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Timabweretsa equation ya linear mu mawonekedwe a matrix

Timabweretsa equation ya linear mu mawonekedwe a matrix

Cholinga cha nkhaniyi ndikupereka chithandizo kwa asayansi oyambira deta. MU nkhani yapita Tafotokoza njira zitatu zothanirana ndi mizere yolumikizira: yankho la analytical, gradient descent, stochastic gradient descent. Ndiye kuti tipeze yankho la analytical tinagwiritsa ntchito fomula Timabweretsa equation ya linear mu mawonekedwe a matrix. M'nkhaniyi, monga momwe mutuwo ukusonyezera, tidzalungamitsa kugwiritsa ntchito ndondomekoyi kapena, mwa kuyankhula kwina, tidzadzipeza tokha.

Chifukwa chake ndizomveka kumvetsera kwambiri chilinganizo Timabweretsa equation ya linear mu mawonekedwe a matrix?

Ndi ndi matrix equation kuti nthawi zambiri munthu amayamba kuzolowerana ndi mzere wobwerera. Panthawi imodzimodziyo, kuwerengera mwatsatanetsatane momwe formula inapangidwira ndizosowa.

Mwachitsanzo, m'maphunziro ophunzirira makina kuchokera ku Yandex, ophunzira akadziwitsidwa kukhazikika, amaperekedwa kuti agwiritse ntchito ntchito kuchokera ku library. dziwa, pomwe palibe mawu omwe amatchulidwa okhudza matrix oyimira algorithm. Ndi panthawiyi pomwe omvera ena angafune kumvetsetsa nkhaniyi mwatsatanetsatane - lembani kachidindo popanda kugwiritsa ntchito zomwe zapangidwa kale. Ndipo kuti muchite izi, choyamba muyenera kuwonetsa equation ndi chokhazikika mu mawonekedwe a matrix. Nkhaniyi ipereka mwayi kwa omwe akufuna kudziwa luso lotere. Tiyeni tiyambe.

Mikhalidwe yoyambirira

Zizindikiro za zomwe mukufuna

Tili ndi zinthu zingapo zomwe tikufuna. Mwachitsanzo, chizindikiro chandamale chikhoza kukhala mtengo wazinthu zilizonse: mafuta, golide, tirigu, dola, ndi zina. Pa nthawi yomweyo, ndi chiwerengero cha chandamale chandamale mfundo timatanthawuza chiwerengero cha zowonera. Zowona zotere zitha kukhala, mwachitsanzo, mitengo yamafuta pamwezi pachaka, ndiye kuti, tidzakhala ndi 12 zomwe tikufuna. Tiyeni tiyambe kuyambitsa notation. Tiyeni tiwone mtengo uliwonse wa chizindikiro chandamale ngati Timabweretsa equation ya linear mu mawonekedwe a matrix. Zonse tili nazo Timabweretsa equation ya linear mu mawonekedwe a matrix zowonera, zomwe zikutanthauza kuti titha kuyimira zomwe tikuwona ngati Timabweretsa equation ya linear mu mawonekedwe a matrix.

Obwerera

Tidzaganiza kuti pali zinthu zomwe zimafotokozera pamlingo wina zomwe zikuwonetsa zomwe mukufuna. Mwachitsanzo, ndalama za dollar / ruble zimakhudzidwa kwambiri ndi mtengo wa mafuta, Federal Reserve rate, etc. Zinthu zoterezi zimatchedwa regressors. Panthawi imodzimodziyo, mtengo uliwonse wamtengo wapatali uyenera kufanana ndi mtengo wa regressor, ndiye kuti, ngati tili ndi zizindikiro 12 za mwezi uliwonse mu 2018, ndiye kuti tiyeneranso kukhala ndi 12 regressor values ​​nthawi yomweyo. Tiyeni tifotokozere za regressor iliyonse ndi Timabweretsa equation ya linear mu mawonekedwe a matrix. Kwa ife pakhale Timabweretsa equation ya linear mu mawonekedwe a matrix regressors (ie. Timabweretsa equation ya linear mu mawonekedwe a matrix zinthu zomwe zimakhudza zilolezo zowunikira). Izi zikutanthauza kuti ma regressors athu atha kuperekedwa motere: kwa 1st regressor (mwachitsanzo, mtengo wamafuta): Timabweretsa equation ya linear mu mawonekedwe a matrix, kwa 2nd regressor (mwachitsanzo, Fed rate): Timabweretsa equation ya linear mu mawonekedwe a matrix, Za "Timabweretsa equation ya linear mu mawonekedwe a matrix-th" regressor: Timabweretsa equation ya linear mu mawonekedwe a matrix

