{"id":37955,"date":"2019-10-31T22:20:47","date_gmt":"2019-10-31T19:20:47","guid":{"rendered":"https:\/\/prohoster.info\/blog\/linejnaya-regressiya-i-metody-eyo-vosstanovleniya\/"},"modified":"2019-10-31T22:20:47","modified_gmt":"2019-10-31T19:20:47","slug":"linejnaya-regressiya-i-metody-eyo-vosstanovleniya","status":"publish","type":"post","link":"https:\/\/prohoster.info\/sq\/blog\/administrirovanie\/linejnaya-regressiya-i-metody-eyo-vosstanovleniya","title":{"rendered":"Regresioni linear dhe metodat e tij t\u00eb rikuperimit","gt_translate_keys":[{"key":"rendered","format":"text"}]},"content":{"rendered":"

\"Regresioni
\nBurimi: xkcd<\/a><\/noindex><\/i><\/p>\n

Regresioni linear \u00ebsht\u00eb nj\u00eb nga algoritmet themelore p\u00ebr shum\u00eb fusha q\u00eb lidhen me analiz\u00ebn e t\u00eb dh\u00ebnave. Arsyeja p\u00ebr k\u00ebt\u00eb \u00ebsht\u00eb e qart\u00eb. \u00cbsht\u00eb nj\u00eb algorit\u00ebm shum\u00eb i thjesht\u00eb dhe i kuptuesh\u00ebm, q\u00eb kontribuon n\u00eb p\u00ebrdorimin e tij t\u00eb gjer\u00eb p\u00ebr shum\u00eb dekada, n\u00ebse jo p\u00ebr shekuj. Ideja \u00ebsht\u00eb q\u00eb ne supozojm\u00eb nj\u00eb var\u00ebsi lineare t\u00eb nj\u00eb variabli nga nj\u00eb grup variablash t\u00eb tjer\u00eb, dhe m\u00eb pas p\u00ebrpiqemi ta rikonstruktojm\u00eb k\u00ebt\u00eb var\u00ebsi.<\/p>\n

Por n\u00eb k\u00ebt\u00eb artikull nuk do t\u00eb flasim p\u00ebr p\u00ebrdorimin e regresionit linear p\u00ebr zgjidhjen e problemeve praktike. K\u00ebtu do t\u00eb shqyrtohen karakteristikat interesante t\u00eb implementimit t\u00eb algoritmeve t\u00eb shp\u00ebrndara p\u00ebr rikonstruksionin e tij, me t\u00eb cilat ne u p\u00ebrball\u00ebm gjat\u00eb shkruajtjes s\u00eb modulit t\u00eb m\u00ebsimit t\u00eb makinerive n\u00eb Apache Ignite<\/a><\/noindex>. Pak matematik\u00eb themelore, bazat e m\u00ebsimit t\u00eb makinerive dhe t\u00eb dh\u00ebnat e shp\u00ebrndara do t\u00eb ndihmojn\u00eb t\u00eb kuptojm\u00eb se si t\u00eb rikonstruktojm\u00eb regresionin linear, edhe n\u00ebse t\u00eb dh\u00ebnat jan\u00eb t\u00eb shp\u00ebrndara mes mij\u00ebra nyjash.
\n<\/a><\/noindex><\/p>\n

\u00c7far\u00eb \u00ebsht\u00eb n\u00eb bised\u00eb?<\/h3>\n

\nNa pritet nj\u00eb detyr\u00eb p\u00ebr rikonstruktimin e var\u00ebsis\u00eb lineare. Si t\u00eb dh\u00ebna hyr\u00ebse jepen shum\u00eb vektor\u00ebsh variablash t\u00eb supozuar si t\u00eb pavarur, t\u00eb cil\u00ebve \u00e7do prej tyre i korrespondon nj\u00eb vler\u00eb e caktuar e variablit t\u00eb varur. K\u00ebto t\u00eb dh\u00ebna mund t\u00eb paraqiten n\u00eb form\u00ebn e dy matricave:<\/p>\n

\"Regresioni<\/p>\n

\nTani, pasi q\u00eb supozohet nj\u00eb var\u00ebsi, dhe madje edhe lineare, do ta shkruajm\u00eb supozimin ton\u00eb n\u00eb form\u00ebn e nj\u00eb producti matricash (p\u00ebr thjesht\u00ebsim, k\u00ebtu dhe m\u00eb tej supozohet q\u00eb pjesa e lir\u00eb e ekuacionit fshihet pas \"Regresioni, dhe kolon\u00eb e fundit e matric\u00ebs \"Regresioni p\u00ebrmban nj\u00ebsi):<\/p>\n

