{"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\/ro\/blog\/news\/privodim-uravnenie-linejnoj-regressii-v-matrichnyj-vid","title":{"rendered":"Transform\u0103m ecua\u021bia regresiei liniare \u00een form\u0103 matricial\u0103","gt_translate_keys":[{"key":"rendered","format":"text"}]},"content":{"rendered":"<p><img decoding=\"async\" alt=\"Transform\u0103m ecua\u021bia regresiei liniare \u00een form\u0103 matricial\u0103\" src=\"\/wp-content\/uploads\/2019\/12\/47507c0ae01db89c623d94f540c22aea.jpg\" style=\"display:block;margin: 0 auto;\" \/><br \/>\n<br \/>\nScopul articolului este de a oferi suport \u00eencep\u0103torilor \u00een data science. \u00cen <noindex><a rel=\"nofollow\" href=\"https:\/\/habr.com\/ru\/post\/474602\/\">articolul anterior<\/a><\/noindex> am explicat pe \u00een\u021belesul tuturor trei metode de rezolvare a ecua\u021biei regresiei liniare: solu\u021bia analitic\u0103, cobor\u00e2rea gradientului \u0219i cobor\u00e2rea stochastic\u0103 a gradientului. Astfel, pentru solu\u021bia analitic\u0103 am aplicat formula <img decoding=\"async\" alt=\"Transform\u0103m ecua\u021bia regresiei liniare \u00een form\u0103 matricial\u0103\" src=\"\/wp-content\/uploads\/2019\/12\/50b913c04204ae6400ca1676cca57d18.jpg\" style=\"display:block;margin: 0 auto;\" \/>. \u00cen acest articol, dup\u0103 cum sugereaz\u0103 titlul, vom justifica aplicarea acestei formule sau, cu alte cuvinte, o vom deduce singuri. <\/p>\n<p>De ce are sens s\u0103 acord\u0103m o aten\u021bie deosebit\u0103 formulei <img decoding=\"async\" alt=\"Transform\u0103m ecua\u021bia regresiei liniare \u00een form\u0103 matricial\u0103\" src=\"\/wp-content\/uploads\/2019\/12\/9d44749d6f8b509b3167a25f0c583f7f.jpg\" style=\"display:block;margin: 0 auto;\" \/>? <\/p>\n<p>Anume cu ecua\u021bia matricial\u0103, \u00een cele mai multe cazuri, \u00eencepe familiarizarea cu regresia liniar\u0103. Totu\u0219i, detaliile despre cum a fost dedus\u0103 formula sunt rare.<br \/>\n<noindex><a rel=\"nofollow\" name=\"habracut\"><\/a><\/noindex><br \/>\nDe exemplu, \u00een cursurile de \u00eenv\u0103\u021bare automat\u0103 de la Yandex, c\u00e2nd ascult\u0103torii sunt familiariza\u021bi cu regularizarea, li se sugereaz\u0103 s\u0103 foloseasc\u0103 func\u021biile din biblioteca <i>sklearn<\/i>, f\u0103r\u0103 a men\u021biona vreun cuv\u00e2nt despre reprezentarea matricial\u0103 a algoritmului. Exact \u00een acel moment, unii dintre ascult\u0103tori ar putea dori s\u0103 aprofundeze aceast\u0103 problem\u0103 \u2014 s\u0103 scrie un cod f\u0103r\u0103 a folosi func\u021bii gata f\u0103cute. \u0218i pentru aceasta, trebuie mai \u00eent\u00e2i s\u0103 ne reprezent\u0103m ecua\u021bia cu regularizatorul \u00een form\u0103 matricial\u0103. Acest articol va permite celor care doresc s\u0103 dob\u00e2ndeasc\u0103 astfel de abilit\u0103\u021bi. S\u0103 \u00eencepem.<\/p>\n<h2>Condi\u021biile ini\u021biale<\/h2>\n<p><\/p>\n<h3>Indicatorii \u021binti\u021bi<\/h3>\n<p>\nAvem o serie de valori ale indicatorului \u021bint\u0103. De exemplu, indicatorul \u021bint\u0103 poate fi pre\u021bul unui activ: petrol, aur, gr\u00e2u, dolar etc. \u00cen acest context, prin seria de valori ale indicatorului \u021bint\u0103 \u00een\u021belegem num\u0103rul de observa\u021bii. Aceste observa\u021bii pot fi, de exemplu, pre\u021burile lunare ale petrolului timp de un an, adic\u0103 vom avea 12 valori ale indicatorului \u021bint\u0103. S\u0103 \u00eencepem s\u0103 introducem denumirile. Vom denumi fiecare valoare a indicatorului \u021bint\u0103 ca <img decoding=\"async\" alt=\"Transform\u0103m ecua\u021bia regresiei liniare \u00een form\u0103 matricial\u0103\" src=\"\/wp-content\/uploads\/2019\/12\/d974b10a558b8a917aca3dd7d16e9a7d.jpg\" style=\"display:block;margin: 0 auto;\" \/>. \u00cen total, avem <img decoding=\"async\" alt=\"Transform\u0103m ecua\u021bia regresiei liniare \u00een form\u0103 matricial\u0103\" src=\"\/wp-content\/uploads\/2019\/12\/5e826824b32c20c20eb15b276c544583.jpg\" style=\"display:block;margin: 0 auto;\" \/> observa\u021bii, a\u0219a c\u0103 putem reprezenta observa\u021biile noastre ca <img decoding=\"async\" alt=\"Transform\u0103m ecua\u021bia regresiei liniare \u00een form\u0103 matricial\u0103\" src=\"\/wp-content\/uploads\/2019\/12\/599bcd56d392e21dcf63b72b07edcbc5.jpg\" style=\"display:block;margin: 0 auto;\" \/>.<\/p>\n<h3>Regresorii<\/h3>\n<p>\nS\u0103 consider\u0103m c\u0103 exist\u0103 factori care, \u00eentr-o anumit\u0103 m\u0103sur\u0103, explic\u0103 valorile indicatorului \u021bint\u0103. De exemplu, \u00een cazul cursului perechii dolar\/rubl\u0103, pre\u021bul petrolului, rata FRS \u0219i altele au un impact semnificativ. Ace\u0219ti factori se numesc regresori. Astfel, fiec\u0103rei valori a indicatorului \u021bint\u0103 trebuie s\u0103 \u00eei corespund\u0103 o valoare a regresorului, adic\u0103, dac\u0103 avem 12 indicatori \u021bint\u0103 pentru fiecare lun\u0103 din 2018, atunci trebuie s\u0103 avem \u0219i 12 valori ale regresorilor pentru aceea\u0219i perioad\u0103. S\u0103 denot\u0103m valorile fiec\u0103rui regresor prin <img decoding=\"async\" alt=\"Transform\u0103m ecua\u021bia regresiei liniare \u00een form\u0103 matricial\u0103\" src=\"\/wp-content\/uploads\/2019\/12\/d49b372bea0c1d50c21681357f85610a.jpg\" style=\"display:block;margin: 0 auto;\" \/>. S\u0103 presupunem c\u0103 \u00een cazul nostru avem <img decoding=\"async\" alt=\"Transform\u0103m ecua\u021bia regresiei liniare \u00een form\u0103 matricial\u0103\" src=\"\/wp-content\/uploads\/2019\/12\/61a94efcc11446ebee920870a684c216.jpg\" style=\"display:block;margin: 0 auto;\" \/> regresori (adic\u0103 <img decoding=\"async\" alt=\"Transform\u0103m ecua\u021bia regresiei liniare \u00een form\u0103 matricial\u0103\" src=\"\/wp-content\/uploads\/2019\/12\/2699691a567d694e61e6fdb288bc9b54.jpg\" style=\"display:block;margin: 0 auto;\" \/> factori care influen\u021beaz\u0103 valorile indicatorului \u021bint\u0103). Astfel, regresorii no\u0219tri pot fi reprezenta\u021bi dup\u0103 cum urmeaz\u0103: pentru 1-regresor (de exemplu, pre\u021bul petrolului): <img decoding=\"async\" alt=\"Transform\u0103m ecua\u021bia regresiei liniare \u00een form\u0103 matricial\u0103\" src=\"\/wp-content\/uploads\/2019\/12\/79434bf84a9b18f83602225f43e68cc7.jpg\" style=\"display:block;margin: 0 auto;\" \/>, pentru 2-regresor (de exemplu, rata FRS): <img decoding=\"async\" alt=\"Transform\u0103m ecua\u021bia regresiei liniare \u00een form\u0103 