Kudalira kwa zizindikiro zomwe zikuyang'aniridwa pa regressors

Tiyerekeze kuti kudalira chizindikiro chandamale Timabweretsa equation ya linear mu mawonekedwe a matrix kuchokera ku regressors "Timabweretsa equation ya linear mu mawonekedwe a matrixth" kuyang'ana kungasonyezedwe kupyolera mu mzere wobwerezabwereza wa mawonekedwe:

Timabweretsa equation ya linear mu mawonekedwe a matrix

kumene Timabweretsa equation ya linear mu mawonekedwe a matrix - "Timabweretsa equation ya linear mu mawonekedwe a matrix-th" mtengo wa regressor kuchokera ku 1 mpaka Timabweretsa equation ya linear mu mawonekedwe a matrix,

Timabweretsa equation ya linear mu mawonekedwe a matrix - chiwerengero cha regressors kuchokera 1 mpaka Timabweretsa equation ya linear mu mawonekedwe a matrix

Timabweretsa equation ya linear mu mawonekedwe a matrix - ma coefficients aang'ono, omwe amaimira kuchuluka komwe chizindikiro chowerengera chidzasintha pafupifupi pamene regressor ikusintha.

Mwanjira ina, ndife a aliyense (kupatulapo Timabweretsa equation ya linear mu mawonekedwe a matrix) ya regressor timasankha "zathu" coefficient Timabweretsa equation ya linear mu mawonekedwe a matrix, kenaka chulukitsani ma coefficients ndi ma regressors "Timabweretsa equation ya linear mu mawonekedwe a matrixth" kuwunika, chifukwa chake timapeza pafupifupi "Timabweretsa equation ya linear mu mawonekedwe a matrix-th" chizindikiro chandamale.

Chifukwa chake, tiyenera kusankha ma coefficients oterowo Timabweretsa equation ya linear mu mawonekedwe a matrix, pomwe mayendedwe athu oyandikira amagwira ntchito Timabweretsa equation ya linear mu mawonekedwe a matrix idzakhala pafupi kwambiri ndi zizindikiro zomwe mukufuna.

Kuwunika mtundu wa ntchito yoyandikira

Tidzazindikira kuwunika kwa ntchito yoyandikira pogwiritsa ntchito njira ya ma squares ochepa. Ntchito yowunika bwino pankhaniyi itenga mawonekedwe awa:

Timabweretsa equation ya linear mu mawonekedwe a matrix

Tiyenera kusankha zikhalidwe zotere za coefficients $w$ zomwe mtengo wake Timabweretsa equation ya linear mu mawonekedwe a matrix adzakhala wamng'ono kwambiri.

