Manufar labarin shine don ba da tallafi ga farkon masana kimiyyar bayanai. IN Mun zayyana hanyoyi guda uku don warware ma'aunin koma baya na linzamin kwamfuta: mafita na nazari, zuriyar gradient, zuriyar gradient na stochastic. Sa'an nan don bayani na nazari mun yi amfani da dabarar . A cikin wannan labarin, kamar yadda take ya nuna, za mu ba da hujjar yin amfani da wannan dabara ko kuma, a wasu kalmomi, za mu samo shi da kanmu.
Me yasa yana da ma'ana don ba da hankali sosai ga dabara ?
Tare da matrix ɗin ne a mafi yawan lokuta mutum ya fara sanin koma bayan layi. A lokaci guda, ƙididdiga dalla-dalla na yadda aka samo dabarar ba safai ba ne.
Misali, a cikin darussan koyon injin daga Yandex, lokacin da aka gabatar da ɗalibai don daidaitawa, ana ba su don amfani da ayyuka daga ɗakin karatu. sklearn, yayin da ba a ambaci kalma ɗaya ba game da wakilcin matrix na algorithm. A wannan lokacin ne wasu masu sauraro na iya son fahimtar wannan batu dalla-dalla - rubuta lambar ba tare da amfani da ayyukan da aka shirya ba. Kuma don yin wannan, dole ne ka fara gabatar da ma'auni tare da mai daidaitawa a cikin sigar matrix. Wannan labarin zai ba wa waɗanda suke so su mallaki irin wannan ƙwarewar. Mu fara.
Yanayin farko
Manufofin manufa
Muna da kewayon ƙimar manufa. Misali, alamar da aka yi niyya na iya zama farashin kowace kadara: mai, zinare, alkama, dala, da sauransu. A lokaci guda, ta yawan ma'auni masu nuna alama muna nufin adadin abubuwan lura. Irin waɗannan abubuwan lura na iya zama, alal misali, farashin mai na wata-wata na shekara, wato, za mu sami ƙima 12 da aka yi niyya. Bari mu fara gabatar da bayanin. Bari mu nuna kowace ƙimar maƙasudin alama azaman . Gabaɗaya muna da lura, wanda ke nufin za mu iya wakiltar abubuwan lura kamar .
Masu koma baya
Za mu ɗauka cewa akwai abubuwan da za su bayyana ƙimar maƙasudin manufa. Misali, farashin dala/ruble yana da tasiri sosai ta hanyar farashin man fetur, da kuɗin Tarayyar Tarayya, da dai sauransu. Irin waɗannan abubuwan ana kiran su regressors. A lokaci guda kuma, kowane ƙimar mai nuna alama dole ne ya dace da ƙimar regressor, wato, idan muna da alamomin manufa guda 12 a kowane wata a cikin 2018, to yakamata mu sami ƙimar regressor 12 na lokaci guda. Bari mu nuna kimar kowane regressor ta . Bari a cikin yanayinmu akwai regressors (i.e. abubuwan da ke tasiri ga ƙima mai nuna alama). Wannan yana nufin za a iya gabatar da namu koma baya kamar haka: ga mai koma baya na farko (misali, farashin mai): , na 2nd regressor (misali, ƙimar Fed): , Don"-th" regressor:
Dogaro da alamomin manufa akan masu sake komawa
Bari mu ɗauka cewa dogara ga mai nuna alama daga regressors"th" za a iya bayyana abin lura ta hanyar ma'auni na koma baya na layin:
inda - "-th" darajar regressor daga 1 zuwa ,
- adadin regressors daga 1 zuwa
- Ƙididdigar angular, waɗanda ke wakiltar adadin da abin da ƙididdigewa mai nuna alama zai canza a matsakaici lokacin da regressor ya canza.
A takaice dai, mu na kowa ne (sai dai ) na regressor mun ƙayyade "mu" coefficient , sa'an nan ninka coefficients da dabi'u na regressors "th" lura, a sakamakon haka mun sami wani kima"-th" mai nuna manufa.
Don haka, muna buƙatar zaɓar irin waɗannan ƙididdiga , wanda darajar mu approximating aiki za a kasance kusa da yiwuwar ƙima mai nuna alama.