matricial\u0103\" src=\"\/wp-content\/uploads\/2019\/12\/378180ee8fdaa8f6aca7bc8300a21207.jpg\" style=\"display:block;margin: 0 auto;\" \/>, pentru &#171;<img decoding=\"async\" alt=\"Transform\u0103m ecua\u021bia regresiei liniare \u00een form\u0103 matricial\u0103\" src=\"\/wp-content\/uploads\/2019\/12\/fc70caea64d3bc2aa5e797257f6627fb.jpg\" style=\"display:block;margin: 0 auto;\" \/>-le&#187; regresor: <img decoding=\"async\" alt=\"Transform\u0103m ecua\u021bia regresiei liniare \u00een form\u0103 matricial\u0103\" src=\"\/wp-content\/uploads\/2019\/12\/90499d34f1577681de33966a024faf2e.jpg\" style=\"display:block;margin: 0 auto;\" \/><\/p>\n<h3>Dependenta indicatorilor \u021bint\u0103 de regresori<\/h3>\n<p>\nPresupunem c\u0103 dependen\u021ba indicatorului \u021bint\u0103 <img decoding=\"async\" alt=\"Transform\u0103m ecua\u021bia regresiei liniare \u00een form\u0103 matricial\u0103\" src=\"\/wp-content\/uploads\/2019\/12\/aa7d30d8fb9ca837144d34423aed5a77.jpg\" style=\"display:block;margin: 0 auto;\" \/> de la regresorii &#171;<img decoding=\"async\" alt=\"Transform\u0103m ecua\u021bia regresiei liniare \u00een form\u0103 matricial\u0103\" src=\"\/wp-content\/uploads\/2019\/12\/97a9aba33249a8b9eb3a61bdfc794287.jpg\" style=\"display:block;margin: 0 auto;\" \/>-le&#187; observa\u021bie poate fi exprimat\u0103 prin ecua\u021bia regresiei liniare de forma:<\/p>\n<p><img decoding=\"async\" alt=\"Transform\u0103m ecua\u021bia regresiei liniare \u00een form\u0103 matricial\u0103\" src=\"\/wp-content\/uploads\/2019\/12\/ff892eb01b0bb830a94cacf62168d142.jpg\" style=\"display:block;margin: 0 auto;\" \/><\/p>\n<p>, unde <img decoding=\"async\" alt=\"Transform\u0103m ecua\u021bia regresiei liniare \u00een form\u0103 matricial\u0103\" src=\"\/wp-content\/uploads\/2019\/12\/27d8f05c058f7b42aac3a4d3fab1db51.jpg\" style=\"display:block;margin: 0 auto;\" \/> \u2014 &#171;<img decoding=\"async\" alt=\"Transform\u0103m ecua\u021bia regresiei liniare \u00een form\u0103 matricial\u0103\" src=\"\/wp-content\/uploads\/2019\/12\/cd438a2e58f8db5e9f1df8b387b18cff.jpg\" style=\"display:block;margin: 0 auto;\" \/>-a&#187; valoare a regresorului de la 1 la <img decoding=\"async\" alt=\"Transform\u0103m ecua\u021bia regresiei liniare \u00een form\u0103 matricial\u0103\" src=\"\/wp-content\/uploads\/2019\/12\/21513049bf0d7b13af73d53605089765.jpg\" style=\"display:block;margin: 0 auto;\" \/>,<\/p>\n<p><img decoding=\"async\" alt=\"Transform\u0103m ecua\u021bia regresiei liniare \u00een form\u0103 matricial\u0103\" src=\"\/wp-content\/uploads\/2019\/12\/268ca408ca89b7846220c920e2078d5d.jpg\" style=\"display:block;margin: 0 auto;\" \/> \u2014 num\u0103rul regresorilor de la 1 la <img decoding=\"async\" alt=\"Transform\u0103m ecua\u021bia regresiei liniare \u00een form\u0103 matricial\u0103\" src=\"\/wp-content\/uploads\/2019\/12\/8be331a5b16bf3796e6200e77ee58c09.jpg\" style=\"display:block;margin: 0 auto;\" \/><\/p>\n<p><img decoding=\"async\" alt=\"Transform\u0103m ecua\u021bia regresiei liniare \u00een form\u0103 matricial\u0103\" src=\"\/wp-content\/uploads\/2019\/12\/49558388879a529fc342848d80b60d32.jpg\" style=\"display:block;margin: 0 auto;\" \/> \u2014 coeficien\u021bii unghiulari, care reprezint\u0103 magnitudinea cu care se va schimba indicatorul \u021bint\u0103 calculat \u00een medie la schimbarea regresorului. <\/p>\n<p>Cu alte cuvinte, pentru fiecare regresor (cu excep\u021bia <img decoding=\"async\" alt=\"Transform\u0103m ecua\u021bia regresiei liniare \u00een form\u0103 matricial\u0103\" src=\"\/wp-content\/uploads\/2019\/12\/fcc00174a0090f9974712c779718f13c.jpg\" style=\"display:block;margin: 0 auto;\" \/>) definim coeficientul \u201epropriu\u201d <img decoding=\"async\" alt=\"Transform\u0103m ecua\u021bia regresiei liniare \u00een form\u0103 matricial\u0103\" src=\"\/wp-content\/uploads\/2019\/12\/c40688d883e7a22b52905be2517a91f9.jpg\" style=\"display:block;margin: 0 auto;\" \/>, apoi \u00eenmul\u021bim coeficien\u021bii cu valorile regresorilor &#171;<img decoding=\"async\" alt=\"Transform\u0103m ecua\u021bia regresiei liniare \u00een form\u0103 matricial\u0103\" src=\"\/wp-content\/uploads\/2019\/12\/48db80e36602c8a55eba1e518b1e3efa.jpg\" style=\"display:block;margin: 0 auto;\" \/>-le&#187; observa\u021bie, rezult\u00e2nd o aproximare &#171;<img decoding=\"async\" alt=\"Transform\u0103m ecua\u021bia regresiei liniare \u00een form\u0103 matricial\u0103\" src=\"\/wp-content\/uploads\/2019\/12\/a182d4ce402fa8b1d8f631ffddd5e63e.jpg\" style=\"display:block;margin: 0 auto;\" \/>-le&#187; indicatorului \u021bint\u0103.<\/p>\n<p>Prin urmare, trebuie s\u0103 g\u0103sim coeficien\u021bi <img decoding=\"async\" alt=\"Transform\u0103m ecua\u021bia regresiei liniare \u00een form\u0103 matricial\u0103\" src=\"\/wp-content\/uploads\/2019\/12\/81edeafa5c9539a4b168d5b40027ffd0.jpg\" style=\"display:block;margin: 0 auto;\" \/>, pentru care valorile func\u021biei noastre de aproximare <img decoding=\"async\" alt=\"Transform\u0103m ecua\u021bia regresiei liniare \u00een form\u0103 matricial\u0103\" src=\"\/wp-content\/uploads\/2019\/12\/84ae78e0b61fd87769ae120f903f2a8a.jpg\" style=\"display:block;margin: 0 auto;\" \/> vor fi plasate c\u00e2t mai aproape de valorile indicatorilor \u021bint\u0103.<\/p>\n<h3>Evaluarea calit\u0103\u021bii func\u021biei de aproximare<\/h3>\n<p>\nVom determina evaluarea calit\u0103\u021bii func\u021biei de aproximare prin metoda celor mai mici p\u0103trate. Func\u021bia de evaluare a calit\u0103\u021bii va avea urm\u0103toarea form\u0103:<\/p>\n<p><img decoding=\"async\" alt=\"Transform\u0103m ecua\u021bia regresiei liniare \u00een form\u0103 matricial\u0103\" src=\"\/wp-content\/uploads\/2019\/12\/d50c5c70fda27cf18308baa31c2366a5.jpg\" style=\"display:block;margin: 0 auto;\" \/><\/p>\n<p>Trebuie s\u0103 g\u0103sim valori ale coeficientelor $w$, pentru care valoarea <img decoding=\"async\" alt=\"Transform\u0103m ecua\u021bia regresiei liniare \u00een form\u0103 matricial\u0103\" src=\"\/wp-content\/uploads\/2019\/12\/e2ad7197c6ca2c9ac986a62fcfef1f7d.jpg\" style=\"display:block;margin: 0 auto;\" \/> va fi cea mai mic\u0103.