{"id":53777,"date":"2019-12-10T00:00:00","date_gmt":"2019-12-09T21:00:00","guid":{"rendered":"https:\/\/prohoster.info\/blog\/blog_prohoster\/privodim-uravnenie-linejnoj-regressii-v-matrichnyj-vid"},"modified":"2020-02-18T14:01:42","modified_gmt":"2020-02-18T11:01:42","slug":"privodim-uravnenie-linejnoj-regressii-v-matrichnyj-vid","status":"publish","type":"post","link":"https:\/\/prohoster.info\/sq\/blog\/news\/privodim-uravnenie-linejnoj-regressii-v-matrichnyj-vid","title":{"rendered":"P\u00ebrkthejm\u00eb ekuacionin e regresionit linear n\u00eb formatin matricor.","gt_translate_keys":[{"key":"rendered","format":"text"}]},"content":{"rendered":"

\"P\u00ebrkthejm\u00eb
\n
\nQ\u00ebllimi i k\u00ebtij artikulli \u00ebsht\u00eb t\u00eb ofroj\u00eb mb\u00ebshtetje p\u00ebr fillestar\u00ebt n\u00eb shkenc\u00ebn e t\u00eb dh\u00ebnave. N\u00eb artikulli i m\u00ebparsh\u00ebm<\/a><\/noindex> ne analizuam n\u00eb detaje tre m\u00ebnyra p\u00ebr t\u00eb zgjidhur ekuacionin e regresionit linear: zgjidhja analitike, zbritja gradiente dhe zbritja gradiente stohastike. At\u00ebher\u00eb, p\u00ebr zgjidhjen analitike, aplikuam formul\u00ebn \"P\u00ebrkthejm\u00eb. N\u00eb k\u00ebt\u00eb artikull, si\u00e7 tregohet n\u00eb titull, do t\u00eb justifikojm\u00eb p\u00ebrdorimin e k\u00ebsaj formule, ose m\u00eb sakt\u00eb, do ta nxjerrim at\u00eb vet\u00eb. <\/p>\n

Pse ka kuptim t\u00eb kushtohet v\u00ebmendje e ve\u00e7ant\u00eb formul\u00ebs \"P\u00ebrkthejm\u00eb? <\/p>\n

Ishte pik\u00ebrisht nga ekuacioni matricor q\u00eb zakonisht fillon njohja me regresionin linear. Megjithat\u00eb, p\u00ebrshkrimet e detajuara t\u00eb m\u00ebnyr\u00ebs se si u nxorr formula jan\u00eb t\u00eb rralla.
\n<\/a><\/noindex>
\nP\u00ebrdoruesit e kurseve t\u00eb m\u00ebsimit t\u00eb makin\u00eb nga Yandex, kurstudent\u00ebt njoftohen me regularizimin, iu sugjerohet t\u00eb p\u00ebrdorin funksionet nga biblioteka sklearn<\/i>, nd\u00ebrsa asnj\u00eb fjal\u00eb nuk p\u00ebrmendet p\u00ebr p\u00ebrfaq\u00ebsimin matricor t\u00eb algoritmit. N\u00eb k\u00ebt\u00eb moment, disa student\u00eb mund t\u00eb ndihen t\u00eb motivuar t\u00eb kuptojn\u00eb k\u00ebt\u00eb \u00e7\u00ebshtje m\u00eb thell\u00eb - duke shkruar kod pa p\u00ebrdorimin e funksioneve t\u00eb gatshme. Dhe p\u00ebr k\u00ebt\u00eb, s\u00eb pari duhet t\u00eb p\u00ebrshkruajm\u00eb ekuacionin me regularizatorin n\u00eb form\u00ebn matricore. Ky artikull \u00ebsht\u00eb sakt\u00ebsisht p\u00ebr ata q\u00eb d\u00ebshirojn\u00eb t\u00eb zot\u00ebrojn\u00eb k\u00ebto aft\u00ebsi. Le t\u00eb fillojm\u00eb.<\/p>\n