Kutembenuza equation kukhala mawonekedwe a matrix

Chiwonetsero cha Vector

Poyamba, kuti moyo wanu ukhale wosavuta, muyenera kulabadira mizere yosinthira equation ndikuzindikira kuti coefficient yoyamba. Timabweretsa equation ya linear mu mawonekedwe a matrix sikuchulukitsidwa ndi regressor iliyonse. Panthawi imodzimodziyo, tikasintha deta kukhala mawonekedwe a matrix, zochitika zomwe tatchulazi zidzasokoneza kwambiri kuwerengera. Pachifukwa ichi, akufunsidwa kuti awonetsere regressor wina wa coefficient yoyamba Timabweretsa equation ya linear mu mawonekedwe a matrix ndikufananiza ndi chimodzi. Kapena, aliyense "Timabweretsa equation ya linear mu mawonekedwe a matrixyerekezerani mtengo wa regressor iyi ndi imodzi - pambuyo pake, ikachulukitsidwa ndi imodzi, palibe chomwe chidzasinthe kuchokera pakuwona zotsatira za kuwerengera, koma kuchokera pamalingaliro a malamulo opangira matrices, mazunzo athu. zidzachepetsedwa kwambiri.

Tsopano, pakadali pano, kuti tifewetse zinthuzo, tiyeni tiyerekeze kuti tili ndi imodzi yokha "Timabweretsa equation ya linear mu mawonekedwe a matrix-th" kuwona. Kenako, taganizirani zamtengo wapatali za regressors "Timabweretsa equation ya linear mu mawonekedwe a matrix-th" zowonera ngati vekitala Timabweretsa equation ya linear mu mawonekedwe a matrix. Vector Timabweretsa equation ya linear mu mawonekedwe a matrix ali ndi dimension Timabweretsa equation ya linear mu mawonekedwe a matrix, ndiko Timabweretsa equation ya linear mu mawonekedwe a matrix mizere ndi 1 ndime:

Timabweretsa equation ya linear mu mawonekedwe a matrix

Tiyeni tiyimire ma coefficients ofunikira ngati vekitala Timabweretsa equation ya linear mu mawonekedwe a matrix, kukhala ndi dimension Timabweretsa equation ya linear mu mawonekedwe a matrix:

Timabweretsa equation ya linear mu mawonekedwe a matrix

Linear regression equation ya "Timabweretsa equation ya linear mu mawonekedwe a matrix-th" kuyang'ana kudzatenga mawonekedwe:

Timabweretsa equation ya linear mu mawonekedwe a matrix

Ntchito yowunika mtundu wamtundu wa mzere itenga mawonekedwe:

Timabweretsa equation ya linear mu mawonekedwe a matrix

Chonde dziwani kuti molingana ndi malamulo akuchulutsa matrix, tidafunikira kusinthira vekitala Timabweretsa equation ya linear mu mawonekedwe a matrix.

Chiwonetsero cha matrix

Chifukwa cha kuchuluka kwa ma vectors, timapeza nambala: Timabweretsa equation ya linear mu mawonekedwe a matrix, zomwe ziyenera kuyembekezera. Nambala iyi ndiye kuyerekezera "Timabweretsa equation ya linear mu mawonekedwe a matrix-th" chizindikiro chandamale. Koma tifunika kuyerekeza osati mtengo umodzi wokha, koma onsewo. Kuti tichite izi, lembani zonse "Timabweretsa equation ya linear mu mawonekedwe a matrix-th" ma regressors mumtundu wa matrix Timabweretsa equation ya linear mu mawonekedwe a matrix. Zotsatira zake zimakhala ndi kukula kwake Timabweretsa equation ya linear mu mawonekedwe a matrix:

Timabweretsa equation ya linear mu mawonekedwe a matrix

Tsopano linear regression equation itenga mawonekedwe:

Timabweretsa equation ya linear mu mawonekedwe a matrix

Tiwonetseni zomwe zikuwonetsa zomwe mukufuna (zonse Timabweretsa equation ya linear mu mawonekedwe a matrix) pa vector Timabweretsa equation ya linear mu mawonekedwe a matrix dimension Timabweretsa equation ya linear mu mawonekedwe a matrix:

Timabweretsa equation ya linear mu mawonekedwe a matrix

Tsopano titha kulemba equation kuti tiwunikire mtundu wa mzere wa mzere mumtundu wa matrix:

Timabweretsa equation ya linear mu mawonekedwe a matrix

Kwenikweni, kuchokera mu formula iyi timapezanso fomula yomwe tikudziwa Timabweretsa equation ya linear mu mawonekedwe a matrix

Kodi zachitika bwanji? Mabakiteriya amatsegulidwa, kusiyanitsa kukuchitika, mawu otulukawo amasinthidwa, etc., ndipo izi ndi zomwe tidzachita tsopano.