Tantance ingancin aikin kusantar
Za mu ƙayyade ƙimar ƙimar aikin kusantar ta amfani da mafi ƙarancin hanyar murabba'i. Aikin tantance ingancin a wannan yanayin zai ɗauki nau'i mai zuwa:
Muna buƙatar zaɓar irin waɗannan dabi'u na ƙididdiga $ w$ wanda ƙimar zai zama mafi ƙanƙanta.
Mayar da ma'auni zuwa sigar matrix
Wakilin vector
Don fara da, don sauƙaƙe rayuwar ku, ya kamata ku kula da ma'auni na koma baya na layi kuma ku lura cewa ƙididdiga ta farko. ba a ninka ta da wani regressor. A lokaci guda, lokacin da muka canza bayanai zuwa sigar matrix, yanayin da aka ambata a sama zai dagula lissafin. Dangane da wannan, an ba da shawarar gabatar da wani regressor don haɗin farko na farko kuma a daidaita shi zuwa daya. Ko dai, kowane"daidaita darajar wannan regressor zuwa daya - bayan haka, idan aka ninka ta daya, babu abin da zai canza daga mahangar sakamakon lissafin, amma daga ra'ayi na ka'idojin samfurin matrices, azabarmu. za a rage muhimmanci.
Yanzu, don lokacin, don sauƙaƙe kayan, bari mu ɗauka cewa muna da ɗaya kawai "-th" lura. Sa'an nan, yi tunanin dabi'u na regressors "-th" lura a matsayin vector . Vector yana da girma , wato layuka da shafi 1:
Bari mu wakilci ƙididdigan da ake buƙata azaman vector , yana da girma :
Ma'aunin koma baya na layi don"-th" lura zai dauki sigar:
Ayyukan tantance ingancin samfurin linzamin zai ɗauki tsari:
Lura cewa bisa ga ƙa'idodin haɓaka matrix, muna buƙatar musanya vector .
Matrix wakilci
Sakamakon yawaitar vectors, muna samun lamba: , wanda ake tsammanin. Wannan lambar ita ce kusanta"-th" mai nuna manufa. Amma muna buƙatar ƙima ba kawai ƙimar manufa ɗaya ba, amma duka. Don yin wannan, bari mu rubuta duk abin da aka rubuta.-th" regressors a cikin tsarin matrix . Sakamakon matrix yana da girma :
Yanzu ma'aunin koma baya na layi zai ɗauki tsari:
Bari mu nuna ƙimar maƙasudin manufa (duk ) kowane vector girma :
Yanzu za mu iya rubuta ma'auni don tantance ingancin samfurin layi a cikin tsarin matrix:
A zahiri, daga wannan dabarar za mu ƙara samun dabarar da aka sani da mu
Yaya ake yi? An buɗe maƙallan, ana aiwatar da bambance-bambance, ana canza maganganun da aka haifar, da dai sauransu, kuma wannan shine ainihin abin da za mu yi a yanzu.
Matrix canje-canje
Bari mu buɗe maƙallan
Bari mu shirya ma'auni don bambanta
Don yin wannan, za mu gudanar da wasu canje-canje. A cikin lissafi na gaba zai zama mafi dacewa a gare mu idan vector za a wakilta a farkon kowane samfur a cikin lissafin.
Juyawa 1
Ta yaya ya faru? Don amsa wannan tambayar, duba kawai girman matrices ɗin da ake ninkawa kuma ku ga cewa a cikin fitarwa muna samun lamba ko akasin haka. .
Bari mu rubuta girman maganganun matrix.
Juyawa 2
Bari mu rubuta shi ta hanyar irin wannan don canzawa 1
A cikin fitarwa muna samun ma'auni wanda dole ne mu bambanta:
Muna bambanta aikin tantance ingancin samfurin
Bari mu bambanta game da vector :
Tambayoyin me yasa bai kamata ba, amma za mu bincika ayyukan don tantance abubuwan da aka samo asali a cikin sauran maganganu biyu dalla-dalla.