Kushtet fillestare<\/h2>\n

<\/p>\n

T\u00eb dh\u00ebnat e synuara<\/h3>\n

\nKemi nj\u00eb seri vlerash t\u00eb t\u00eb dh\u00ebnave synuese. P\u00ebr shembull, nj\u00eb t\u00eb dh\u00ebn\u00eb synuese mund t\u00eb jet\u00eb \u00e7mimi i nj\u00eb aktive: naft\u00eb, ar, grur\u00eb, dollar etj. N\u00eb k\u00ebt\u00eb rast, me nj\u00eb seri vlerash t\u00eb t\u00eb dh\u00ebnave synuese kuptojm\u00eb numrin e v\u00ebzhgimeve. V\u00ebzhgimet e tilla mund t\u00eb jen\u00eb, p\u00ebr shembull, \u00e7mimet mujore t\u00eb naft\u00ebs p\u00ebr nj\u00eb vit, q\u00eb do t\u00eb thot\u00eb se do t\u00eb kemi 12 vlera t\u00eb t\u00eb dh\u00ebnave synuese. Le t\u00eb fillojm\u00eb t\u00eb prezantojm\u00eb shenjat. Le t\u00eb caktosh \u00e7do vler\u00eb t\u00eb t\u00eb dh\u00ebn\u00ebs synuese si \"P\u00ebrkthejm\u00eb. N\u00eb total, kemi \"P\u00ebrkthejm\u00eb v\u00ebzhgime, k\u00ebshtu q\u00eb mund t\u00eb p\u00ebrshkruajm\u00eb v\u00ebzhgimet tona si \"P\u00ebrkthejm\u00eb.<\/p>\n

Regressor\u00ebt<\/h3>\n

\nLe t\u00eb supozojm\u00eb se ekzistojn\u00eb faktor\u00eb q\u00eb n\u00eb nj\u00eb mas\u00eb t\u00eb caktuar shpjegojn\u00eb vlerat e t\u00eb dh\u00ebnave synuese. P\u00ebr shembull, n\u00eb kursin e \u00e7iftit dollar\/rub\u00ebl, \u00e7mimi i naft\u00ebs, norma e FRS dhe t\u00eb tjer\u00eb kan\u00eb nj\u00eb ndikim t\u00eb r\u00ebnd\u00ebsish\u00ebm. K\u00ebta faktor\u00eb quhen regressor\u00eb. N\u00eb k\u00ebt\u00eb rast, p\u00ebr \u00e7do vler\u00eb t\u00eb t\u00eb dh\u00ebn\u00ebs synuese duhet t\u00eb p\u00ebrputhet me vler\u00ebn e regressor\u00ebve, dometh\u00ebn\u00eb, n\u00ebse kemi 12 t\u00eb dh\u00ebna synuese p\u00ebr \u00e7do muaj n\u00eb vitin 2018, at\u00ebher\u00eb do t\u00eb kemi gjithashtu 12 vlera t\u00eb regressor\u00ebve p\u00ebr t\u00eb nj\u00ebjtin periudh\u00eb. Le t\u00eb caktosh vlerat e \u00e7do regressor si \"P\u00ebrkthejm\u00eb. Le t\u00eb supozojm\u00eb se n\u00eb rastin ton\u00eb kemi \"P\u00ebrkthejm\u00eb regressor\u00eb (dmth. \"P\u00ebrkthejm\u00eb faktor\u00eb q\u00eb ndikojn\u00eb n\u00eb vlerat e t\u00eb dh\u00ebnave synuese). K\u00ebshtu, regressor\u00ebt tan\u00eb mund t\u00eb p\u00ebrshkruhen si: p\u00ebr regressor-in e par\u00eb (p\u00ebr shembull, \u00e7mimi i naft\u00ebs): \"P\u00ebrkthejm\u00eb, p\u00ebr regressor-in e dyt\u00eb (p\u00ebr shembull, norma e FRS): \"P\u00ebrkthejm\u00eb, p\u00ebr «\"P\u00ebrkthejm\u00eb-tin» regresorit: \"P\u00ebrkthejm\u00eb<\/p>\n

Var\u00ebsia e treguesve t\u00eb synuar nga regresor\u00ebt<\/h3>\n

\nLe t\u00eb supozojm\u00eb se var\u00ebsia e t\u00eb dh\u00ebn\u00ebs synuese \"P\u00ebrkthejm\u00eb nga regresor\u00ebt «\"P\u00ebrkthejm\u00eb-t\u00eb» v\u00ebzhgimit mund t\u00eb shprehet p\u00ebrmes ekuacionit t\u00eb regresionit linear t\u00eb formatit:<\/p>\n