<\/p>\n<h2>Transform\u0103m ecua\u021bia \u00eentr-o form\u0103 matriceal\u0103<\/h2>\n<p><\/p>\n<h3>Reprezentare vectorial\u0103<\/h3>\n<p>\nPentru \u00eenceput, pentru a ne u\u0219ura via\u021ba, ar trebui s\u0103 ne \u00eendrept\u0103m aten\u021bia c\u0103tre ecua\u021bia de regresie liniar\u0103 \u0219i s\u0103 observ\u0103m c\u0103 primul coeficient <img decoding=\"async\" alt=\"Transform\u0103m ecua\u021bia regresiei liniare \u00een form\u0103 matricial\u0103\" src=\"\/wp-content\/uploads\/2019\/12\/4b34e4c0e260b7bc143e95e83169fcc9.jpg\" style=\"display:block;margin: 0 auto;\" \/> nu este multiplicat cu niciun regresor. \u00cen acest context, atunci c\u00e2nd vom transforma datele \u00eentr-o form\u0103 matricial\u0103, circumstan\u021ba men\u021bionat\u0103 anterior va complica serios calculele. \u00cen acest sens, se propune introducerea unui alt regresor pentru primul coeficient. <img decoding=\"async\" alt=\"Transform\u0103m ecua\u021bia regresiei liniare \u00een form\u0103 matricial\u0103\" src=\"\/wp-content\/uploads\/2019\/12\/6b6c662ff0c499870d0f119589ae2b44.jpg\" style=\"display:block;margin: 0 auto;\" \/> \u0219i s\u0103-l egaliz\u0103m cu unul. Mai precis, fiecare &#171;<img decoding=\"async\" alt=\"Transform\u0103m ecua\u021bia regresiei liniare \u00een form\u0103 matricial\u0103\" src=\"\/wp-content\/uploads\/2019\/12\/7c0aa6c3c797066208fb19a638ef513f.jpg\" style=\"display:block;margin: 0 auto;\" \/>-a&#187; valoare a acestui regresor trebuie s\u0103 fie egalizat\u0103 cu unul \u2014 pentru c\u0103, \u00een urma \u00eenmul\u021birii cu unul, din punctul de vedere al rezultatelor calculului, nimic nu se va schimba, iar din perspectiva regulilor produsului matricial, ne va u\u0219ura semnificativ eforturile. <\/p>\n<p>Acum, pentru o perioad\u0103, \u00een scopul simplific\u0103rii materialului, s\u0103 presupunem c\u0103 avem doar o &#171;<img decoding=\"async\" alt=\"Transform\u0103m ecua\u021bia regresiei liniare \u00een form\u0103 matricial\u0103\" src=\"\/wp-content\/uploads\/2019\/12\/eac0db64e38d3ce6b5a0303393704fab.jpg\" style=\"display:block;margin: 0 auto;\" \/>-a&#187; observa\u021bie. Atunci, s\u0103 reprezent\u0103m valorile regresorilor &#171;<img decoding=\"async\" alt=\"Transform\u0103m ecua\u021bia regresiei liniare \u00een form\u0103 matricial\u0103\" src=\"\/wp-content\/uploads\/2019\/12\/ca79caf5f669c549b97d51ce1cb6f7cd.jpg\" style=\"display:block;margin: 0 auto;\" \/>-le&#187; observa\u021bie ca un vector <img decoding=\"async\" alt=\"Transform\u0103m ecua\u021bia regresiei liniare \u00een form\u0103 matricial\u0103\" src=\"\/wp-content\/uploads\/2019\/12\/63e0cf2d3b09577dfd6d76718864e56a.jpg\" style=\"display:block;margin: 0 auto;\" \/>Vectorul <img decoding=\"async\" alt=\"Transform\u0103m ecua\u021bia regresiei liniare \u00een form\u0103 matricial\u0103\" src=\"\/wp-content\/uploads\/2019\/12\/bff53727ebff61bbbe97ca02a132338d.jpg\" style=\"display:block;margin: 0 auto;\" \/> are dimensiunea <img decoding=\"async\" alt=\"Transform\u0103m ecua\u021bia regresiei liniare \u00een form\u0103 matricial\u0103\" src=\"\/wp-content\/uploads\/2019\/12\/f87f7646f1f53f041e53c8f63bb30405.jpg\" style=\"display:block;margin: 0 auto;\" \/>, adic\u0103 <img decoding=\"async\" alt=\"Transform\u0103m ecua\u021bia regresiei liniare \u00een form\u0103 matricial\u0103\" src=\"\/wp-content\/uploads\/2019\/12\/bda60053e47e7a5324106d5e6cdedbab.jpg\" style=\"display:block;margin: 0 auto;\" \/> linii \u0219i 1 coloan\u0103:<\/p>\n<p><img decoding=\"async\" alt=\"Transform\u0103m ecua\u021bia regresiei liniare \u00een form\u0103 matricial\u0103\" src=\"\/wp-content\/uploads\/2019\/12\/fc1a58c18e118375c3bee17e1a3143d9.jpg\" style=\"display:block;margin: 0 auto;\" \/><\/p>\n<p>Coeficientii c\u0103uta\u021bi i-am reprezentat sub form\u0103 de vector <img decoding=\"async\" alt=\"Transform\u0103m ecua\u021bia regresiei liniare \u00een form\u0103 matricial\u0103\" src=\"\/wp-content\/uploads\/2019\/12\/4c6805c97148d50d3233e183a4690b47.jpg\" style=\"display:block;margin: 0 auto;\" \/>, av\u00e2nd dimensiunea <img decoding=\"async\" alt=\"Transform\u0103m ecua\u021bia regresiei liniare \u00een form\u0103 matricial\u0103\" src=\"\/wp-content\/uploads\/2019\/12\/8cbddba5ac5f8c0e8f5d80c5dd5a4c6f.jpg\" style=\"display:block;margin: 0 auto;\" \/>:<\/p>\n<p><img decoding=\"async\" alt=\"Transform\u0103m ecua\u021bia regresiei liniare \u00een form\u0103 matricial\u0103\" src=\"\/wp-content\/uploads\/2019\/12\/ebcf96a3aa612fbdf10864c010787d0e.jpg\" style=\"display:block;margin: 0 auto;\" \/><\/p>\n<p>Ecua\u021bia regresiei liniare pentru &#171;<img decoding=\"async\" alt=\"Transform\u0103m ecua\u021bia regresiei liniare \u00een form\u0103 matricial\u0103\" src=\"\/wp-content\/uploads\/2019\/12\/6a93855293a06ea44f48077c315dde79.jpg\" style=\"display:block;margin: 0 auto;\" \/>-le&#187; observa\u021bie va avea forma: <\/p>\n<p><img decoding=\"async\" alt=\"Transform\u0103m ecua\u021bia regresiei liniare \u00een form\u0103 matricial\u0103\" src=\"\/wp-content\/uploads\/2019\/12\/e73499da0077212806ea3ce5d639bb9c.jpg\" style=\"display:block;margin: 0 auto;\" \/><\/p>\n<p>Func\u021bia de evaluare a calit\u0103\u021bii modelului liniare va lua urm\u0103toarea form\u0103:<\/p>\n<p><img decoding=\"async\" alt=\"Transform\u0103m ecua\u021bia regresiei liniare \u00een form\u0103 matricial\u0103\" src=\"\/wp-content\/uploads\/2019\/12\/a33c79a345dccc9e1fce3a32d53b2164.jpg\" style=\"display:block;margin: 0 auto;\" \/><\/p>\n<p>S\u0103 observ\u0103m c\u0103, \u00een conformitate cu regulile multiplic\u0103rii matricelor, a fost necesar s\u0103 transpunem vectorul <img decoding=\"async\" alt=\"Transform\u0103m ecua\u021bia regresiei liniare \u00een form\u0103 matricial\u0103\" src=\"\/wp-content\/uploads\/2019\/12\/3e551ab86fc872959afe9ea10344549e.jpg\" style=\"display:block;margin: 0 auto;\" \/>.<\/p>\n<h3>Reprezentarea matricial\u0103<\/h3>\n<p>\nCa rezultat al multiplic\u0103rii vectorilor, vom ob\u021bine un num\u0103r: <img decoding=\"async\" alt=\"Transform\u0103m ecua\u021bia regresiei liniare \u00een form\u0103 matricial\u0103\" src=\"\/wp-content\/uploads\/2019\/12\/1889e5ba7053a652c7ad1ea6b37627ab.jpg\" style=\"display:block;margin: 0 auto;\" \/>, ceea ce era de a\u0219teptat. Acest num\u0103r este aproxima\u021bia &#171;<img decoding=\"async\" alt=\"Transform\u0103m ecua\u021bia regresiei liniare \u00een form\u0103 matricial\u0103\" src=\"\/wp-content\/uploads\/2019\/12\/255557422b97a46c2575f2279c656bd0.jpg\" style=\"display:block;margin: 0 auto;\" \/>-le&#187; indicatorului \u021bint\u0103. Dar ceea ce ne trebuie este aproxima\u021bia nu a unei singure valori a indicatorului \u021bint\u0103, ci a tuturor. Pentru aceasta, vom scrie to\u021bi &#171;<img decoding=\"async\" alt=\"Transform\u0103m ecua\u021bia regresiei liniare \u00een form\u0103 