\"Regresioni<\/p>\n

\nShum\u00eb e ngjashme me nj\u00eb sistem ekuacionesh lineare, apo jo? Ngjan, por ka t\u00eb ngjar\u00eb q\u00eb ky sistem ekuacionesh t\u00eb mos ket\u00eb zgjidhje. Arsyeja p\u00ebr k\u00ebt\u00eb \u00ebsht\u00eb zhurma, e cila \u00ebsht\u00eb e pranishme praktikisht n\u00eb t\u00eb dh\u00ebnat reale. Nj\u00eb tjet\u00ebr arsye mund t\u00eb jet\u00eb mungesa e var\u00ebsis\u00eb lineare si e till\u00eb, me t\u00eb cil\u00ebn mund t\u00eb p\u00ebrpiqemi t\u00eb p\u00ebrballojm\u00eb duke futur variablat shtes\u00eb, q\u00eb varen n\u00eb m\u00ebnyr\u00eb jo lineare nga t\u00eb dh\u00ebnat fillestare. Le t\u00eb shqyrtojm\u00eb shembullin e m\u00ebposht\u00ebm:
\n\"Regresioni
\nBurimi: Wikipedia<\/a><\/noindex><\/i><\/p>\n

Ky \u00ebsht\u00eb nj\u00eb shembull i thjesht\u00eb i regresionit linear, i cili demonstruar var\u00ebsin\u00eb e nj\u00eb variabli (n\u00eb boshtin \"Regresioni) nga variabli tjet\u00ebr (n\u00eb boshtin \"Regresioni). P\u00ebr t\u00eb pasur nj\u00eb sistem linjar ekuacionesh n\u00eb p\u00ebrputhje me k\u00ebt\u00eb shembull, t\u00eb gjith\u00eb pikat duhet t\u00eb bien sakt\u00ebsisht n\u00eb nj\u00eb vij\u00eb. Por nuk \u00ebsht\u00eb k\u00ebshtu. Ato nuk bien n\u00eb nj\u00eb vij\u00eb pik\u00ebrisht p\u00ebr shkak t\u00eb zhurm\u00ebs (apo p\u00ebr shkak se supozimi i pranish\u00ebm i var\u00ebsis\u00eb linjare ishte i gabuar). Pra, p\u00ebr t\u00eb rikuperuar var\u00ebsin\u00eb linjare nga t\u00eb dh\u00ebnat reale zakonisht k\u00ebrkohet t\u00eb futet nj\u00eb supozim tjet\u00ebr: t\u00eb dh\u00ebnat p\u00ebrmbajn\u00eb zhurm\u00eb dhe kjo zhurm\u00eb ka shp\u00ebrndarje normale<\/a><\/noindex>. Mund t\u00eb b\u00ebhen supozime dhe p\u00ebr tipe t\u00eb tjera shp\u00ebrndarjesh zhurme, por n\u00eb shumic\u00ebn d\u00ebrrmuese t\u00eb rasteve shqyrtohet pik\u00ebrisht shp\u00ebrndarja normale, p\u00ebr t\u00eb cil\u00ebn do t\u00eb flitet m\u00eb tej.<\/p>\n

Metoda e maksimumit t\u00eb mund\u00ebsive<\/h3>\n

\nPra, ne supozuam prani t\u00eb zhurm\u00ebs rast\u00ebsore me shp\u00ebrndarje normale. \u00c7far\u00eb t\u00eb b\u00ebjm\u00eb n\u00eb nj\u00eb situat\u00eb t\u00eb till\u00eb? P\u00ebr k\u00ebt\u00eb rast, n\u00eb matematik\u00eb ekziston dhe p\u00ebrdoret gjer\u00ebsisht metod\u00ebn e maksimalit t\u00eb mund\u00ebsis\u00eb<\/a><\/noindex>. N\u00ebse e p\u00ebrmbledhim, thelbi i tij \u00ebsht\u00eb zgjedhja funksionit t\u00eb mund\u00ebsive<\/a><\/noindex> dhe maksimizimi i saj t\u00eb m\u00ebtejsh\u00ebm.<\/p>\n