\"P\u00ebrkthejm\u00eb<\/p>\n

, ku \"P\u00ebrkthejm\u00eb \u2014 «\"P\u00ebrkthejm\u00eb-ta» vler\u00eb regresori nga 1 deri n\u00eb \"P\u00ebrkthejm\u00eb,<\/p>\n

\"P\u00ebrkthejm\u00eb - coeficient\u00ebt k\u00ebndor\u00eb q\u00eb p\u00ebrfaq\u00ebsojn\u00eb madh\u00ebsin\u00eb, me t\u00eb cil\u00ebn do t\u00eb ndryshoj\u00eb vler\u00ebn e llogaritur t\u00eb dh\u00ebnave synuese n\u00eb mesatare kur ndryshon regressor-i. \"P\u00ebrkthejm\u00eb<\/p>\n

\"P\u00ebrkthejm\u00eb - jan\u00eb koeficient\u00ebt k\u00ebndor\u00eb, t\u00eb cil\u00ebt paraqesin madh\u00ebsin\u00eb me t\u00eb cil\u00ebn do t\u00eb ndryshoj\u00eb treguesi i llogaritur i synuar mesatarisht me ndryshimin e regresorit. <\/p>\n

N\u00eb m\u00ebnyr\u00eb t\u00eb tjera, p\u00ebr \u00e7do regressor (p\u00ebrve\u00e7 \"P\u00ebrkthejm\u00eb) ne p\u00ebrcaktojm\u00eb \"coeficientin\" ton\u00eb \"P\u00ebrkthejm\u00eb, m\u00eb pas shumojm\u00eb koeficient\u00ebt me vlerat e regresor\u00ebve «\"P\u00ebrkthejm\u00eb-t\u00eb» v\u00ebzhgimit, si rezultat marrim nj\u00eb p\u00ebrafrim «\"P\u00ebrkthejm\u00eb-t\u00eb» treguesit target.<\/p>\n

Prandaj, na nevojitet t\u00eb gjenden coeficient\u00ebt t\u00eb till\u00eb \"P\u00ebrkthejm\u00eb, p\u00ebr t\u00eb cilat vlerat e funksionit ton\u00eb p\u00ebraf\u00ebrues \"P\u00ebrkthejm\u00eb do t\u00eb jen\u00eb t\u00eb pozicionuara sa m\u00eb af\u00ebr me vlerat e t\u00eb dh\u00ebnave synuese.<\/p>\n

Vler\u00ebsimi i cil\u00ebsis\u00eb s\u00eb funksionit t\u00eb aproksimuar<\/h3>\n

\nVler\u00ebsimi i cil\u00ebsis\u00eb s\u00eb funksionit p\u00ebraf\u00ebrues<\/p>\n

\"P\u00ebrkthejm\u00eb<\/p>\n

Ne do t\u00eb p\u00ebrcaktojm\u00eb cil\u00ebsin\u00eb e funksionit p\u00ebraf\u00ebrues p\u00ebrmes metod\u00ebs s\u00eb katror\u00ebve t\u00eb vegj\u00ebl. Funksioni i vler\u00ebsimit t\u00eb cil\u00ebsis\u00eb n\u00eb k\u00ebt\u00eb rast do t\u00eb marr\u00eb form\u00ebn e m\u00ebposhtme: \"P\u00ebrkthejm\u00eb do t\u00eb jet\u00eb sa m\u00eb e vog\u00ebl.<\/p>\n

Na nevojitet t\u00eb gjejm\u00eb vlerat e coeficient\u00ebve $w$, p\u00ebr t\u00eb cilat vlera<\/h2>\n