matricial\u0103\" src=\"\/wp-content\/uploads\/2019\/12\/63289920f39f55a57a48b48255ad96dd.jpg\" style=\"display:block;margin: 0 auto;\" \/>-ii&#187; regresori \u00een format matrice <img decoding=\"async\" alt=\"Transform\u0103m ecua\u021bia regresiei liniare \u00een form\u0103 matricial\u0103\" src=\"\/wp-content\/uploads\/2019\/12\/3a003dcab41a7c3c5cff534fd0b91e3b.jpg\" style=\"display:block;margin: 0 auto;\" \/>Matricea ob\u021binut\u0103 are dimensiunea <img decoding=\"async\" alt=\"Transform\u0103m ecua\u021bia regresiei liniare \u00een form\u0103 matricial\u0103\" src=\"\/wp-content\/uploads\/2019\/12\/3435a6daffd0abb5961629671c8fa441.jpg\" style=\"display:block;margin: 0 auto;\" \/>:<\/p>\n<p><img decoding=\"async\" alt=\"Transform\u0103m ecua\u021bia regresiei liniare \u00een form\u0103 matricial\u0103\" src=\"\/wp-content\/uploads\/2019\/12\/3901fc083e9dcf646d9d3da4c06de85d.jpg\" style=\"display:block;margin: 0 auto;\" \/><\/p>\n<p>Acum ecua\u021bia regresiei liniare va ar\u0103ta astfel:<\/p>\n<p><img decoding=\"async\" alt=\"Transform\u0103m ecua\u021bia regresiei liniare \u00een form\u0103 matricial\u0103\" src=\"\/wp-content\/uploads\/2019\/12\/73e7317b0815f3d8bdf514907d0c0a90.jpg\" style=\"display:block;margin: 0 auto;\" \/><\/p>\n<p>S\u0103 not\u0103m valorile indicatorilor \u021bint\u0103 (toate <img decoding=\"async\" alt=\"Transform\u0103m ecua\u021bia regresiei liniare \u00een form\u0103 matricial\u0103\" src=\"\/wp-content\/uploads\/2019\/12\/e3b9482a385dd9be37c568c4e738883e.jpg\" style=\"display:block;margin: 0 auto;\" \/>) prin vectorul <img decoding=\"async\" alt=\"Transform\u0103m ecua\u021bia regresiei liniare \u00een form\u0103 matricial\u0103\" src=\"\/wp-content\/uploads\/2019\/12\/55cdadb0e1c9c9a1f39d918160f51c07.jpg\" style=\"display:block;margin: 0 auto;\" \/> de dimensiune <img decoding=\"async\" alt=\"Transform\u0103m ecua\u021bia regresiei liniare \u00een form\u0103 matricial\u0103\" src=\"\/wp-content\/uploads\/2019\/12\/62d6246fd9a76e9da4bea15d44456d68.jpg\" style=\"display:block;margin: 0 auto;\" \/>:<\/p>\n<p><img decoding=\"async\" alt=\"Transform\u0103m ecua\u021bia regresiei liniare \u00een form\u0103 matricial\u0103\" src=\"\/wp-content\/uploads\/2019\/12\/8b7626b09765e9e059c88d21b4593c31.jpg\" style=\"display:block;margin: 0 auto;\" \/><\/p>\n<p>Acum putem scrie \u00een format matricial ecua\u021bia evalu\u0103rii calit\u0103\u021bii modelului liniare:<\/p>\n<p><img decoding=\"async\" alt=\"Transform\u0103m ecua\u021bia regresiei liniare \u00een form\u0103 matricial\u0103\" src=\"\/wp-content\/uploads\/2019\/12\/394b3b15cc03bb921d7ca55c2ec3db6c.jpg\" style=\"display:block;margin: 0 auto;\" \/><\/p>\n<p>Practic, din aceast\u0103 formul\u0103 ob\u021binem formula bine cunoscut\u0103 nou\u0103. <img decoding=\"async\" alt=\"Transform\u0103m ecua\u021bia regresiei liniare \u00een form\u0103 matricial\u0103\" src=\"\/wp-content\/uploads\/2019\/12\/70f26b6f641c29eb4de0211778b0b207.jpg\" style=\"display:block;margin: 0 auto;\" \/><\/p>\n<p>Cum se face acest lucru? Se deschid parantezele, se realizeaz\u0103 derivarea, se transform\u0103 expresiile ob\u021binute etc., iar exact asta ne vom ocupa acum.<\/p>\n<h2>Transform\u0103ri matriciale<\/h2>\n<p><\/p>\n<h3>S\u0103 deschidem parantezele<\/h3>\n<p>\n<img decoding=\"async\" alt=\"Transform\u0103m ecua\u021bia regresiei liniare \u00een form\u0103 matricial\u0103\" src=\"\/wp-content\/uploads\/2019\/12\/e7d2ff15007e174e94cf65793538f6df.jpg\" style=\"display:block;margin: 0 auto;\" \/><\/p>\n<p><img decoding=\"async\" alt=\"Transform\u0103m ecua\u021bia regresiei liniare \u00een form\u0103 matricial\u0103\" src=\"\/wp-content\/uploads\/2019\/12\/2ef9bcd45ff0b3268aa40445d38b3ec0.jpg\" style=\"display:block;margin: 0 auto;\" \/><\/p>\n<h3>S\u0103 preg\u0103tim ecua\u021bia pentru derivare<\/h3>\n<p>\nPentru aceasta, vom efectua unele transform\u0103ri. \u00cen calculele viitoare, ne va fi mai convenabil dac\u0103 vectorul <img decoding=\"async\" alt=\"Transform\u0103m ecua\u021bia regresiei liniare \u00een form\u0103 matricial\u0103\" src=\"\/wp-content\/uploads\/2019\/12\/e14c5e74d164b3fc53f654d1f5b0be41.jpg\" style=\"display:block;margin: 0 auto;\" \/> va fi reprezentat la \u00eenceputul fiec\u0103rui produs din ecua\u021bie.<\/p>\n<h4>Transformarea 1<\/h4>\n<p>\n<img decoding=\"async\" alt=\"Transform\u0103m ecua\u021bia regresiei liniare \u00een form\u0103 matricial\u0103\" src=\"\/wp-content\/uploads\/2019\/12\/f2a2bf00b2ebbd266789420b0aa33404.jpg\" style=\"display:block;margin: 0 auto;\" \/><\/p>\n<p>Cum s-a \u00eent\u00e2mplat asta? Pentru a r\u0103spunde la aceast\u0103 \u00eentrebare, este suficient s\u0103 ne uit\u0103m la dimensiunile matricelor \u00eenmul\u021bite \u0219i s\u0103 vedem c\u0103 la ie\u0219ire ob\u021binem un num\u0103r sau altfel <img decoding=\"async\" alt=\"Transform\u0103m ecua\u021bia regresiei liniare \u00een form\u0103 matricial\u0103\" src=\"\/wp-content\/uploads\/2019\/12\/e6e75542a7ab114c0dc7277593ccc86d.jpg\" style=\"display:block;margin: 0 auto;\" \/>.<\/p>\n<p>S\u0103 not\u0103m dimensiunile expresiilor matriciale.<\/p>\n<p><img decoding=\"async\" alt=\"Transform\u0103m ecua\u021bia regresiei liniare \u00een form\u0103 matricial\u0103\" src=\"\/wp-content\/uploads\/2019\/12\/9913d89c711c919ea5dcc5f6c2beaadf.jpg\" style=\"display:block;margin: 0 auto;\" \/><\/p>\n<p><img decoding=\"async\" alt=\"Transform\u0103m ecua\u021bia regresiei liniare \u00een form\u0103 matricial\u0103\" src=\"\/wp-content\/uploads\/2019\/12\/b8b65ace6ac8a6b5218b9d79f6eceff1.jpg\" style=\"display:block;margin: 0 auto;\" \/><\/p>\n<p><img decoding=\"async\" alt=\"Transform\u0103m ecua\u021bia regresiei liniare \u00een form\u0103 matricial\u0103\" src=\"\/wp-content\/uploads\/2019\/12\/b1982bd25dc54b29220c1021c05880ef.jpg\" style=\"display:block;margin: 0 auto;\" \/><\/p>\n<h4>Transformare 2<\/h4>\n<p>\n<img decoding=\"async\" alt=\"Transform\u0103m ecua\u021bia regresiei liniare \u00een form\u0103 matricial\u0103\" src=\"\/wp-content\/uploads\/2019\/12\/3801b847f4e69894dc0e19ceefb5de55.jpg\" style=\"display:block;margin: 0 auto;\" \/><\/p>\n<p>S\u0103 detaliem \u00een mod similar transformarea 1<\/p>\n<p><img decoding=\"async\" alt=\"Transform\u0103m ecua\u021bia regresiei liniare \u00een form\u0103 matricial\u0103\" src=\"\/wp-content\/uploads\/2019\/12\/4bb7dbb1211c7e15ba18a8b2a0f63191.jpg\" style=\"display:block;margin: 0 auto;\" \/><\/p>\n<p><img decoding=\"async\" alt=\"Transform\u0103m ecua\u021bia regresiei liniare \u00een form\u0103 