Injongo yenqaku kukubonelela ngenkxaso ekuqaleni kwezazinzulu zedatha. IN Sichaze iindlela ezintathu zokusombulula i-equation yohlengahlengiso lomgca: isisombululo sohlahlelo, ukwehla kwe-gradient, ukuhla kwe-stochastic. Emva koko kwisisombululo sohlalutyo sisebenzise ifomula . Kweli nqaku, njengoko isihloko sicebisa, siya kuthethelela ukusetyenziswa kwale fomula okanye, ngamanye amazwi, siya kuzifumana ngokwethu.
Kutheni kunengqiqo ukunikela ingqwalasela eyongezelelweyo kwifomula ?
Kungenxa ye-matrix equation ukuba kwiimeko ezininzi umntu uqala ukuqhelana nokuhlehla komgca. Kwangaxeshanye, izibalo ezineenkcukacha zendlela efunyenwe ngayo ifomula zinqabile.
Ngokomzekelo, kwiikhosi zokufunda zoomatshini ezivela kwiYandex, xa abafundi bekwaziswa ngokuphindaphindiweyo, banikezelwa ukuba basebenzise imisebenzi evela kwilayibrari. funda, ngelixa kungekho gama likhankanyiweyo malunga nokumelwa kwe-matrix ye-algorithm. Kungoku nje abanye abaphulaphuli banokufuna ukuwuqonda ngakumbi lo mbandela - bhala ikhowudi ngaphandle kokusebenzisa imisebenzi esele yenziwe. Kwaye ukwenza oku, kufuneka uqale ubonise i-equation kunye ne-regularizer kwifom ye-matrix. Eli nqaku liza kuvumela abo banqwenela ukwazi ubuchule obunjalo. Masiqalise.
Iimeko zokuqala
Izalathi ekujoliswe kuzo
Sinoluhlu lwamaxabiso ekujoliswe kuwo. Ngokomzekelo, isalathisi esijoliswe kuyo ingaba yixabiso layo nayiphi na i-asethi: ioli, igolide, ingqolowa, idola, njl. Kwangaxeshanye, ngenani lamaxabiso ekujoliswe kuwo sithetha inani lokuqwalaselwa. Ukuqwalaselwa okunjalo kunokuba, umzekelo, amaxabiso e-oyile ngenyanga ngonyaka, oko kukuthi, siya kuba ne-12 ekujoliswe kuyo. Masiqale ukwazisa inqaku. Masibonise ixabiso ngalinye lesalathisi ekujoliswe kuso njenge . Lilonke sinalo imigqaliselo, nto leyo ethetha ukuba sinokumela imigqaliselo yethu njenge .
IiRegressors
Siya kucinga ukuba kukho izinto ezichaza amaxabiso esalathisi ekujoliswe kuso ukuya kuthi ga kwinqanaba elithile. Ngokomzekelo, i-dollar / i-ruble yokutshintshiselana ifuthe kakhulu ngexabiso leoli, i-Federal Reserve rate, njl. Izinto ezinjalo zibizwa ngokuba yi-regressors. Ngexesha elifanayo, ixabiso lesalathisi ngasinye kufuneka lihambelane nexabiso le-regressor, oko kukuthi, ukuba sinezikhombisi ezili-12 ekujoliswe kuzo kwinyanga nganye ngo-2018, ngoko kufuneka sibe ne-12 ye-regressor values ββkwixesha elifanayo. Masibonise amaxabiso e-regressor nganye ngo . Makubekho kwimeko yethu regressors (okt. izinto eziphembelela amaxabiso esalathisi ekujoliswe kuko). Oku kuthetha ukuba i-regressors yethu inokuboniswa ngolu hlobo lulandelayo: kwi-1st regressor (umzekelo, ixabiso leoli): , ye-2nd regressor (umzekelo, ireyithi yeFed): , Kuba "-th" ukubuyisela:
Ukuxhomekeka kwezalathi ekujoliswe kuzo kwii-regressors
Makhe sicinge ukuba ukuxhomekeka kwesalathisi esijoliswe kuyo ukusuka kwi-regressors "th" uqwalaselo lunokubonakaliswa ngomgca wobuyiselo lwe-equation kwifom:
phi - "-th" ixabiso lokubuyisela ukusuka ku-1 ukuya ,
- inani le-regressors ukusuka kwi-1 ukuya
- i-angular coefficients, emele inani apho isalathisi esijoliswe kuyo siya kutshintsha ngokomyinge xa ββi-regressor itshintsha.
Ngamanye amazwi, singabantu bonke (ngaphandle ) yeregressor simisela i-coefficient "yethu". , emva koko phinda-phinda i-coefficients ngamaxabiso e-regressors "th" uqwalaselo, ngenxa yoko sifumana uqikelelo oluthile "-th" isalathisi ekujoliswe kuso.