<\/p>\n

do t\u00eb jet\u00eb m\u00eb e vog\u00ebl.<\/h3>\n

\nT\u00eb kthejm\u00eb ekuacionin n\u00eb form\u00eb matricore \"P\u00ebrkthejm\u00eb Prezantimi vektorial \"P\u00ebrkthejm\u00eb dhe ta barazojm\u00eb me nj\u00eb. M\u00eb sakt\u00eb, \u00e7do «\"P\u00ebrkthejm\u00eb-ta» vler\u00eb e k\u00ebtij regresori ta barazojm\u00eb me nj\u00eb \u2014 sepse duke e shum\u00ebzuar me nj\u00eb, nga pik\u00ebpamja e rezultateve t\u00eb llogaritjeve nuk ndryshon asgj\u00eb, nd\u00ebrsa nga pik\u00ebpamja e rregullave t\u00eb shum\u00ebzimit t\u00eb matriceve, do t\u00eb reduktoj\u00eb ndjesh\u00ebm vuajtjet tona. <\/p>\n

Tani, p\u00ebr nj\u00eb koh\u00eb t\u00eb caktuar, me q\u00ebllim t\u00eb thjeshtimit t\u00eb materialit, le t\u00eb supozojm\u00eb se kemi vet\u00ebm nj\u00eb «\"P\u00ebrkthejm\u00eb-ta» v\u00ebzhgim. At\u00ebher\u00eb, le t\u00eb perceptojm\u00eb vlerat e regresor\u00ebve «\"P\u00ebrkthejm\u00eb-t\u00eb» v\u00ebzhgimit si nj\u00eb vektor \"P\u00ebrkthejm\u00eb. Vektori \"P\u00ebrkthejm\u00eb ka dimensionalitetin \"P\u00ebrkthejm\u00eb, q\u00eb do t\u00eb thot\u00eb \"P\u00ebrkthejm\u00eb rreshtave dhe 1 kolon\u00eb:<\/p>\n

\"P\u00ebrkthejm\u00eb<\/p>\n

Koeficient\u00ebt e k\u00ebrkuar le t\u00eb paraqiten si nj\u00eb vektor \"P\u00ebrkthejm\u00eb, q\u00eb ka dimensionalitetin \"P\u00ebrkthejm\u00eb:<\/p>\n

\"P\u00ebrkthejm\u00eb<\/p>\n

Ekuacioni i regresionit linear p\u00ebr «\"P\u00ebrkthejm\u00eb-t\u00eb» v\u00ebzhgimit do t\u00eb marr\u00eb form\u00ebn: <\/p>\n

\"P\u00ebrkthejm\u00eb<\/p>\n

Funksioni i vler\u00ebsimit t\u00eb cil\u00ebsis\u00eb s\u00eb modelit linear do t\u00eb ket\u00eb k\u00ebt\u00eb form\u00eb:<\/p>\n

\"P\u00ebrkthejm\u00eb<\/p>\n

Vlen t\u00eb theksohet se, n\u00eb p\u00ebrputhje me rregullat e shum\u00ebzimit t\u00eb matriceve, na nevojitet t\u00eb transponojm\u00eb vektorin \"P\u00ebrkthejm\u00eb.<\/p>\n

Pr\u00e4sentimi matricor<\/h3>\n

\nSi rezultat i shum\u00ebzimit t\u00eb vektor\u00ebve, do t\u00eb marrim nj\u00eb num\u00ebr: \"P\u00ebrkthejm\u00eb, si\u00e7 pritej. Ky num\u00ebr \u00ebsht\u00eb p\u00ebrafrimi «\"P\u00ebrkthejm\u00eb-t\u00eb» treguesit target. Por ne na nevojitet nj\u00eb p\u00ebrafrim jo p\u00ebr nj\u00eb vler\u00eb t\u00eb vet\u00ebm t\u00eb treguesit target, por p\u00ebr t\u00eb gjith\u00eb. P\u00ebr k\u00ebt\u00eb, do t\u00eb shkruajm\u00eb t\u00eb gjith\u00eb «\"P\u00ebrkthejm\u00eb-t\u00eb» regresor\u00ebt n\u00eb formatin e matric\u00ebs \"P\u00ebrkthejm\u00eb. Matrica e marre ka dimensionalitetin \"P\u00ebrkthejm\u00eb:<\/p>\n

\"P\u00ebrkthejm\u00eb<\/p>\n

Tani ekuacioni i regresionit linear do t\u00eb ket\u00eb k\u00ebt\u00eb form\u00eb:<\/p>\n