matricial\u0103\" src=\"\/wp-content\/uploads\/2019\/12\/60e7b41a69d3fb229e36f84960527769.jpg\" style=\"display:block;margin: 0 auto;\" \/><\/p>\n<p>La ie\u0219ire ob\u021binem ecua\u021bia pe care trebuie s\u0103 o deriv\u0103m:<br \/>\n<img decoding=\"async\" alt=\"Transform\u0103m ecua\u021bia regresiei liniare \u00een form\u0103 matricial\u0103\" src=\"\/wp-content\/uploads\/2019\/12\/2c3b23bef5eba43159baf5be3effe604.jpg\" style=\"display:block;margin: 0 auto;\" \/><\/p>\n<h2>Deriv\u0103m func\u021bia de evaluare a modelului<\/h2>\n<p>\nDeriv\u0103m dup\u0103 vector <img decoding=\"async\" alt=\"Transform\u0103m ecua\u021bia regresiei liniare \u00een form\u0103 matricial\u0103\" src=\"\/wp-content\/uploads\/2019\/12\/7f408dd080459692d93f343528219bd6.jpg\" style=\"display:block;margin: 0 auto;\" \/>:<\/p>\n<p><img decoding=\"async\" alt=\"Transform\u0103m ecua\u021bia regresiei liniare \u00een form\u0103 matricial\u0103\" src=\"\/wp-content\/uploads\/2019\/12\/fb6025293b318051b926913bc8e5849c.jpg\" style=\"display:block;margin: 0 auto;\" \/><\/p>\n<p>\n<img decoding=\"async\" alt=\"Transform\u0103m ecua\u021bia regresiei liniare \u00een form\u0103 matricial\u0103\" src=\"\/wp-content\/uploads\/2019\/12\/d5d089a76ceeda9dae631e9af478df36.jpg\" style=\"display:block;margin: 0 auto;\" \/><\/p>\n<p><img decoding=\"async\" alt=\"Transform\u0103m ecua\u021bia regresiei liniare \u00een form\u0103 matricial\u0103\" src=\"\/wp-content\/uploads\/2019\/12\/4dca6e16f82114a777ec2d5dbd6df998.jpg\" style=\"display:block;margin: 0 auto;\" \/><\/p>\n<p><img decoding=\"async\" alt=\"Transform\u0103m ecua\u021bia regresiei liniare \u00een form\u0103 matricial\u0103\" src=\"\/wp-content\/uploads\/2019\/12\/efb26c3a7e8b40de68d8aba115f09feb.jpg\" style=\"display:block;margin: 0 auto;\" \/><\/p>\n<p>\u00centreb\u0103rile de ce <img decoding=\"async\" alt=\"Transform\u0103m ecua\u021bia regresiei liniare \u00een form\u0103 matricial\u0103\" src=\"\/wp-content\/uploads\/2019\/12\/e814552115f81a72fa60a988b7644af6.jpg\" style=\"display:block;margin: 0 auto;\" \/> nu ar trebui s\u0103 existe, dar opera\u021biile de definire a derivatelor \u00een celelalte dou\u0103 expresii le vom analiza mai detaliat.<\/p>\n<h3>Derivare 1<\/h3>\n<p>\nS\u0103 detaliem derivarea: <img decoding=\"async\" alt=\"Transform\u0103m ecua\u021bia regresiei liniare \u00een form\u0103 matricial\u0103\" src=\"\/wp-content\/uploads\/2019\/12\/6e00a4d10e726babe1d058c9d2ce7723.jpg\" style=\"display:block;margin: 0 auto;\" \/><\/p>\n<p>Pentru a determina derivata unei matrice sau a unui vector, trebuie s\u0103 ne uit\u0103m la ce au \u00een interior. S\u0103 ne uit\u0103m:<\/p>\n<p><img decoding=\"async\" alt=\"Transform\u0103m ecua\u021bia regresiei liniare \u00een form\u0103 matricial\u0103\" src=\"\/wp-content\/uploads\/2019\/12\/edccede10d4d013897fefdf65e81add5.jpg\" style=\"display:block;margin: 0 auto;\" \/><\/p>\n<p><img decoding=\"async\" alt=\"Transform\u0103m ecua\u021bia regresiei liniare \u00een form\u0103 matricial\u0103\" src=\"\/wp-content\/uploads\/2019\/12\/2263c941aa42adc1ef20031fc2facf17.jpg\" style=\"display:block;margin: 0 auto;\" \/><\/p>\n<p><img decoding=\"async\" alt=\"Transform\u0103m ecua\u021bia regresiei liniare \u00een form\u0103 matricial\u0103\" src=\"\/wp-content\/uploads\/2019\/12\/cd4c1ec8f6f5934f24fa7c21ebe51358.jpg\" style=\"display:block;margin: 0 auto;\" \/> <img decoding=\"async\" alt=\"Transform\u0103m ecua\u021bia regresiei liniare \u00een form\u0103 matricial\u0103\" src=\"\/wp-content\/uploads\/2019\/12\/d75a93eaae356ef96390b4e95e2d5cb9.jpg\" style=\"display:block;margin: 0 auto;\" \/><\/p>\n<p>S\u0103 denot\u0103m produsul matricelor <img decoding=\"async\" alt=\"Transform\u0103m ecua\u021bia regresiei liniare \u00een form\u0103 matricial\u0103\" src=\"\/wp-content\/uploads\/2019\/12\/75083779f507ad45545f6e10abb3902d.jpg\" style=\"display:block;margin: 0 auto;\" \/> prin matricea <img decoding=\"async\" alt=\"Transform\u0103m ecua\u021bia regresiei liniare \u00een form\u0103 matricial\u0103\" src=\"\/wp-content\/uploads\/2019\/12\/10290d0fb2886757f5060bc713e0f666.jpg\" style=\"display:block;margin: 0 auto;\" \/>. Matricea <img decoding=\"async\" alt=\"Transform\u0103m ecua\u021bia regresiei liniare \u00een form\u0103 matricial\u0103\" src=\"\/wp-content\/uploads\/2019\/12\/d8c8c636d3f31d64f2c11b723a61ad76.jpg\" style=\"display:block;margin: 0 auto;\" \/> este p\u0103trat\u0103 \u0219i mai mult dec\u00e2t at\u00e2t, este simetric\u0103. Aceste propriet\u0103\u021bi ne vor fi utile mai departe, s\u0103 le re\u021binem. Matricea <img decoding=\"async\" alt=\"Transform\u0103m ecua\u021bia regresiei liniare \u00een form\u0103 matricial\u0103\" src=\"\/wp-content\/uploads\/2019\/12\/a0bb9002c1453028bcef0d34dfc23573.jpg\" style=\"display:block;margin: 0 auto;\" \/> are dimensiunea <img decoding=\"async\" alt=\"Transform\u0103m ecua\u021bia regresiei liniare \u00een form\u0103 matricial\u0103\" src=\"\/wp-content\/uploads\/2019\/12\/ee3eb39bf58b2fe9c6a5b69b55ff140e.jpg\" style=\"display:block;margin: 0 auto;\" \/>:<\/p>\n<p><img decoding=\"async\" alt=\"Transform\u0103m ecua\u021bia regresiei liniare \u00een form\u0103 matricial\u0103\" src=\"\/wp-content\/uploads\/2019\/12\/0ce2404bd76804e036b1e5a82dabe22a.jpg\" style=\"display:block;margin: 0 auto;\" \/><\/p>\n<p>Acum, sarcina noastr\u0103 este s\u0103 \u00eenmul\u021bim corect vectorii cu matricea \u0219i s\u0103 nu ob\u021binem \u201edoi ori doi este cinci\u201d, a\u0219a c\u0103 s\u0103 ne concentr\u0103m \u0219i s\u0103 fim extrem de aten\u021bi.