Ngoko ke, kufuneka sikhethe ii-coefficients ezinjalo , apho amaxabiso emisebenzi yethu esondeleyo iya kuba kufutshane kangangoko kumaxabiso esalathisi ekujoliswe kuko.
Ukuvavanya umgangatho womsebenzi osondeleyo
Siya kumisela uvavanyo lomgangatho womsebenzi oqikelelweyo sisebenzisa eyona ndlela incinci yesikwere. Umsebenzi wovavanyo lomgangatho kule meko uya kuba ngolu hlobo lulandelayo:
Kufuneka sikhethe amaxabiso anjalo ee-coefficients $w$ apho ixabiso iya kuba ngoyena mncinane.
Ukuguqula i-equation kwifom ye-matrix
Vector ukumelwa
Ukuqala, ukwenza ubomi bakho bube lula, kuya kufuneka ubeke ingqalelo kwi-equation yohlengahlengiso kwaye uqaphele ukuba i-coefficient yokuqala. ayiphinda-phindiswe nasiphi na isibuyiseli. Kwangaxeshanye, xa siguqula idatha kwifom ye-matrix, le meko ikhankanywe ngasentla iya kwenza ukuba izibalo zibe nzima kakhulu. Kule nkalo, kucetywayo ukwazisa enye i-regressor ye-coefficient yokuqala kwaye uyilinganise kwenye. Okanye, yonke into "linganisa ixabiso le-th yale regressor kwenye - emva kwayo yonke into, xa iphindaphindwe enye, akukho nto iya kutshintsha ukusuka kwindawo yokujonga isiphumo sokubala, kodwa ukusuka kwindawo yokujonga imithetho yemveliso yeematriki, intuthumbo yethu. iya kuncitshiswa kakhulu.
Ngoku, okwangoku, ukwenza lula izinto, makhe sicinge ukuba sinenye kuphela "-th" ukujonga. Emva koko, khawucinge amaxabiso e-regressors "-th" uqwalaselo njenge vector . IVector inomlinganiselo , oko kukuthi imiqolo kunye noluhlu olu-1:
Masimele i-coefficients efunekayo njengevektha , enomlinganiselo :
Inxaki yobuyiselo ngomgca ye "-th" ukujonga kuya kuthatha ifom:
Umsebenzi wokuvavanya umgangatho wemodeli yomgca uya kuba ngolu hlobo:
Nceda uqaphele ukuba ngokuhambelana nemithetho yokuphindaphinda kwe-matrix, bekufuneka sitshintshe i-vector .
Ukumelwa kweMatrix
Njengesiphumo sokuphindaphinda ii-vectors, sifumana inani: , into elindelekileyo. Eli nani liqikelelo "-th" isalathisi ekujoliswe kuso. Kodwa sifuna uqikelelo lwexabiso elinye ekujoliswe kulo, kodwa zonke. Ukwenza oku, masibhale phantsi yonke into ""th" iiregressors kwifomathi yematrix . I-matrix enesiphumo inomlinganiselo :
Ngoku i-linear regresse equation iyakuthatha imo:
Masibonise amaxabiso ezalathi ekujoliswe kuzo (zonke ) ngevektha nganye ubukhulu :
Ngoku sinokubhala i-equation yokuvavanya umgangatho wemodeli yomgca kwifomathi yematrix:
Ngokwenyani, kule fomula sifumana ngakumbi ifomula eyaziwayo kuthi
Yenziwa njani? Izibiyeli zivuliwe, ukwahlula kuyenziwa, iintetho eziphumelayo zitshintshwa, njalo njalo, kwaye yile nto kanye esiya kuyenza ngoku.
Ukuguqulwa kweMatrix
Masivule izibiyeli
Masilungiselele i-equation yomahluko
Ukwenza oku, siya kwenza iinguqu ezithile. Kwizibalo ezilandelayo kuya kuba lula kuthi ukuba i-vector iya kumelwa ekuqaleni kwemveliso nganye kwinxaki.
Uguqulo 1
Ingaba yenzeke njani? Ukuphendula lo mbuzo, jonga nje ubungakanani beematriki eziphindaphindwayo kwaye ubone ukuba kwisiphumo sifumana inani okanye ngenye indlela. .
Masibhale phantsi ubungakanani bentetho ye-matrix.
Uguqulo 2
Masiyibhale ngendlela efanayo kwinguqu yoku-1
Kwimveliso sifumana i-equation ekufuneka siyahlule:
Siyahlula umsebenzi wovavanyo lomgangatho wemodeli
Masihlule ngokubhekiselele kwi-vector :
Imibuzo ukuba kutheni Akumele kubekho, kodwa siya kuphonononga imisebenzi yokumisela izinto eziphuma kwamanye amabinzana amabini ngokweenkcukacha ezithe vetshe.