\"P\u00ebrkthejm\u00eb<\/p>\n

T\u00eb em\u00ebrtojm\u00eb vlerat e indikator\u00ebve t\u00eb synuar (t\u00eb gjitha \"P\u00ebrkthejm\u00eb) si vektor \"P\u00ebrkthejm\u00eb me dimensionalitetin \"P\u00ebrkthejm\u00eb:<\/p>\n

\"P\u00ebrkthejm\u00eb<\/p>\n

Tani mund t\u00eb shkruajm\u00eb n\u00eb format matricor ekuacionin e vler\u00ebsimit t\u00eb cil\u00ebsis\u00eb s\u00eb modelit linear:<\/p>\n

\"P\u00ebrkthejm\u00eb<\/p>\n

N\u00eb fakt, nga kjo formul\u00eb m\u00eb pas arrijm\u00eb n\u00eb formul\u00ebn e njohur \"P\u00ebrkthejm\u00eb<\/p>\n

Si b\u00ebhet kjo? Shk\u00ebputen kllap\u00ebt, kryhet diferencimi, shtransformohen shprehjet e marra etj., dhe k\u00ebt\u00eb ne do ta b\u00ebjm\u00eb tani.<\/p>\n

Transformimet matricore<\/h2>\n

<\/p>\n

Shk\u00ebputim kllap\u00ebt<\/h3>\n

\n\"P\u00ebrkthejm\u00eb<\/p>\n

\"P\u00ebrkthejm\u00eb<\/p>\n

T\u00eb p\u00ebrgatisim ekuacionin p\u00ebr diferencim<\/h3>\n

\nP\u00ebr k\u00ebt\u00eb do t\u00eb kryejm\u00eb disa transformime. N\u00eb llogaritjet e m\u00ebvonshme do t\u00eb jet\u00eb m\u00eb e leht\u00eb n\u00ebse vektori \"P\u00ebrkthejm\u00eb do t\u00eb paraqitet n\u00eb fillim t\u00eb \u00e7do shum\u00ebzimi n\u00eb ekuacion.<\/p>\n

Transformimi 1<\/h4>\n

\n\"P\u00ebrkthejm\u00eb<\/p>\n

Si erdh\u00ebn n\u00eb k\u00ebt\u00eb rezultat? P\u00ebr t\u00eb p\u00ebrgjigjur n\u00eb k\u00ebt\u00eb pyetje, \u00ebsht\u00eb e mjaftueshme t\u00eb shohim dimensionet e matricave q\u00eb shum\u00ebzohen dhe t\u00eb shohim se nga rezultatet ne marrim nj\u00eb num\u00ebr, ose ndryshe \"P\u00ebrkthejm\u00eb.<\/p>\n

Le t\u00eb shkruajm\u00eb dimensionet e shprehjeve matricore.<\/p>\n

\"P\u00ebrkthejm\u00eb<\/p>\n

\"P\u00ebrkthejm\u00eb<\/p>\n

\"P\u00ebrkthejm\u00eb<\/p>\n

Transformimi 2<\/h4>\n

\n\"P\u00ebrkthejm\u00eb<\/p>\n

Le t\u00eb zhvillojm\u00eb n\u00eb m\u00ebnyr\u00eb t\u00eb ngjashme si n\u00eb transformimin 1<\/p>\n

\"P\u00ebrkthejm\u00eb<\/p>\n

\"P\u00ebrkthejm\u00eb<\/p>\n

Dhe si rezultat marrim ekuacionin, t\u00eb cilin do ta diferencojm\u00eb:
\n\"P\u00ebrkthejm\u00eb<\/p>\n

Diferencojm\u00eb funksionin e vler\u00ebsimit t\u00eb cil\u00ebsis\u00eb s\u00eb modelit<\/h2>\n

\nDiferencojm\u00eb sipas vektorit \"P\u00ebrkthejm\u00eb:<\/p>\n

\"P\u00ebrkthejm\u00eb<\/p>\n

\n\"P\u00ebrkthejm\u00eb<\/p>\n

\"P\u00ebrkthejm\u00eb<\/p>\n

\"P\u00ebrkthejm\u00eb<\/p>\n

Nuk duhet t\u00eb ket\u00eb pyetje p\u00ebrse \"P\u00ebrkthejm\u00eb n\u00eb dy shprehjet e tjera ne do t\u00eb shqyrtojm\u00eb m\u00eb me detaje procedurat e p\u00ebrcaktimit t\u00eb derivatave.<\/p>\n