<\/p>\n<p><img decoding=\"async\" alt=\"Transform\u0103m ecua\u021bia regresiei liniare \u00een form\u0103 matricial\u0103\" src=\"\/wp-content\/uploads\/2019\/12\/2fc1ec5f4e6ce5cb08d8790149129621.jpg\" style=\"display:block;margin: 0 auto;\" \/><\/p>\n<p><img decoding=\"async\" alt=\"Transform\u0103m ecua\u021bia regresiei liniare \u00een form\u0103 matricial\u0103\" src=\"\/wp-content\/uploads\/2019\/12\/b9b27f264a319fb9599597a7fffd1ac9.jpg\" style=\"display:block;margin: 0 auto;\" \/><\/p>\n<p><img decoding=\"async\" alt=\"Transform\u0103m ecua\u021bia regresiei liniare \u00een form\u0103 matricial\u0103\" src=\"\/wp-content\/uploads\/2019\/12\/fe86080e1f0ae686df40eda6b67946db.jpg\" style=\"display:block;margin: 0 auto;\" \/><\/p>\n<p><img decoding=\"async\" alt=\"Transform\u0103m ecua\u021bia regresiei liniare \u00een form\u0103 matricial\u0103\" src=\"\/wp-content\/uploads\/2019\/12\/281d7e0888baae8417b485f6ccc7630c.jpg\" style=\"display:block;margin: 0 auto;\" \/><\/p>\n<p>Cu toate acestea, am avut o expresie complicat\u0103! De fapt, am ob\u021binut un num\u0103r \u2014 scalar. \u0218i acum, cu adev\u0103rat, trecem la derivare. Este necesar s\u0103 g\u0103sim derivata expresiei ob\u021binute dup\u0103 fiecare coeficient <img decoding=\"async\" alt=\"Transform\u0103m ecua\u021bia regresiei liniare \u00een form\u0103 matricial\u0103\" src=\"\/wp-content\/uploads\/2019\/12\/b4acde2261a1ef82fdd4bcb851d3d54f.jpg\" style=\"display:block;margin: 0 auto;\" \/> \u0219i s\u0103 ob\u021binem un vector de dimensiune <img decoding=\"async\" alt=\"Transform\u0103m ecua\u021bia regresiei liniare \u00een form\u0103 matricial\u0103\" src=\"\/wp-content\/uploads\/2019\/12\/9f5b8adf1e914ea3c6181a627b355d40.jpg\" style=\"display:block;margin: 0 auto;\" \/>. Ca o precau\u021bie, voi scrie procedurile pas cu pas:<\/p>\n<p>1) deriv\u0103m dup\u0103 <img decoding=\"async\" alt=\"Transform\u0103m ecua\u021bia regresiei liniare \u00een form\u0103 matricial\u0103\" src=\"\/wp-content\/uploads\/2019\/12\/2522e823e4dee25af7d0268ea467ad75.jpg\" style=\"display:block;margin: 0 auto;\" \/>, ob\u021binem: <img decoding=\"async\" alt=\"Transform\u0103m ecua\u021bia regresiei liniare \u00een form\u0103 matricial\u0103\" src=\"\/wp-content\/uploads\/2019\/12\/7e89620225eaf5e85348ba0f590ecae1.jpg\" style=\"display:block;margin: 0 auto;\" \/><\/p>\n<p>2) deriv\u0103m dup\u0103 <img decoding=\"async\" alt=\"Transform\u0103m ecua\u021bia regresiei liniare \u00een form\u0103 matricial\u0103\" src=\"\/wp-content\/uploads\/2019\/12\/7e5c21598d060f32fee5bd8c46851799.jpg\" style=\"display:block;margin: 0 auto;\" \/>, ob\u021binem: <img decoding=\"async\" alt=\"Transform\u0103m ecua\u021bia regresiei liniare \u00een form\u0103 matricial\u0103\" src=\"\/wp-content\/uploads\/2019\/12\/e274a91726e02c61ab075625c10373fa.jpg\" style=\"display:block;margin: 0 auto;\" \/><\/p>\n<p>3) deriv\u0103m dup\u0103 <img decoding=\"async\" alt=\"Transform\u0103m ecua\u021bia regresiei liniare \u00een form\u0103 matricial\u0103\" src=\"\/wp-content\/uploads\/2019\/12\/3aab71b33e32b9ce0dee2bcd94c32f14.jpg\" style=\"display:block;margin: 0 auto;\" \/>, ob\u021binem: <img decoding=\"async\" alt=\"Transform\u0103m ecua\u021bia regresiei liniare \u00een form\u0103 matricial\u0103\" src=\"\/wp-content\/uploads\/2019\/12\/abb3d3101af706ea22c98d3cde64dd03.jpg\" style=\"display:block;margin: 0 auto;\" \/><\/p>\n<p>La ie\u0219ire \u2014 vectorul promis de dimensiune <img decoding=\"async\" alt=\"Transform\u0103m ecua\u021bia regresiei liniare \u00een form\u0103 matricial\u0103\" src=\"\/wp-content\/uploads\/2019\/12\/2c50df26519c9627e07598715b132c72.jpg\" style=\"display:block;margin: 0 auto;\" \/>:<\/p>\n<p><img decoding=\"async\" alt=\"Transform\u0103m ecua\u021bia regresiei liniare \u00een form\u0103 matricial\u0103\" src=\"\/wp-content\/uploads\/2019\/12\/2312ff08655bffec14f9b969d984ffc2.jpg\" style=\"display:block;margin: 0 auto;\" \/><\/p>\n<p>Dac\u0103 ne uit\u0103m atent la vector, putem observa c\u0103 elementele din st\u00e2nga \u0219i cele corespunz\u0103toare din dreapta ale vectorului pot fi grupate \u00een a\u0219a fel \u00eenc\u00e2t, \u00een final, din vectorul prezentat s\u0103 putem desprinde vectorul <img decoding=\"async\" alt=\"Transform\u0103m ecua\u021bia regresiei liniare \u00een form\u0103 matricial\u0103\" src=\"\/wp-content\/uploads\/2019\/12\/c0dfe1a9a170f0893dd698aaf9ba9701.jpg\" style=\"display:block;margin: 0 auto;\" \/> de dimensiune <img decoding=\"async\" alt=\"Transform\u0103m ecua\u021bia regresiei liniare \u00een form\u0103 matricial\u0103\" src=\"\/wp-content\/uploads\/2019\/12\/a813524108fffe275a91842c79f8bda8.jpg\" style=\"display:block;margin: 0 auto;\" \/>. De exemplu, <img decoding=\"async\" alt=\"Transform\u0103m ecua\u021bia regresiei liniare \u00een form\u0103 matricial\u0103\" src=\"\/wp-content\/uploads\/2019\/12\/afdf43758b7923571006e70b1e5de969.jpg\" style=\"display:block;margin: 0 auto;\" \/> (elementul din st\u00e2nga al primei linii a vectorului) <img decoding=\"async\" alt=\"Transform\u0103m ecua\u021bia regresiei liniare \u00een form\u0103 matricial\u0103\" src=\"\/wp-content\/uploads\/2019\/12\/59d90b282c4c6461b19829a1191330d1.jpg\" style=\"display:block;margin: 0 auto;\" \/> (elementul din dreapta al primei linii a vectorului) poate fi reprezentat ca <img decoding=\"async\" alt=\"Transform\u0103m ecua\u021bia regresiei liniare \u00een form\u0103 matricial\u0103\" src=\"\/wp-content\/uploads\/2019\/12\/4b02117662623ab30edd4f1ea7b1be6c.jpg\" style=\"display:block;margin: 0 auto;\" \/>, iar <img decoding=\"async\" alt=\"Transform\u0103m ecua\u021bia regresiei liniare \u00een form\u0103 matricial\u0103\" src=\"\/wp-content\/uploads\/2019\/12\/219fe99a3a91a27a630bc99feccb3b55.jpg\" style=\"display:block;margin: 0 auto;\" \/> \u2014 ca <img decoding=\"async\" alt=\"Transform\u0103m ecua\u021bia regresiei liniare \u00een form\u0103 matricial\u0103\" src=\"\/wp-content\/uploads\/2019\/12\/3179a9d3ec503338de84e5acccc1b4cf.jpg\" style=\"display:block;margin: 0 auto;\" \/> \u0219i a\u0219a mai departe pentru fiecare linie. S\u0103 grup\u0103m:<\/p>\n<p><img decoding=\"async\" alt=\"Transform\u0103m ecua\u021bia regresiei liniare \u00een form\u0103 matricial\u0103\" src=\"\/wp-content\/uploads\/2019\/12\/07ca35f821e7d5af53d9e0209d007646.jpg\" style=\"display:block;margin: 0 auto;\" \/><\/p>\n<p>S\u0103 scoatem vectorul <img decoding=\"async\" alt=\"Transform\u0103m ecua\u021bia regresiei liniare \u00een form\u0103 matricial\u0103\" src=\"\/wp-content\/uploads\/2019\/12\/db4e2686a837b8e1c916c267df9faf46.jpg\" style=\"display:block;margin: 0 auto;\" \/> \u0219i la ie\u0219ire vom ob\u021bine:<\/p>\n<p><img decoding=\"async\" alt=\"Transform\u0103m ecua\u021bia regresiei liniare \u00een form\u0103 matricial\u0103\" src=\"\/wp-content\/uploads\/2019\/12\/54e24819b98dd56d38542e64d977a8bb.jpg\" style=\"display:block;margin: 0 auto;\" \/><\/p>\n<p>Acum, s\u0103 ne uit\u0103m la matricea ob\u021binut\u0103. Matricea reprezint\u0103 suma a dou\u0103 matrice <img decoding=\"async\" alt=\"Transform\u0103m ecua\u021bia regresiei liniare \u00een form\u0103 matricial\u0103\" src=\"\/wp-content\/uploads\/2019\/12\/2676c70f3a1eff48e5fc831a75440194.jpg\" style=\"display:block;margin: 0 auto;\" \/>:<\/p>\n<p><img decoding=\"async\" alt=\"Transform\u0103m ecua\u021bia regresiei liniare \u00een form\u0103 