Kthehemi n\u00eb rikuperimin e var\u00ebsis\u00eb linjare nga t\u00eb dh\u00ebnat me zhurm\u00eb normale. V\u00ebrejm\u00eb se var\u00ebsia linjare e supozuar \u00ebsht\u00eb nj\u00eb pr \u043e\u0436\u0438\u0434\u0430\u043d\u0438\u0435 matematikore \"Regresioni e shp\u00ebrndarjes normale t\u00eb pranishme. N\u00eb t\u00eb nj\u00ebjt\u00ebn koh\u00eb, probabiliteti q\u00eb \"Regresioni merr vler\u00ebn e caktuar, ndonj\u00ebher\u00eb n\u00eb kushtet e pranis\u00eb s\u00eb observimeve \"Regresioni, duket si m\u00eb posht\u00eb:<\/p>\n

\"Regresioni<\/p>\n

\nTani le t\u00eb vendosim n\u00eb vend t\u00eb \"Regresioni dhe \"Regresioni variablat q\u00eb na duhen:<\/p>\n

\"Regresioni<\/p>\n

\nVazhdon t\u00eb mbetet vet\u00ebm t\u00eb gjejm\u00eb vektorin \"Regresioni, p\u00ebr t\u00eb cilin ky probabilitet \u00ebsht\u00eb maksimal. P\u00ebr t\u00eb maksimizuar nj\u00eb funksion t\u00eb till\u00eb, \u00ebsht\u00eb e p\u00ebrshtatshme s\u00eb pari ta prologarit\u00ebsh at\u00eb (logaritemi i funksionit do t\u00eb arrij\u00eb maksimumin n\u00eb t\u00eb nj\u00ebjtin pik\u00eb si vet\u00eb funksioni):<\/p>\n

\"Regresioni<\/p>\n

\n\u00c7ka, nga ana tjet\u00ebr, reduktonet n\u00eb minimizimin e funksionit t\u00eb m\u00ebposht\u00ebm:<\/p>\n

\"Regresioni<\/p>\n

\nDuket, kjo quhet metoda e katror\u00ebve t\u00eb vegj\u00ebl<\/a><\/noindex>. Shpesh t\u00eb gjitha k\u00ebto konsiderata t\u00eb m\u00ebsip\u00ebrme kalohen dhe p\u00ebrdoren thjesht kjo metod\u00eb.<\/p>\n

QR dekompozimi<\/h3>\n

\nMinimalen e funksionit t\u00eb m\u00ebsip\u00ebrm mund ta gjejm\u00eb, n\u00ebse gjejm\u00eb pik\u00ebn ku gradienti i k\u00ebtij funksioni \u00ebsht\u00eb zero. Dhe gradienti do t\u00eb regjistrohet n\u00eb k\u00ebt\u00eb m\u00ebnyr\u00eb:<\/p>\n

\"Regresioni<\/p>\n

\nQR dekompozimi<\/a><\/noindex> \u00ebsht\u00eb nj\u00eb metod\u00eb matricore p\u00ebr zgjidhjen e problemit t\u00eb minimizimit t\u00eb p\u00ebrdorur n\u00eb metod\u00ebn e katror\u00ebve t\u00eb vegj\u00ebl. P\u00ebr k\u00ebt\u00eb arsye, le t\u00eb shkruajm\u00eb ekuacionin n\u00eb form\u00eb matricore:<\/p>\n

\"Regresioni<\/p>\n

\nPra, ne po ndajm\u00eb matric\u00ebn \"Regresioni n\u00eb matrica \"Regresioni dhe \"Regresioni dhe kryejm\u00eb nj\u00eb s\u00ebr\u00eb transformimesh (algoritmi i QR dekompozitimit nuk do t\u00eb diskutohet k\u00ebtu, vet\u00ebm p\u00ebrdorimi i tij n\u00eb lidhje me problemin e caktuar):<\/p>\n

\"Regresioni<\/p>\n

\nMatrica \"Regresioni \u00ebsht\u00eb ortogonale. Kjo na lejon t\u00eb heqim dor\u00eb nga produkti \"Regresioni:<\/p>\n

\"Regresioni<\/p>\n

\nDhe n\u00ebse z\u00ebvend\u00ebsojm\u00eb \"Regresioni n\u00eb \"Regresioni, do t\u00eb rezultoj\u00eb n\u00eb \"Regresioni. Duke marr\u00eb parasysh se \"Regresioni \u00ebsht\u00eb nj\u00eb matric\u00eb e sip\u00ebrme triangulare, kjo duket k\u00ebshtu:<\/p>\n

\"Regresioni<\/p>\n

\nKjo mund t\u00eb zgjidhet me metod\u00ebn e z\u00ebvend\u00ebsimit. Elementi \"Regresioni ndodhet si \"Regresioni, elementi i m\u00ebparsh\u00ebm \"Regresioni ndodhet si \"Regresioni dhe k\u00ebshtu me radh\u00eb.<\/p>\n