Diferencimi 1<\/h3>\n

\nShk\u00ebputim diferencimin: \"P\u00ebrkthejm\u00eb<\/p>\n

P\u00ebr t\u00eb p\u00ebrcaktuar derivat\u00ebn nga nj\u00eb matric\u00eb ose vektor duhet t\u00eb shikojm\u00eb se \u00e7far\u00eb ka brenda. Shikojm\u00eb:<\/p>\n

\"P\u00ebrkthejm\u00eb<\/p>\n

\"P\u00ebrkthejm\u00eb<\/p>\n

\"P\u00ebrkthejm\u00eb \"P\u00ebrkthejm\u00eb<\/p>\n

T\u00eb em\u00ebrtojm\u00eb shum\u00ebzimin e matricave \"P\u00ebrkthejm\u00eb si matric\u00ebn \"P\u00ebrkthejm\u00eb. Matrica \"P\u00ebrkthejm\u00eb \u00ebsht\u00eb katrore dhe p\u00ebr m\u00eb tep\u00ebr, ajo \u00ebsht\u00eb simetrike. K\u00ebta veti do t'i nevojitin m\u00eb von\u00eb, t'i mbajm\u00eb mend. Matrica \"P\u00ebrkthejm\u00eb ka dimensionalitetin \"P\u00ebrkthejm\u00eb:<\/p>\n

\"P\u00ebrkthejm\u00eb<\/p>\n

Tani detyra jon\u00eb \u00ebsht\u00eb t\u00eb b\u00ebjm\u00eb shum\u00ebzimin e vektor\u00ebve me matric\u00ebn dhe t\u00eb mos marrim \u00abdy her\u00eb dy pes\u00eb\u00bb, k\u00ebshtu q\u00eb t\u00eb p\u00ebrqendrohemi dhe t\u00eb tregojm\u00eb shum\u00eb kujdes.<\/p>\n

\"P\u00ebrkthejm\u00eb<\/p>\n

\"P\u00ebrkthejm\u00eb<\/p>\n

\"P\u00ebrkthejm\u00eb<\/p>\n

\"P\u00ebrkthejm\u00eb<\/p>\n

Megjithat\u00eb, shprehja e nd\u00ebrlikuar na doli! N\u00eb t\u00eb v\u00ebrtet\u00eb kemi marr\u00eb nj\u00eb num\u00ebr \u2014 skalar. Dhe tani, v\u00ebrtet, kalojm\u00eb te diferencimi. Duhet t\u00eb gjejm\u00eb derivat\u00ebn e shprehjes s\u00eb marr\u00eb p\u00ebr \u00e7do koeficient \"P\u00ebrkthejm\u00eb dhe t\u00eb marrim si rezultat vektorin me dimensionalitetin \"P\u00ebrkthejm\u00eb. N\u00eb \u00e7do rast, do t\u00eb p\u00ebrshkruaj procedurat e veprimeve:<\/p>\n

1) do t\u00eb diferencojm\u00eb sipas \"P\u00ebrkthejm\u00eb, do t\u00eb marrim: \"P\u00ebrkthejm\u00eb<\/p>\n

2) do t\u00eb diferencojm\u00eb sipas \"P\u00ebrkthejm\u00eb, do t\u00eb marrim: \"P\u00ebrkthejm\u00eb<\/p>\n

3) do t\u00eb diferencojm\u00eb sipas \"P\u00ebrkthejm\u00eb, do t\u00eb marrim: \"P\u00ebrkthejm\u00eb<\/p>\n

Si rezultat \u2014 vektori i premtuar me dimension \"P\u00ebrkthejm\u00eb:<\/p>\n