matricial\u0103\" src=\"\/wp-content\/uploads\/2019\/12\/68e49b0efd5cea53f742b7615c77d639.jpg\" style=\"display:block;margin: 0 auto;\" \/><\/p>\n<p>S\u0103 ne amintim c\u0103, cu pu\u021bin timp \u00een urm\u0103, am men\u021bionat o proprietate important\u0103 a matricei <img decoding=\"async\" alt=\"Transform\u0103m ecua\u021bia regresiei liniare \u00een form\u0103 matricial\u0103\" src=\"\/wp-content\/uploads\/2019\/12\/78f2e8bc8a9cb907a8317f5859036f71.jpg\" style=\"display:block;margin: 0 auto;\" \/> \u2014 este simetric\u0103. Pe baza acestei propriet\u0103\u021bi, putem afirma cu \u00eencredere c\u0103 expresia <img decoding=\"async\" alt=\"Transform\u0103m ecua\u021bia regresiei liniare \u00een form\u0103 matricial\u0103\" src=\"\/wp-content\/uploads\/2019\/12\/60fe7f4b8c34c3dc15fc7c87eb8aaf61.jpg\" style=\"display:block;margin: 0 auto;\" \/> este egal\u0103 cu <img decoding=\"async\" alt=\"Transform\u0103m ecua\u021bia regresiei liniare \u00een form\u0103 matricial\u0103\" src=\"\/wp-content\/uploads\/2019\/12\/65f7e5e232d7e3062e17dfbacc979ea7.jpg\" style=\"display:block;margin: 0 auto;\" \/>. Este u\u0219or de verificat, deschiz\u00e2nd produsul matricelor pe elemente. <img decoding=\"async\" alt=\"Transform\u0103m ecua\u021bia regresiei liniare \u00een form\u0103 matricial\u0103\" src=\"\/wp-content\/uploads\/2019\/12\/d4684c23eba61d9ddf5b05f59cc5b8f8.jpg\" style=\"display:block;margin: 0 auto;\" \/>. Nu vom face asta aici, cei dornici pot efectua verificarea singuri. <\/p>\n<p>S\u0103 ne \u00eentoarcem la expresia noastr\u0103. Dup\u0103 transform\u0103rile noastre, a rezultat exact a\u0219a cum voiam s\u0103-l vedem:<\/p>\n<p><img decoding=\"async\" alt=\"Transform\u0103m ecua\u021bia regresiei liniare \u00een form\u0103 matricial\u0103\" src=\"\/wp-content\/uploads\/2019\/12\/01349d32efab108cc0208b4ff732f5b4.jpg\" style=\"display:block;margin: 0 auto;\" \/><\/p>\n<p>Deci, am reu\u0219it cu prima diferen\u021biere. Trecem la a doua expresie.<\/p>\n<h3>Diferen\u021biere 2<\/h3>\n<p>\n<img decoding=\"async\" alt=\"Transform\u0103m ecua\u021bia regresiei liniare \u00een form\u0103 matricial\u0103\" src=\"\/wp-content\/uploads\/2019\/12\/5774072b36c83a305fd525b0fdffd59b.jpg\" style=\"display:block;margin: 0 auto;\" \/><\/p>\n<p>S\u0103 urm\u0103m calea trasat\u0103. Aceasta va fi mult mai scurt\u0103 dec\u00e2t precedentul, a\u0219a c\u0103 nu v\u0103 \u00eendep\u0103rta\u021bi prea mult de ecran.<\/p>\n<p>S\u0103 descompunem elementele vectorilor \u0219i matricea:<\/p>\n<p><img decoding=\"async\" alt=\"Transform\u0103m ecua\u021bia regresiei liniare \u00een form\u0103 matricial\u0103\" src=\"\/wp-content\/uploads\/2019\/12\/d6f1f2520c683c34c93947fdc1d74c7d.jpg\" style=\"display:block;margin: 0 auto;\" \/><\/p>\n<p><img decoding=\"async\" alt=\"Transform\u0103m ecua\u021bia regresiei liniare \u00een form\u0103 matricial\u0103\" src=\"\/wp-content\/uploads\/2019\/12\/a15e0b5d35596a2eb210d4467426ebda.jpg\" style=\"display:block;margin: 0 auto;\" \/><\/p>\n<p><img decoding=\"async\" alt=\"Transform\u0103m ecua\u021bia regresiei liniare \u00een form\u0103 matricial\u0103\" src=\"\/wp-content\/uploads\/2019\/12\/6f4f7ebd2f1e9cfa5d5f96c1b5298c3a.jpg\" style=\"display:block;margin: 0 auto;\" \/><\/p>\n<p>Pentru un timp vom elimina din calcule cifra dou\u0103 - aceasta nu are un rol semnificativ, o vom aduce \u00eenapoi mai t\u00e2rziu. Vom \u00eenmul\u021bi vectorii cu matricea. \u00cen primul r\u00e2nd, vom \u00eenmul\u021bi matricea <img decoding=\"async\" alt=\"Transform\u0103m ecua\u021bia regresiei liniare \u00een form\u0103 matricial\u0103\" src=\"\/wp-content\/uploads\/2019\/12\/8f09bef67ae021b46e878b4ae32e122d.jpg\" style=\"display:block;margin: 0 auto;\" \/> cu vectorul <img decoding=\"async\" alt=\"Transform\u0103m ecua\u021bia regresiei liniare \u00een form\u0103 matricial\u0103\" src=\"\/wp-content\/uploads\/2019\/12\/1db14935f79198aebd01b753ff3b697b.jpg\" style=\"display:block;margin: 0 auto;\" \/>, aici nu avem nicio restric\u021bie. Vom ob\u021bine un vector de dimensiune <img decoding=\"async\" alt=\"Transform\u0103m ecua\u021bia regresiei liniare \u00een form\u0103 matricial\u0103\" src=\"\/wp-content\/uploads\/2019\/12\/a962a34aa48787f3a5befd3bbfd0ed8c.jpg\" style=\"display:block;margin: 0 auto;\" \/>:<\/p>\n<p><img decoding=\"async\" alt=\"Transform\u0103m ecua\u021bia regresiei liniare \u00een form\u0103 matricial\u0103\" src=\"\/wp-content\/uploads\/2019\/12\/5612d20b4c5523cf0e34c409ba424d0e.jpg\" style=\"display:block;margin: 0 auto;\" \/><\/p>\n<p>Vom efectua urm\u0103toarea ac\u021biune - vom \u00eenmul\u021bi vectorul <img decoding=\"async\" alt=\"Transform\u0103m ecua\u021bia regresiei liniare \u00een form\u0103 matricial\u0103\" src=\"\/wp-content\/uploads\/2019\/12\/6b8566eabdfe892618ca9c26a9ac4faf.jpg\" style=\"display:block;margin: 0 auto;\" \/> cu vectorul ob\u021binut. La ie\u0219ire ne va a\u0219tepta un num\u0103r:<\/p>\n<p><img decoding=\"async\" alt=\"Transform\u0103m ecua\u021bia regresiei liniare \u00een form\u0103 matricial\u0103\" src=\"\/wp-content\/uploads\/2019\/12\/50698c81b18f0273550fdf34ba6626fd.jpg\" style=\"display:block;margin: 0 auto;\" \/><\/p>\n<p>Exact acesta \u00eel vom diferen\u021bia. La ie\u0219ire vom ob\u021bine un vector de dimensiune <img decoding=\"async\" alt=\"Transform\u0103m ecua\u021bia regresiei liniare \u00een form\u0103 matricial\u0103\" src=\"\/wp-content\/uploads\/2019\/12\/73a13a4ef020e6d67551dd2dd66544e0.jpg\" style=\"display:block;margin: 0 auto;\" \/>:<\/p>\n<p><img decoding=\"async\" alt=\"Transform\u0103m ecua\u021bia regresiei liniare \u00een form\u0103 matricial\u0103\" src=\"\/wp-content\/uploads\/2019\/12\/d63b2486e7186c4a7e11d21a769c0aa5.jpg\" style=\"display:block;margin: 0 auto;\" \/><\/p>\n<p>\u00ce\u021bi pare cunoscut? Precis! Aceasta este produsul matricei <img decoding=\"async\" alt=\"Transform\u0103m ecua\u021bia regresiei liniare \u00een form\u0103 matricial\u0103\" src=\"\/wp-content\/uploads\/2019\/12\/67146a714d3889b5e9fcaf19e7e2407d.jpg\" style=\"display:block;margin: 0 auto;\" \/> cu vectorul <img decoding=\"async\" alt=\"Transform\u0103m ecua\u021bia regresiei liniare \u00een form\u0103 matricial\u0103\" src=\"\/wp-content\/uploads\/2019\/12\/8c3e90abfd5f687c1b5852980c65b0a8.jpg\" style=\"display:block;margin: 0 auto;\" \/>. <\/p>\n<p>Astfel, a doua diferen\u021biere a fost finalizat\u0103 cu succes.