K\u00ebtu duhet t\u00eb theksojm\u00eb se kompleksiteti i algoritmit t\u00eb fituar p\u00ebr shkak t\u00eb p\u00ebrdorimit t\u00eb QR dekompozitimit \u00ebsht\u00eb i barabart\u00eb me \"Regresioni. Nd\u00ebrkoh\u00eb, pavar\u00ebsisht se operacioni i shumzimit t\u00eb matricave mund t\u00eb paralelizohet mir\u00eb, nuk duket e mundur t\u00eb shkruhen nj\u00eb version efektiv i distribuar i k\u00ebtij algoritmi.<\/p>\n

Zbritja gradiente<\/h3>\n

\nDuke folur p\u00ebr minimizimin e nj\u00eb funksioni t\u00eb caktuar, gjithmon\u00eb ia vlen t\u00eb p\u00ebrmendet metoda e (stohastik) gradienteve t\u00eb zbritjes. Kjo \u00ebsht\u00eb nj\u00eb metod\u00eb e thjesht\u00eb dhe efektive e minimizimit, e bazuar n\u00eb llogaritjen iteruese t\u00eb gradientit t\u00eb funksionit n\u00eb nj\u00eb pik\u00eb dhe l\u00ebvizjen e m\u00ebpasshme n\u00eb drejtimin e kund\u00ebrt t\u00eb gradientit. \u00c7do hap i till\u00eb afrohet zgjidhjes n\u00eb minimum. Gradienti n\u00eb k\u00ebt\u00eb rast duket ende k\u00ebshtu:<\/p>\n

\"Regresioni<\/p>\n

Gjithashtu, kjo metod\u00eb paralelizohet dhe shp\u00ebrndahet mir\u00eb p\u00ebr shkak t\u00eb pronave lineare t\u00eb operatorit t\u00eb gradientit. V\u00ebrejm\u00eb se n\u00eb formul\u00ebn e m\u00ebsip\u00ebrme n\u00ebn shenj\u00ebn e shum\u00ebs jan\u00eb terma t\u00eb pavarur. Me fjal\u00eb t\u00eb tjera, ne mund t\u00eb llogarisim gradientin n\u00eb m\u00ebnyr\u00eb t\u00eb pavarur p\u00ebr t\u00eb gjitha indeksat \"Regresioni nga i pari deri te \"Regresioni, nd\u00ebrkoh\u00eb q\u00eb paralelisht llogarisim gradientin p\u00ebr indeksat nga \"Regresioni n\u00eb \"Regresioni. Pastaj, do t\u00eb mbledhim gradient\u00ebt e marra. Rezultati i mbledhjes do t\u00eb jet\u00eb i nj\u00ebjt\u00eb me at\u00eb n\u00ebse do t\u00eb llogarisnim menj\u00ebher\u00eb gradientin p\u00ebr indeksat nga i pari deri te \"Regresioni. K\u00ebshtu, n\u00ebse t\u00eb dh\u00ebnat jan\u00eb t\u00eb shp\u00ebrndara midis disa pjes\u00ebve t\u00eb dh\u00ebnash, gradienti mund t\u00eb llogaritet n\u00eb m\u00ebnyr\u00eb t\u00eb pavarur n\u00eb secil\u00ebn pjes\u00eb, e m\u00eb pas rezultatet e k\u00ebtyre llogaritjeve mund t\u00eb mbledhen p\u00ebr t\u00eb marr\u00eb rezultatin p\u00ebrfundimtar:<\/p>\n

\"Regresioni<\/p>\n

Nga pik\u00ebpamja e implementimit, kjo p\u00ebrputhet me paradigm\u00ebn MapReduce<\/a><\/noindex>N\u00eb \u00e7do hap t\u00eb gradientes s\u00eb zbritjes, nj\u00eb detyr\u00eb p\u00ebr t\u00eb llogaritur gradientin d\u00ebrgohet n\u00eb \u00e7do nyje t\u00eb dh\u00ebnash, m\u00eb pas gradientet e llogaritura mblidhen s\u00eb bashku, dhe rezultati i mbledhjes s\u00eb tyre p\u00ebrdoret p\u00ebr t\u00eb p\u00ebrmir\u00ebsuar rezultatin.<\/p>\n