Bambance-bambance 1
Bari mu fadada akan bambancin:
Domin sanin abin da aka samu na matrix ko vector, kuna buƙatar duba abin da ke cikin su. Mu duba:
Bari mu nuna samfurin matrices ta hanyar matrix . Matrix murabba'i kuma haka ma, yana da simmetrical. Wadannan kaddarorin za su kasance masu amfani a gare mu daga baya, bari mu tuna da su. Matrix yana da girma :
Yanzu aikinmu shine mu ninka madaidaicin matrix kuma kada mu sami "biyu biyu biyar ne," don haka bari mu mai da hankali kuma mu yi taka tsantsan.
Duk da haka, mun cimma maƙasudin magana! A gaskiya ma, mun sami lamba - scalar. Kuma yanzu, a zahiri, mun ci gaba zuwa rarrabuwa. Wajibi ne a nemo abin da aka samo asali daga bayanin da aka samu don kowane ƙididdiga kuma sami girman vector azaman fitarwa . Kawai idan, zan rubuta hanyoyin ta hanyar aiki:
1) bambanta ta , mun samu:
2) bambanta ta , mun samu:
3) bambanta ta , mun samu:
Fitowar ita ce girman da aka yi alkawarinsa :
Idan ka kalli vector da kyau, za ka lura cewa ana iya haɗawa da hagu da daidaitattun abubuwan da ke cikin vector ta yadda, a sakamakon haka, za a iya ware vector daga abin da aka gabatar. size . Misali (bangaren hagu na saman layin vector) (dama kashi na saman layi na vector) ana iya wakilta azaman da kuma - kamar yadda da dai sauransu. akan kowane layi. Mu yi group:
Bari mu fitar da vector kuma a cikin fitarwa muna samun:
Yanzu, bari mu yi la'akari a kusa da sakamakon matrix. Matrix shine jimlar matrix biyu :
Bari mu tuna cewa kadan a baya mun lura da wani muhimmin dukiya na matrix - yana da simmetrical. Bisa ga wannan dukiya, za mu iya amincewa da cewa magana daidai . Ana iya tabbatar da wannan cikin sauƙi ta hanyar faɗaɗa samfurin matrices ta kashi . Ba za mu yi wannan a nan ba; masu sha'awar za su iya duba shi da kansu.
Mu koma ga maganar mu. Bayan canje-canjen mu, ya zama kamar yadda muke son ganinsa:
Don haka, mun kammala bambancin farko. Mu ci gaba zuwa magana ta biyu.
Bambance-bambance 2
Mu bi hanyar tsiya. Zai fi guntu da yawa fiye da na baya, don haka kada ku yi nisa da allon.
Bari mu fadada vectors da matrix element ta kashi:
Bari mu cire biyun daga lissafin na ɗan lokaci - ba ya taka rawa sosai, sannan za mu mayar da shi a matsayinsa. Bari mu ninka vectors ta matrix. Da farko, bari mu ninka matrix ku vector , ba mu da hani a nan. Muna samun girman vector :
Bari mu yi aikin mai zuwa - ninka vector zuwa sakamakon vector. A wurin fita lambar za ta jira mu:
Sa'an nan kuma za mu bambanta shi. A fitarwa muna samun vector na girma :
Tuna da ni wani abu? Haka ne! Wannan shine samfurin matrix ku vector .
Don haka, an kammala bambance-bambancen na biyu cikin nasara.
Maimakon a ƙarshe
Yanzu mun san yadda daidaito ya kasance .
A ƙarshe, za mu bayyana hanya mai sauri don canza tsarin ƙira.
Bari mu kimanta ingancin samfurin daidai da mafi ƙarancin hanyar murabba'ai:
Bari mu bambanta bayanin da aka samu:
Litattafai
Tushen Intanet:
1)
2)
3)
4)
Littattafan karatu, tarin matsaloli:
1) Bayanan kula da lacca akan manyan mathematics: cikakken kwas / D.T. An rubuta - 4th ed. - M.: Iris-press, 2006
2) Ana amfani da nazarin koma baya / N. Draper, G. Smith - 2nd ed. – M.: Kudi da Kididdigar, 1986 (fassara daga Turanci)
3) Matsaloli don warware matrix equations:
source: www.habr.com