Kusintha kwa matrix

Tiyeni titsegule mabulaketi

Timabweretsa equation ya linear mu mawonekedwe a matrix

Timabweretsa equation ya linear mu mawonekedwe a matrix

Tiyeni tikonzekere equation kuti tisiyanitse

Kuti tichite izi, tidzasintha zina. M'mawerengedwe otsatirawa zidzakhala zosavuta kwa ife ngati vekitala Timabweretsa equation ya linear mu mawonekedwe a matrix idzayimiridwa kumayambiriro kwa chinthu chilichonse mu equation.

Kutembenuka 1

Timabweretsa equation ya linear mu mawonekedwe a matrix

Zinachitika bwanji? Kuti muyankhe funsoli, ingoyang'anani kukula kwa matrices omwe akuchulukitsidwa ndikuwona kuti pakutulutsa timapeza nambala kapena ayi. Timabweretsa equation ya linear mu mawonekedwe a matrix.

Tiyeni tilembe kukula kwa mawu a matrix.

Timabweretsa equation ya linear mu mawonekedwe a matrix

Timabweretsa equation ya linear mu mawonekedwe a matrix

Timabweretsa equation ya linear mu mawonekedwe a matrix

Kutembenuka 2

Timabweretsa equation ya linear mu mawonekedwe a matrix

Tiyeni tilembe mofanana ndi kusintha 1

Timabweretsa equation ya linear mu mawonekedwe a matrix

Timabweretsa equation ya linear mu mawonekedwe a matrix

Pazotulutsa timapeza equation yomwe tiyenera kusiyanitsa:
Timabweretsa equation ya linear mu mawonekedwe a matrix

Timasiyanitsa ntchito yowunikira khalidwe lachitsanzo

Tiyeni tisiyanitse polemekeza vekitala Timabweretsa equation ya linear mu mawonekedwe a matrix:

Timabweretsa equation ya linear mu mawonekedwe a matrix

Timabweretsa equation ya linear mu mawonekedwe a matrix

Timabweretsa equation ya linear mu mawonekedwe a matrix

Timabweretsa equation ya linear mu mawonekedwe a matrix

Mafunso chifukwa Timabweretsa equation ya linear mu mawonekedwe a matrix sipayenera kukhala, koma tiwona momwe tingagwiritsire ntchito kudziwa zotumphukira m'mawu ena awiriwo mwatsatanetsatane.

Kusiyana 1

Tiyeni tiwonjezere pa kusiyanasiyana: Timabweretsa equation ya linear mu mawonekedwe a matrix

Kuti mudziwe zomwe zimachokera ku matrix kapena vector, muyenera kuyang'ana zomwe zili mkati mwake. Tiyeni tiwone:

Timabweretsa equation ya linear mu mawonekedwe a matrix

Timabweretsa equation ya linear mu mawonekedwe a matrix

Timabweretsa equation ya linear mu mawonekedwe a matrix Timabweretsa equation ya linear mu mawonekedwe a matrix

Tiyeni tifotokozere za matrices Timabweretsa equation ya linear mu mawonekedwe a matrix kudzera pa matrix Timabweretsa equation ya linear mu mawonekedwe a matrix. Matrix Timabweretsa equation ya linear mu mawonekedwe a matrix square ndi kuwonjezera apo, ndi symmetrical. Zinthu izi zitha kukhala zothandiza kwa ife pambuyo pake, tizikumbukira. Matrix Timabweretsa equation ya linear mu mawonekedwe a matrix ali ndi dimension Timabweretsa equation ya linear mu mawonekedwe a matrix:

Timabweretsa equation ya linear mu mawonekedwe a matrix

Tsopano ntchito yathu ndikuchulutsa ma vectors moyenera ndi matrix osapeza "kawiri kawiri ndi kasanu," kotero tiyeni tilingalire ndikusamala kwambiri.