Pavar\u00ebsisht thjesht\u00ebsis\u00eb s\u00eb zbatimit dhe mund\u00ebsis\u00eb s\u00eb ekzekutimit n\u00eb paradigm\u00ebn MapReduce, gradient descent ka gjithashtu disavantazhet e tij. N\u00eb ve\u00e7anti, numri i hapave t\u00eb nevojsh\u00ebm p\u00ebr t\u00eb arritur konvergjenc\u00ebn \u00ebsht\u00eb ndjesh\u00ebm m\u00eb i lart\u00eb krahasuar me metodat e tjera m\u00eb t\u00eb specializuara.<\/p>\n

LSQR<\/h3>\n

\nLSQR<\/a><\/noindex> \u2014 nj\u00eb tjet\u00ebr metod\u00eb zgjidhje e cila \u00ebsht\u00eb e p\u00ebrshtatshme si p\u00ebr rikonstruktimin e regresionit linear, ashtu edhe p\u00ebr zgjidhjen e sistemeve t\u00eb ekuacioneve lineare. Karakteristika e saj kryesore \u00ebsht\u00eb se kombinon avantazhet e metodave matricore dhe t\u00eb qasjes iteruese. Implementimet e k\u00ebsaj metode mund t\u00eb gjenden si n\u00eb bibliotekat SciPy<\/a><\/noindex>, ashtu edhe n\u00eb MATLAB<\/a><\/noindex>. P\u00ebrshkrimi i k\u00ebsaj metode nuk do t\u00eb jepet k\u00ebtu (mund ta gjeni n\u00eb artikullin LSQR: An algorithm for sparse linear equations and sparse least squares<\/a><\/noindex>). N\u00eb vend t\u00eb k\u00ebsaj do t\u00eb demonstrohet nj\u00eb qasje q\u00eb lejon adaptimin e LSQR p\u00ebr t\u00eb funksionuar n\u00eb nj\u00eb ambient t\u00eb shp\u00ebrndar\u00eb.<\/p>\n

Baza e metod\u00ebs LSQR \u00ebsht\u00eb procedura e bidiagonalizimit<\/a><\/noindex>. Kjo \u00ebsht\u00eb nj\u00eb procedur\u00eb iteruese, ku \u00e7do iteracion p\u00ebrb\u00ebhet nga hapat e m\u00ebposht\u00ebm:
\n\"Regresioni
\n
\nPor n\u00ebse merret parasysh se matrica \"Regresioni \u00ebsht\u00eb e particionuar horizontalisht, \u00e7do iteracion mund t\u00eb paraqitet si dy hapa MapReduce. N\u00eb k\u00ebt\u00eb m\u00ebnyr\u00eb minimizohen d\u00ebrgesat e t\u00eb dh\u00ebnave gjat\u00eb \u00e7do iteracioni (vet\u00ebm vektor\u00eb me nj\u00eb gjat\u00ebsi t\u00eb barabart\u00eb me numrin e t\u00eb panjohurave):<\/p>\n

\"Regresioni
\n
\nKjo qasje p\u00ebrdoret p\u00ebr implementimin e regresionit linear n\u00eb Apache Ignite ML<\/a><\/noindex>.<\/p>\n

P\u00ebrfundim<\/h3>\n

\nEkzistojn\u00eb shum\u00eb algoritma p\u00ebr rikonstruktimin e regresionit linear, por jo t\u00eb gjith\u00eb mund t\u00eb aplikohen n\u00eb \u00e7do kushte. K\u00ebshtu, decompozimi QR \u00ebsht\u00eb shum\u00eb i p\u00ebrshtatsh\u00ebm p\u00ebr zgjidhje t\u00eb sakta n\u00eb grupe t\u00eb vogla t\u00eb dh\u00ebnash. Gradient descent \u00ebsht\u00eb leht\u00ebsisht i implementuesh\u00ebm dhe lejon t\u00eb gjejm\u00eb shpejt nj\u00eb zgjidhje t\u00eb p\u00ebraf\u00ebrt. Nd\u00ebrsa LSQR kombinon vetit\u00eb m\u00eb t\u00eb mira t\u00eb dy algoritmeve t\u00eb m\u00ebparshme, pasi mund t\u00eb shp\u00ebrndahet, konvergon m\u00eb shpejt krahasuar me gradient descent, si dhe lejon ndalimin e hersh\u00ebm t\u00eb algoritmit n\u00eb krahasim me decompozimin QR p\u00ebr t\u00eb gjetur nj\u00eb zgjidhje t\u00eb p\u00ebraf\u00ebrt.
\n
Burimi: habr.com<\/a><\/p>","protected":false,"gt_translate_keys":[{"key":"rendered","format":"html"}]},"excerpt":{"rendered":"

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