<\/p>\n<h2>\u00cen concluzie<\/h2>\n<p>\nAcum \u0219tim cum s-a ob\u021binut egalitatea <img decoding=\"async\" alt=\"Transform\u0103m ecua\u021bia regresiei liniare \u00een form\u0103 matricial\u0103\" src=\"\/wp-content\/uploads\/2019\/12\/8dd8f1dae6629da173b7e71f569a18a1.jpg\" style=\"display:block;margin: 0 auto;\" \/>.<\/p>\n<p>\u00cen final, s\u0103 descriem o cale rapid\u0103 de transformare a formelor de baz\u0103.<\/p>\n<p><b>S\u0103 evalu\u0103m calitatea modelului conform metodei celor mai mici p\u0103trate:<\/b><br \/>\n<img decoding=\"async\" alt=\"Transform\u0103m ecua\u021bia regresiei liniare \u00een form\u0103 matricial\u0103\" src=\"\/wp-content\/uploads\/2019\/12\/e4f1216018252f75c89bb283dbd165c0.jpg\" style=\"display:block;margin: 0 auto;\" \/><\/p>\n<p><img decoding=\"async\" alt=\"Transform\u0103m ecua\u021bia regresiei liniare \u00een form\u0103 matricial\u0103\" src=\"\/wp-content\/uploads\/2019\/12\/18c71ca2156610ab70a7776e0d660560.jpg\" style=\"display:block;margin: 0 auto;\" \/><\/p>\n<p><b>Diferen\u021biem expresia ob\u021binut\u0103:<\/b><br \/>\n<img decoding=\"async\" alt=\"Transform\u0103m ecua\u021bia regresiei liniare \u00een form\u0103 matricial\u0103\" src=\"\/wp-content\/uploads\/2019\/12\/ef0989a3750d444845d96cd7b4c26568.jpg\" style=\"display:block;margin: 0 auto;\" \/> <img decoding=\"async\" alt=\"Transform\u0103m ecua\u021bia regresiei liniare \u00een form\u0103 matricial\u0103\" src=\"\/wp-content\/uploads\/2019\/12\/2f0788dce58d900c2b11b826943785fb.jpg\" style=\"display:block;margin: 0 auto;\" \/><\/p>\n<p><img decoding=\"async\" alt=\"Transform\u0103m ecua\u021bia regresiei liniare \u00een form\u0103 matricial\u0103\" src=\"\/wp-content\/uploads\/2019\/12\/fc6183460bbf00ef4c040502055e032a.jpg\" style=\"display:block;margin: 0 auto;\" \/><\/p>\n<h2>Literatur\u0103<\/h2>\n<p>\n<u><b>Sursele online:<\/b><\/u><\/p>\n<p>1) <i><noindex><a rel=\"nofollow\" href=\"https:\/\/habr.com\/ru\/post\/278513\/\">habr.com\/ru\/post\/278513<\/a><\/noindex><\/i><br \/>\n2) <i><noindex><a rel=\"nofollow\" href=\"https:\/\/habr.com\/ru\/company\/ods\/blog\/322076\/\">habr.com\/ru\/company\/ods\/blog\/322076<\/a><\/noindex><\/i><br \/>\n3) <i><noindex><a rel=\"nofollow\" href=\"https:\/\/habr.com\/ru\/post\/307004\/\">habr.com\/ru\/post\/307004<\/a><\/noindex><\/i><br \/>\n4) <i><noindex><a rel=\"nofollow\" href=\"http:\/\/nabatchikov.com\/blog\/view\/matrix_der\">nabatchikov.com\/blog\/view\/matrix_der<\/a><\/noindex><\/i><\/p>\n<p><u><b>C\u0103r\u021bi, culegeri de probleme:<\/b><\/u><\/p>\n<p>1) Curs de predare a matematicii superioare: curs complet \/ D.T. Pisemenyi \u2013 edi\u021bia a 4-a. \u2013 Moscova: Iris-Press, 2006<br \/>\n2) Analiza regresional\u0103 aplicat\u0103 \/ N. Draper, G. Smith \u2013 edi\u021bia a 2-a. \u2013 Moscova: Finan\u021be \u0219i statistic\u0103, 1986 (traducere din englez\u0103)<br \/>\n3) Probleme de solu\u021bionare a ecua\u021biilor matrice: <br \/>\n<i><noindex><a rel=\"nofollow\" href=\"https:\/\/function-x.ru\/matrix_equations.html\">function-x.ru\/matrix_equations.html<\/a><\/noindex><br \/>\n<noindex><a rel=\"nofollow\" href=\"http:\/\/mathprofi.ru\/deistviya_s_matricami.html\">mathprofi.ru\/deistviya_s_matricami.html<\/a><\/noindex><\/i><br \/>\n<br \/>Sursa: <a content=\"nofollow\" rel=\"nofollow\" href=\"https:\/\/habr.com\/ru\/post\/479398\/\">habr.com<\/a><\/p>","protected":false,"gt_translate_keys":[{"key":"rendered","format":"html"}]},"excerpt":{"rendered":"<p>\u0426\u0435\u043b\u044c \u0441\u0442\u0430\u0442\u044c\u0438 \u2014 \u043e\u043a\u0430\u0437\u0430\u043d\u0438\u0435 \u043f\u043e\u0434\u0434\u0435\u0440\u0436\u043a\u0438 \u043d\u0430\u0447\u0438\u043d\u0430\u044e\u0449\u0438\u043c \u0434\u0430\u0442\u0430\u0441\u0430\u0439\u043d\u0442\u0438\u0441\u0442\u0430\u043c. \u0412 \u043f\u0440\u0435\u0434\u044b\u0434\u0443\u0449\u0435\u0439 \u0441\u0442\u0430\u0442\u044c\u0435 \u043c\u044b \u043d\u0430 \u043f\u0430\u043b\u044c\u0446\u0430\u0445 \u0440\u0430\u0437\u043e\u0431\u0440\u0430\u043b\u0438 \u0442\u0440\u0438 \u0441\u043f\u043e\u0441\u043e\u0431\u0430 \u0440\u0435\u0448\u0435\u043d\u0438\u044f \u0443\u0440\u0430\u0432\u043d\u0435\u043d\u0438\u044f \u043b\u0438\u043d\u0435\u0439\u043d\u043e\u0439 \u0440\u0435\u0433\u0440\u0435\u0441\u0441\u0438\u0438: \u0430\u043d\u0430\u043b\u0438\u0442\u0438\u0447\u0435\u0441\u043a\u043e\u0435 \u0440\u0435\u0448\u0435\u043d\u0438\u0435, \u0433\u0440\u0430\u0434\u0438\u0435\u043d\u0442\u043d\u044b\u0439 \u0441\u043f\u0443\u0441\u043a, \u0441\u0442\u043e\u0445\u0430\u0441\u0442\u0438\u0447\u0435\u0441\u043a\u0438\u0439 \u0433\u0440\u0430\u0434\u0438\u0435\u043d\u0442\u043d\u044b\u0439 \u0441\u043f\u0443\u0441\u043a. \u0422\u043e\u0433\u0434\u0430 \u0434\u043b\u044f \u0430\u043d\u0430\u043b\u0438\u0442\u0438\u0447\u0435\u0441\u043a\u043e\u0433\u043e \u0440\u0435\u0448\u0435\u043d\u0438\u044f \u043c\u044b \u043f\u0440\u0438\u043c\u0435\u043d\u0438\u043b\u0438 \u0444\u043e\u0440\u043c\u0443\u043b\u0443 . \u0412 \u044d\u0442\u043e\u0439 \u0441\u0442\u0430\u0442\u044c\u0435, \u043a\u0430\u043a \u0441\u043b\u0435\u0434\u0443\u0435\u0442 \u0438\u0437 \u0437\u0430\u0433\u043e\u043b\u043e\u0432\u043a\u0430, \u043c\u044b \u043e\u0431\u043e\u0441\u043d\u0443\u0435\u043c \u043f\u0440\u0438\u043c\u0435\u043d\u0435\u043d\u0438\u0435 \u0434\u0430\u043d\u043d\u043e\u0439 \u0444\u043e\u0440\u043c\u0443\u043b\u044b \u0438\u043b\u0438 \u0434\u0440\u0443\u0433\u0438\u043c\u0438 \u0441\u043b\u043e\u0432\u0430\u043c\u0438, \u0441\u0430\u043c\u043e\u0441\u0442\u043e\u044f\u0442\u0435\u043b\u044c\u043d\u043e \u0435\u0435 \u0432\u044b\u0432\u0435\u0434\u0435\u043c. \u041f\u043e\u0447\u0435\u043c\u0443 \u0438\u043c\u0435\u0435\u0442 [&hellip;]<\/p>\n","protected":false,"gt_translate_keys":[{"key":"rendered","format":"html"}]},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[702],"tags":[],"class_list":["post-53777","post","type-post","status-publish","format-standard","hentry","category-news"],"aioseo_notices":[],"aioseo_head":"\n\t\t<!-- All in One SEO 5.0.2.1 - aioseo.com -->\n\t<meta name=\"description\" content=\"\u0426\u0435\u043b\u044c \u0441\u0442\u0430\u0442\u044c\u0438 \u2014 \u043e\u043a\u0430\u0437\u0430\u043d\u0438\u0435 \u043f\u043e\u0434\u0434\u0435\u0440\u0436\u043a\u0438.\" \/>\n\t<meta name=\"robots\" content=\"max-image-preview:large\" \/>\n\t<meta name=\"author\" content=\"Yuri Gagarin\"\/>\n\t<link rel=\"canonical\" href=\"https:\/\/prohoster.info\/ro\/blog\/news\/privodim-uravnenie-linejnoj-regressii-v-matrichnyj-vid\" \/>\n\t<meta name=\"generator\" 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