Timabweretsa equation ya linear mu mawonekedwe a matrix

Timabweretsa equation ya linear mu mawonekedwe a matrix

Timabweretsa equation ya linear mu mawonekedwe a matrix

Timabweretsa equation ya linear mu mawonekedwe a matrix

Komabe, tapeza mawu ovuta! M'malo mwake, tili ndi nambala - scalar. Ndipo tsopano, kwenikweni, timapita ku kusiyanitsa. Ndikofunikira kupeza chotengera cha mawu omwe amachokera pa coefficient iliyonse Timabweretsa equation ya linear mu mawonekedwe a matrix ndi kupeza dimension vector monga zotuluka Timabweretsa equation ya linear mu mawonekedwe a matrix. Zikatero, ndilemba ndondomeko ndikuchita:

1) kusiyanitsa ndi Timabweretsa equation ya linear mu mawonekedwe a matrix, timapeza: Timabweretsa equation ya linear mu mawonekedwe a matrix

2) kusiyanitsa ndi Timabweretsa equation ya linear mu mawonekedwe a matrix, timapeza: Timabweretsa equation ya linear mu mawonekedwe a matrix

3) kusiyanitsa ndi Timabweretsa equation ya linear mu mawonekedwe a matrix, timapeza: Timabweretsa equation ya linear mu mawonekedwe a matrix

Zomwe zimatuluka ndi vector yolonjezedwa ya kukula Timabweretsa equation ya linear mu mawonekedwe a matrix:

Timabweretsa equation ya linear mu mawonekedwe a matrix

Ngati muyang'ana kwambiri vekitalayo, mudzazindikira kuti zinthu zakumanzere ndi zofananira za vector zitha kuikidwa m'magulu kuti, chifukwa chake, vekitala ikhoza kupatulidwa kuchokera ku vector yomwe yaperekedwa. Timabweretsa equation ya linear mu mawonekedwe a matrix kukula Timabweretsa equation ya linear mu mawonekedwe a matrix. Mwachitsanzo, Timabweretsa equation ya linear mu mawonekedwe a matrix (chinthu chakumanzere cha mzere wapamwamba wa vector) Timabweretsa equation ya linear mu mawonekedwe a matrix (chinthu choyenera cha mzere wapamwamba wa vekitala) chikhoza kuimiridwa ngati Timabweretsa equation ya linear mu mawonekedwe a matrixndi Timabweretsa equation ya linear mu mawonekedwe a matrix - monga Timabweretsa equation ya linear mu mawonekedwe a matrix ndi zina. pa mzere uliwonse. Tiyeni tipange gulu:

Timabweretsa equation ya linear mu mawonekedwe a matrix

Tiyeni titulutse vekitala Timabweretsa equation ya linear mu mawonekedwe a matrix ndipo pamapeto pake timapeza:

Timabweretsa equation ya linear mu mawonekedwe a matrix

Tsopano, tiyeni tione mwatsatanetsatane zotsatira zake masanjidwewo. Matrix ndi kuchuluka kwa matrices awiri Timabweretsa equation ya linear mu mawonekedwe a matrix:

Timabweretsa equation ya linear mu mawonekedwe a matrix

Tikumbukire kuti m'mbuyomu tidawona chinthu chimodzi chofunikira cha matrix Timabweretsa equation ya linear mu mawonekedwe a matrix - ndi symmetrical. Zochokera katundu, tikhoza kunena molimba mtima kuti mawu Timabweretsa equation ya linear mu mawonekedwe a matrix ofanana Timabweretsa equation ya linear mu mawonekedwe a matrix. Izi zitha kutsimikiziridwa mosavuta ndikukulitsa zinthu za matrices ndi chinthu Timabweretsa equation ya linear mu mawonekedwe a matrix. Sitidzachita izi pano; omwe akufuna azitha kuzifufuza okha.

