malika paddhati.
suruvatichya kamat voltera malikevar aadharit paddhatinche varchasva hote, jeey swatantra velechya babatit asey vyakt kele jaau shakate
jethe "u" ("k"), "y" ("k"); "k" = 1,2,3,... heey anukrame mojalele input aanee output aahet aanee sootra _ 2 heey "l" th-aurder voltera curnell kinwa "l" th-aurder nonlinear impulse pratisad aahe. voltera malika haah rekheey convolution avibhajya bhagacha vistar aahe. purvichya bahutek olakh algoridammadhe asey gruheet dharale gele hote qee fakt pahilya done, resheey aanee varg, voltera curnell upasthit aahet aanee done voltera curnell olakhanyasathi gausian pandhara aavaj aanee parasparsambandh paddhati yasarkhya vishesh inputcha vapar keela jaato. yapaiki bahutek paddhatinmadhye input gausian aanee pandhare asane aavashyak aahe jeey anek vastavik prakriyansathi eka gambhir nirbandh aahe. heey parinam nantar pahilya teen voltera karnalcha samavesh karnyasathi, vegvegalya inputla parvanagi denyasathi aanee winer malikesah itar sambandhit vikasasathi vistarit kele gele. 1940 teey 1960 chya dashakat viener, lii, boses aanee mitramandalatil mitramandalatil prasiddha lii aanee sketzen paddhatisah eka atishay mahattvacha upkram vikasit keela. yaa paddhatincha aajahi sakriyapane abhyas keela jaat asatana anek moolbhut nirbandh aahet. yamadhye voltera maliketil pradhanyachya sandnyanchi sankhya jaanun ghene, vishesh aadanancha vapar aanee olakhane aavashyak asalelya andajanchi mothee sankhya jaanun ghene aavashyak aahe. udaharanarth, jyaa pranalimadhye pahilya kramankachya voltera karnalche varnan 30 namunyandware kele jaate, tyaa pranalisathi yaa kramankachya karnalsathi 30x30 guna, guna kramankachya karnalsathi 30x30 guna aanee asha prakare aavashyak asateel aanee tyamule changle andaj denyasathi aavashyak asalelya detache praman khoop mothe hote. vishisht samamiticha vapar karoon yaa sankhya qamee kelya jaau shakatat parantu olakh patavanyasathi konata algoridam vaparla jaato yachee parva naan karata aavashyakta ajunahi jaast aahet.
block-sanrachit pranali.
voltera models olakhanyachya samasyanmule, nonlinear pranalinsathi pranali olakhanyacha aadhar mhanoon itar model formchi tapasani kelee gelee. block sanrachit nonlinear modelche vividh prakar sadar kele gele aahet kinwa punha sadar kele gele aahet. hammerstein modelmadhye eka sthir ekal moolya asalela nonlinear ghatak asato, tyanantar eka resheey gatishil ghatak asato. winer model heey yaa sanyojanachya ulat aahe jenekarun rekheey ghatak sthir nonlinear vaishishtyachya aadhi udbhavel. winer-hammerstein modelmadhye done gatishil resheey ghatakanmadhil sandwich kelela sthir nonlinear ghatak asato aanee itar anek model prakar upalabdh aahet. hammerstein-viener modelmadhye done sthir nonlinear blocks darmyan sandwich kelela eka resheey gatishil block asato. urisohan model itar block modelpeksha vegale aahe, tyat anukram rekheey aanee araikhik blocks nasatat, parantu operatorchya karnalchya abhivyaktimadhye gatishil aanee sthir asha donhi araikhikta varnan karatat. yaa serve modelsche pratinidhitva voltera malikedware kele jaau shakate parantu yaa prakaranat voltera curnell pratyek prakaranat eka vishesh swaroop ghetat. olakhanyat parasparsambandh aadharit aanee mapdand andaj paddhati asatat. parasparsambandhachya paddhati yaa pranalinchya vishisht gunadharmancha vapar karatat, jyacha arth assa aahe qee jarr vishisht input vaparale gele, tarr bahutekda pandhara gossian aavaj, vaiyaktik ghatak eka velee eka olakhale jaau shakatat. yamule vyavasthapit karata yenyajogya detachya aavashyakta nirman hotat aanee vaiyaktik khand kahivela abhyasat asalelya pranalitil ghatkanshi sambandhit asoo shakatat.
adhik alikadil parinam heey mapdand andaj aanee majjatantu jaalyavar aadharit upayanvar aadharit aahet. anek parinam sadar karnyat aale aahet aanee yaa pranalincha sakhol abhyas suru aahe. eka samasya ashee aahe qee yaa paddhati pratyek prakaranat keval eka atishay vishesh prakarchya pratimala lagoo hotat aanee sahasa haah pratiman swaroop olakhanyapurvi jaanun ghene aavashyak asate.