\"P\u00ebrkthejm\u00eb<\/p>\n

N\u00ebse e shikoni vektorin m\u00eb me v\u00ebmendje, do t\u00eb v\u00ebreni se elementet e majt\u00eb dhe t\u00eb djatht\u00eb t\u00eb vektorit mund t\u00eb grupohen n\u00eb at\u00eb m\u00ebnyr\u00eb sa q\u00eb n\u00eb fund nga vektori i paraqitur mund t\u00eb dalloni vektorin \"P\u00ebrkthejm\u00eb me dimension \"P\u00ebrkthejm\u00eb. P\u00ebr shembull, \"P\u00ebrkthejm\u00eb (elementi i majt\u00eb i rreshtit t\u00eb sip\u00ebrm t\u00eb vektorit) \"P\u00ebrkthejm\u00eb (elementi i djatht\u00eb i rreshtit t\u00eb sip\u00ebrm t\u00eb vektorit) mund t\u00eb paraqitet si \"P\u00ebrkthejm\u00eb, dhe \"P\u00ebrkthejm\u00eb \u2014 si \"P\u00ebrkthejm\u00eb etj. p\u00ebr \u00e7do rresht. Le t\u00eb grupojm\u00eb:<\/p>\n

\"P\u00ebrkthejm\u00eb<\/p>\n

Le t\u00eb nxjerrim vektorin \"P\u00ebrkthejm\u00eb dhe si rezultat do t\u00eb marrim:<\/p>\n

\"P\u00ebrkthejm\u00eb<\/p>\n

Tani, le t\u00eb shikojm\u00eb matric\u00ebn e marr\u00eb. Matrica p\u00ebrb\u00ebhet nga shuma e dy matriceve \"P\u00ebrkthejm\u00eb:<\/p>\n

\"P\u00ebrkthejm\u00eb<\/p>\n

Le t\u00eb kujtojm\u00eb se disa koh\u00eb m\u00eb par\u00eb e theksuam nj\u00eb pron\u00eb t\u00eb r\u00ebnd\u00ebsishme t\u00eb matric\u00ebs \"P\u00ebrkthejm\u00eb \u2014 ajo \u00ebsht\u00eb simetrike. Duke e marr\u00eb parasysh k\u00ebt\u00eb pron\u00eb, ne mund t\u00eb deklarojm\u00eb me siguri se shprehja \"P\u00ebrkthejm\u00eb \u00ebsht\u00eb e barabart\u00eb me \"P\u00ebrkthejm\u00eb. Kjo \u00ebsht\u00eb leht\u00eb p\u00ebr t'u verifikuar, duke hapur produktin e matricave p\u00ebr kat\u00ebr elemente. \"P\u00ebrkthejm\u00eb. Ne nuk do ta b\u00ebjm\u00eb k\u00ebt\u00eb k\u00ebtu, ata q\u00eb d\u00ebshirojn\u00eb mund ta verifikojn\u00eb vet\u00eb. <\/p>\n

T\u00eb kthehemi te shprehja jon\u00eb. Pas transformimeve tona, ajo rezultoi ashtu si\u00e7 ne donim ta shihnim:<\/p>\n

\"P\u00ebrkthejm\u00eb<\/p>\n

Pra, me diferencimin e par\u00eb e p\u00ebrfunduam. Kalojm\u00eb te shprehja e dyt\u00eb.<\/p>\n

Diferencimi 2<\/h3>\n

\n\"P\u00ebrkthejm\u00eb<\/p>\n

T\u00eb vazhdojm\u00eb n\u00eb rrug\u00ebn e trotuar. Ajo do t\u00eb jet\u00eb shum\u00eb m\u00eb e shkurt\u00ebr se e m\u00ebparshmja, k\u00ebshtu q\u00eb mos shkoni larg nga ekrani.<\/p>\n

Do t\u00eb zbulojm\u00eb element p\u00ebr element vektor\u00ebt dhe matric\u00ebn:<\/p>\n

\"P\u00ebrkthejm\u00eb<\/p>\n

\"P\u00ebrkthejm\u00eb<\/p>\n

\"P\u00ebrkthejm\u00eb<\/p>\n

P\u00ebr nj\u00eb koh\u00eb do ta heqim nga p\u00ebrllogaritjet numrin dy \u2014 ai nuk luan nj\u00eb rol t\u00eb madh, m\u00eb von\u00eb do ta kthejm\u00eb n\u00eb vend. Do t\u00eb shumzojm\u00eb vektor\u00ebt me matric\u00ebn. S\u00eb pari, do t\u00eb shumzojm\u00eb matric\u00ebn \"P\u00ebrkthejm\u00eb me vektorin \"P\u00ebrkthejm\u00eb, k\u00ebtu nuk kemi asnj\u00eb kufizim. Do t\u00eb marrim nj\u00eb vektor me madh\u00ebsi \"P\u00ebrkthejm\u00eb:<\/p>\n