Umahluko 1
Makhe sandise umahluko:
Ukuze unqume i-derivative ye-matrix okanye i-vector, kufuneka ujonge into engaphakathi kuyo. Masijonge:
Makhe sibonise imveliso yeematriki ngematrix . Imatrix isikwere kwaye ngaphezu koko, i-symmetrical. Ezi zakhiwo ziya kuba luncedo kuthi kamva, masizikhumbule. Imatrix inomlinganiselo :
Ngoku umsebenzi wethu kukuphinda-phinda ngokuchanekileyo ii-vectors ngematrix kwaye ungafumani "isibini kabini sisihlanu," ke masigxininise kwaye silumke kakhulu.
Nangona kunjalo, siye safumana intetho entsonkothileyo! Enyanisweni, sifumene inani - i-scalar. Kwaye ngoku, ngokwenene, siqhubela phambili ekuhlukaniseni. Kuyimfuneko ukufumana i-derivative yesiphumo sokuchazwa kwe-coefficient nganye kwaye ufumane i-vector yobungakanani njengemveliso . Ukuba kunokwenzeka, ndiza kubhala phantsi iinkqubo ngesenzo:
1) yahlula nge , sifumana:
2) yahlula nge , sifumana:
3) yahlula nge , sifumana:
Isiphumo yivector ethenjisiweyo yobungakanani :
Ukuba ujonga i-vector ngokusondeleyo ngakumbi, uya kuqaphela ukuba izinto ezisekhohlo kunye nezihambelanayo zasekunene ze-vector zinokudityaniswa ngendlela yokuba, ngenxa yoko, i-vector inokwahlulwa kwivector ebonisiweyo. ubungakanani . Umzekelo (isici sasekhohlo somgca ophezulu wevektha) (into efanelekileyo yomgca ophezulu we-vector) inokubonakaliswa njenge , kwaye - njengoko njl. kumgca ngamnye. Masenze iqela:
Masikhuphe i-vector kwaye kwisiphumo sifumana:
Ngoku, makhe sijonge ngakumbi kwisiphumo sematrix. I-matrix sisimbuku seematriki ezimbini :
Masikhumbule ukuba ngaphambilana siqaphele ipropathi enye ebalulekileyo ye-matrix -I-symmetrical. Ngokusekelwe kule propati, sinokuthi ngokuzithemba ukuba ibinzana lingana . Oku kunokuqinisekiswa ngokulula ngokwandisa imveliso ye-matrices element by element . Asizukwenza oku apha; abo banomdla banokuzijonga ngokwabo.
Masibuyele kwintetho yethu. Emva kweenguqu zethu, kuye kwavela ngendlela ebesifuna ukuyibona ngayo:
Ngoko ke, siye sagqiba umahluko wokuqala. Masiqhubele phambili kwintetho yesibini.
Umahluko 2
Masilandele indlela ebethiwe. Iya kuba mfutshane kakhulu kuneyangaphambili, ke musa ukuya kude kakhulu nesikrini.
Masikhulise i-vectors kunye ne-matrix element ngento:
Masisuse zombini kwizibalo kwithuba elithile - ayidlali indima enkulu, ngoko siya kuyibuyisela endaweni yayo. Masiphindaphinde iivektha ngematrix. Okokuqala, masiphindaphinde imatrix kwivektha , asinazithintelo apha. Sifumana ubungakanani bevector :
Masenze eli nyathelo lilandelayo- phinda-phinda i-vector kwivektha yesiphumo. Ekuphumeni inombolo izakube isilindile:
Emva koko siya kuyahlula. Kwimveliso sifumana i-vector ye-dimension :
Undikhumbuza into? Ilungile lo nto! Le yimveliso ye-matrix kwivektha .
Ngaloo ndlela, ulwahlulo lwesibini lugqitywe ngempumelelo.
Endaweni yesiphelo
Ngoku siyazi ukuba ukulingana kwenzeka njani .
Ekugqibeleni, siya kuchaza indlela ekhawulezayo yokuguqula iifomula ezisisiseko.
Masivavanye umgangatho wemodeli ngokuhambelana neyona ndlela incinci yesikwere:
Masenze umahluko kwisiphumo sentetho:
Iincwadi
Imithombo ye-Intanethi:
1)
2)
3)
4)
Iincwadi zezifundo, ingqokelela yeengxaki:
1) Amanqaku esifundo kwimathematika ephezulu: ikhosi epheleleyo / D.T. Kubhaliwe - 4th ed. -M.: Iris-press, 2006
2) Uhlalutyo olusetyenzisiweyo lokuhlehla / N. Draper, G. Smith - 2nd ed. β M.: Finance and Statistics, 1986 (inguqulelo esuka kwisiNgesi)
3) Iingxaki zokusombulula iequation zematrix:
umthombo: www.habr.com