neural network.
krutrim majjatantuche jaale mendutil majjatantuchya jaaleche anukaran karnyacha prayatna karatat, jithe mothya sankhyene sopya prakriya ghatkandware ganana hote. eka vishisht majjatantu jaalyamadhye eka jatil jaal tayar karnyasathi ekmekanshi jodalelya anek sopya prakriya ekkancha samavesh asato. asha ekkanche thurr asha prakare mandanyat yetat jenekarun deta input tharavar pravisht keela jaeel aanee output tharaparyant pohochanyapurvi ektar eka kinwa anek madhyavarti tharanmadhun jaeel. paryavekshit shikshanamadhye, jaale heey vastavik utpadan aanee jaaleche ichchit utpadan, andaj truti, yanchyatil farkavar karya karoon, nodesmadhil jodanichi takad badalanyasathi prashikshit kele jaate. output truti swikararh pataliparyant pohocheparyant vajan punaravrutti karoon sudharit kele jaate. yaa prakriyela machine learning mhanatat kaaran jaale vajananche samayojan karate jenekarun utpadan namuna punarutpadit keela jaeel.
majjasanstheche jaale vyapakpane abhyasale gele aahe aanee sarvasadharanpane yaa vishayasathi samarpit anek utkrusht pathyapustake aahet aanee adhik kendrit pathyapustake aahet jee niyantran aanee pranali anuprayoganvar bharr detat.
done mukhya samasyanche prakar aahet jyancha abhyas majjatantu jaalyancha vapar karoon keela jaau shakatoh sthir samasya aanee gatishil samasya. sthir samasyanmadhye namuna olakhane, vargikaran aanee andaj yancha samavesh hoto. gatishil samasyanmadhye mage rahilelya chalancha samavesh asato aanee pranali olakh aanee sambandhit anuprayogansathi adhik yogya asatat. jaalyachya sanrachanenusar prashikshan samasya ektar nonlinear-inn-daad-parameters asoo shakate jyamadhye optimization kinwa linier-inn-daad-parameterscha samavesh asato jeey shastriy drushtikonancha vapar karoon sodavale jaau shakatat. prashikshan algoridamche paryavekshit, dekhrekh nasalele kinwa majbutikaran shikshanat vargikaran kele jaau shakate. neural networkmadhye utkrusht andaje gunadharm asatat parantu heey sahasa pramanit karya andaje parinamanvar aadharit asatat, udaharanarth, warestrass pramey jeey bahupadi, tarksangat karye aanee itar suprasiddha modelsna titkech changle lagoo hote. 
pranali olakhanyachya samasyanvar neural networkcha mothya pramanat vapar keela gela aahe jyat nonlinear aanee dynamic sambandhancha samavesh aahe. tathapi, shastriy majjatantu jaale heey purnapane sthool sthir andaje yantra aahet. jaalyamadhye konatihi gatishilata nahee. mhanoon gatishil models basavatana, jaalyachya input layerla mage rahilele input aanee output vatap kelyane serve gatishilata udbhavate. prashikshan prakriya nantar sarvottam sthir andaj tayar karate jee input nodesla niyukt kelelya mage rahilelya chalancha outputshee sambandh thevate. punaravartit jaalyansah adhik jatil jaal sanrachane aahet, jee input nodesmadhye mage padalelya chalanchya vadhatya kramancha parichay karoon deuun gatishilata nirman karatat. parantu yaa prakarananmadhye vilamb jaast spasht karane khoop sope aahe aanee yamule atee fitting aanee kharab samanyikaran gunadharm houu shakatat. 
majjasanstheche anek fayade aahet; teey sankalpanatmakdrushtya sope, prashikshit karane aanee vaparnyas sope aahet, utkrusht andaj gunadharm aahet, sthanik aanee samantar prakriyechi sankalpana mahattvachi aahe aanee heey ekatmata aanee dosh sahishnu vartan pradan karate. shastriy majjatantu jaale pratikrutinvar sarvat mothee teeka ashee aahe qee tayar keleli pratikruti purnapane apardarshak asatat aanee sahasa tee lihoon kinwa vishleshan karoon gheta yet naheet. tyamule kaay kaaran aahe heey jaanun ghene, pratimeche vishleshan karane kinwa pratimetil gatishil vaishishtyanchi ganana karane khoop katheen aahe. yapaiki kahee mudde serve anuprayoganshi sambandhit nastil parantu teey gatishil modelingsathi aahet.
nermax paddhati.