\"P\u00ebrkthejm\u00eb<\/p>\n

Do t\u00eb realizojm\u00eb veprimin e m\u00ebposht\u00ebm \u2014 do t\u00eb shumzojm\u00eb vektorin \"P\u00ebrkthejm\u00eb me vektorin e marr\u00eb. N\u00eb dalje, na pret nj\u00eb num\u00ebr:<\/p>\n

\"P\u00ebrkthejm\u00eb<\/p>\n

Ky \u00ebsht\u00eb numri q\u00eb do ta diferencojm\u00eb. N\u00eb dalje do t\u00eb marrim nj\u00eb vektor me p\u00ebrmas\u00eb \"P\u00ebrkthejm\u00eb:<\/p>\n

\"P\u00ebrkthejm\u00eb<\/p>\n

A ju kujton di\u00e7ka? \u00cbsht\u00eb e sakt\u00eb! Kjo \u00ebsht\u00eb produkti i matric\u00ebs \"P\u00ebrkthejm\u00eb me vektorin \"P\u00ebrkthejm\u00eb. <\/p>\n

K\u00ebshtu, diferencimi i dyt\u00eb ka p\u00ebrfunduar me sukses.<\/p>\n

N\u00eb vend t\u00eb p\u00ebrfundimit<\/h2>\n

\nTani e dim\u00eb se si u arrit t\u00eb barazojm\u00eb \"P\u00ebrkthejm\u00eb.<\/p>\n

N\u00eb fund, do t\u00eb p\u00ebrshkruajm\u00eb nj\u00eb rrug\u00eb t\u00eb shpejt\u00eb t\u00eb transformimeve t\u00eb formulave kryesore.<\/p>\n

Do t\u00eb vler\u00ebsojm\u00eb cil\u00ebsin\u00eb e modelit sipas metod\u00ebs s\u00eb m\u00eb t\u00eb voglave katrore:<\/b>
\n\"P\u00ebrkthejm\u00eb<\/p>\n

\"P\u00ebrkthejm\u00eb<\/p>\n

Do t\u00eb diferencojm\u00eb shprehjen e marr\u00eb:<\/b>
\n\"P\u00ebrkthejm\u00eb \"P\u00ebrkthejm\u00eb<\/p>\n

\"P\u00ebrkthejm\u00eb<\/p>\n

Literatura<\/h2>\n

\nBurimet n\u00eb internet:<\/b><\/u><\/p>\n

1) habr.com\/ru\/post\/278513<\/a><\/noindex><\/i>
\n2) habr.com\/ru\/company\/ods\/blog\/322076<\/a><\/noindex><\/i>
\n3) habr.com\/ru\/post\/307004<\/a><\/noindex><\/i>
\n4) nabatchikov.com\/blog\/view\/matrix_der<\/a><\/noindex><\/i><\/p>\n

Shkollat, p\u00ebrmbledhjet e detyrave:<\/b><\/u><\/p>\n

1) Sh\u00ebnime t\u00eb ligj\u00ebratave mbi matematik\u00ebn superiore: kursi i plot\u00eb \/ D.T. Pismenny \u2013 Botimi i 4-t\u00eb \u2013 Mosk\u00eb: Airis-press, 2006
\n2) Analiza regresive e aplikuar \/ N. Draper, G. Smith \u2013 Botimi i 2-t\u00eb \u2013 Mosk\u00eb: Financa dhe statistika, 1986 (p\u00ebrkthimi nga anglishtja)
\n3) Detyra mbi zgjidhjen e ekuacioneve matricore:
\nfunction-x.ru\/matrix_equations.html<\/a><\/noindex>
\nmathprofi.ru\/deistviya_s_matricami.html<\/a><\/noindex><\/i>
\n
Burimi: habr.com<\/a><\/p>","protected":false,"gt_translate_keys":[{"key":"rendered","format":"html"}]},"excerpt":{"rendered":"

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