Tiyeni tibwerere ku mafotokozedwe athu. Pambuyo pa kusintha kwathu, zidakhala momwe timafunira kuziwonera:

Timabweretsa equation ya linear mu mawonekedwe a matrix

Kotero, tatsiriza kusiyanitsa koyamba. Tiyeni tipitirire ku mawu achiwiri.

Kusiyana 2

Timabweretsa equation ya linear mu mawonekedwe a matrix

Tiyeni titsatire njira yopunthidwa. Idzakhala yayifupi kwambiri kuposa yapitayi, kotero musapite kutali kwambiri ndi chophimba.

Tiyeni tikulitse ma vectors ndi matrix element ndi chinthu:

Timabweretsa equation ya linear mu mawonekedwe a matrix

Timabweretsa equation ya linear mu mawonekedwe a matrix

Timabweretsa equation ya linear mu mawonekedwe a matrix

Tiyeni tichotse awiriwa pa mawerengedwe kwa kanthawi - sizimasewera gawo lalikulu, ndiye tidzabwezeretsa m'malo mwake. Tiyeni tichulukitse ma vectors ndi matrix. Choyamba, tiyeni tichulukitse matrix Timabweretsa equation ya linear mu mawonekedwe a matrix ku vector Timabweretsa equation ya linear mu mawonekedwe a matrix, tilibe zoletsa pano. Timapeza kukula kwa vector Timabweretsa equation ya linear mu mawonekedwe a matrix:

Timabweretsa equation ya linear mu mawonekedwe a matrix

Tiyeni tichite izi - chulukitsa vekitala Timabweretsa equation ya linear mu mawonekedwe a matrix ku vekitala yotuluka. Potuluka nambala ikhala ikutidikirira:

Timabweretsa equation ya linear mu mawonekedwe a matrix

Ndiye tidzasiyanitsa. Pazotulutsa timapeza vector of dimension Timabweretsa equation ya linear mu mawonekedwe a matrix:

Timabweretsa equation ya linear mu mawonekedwe a matrix

Mukundikumbutsa chinachake? Ndichoncho! Izi ndizopangidwa ndi matrix Timabweretsa equation ya linear mu mawonekedwe a matrix ku vector Timabweretsa equation ya linear mu mawonekedwe a matrix.

Motero, kusiyanitsa kwachiwiri kumatsirizidwa bwino.

M'malo mapeto

Tsopano tikudziwa momwe kufanana kunayambira Timabweretsa equation ya linear mu mawonekedwe a matrix.

Pomaliza, tifotokoza njira yachangu yosinthira mafomu oyambira.

Tiyeni tiwunikire mtundu wachitsanzo molingana ndi njira yocheperako:
Timabweretsa equation ya linear mu mawonekedwe a matrix

Timabweretsa equation ya linear mu mawonekedwe a matrix

Tiyeni tisiyanitse mawu otsatirawa:
Timabweretsa equation ya linear mu mawonekedwe a matrix Timabweretsa equation ya linear mu mawonekedwe a matrix

Timabweretsa equation ya linear mu mawonekedwe a matrix

Mabuku

Zochokera pa intaneti:

1) habr.com/ru/post/278513
2) habr.com/ru/company/ods/blog/322076
3) habr.com/ru/post/307004
4) nabatchikov.com/blog/view/matrix_der

Mabuku, zosonkhanitsira zovuta:

1) Zolemba pamasamu apamwamba: maphunziro athunthu / D.T. Wolemba - 4th ed. - M.: Iris-press, 2006
2) Kugwiritsidwa ntchito kwa regression analysis / N. Draper, G. Smith - 2nd ed. - M.: Finance and Statistics, 1986 (kumasulira kuchokera ku Chingerezi)
3) Mavuto pakuthana ndi ma equation a matrix:
function-x.ru/matrix_equations.html
mathprofi.ru/deistviya_s_matricami.html

Source: www.habr.com

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