bahya inputs (nermax model) nonlinear autoregressive moving everage model nonlinear pranalinchya vistrut vargache pratinidhitva karoo shakate aanee tyachi vyakhya ashee aaheh
jethe "y" ("k"), "u" ("k") aanee "e" ("k") heey anukrame pranaliche utpadan, input aanee dhwani anukram aahet; sootra _ 4, sootra _ 5 aanee sootra _ 6 heey pranaliche utpadan, input aanee dhvanisathi kamal vilamb aahet; f [] heey kahee nonlinear karya aahe, d haah eka vel vilamb aahe joou samanyatah "d" = 1.the varr seet keela jaato model moolatah mageel input, output aanee dhwani sandnyancha vistar aahe. kaaran aavaj spashtapane pratirupit keela jaato, pranali modelche nihpakshapati andaj heey nirikshan naan kelelya atyant parasparsambandhit aanee araikhik aavajachya upasthitit prapt kele jaau shakatat.
voltera, block sanrachit models aanee anek neural network architecture heey serve nermax modelche upasanch manale jaau shakatat. nermax sadar jhalyapasun, yaa modeldware konatya prakarchya araikhik pranalinche pratinidhitva kele jaau shakate heey siddha karoon, yaa varnanachya aadhare anek parinam aanee algoridam prapt kele gele aahet. suruvatiche bahutek kaam nermax modelchya bahupadi vistaravar aadharit hote. yaa aajahi sarvat lokpriy paddhati aahet parantu tivrapane araikhik aanee atyant jatil araikhik pranalinche pratinidhitva karnyasathi wavelets aanee itar vistaranvar aadharit itar adhik jatil prakar sadar kele gele aahet. gairrekheey pranalinche lakshaniy praman nermax modeldware darshavile jaau shakate, jyat arajakta, dwibhajan aanee subharmonics yasarkhya videshi vartanansah pranalincha samavesh aahe.
nermaxchee suruvat eka pratimeche naav mhanoon jhaali, parantu aata teey nonlinear pranali olakhanyachya tatvadnyanat vikasit jhaale aahe. nermaxchya drushtikonat anek anek asatatah
rachana shodhane haah nermaxcha sarvat moolbhut bhag aahe. udaharanarth, eka lagged input aanee eka lagged output term, teen lagged noise terms, cubic polynomial mhanoon vistarit asalelya nermax modelmadhye 82 sambhavya umedvar terms asateel. umedvaranchya atinchi hii sankhya udbhavate kaaran vyakhyenusar vistaramadhye ghan vistaramadhye serve sambhavya sanyojanancha samavesh asato. nishkapatapane yaa serve atincha samavesh asalelya pratimecha andaj lavnyasathi pudhe jaane aanee nantar chhatani kelyane sankhyatmak aanee sanganakeey samasya udbhavatil aanee tyaa nehmi talhalya pahijet. tathapi, modelmadhye fakt kahee sandnya sandnya mahattvachya asatat. eka velee eka sandnya nivadanyache uddisht asalelya sanrachanecha shodh ghene, tyamule gambhirpane mahatvache aahe. eka velee eka karoon nermax modelchya sandnya nivadanyasathi orthogonal lest squares algoridam aanee tyache derivatives vaparoon hii uddishte sahajpane sadhya kelee jaau shakatat. yaa kalpana namuna olakh aanee vaishishtyanchya nivadisathi dekhil swikarlya jaau shakatat aanee mukhya ghatak vishleshanala paryay pradan karoo shakatat parantu yaa fayadhyasah qee vaishishtye mool samasyeshi sahajpane sambandhit asalelya aadhar karye mhanoon ughad kelee jaatat. 
nermax paddhati sarvottam andaje model shodhanyapeksha adhik karnyasathi tayar kelya aahet. pranali olakh done uddishtanmadhye vibhagali jaau shakate. pahilya tappyat andaj samavisht aahe, jyamadhye deta sanch andaje karnare model vikasit karane heey mukhya uddisht aahe, jenekarun changle andaj lavale jaau shaktil. asey anek anuprayog aahet jethe haah drushtikon yogya aahe, udaharanarth hawaman, stockchya kimmati, bhashan, lakshya magova, namuna vargikaran ityadincha kalshreni andaj. asha anuprayoganmadhye pratikrutiche swaroop titke mahattvache nasate. andajit kiman truti nirman karnari andaj yojana shodhane haah uddesh aahe. pranali olakhanyache dusare uddisht, jyamadhye upasanch mhanoon pahile uddisht samavisht aahe, tyat sarvottam sarasari varg truti sadhya karnyasathi keval eka model shodhanyapeksha barech kahee samavisht aahe. haah dusra uddesh mhanje nermax tattvadnyan vikasit qaa kele gele aanee sarvat sopee pratikruti rachana shodhanyachya kalpaneshi jodalele aahe. antarnihit pranalichi gatishil vaishishtye punarutpadit karnari, shakya titke sope model shodhanyasathi aanee shakya asalyas abhyasat asalelya pranaliche ghatak aanee vartananshi yacha sambandh jodanari model vikasit karane haah yetheel uddesh aahe. tyamule olakh patavanyachya yaa yaa drushtikonacha mukhya uddesh pranaliche pratinidhitva karnara niyam olakhane aanee ughad karane haah aahe. hii uddishte pratikruti anukaran aanee niyantran pranalichya rachneshi sambandhit aahet, parantu aushadh, majjatantu vidnyan aanee jeevan vidnyanatil anuprayoganshi adhikadhik sambandhit aahet. yethe, anekada nonlinear models olakhane haah uddesh aahe, jyacha vapar yaa pranali kasha chalatat aanee kase vagatat yachee moolbhut yantrana samajoon ghenyasathi keela jaau shakato, jenekarun aapan hyanmadhye ferfar karoo shakoo aanee tyancha vapar karoo shakoo. varanvarta aanee sthanik-tatpuratya kshetranmadhyehi nermax paddhati vikasit kelya gelya aahet.
yadruchchik araikhik pratikruti.
samanya paristhitit, asey houu shakate qee kahee bahya anishchit adthala nonlinear dynamixmadhun jaato aanee outputwar prabhav padato. hii paristhiti samajoon ghenyasathi puresa samanya asalela eka model varg mhanje yadruchchik araikhik rajya-antaral modelcha varg. yantrik, vidyut kinwa ushmagatikiy bhautik niyam yasarkhya pahilya tattvanchya niyamancha vapar karoon samanyatah rajya-antaral model prapt kele jaate aanee olakhalya olakhalya mapdandanna sahasa kahee bhautik arth kinwa mahatva asate.
eka swatantra-vel sthitee-avkash model bhinn samikaranandware paribhashit kele jaau shakateh
jyaa sutramadhye _ 8 haah velecha sandarbh denara sakaratmak purnank aahe. sootra _ 9 aanee sootra _ 10 hii karye samanya araikhik karye aahet. pahile samikaran sthitee samikaran mhanoon olakhale jaate aanee dusare output samikaran mhanoon olakhale jaate. serve sanket yadruchchik prakriyancha vapar karoon tayar kele jaatat. prakriya sootra _ 11 laa avastha prakriya mhanoon olakhale jaate, sootra _ 12 aanee sootra _ 13 sahasa swatantra aanee paraspar swatantra manale jaatat jase qee sootra _ 14. parameter sootra _ 15 haah samanyatah andaj lavnyasathi (prayogik deta vaparoon) maryadit-aayami (vastavik) parameter asato. sthitee prakriya bhautik sanket asanyachi garaj nahee aanee tee samanyatah naan pahileli asate (mojali jaat nahee) heey lakshat ghya. kahee maryadit sakaratmak purnank moolya sootra _ 18 sathee sootra _ 17 sathee input-output jodya sootra _ 16 chaa sanch mhanoon deta sanch deela jaato.
durdaivane, naan pahilelya yadruchchik chalanchya araikhik parivartanamule, parinamanche sambhavyata karya vishleshanatmakdrushtya klisht aahe; teey bahuayami marginalization integralchya drushtine dile aahe. parinami, samanyatah vaparlya jaanarya parameter andaj paddhati jase qee kamal sambhavyata paddhat kinwa ishtatam eka-paul pudhe bhavishyavanivar aadharit andaj truti paddhat vishleshanatmakdrushtya klisht aahet. alikade, anukramik monte carlo paddhatinvar aadharit algoridamcha vapar outputchya sashart sarasaricha andaj lavnyasathi kinwa apeksha-kamal algoridamchya sanyogane, kamal sambhavyata andajak andaje karnyasathi keela gela aahe. yaa paddhati, jaree lakshanrahitdrushtya ishtatam asalya taree, sangankiyadrushtya magani asalelya aahet aanee tyancha vapar vishisht prakarananparyant maryadit aahe jithe karyarat kann filterchya moolbhut maryada talhalya jaau shakatat. eka paryayi upay mhanje up-ishtatam bhavishyasuchak vaparoon andaj truti paddhat lagoo karane. parinami andajak jordar susangat aanee lakshanrahitpane samanya asalyache darshavile jaau shakate aanee tulanene sope algoridam vaparoon tyache moolyankan kele jaau shakate.

