{"id":318,"date":"2023-08-02T13:13:50","date_gmt":"2023-08-02T13:13:50","guid":{"rendered":"https:\/\/statorials.org\/pl\/rownanie-regresji\/"},"modified":"2023-08-02T13:13:50","modified_gmt":"2023-08-02T13:13:50","slug":"rownanie-regresji","status":"publish","type":"post","link":"https:\/\/statorials.org\/pl\/rownanie-regresji\/","title":{"rendered":"R\u00f3wnanie regresji"},"content":{"rendered":"<p>W tym artykule wyja\u015bniono, czym jest r\u00f3wnanie regresji i do czego si\u0119 go stosuje. Podobnie dowiesz si\u0119, jak znale\u017a\u0107 r\u00f3wnanie regresji, rozwi\u0105zane \u0107wiczenie i wreszcie kalkulator online umo\u017cliwiaj\u0105cy obliczenie r\u00f3wnania regresji dla dowolnego zbioru danych. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"%c2%bfcual-es-la-ecuacion-de-regresion\"><\/span> Co to jest r\u00f3wnanie regresji?<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> <strong>R\u00f3wnanie regresji<\/strong> to r\u00f3wnanie, kt\u00f3re najlepiej pasuje do wykresu punktowego, co oznacza, \u017ce r\u00f3wnanie regresji jest najlepszym przybli\u017ceniem zbioru danych.<\/p>\n<p> R\u00f3wnanie regresji ma posta\u0107 y=\u03b2 <sub>0<\/sub> + \u03b2 <sub>1<\/sub> x, gdzie \u03b2 <sub>0<\/sub> jest sta\u0142\u0105 r\u00f3wnania, a \u03b2 <sub>1<\/sub> jest nachyleniem r\u00f3wnania.<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/statorials.org\/wp-content\/ql-cache\/quicklatex.com-83e43f626a469e9de3d5ecfed9a216ac_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y=\\beta_0+\\beta_1x\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"100\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p> Je\u015bli spojrzysz na r\u00f3wnanie regresji, jest to r\u00f3wnanie linii. Oznacza to, \u017ce zwi\u0105zek pomi\u0119dzy zmienn\u0105 niezale\u017cn\u0105 X i zmienn\u0105 zale\u017cn\u0105 Y jest modelowany jako zale\u017cno\u015b\u0107 liniowa, poniewa\u017c prosta reprezentuje zale\u017cno\u015b\u0107 liniow\u0105.<\/p>\n<p> Zatem r\u00f3wnanie regresji pozwala nam matematycznie powi\u0105za\u0107 zmienn\u0105 niezale\u017cn\u0105 i zmienn\u0105 zale\u017cn\u0105 zbioru danych. Chocia\u017c r\u00f3wnanie regresji na og\u00f3\u0142 nie jest w stanie precyzyjnie okre\u015bli\u0107 warto\u015bci ka\u017cdej obserwacji, niemniej jednak s\u0142u\u017cy do uzyskania przybli\u017cenia jej warto\u015bci. <\/p>\n<figure class=\"wp-block-image aligncenter size-full is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/statorials.org\/wp-content\/uploads\/2023\/08\/ligne-de-regression.png\" alt=\"r\u00f3wnanie regresji\" class=\"wp-image-6708\" width=\"235\" height=\"219\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<p> Jak wida\u0107 na poprzednim wykresie, r\u00f3wnanie regresji pomaga nam zobaczy\u0107 trend zbioru danych oraz rodzaj zwi\u0105zku pomi\u0119dzy zmienn\u0105 niezale\u017cn\u0105 a zmienn\u0105 zale\u017cn\u0105. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"como-calcular-la-ecuacion-de-regresion\"><\/span> Jak obliczy\u0107 r\u00f3wnanie regresji<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> <strong>Wzory do obliczania wsp\u00f3\u0142czynnik\u00f3w prostego r\u00f3wnania regresji liniowej<\/strong> s\u0105 nast\u0119puj\u0105ce:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/statorials.org\/wp-content\/ql-cache\/quicklatex.com-459281504d26f92756115054ef567021_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{array}{c}\\beta_1=\\cfrac{\\displaystyle \\sum_{i=1}^n (x_i-\\overline{x})(y_i-\\overline{y})}{\\displaystyle \\sum_{i=1}^n (x_i-\\overline{x})^2}\\\\[12ex]\\beta_0=\\overline{y}-\\beta_1\\overline{x}\\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"176\" width=\"192\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p style=\"margin-bottom:5px\"> Z\u0142oto:<\/p>\n<ul style=\"color:#FF8A05; font-weight: bold;\">\n<li style=\"margin-bottom:5px\">\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/statorials.org\/wp-content\/ql-cache\/quicklatex.com-c5ba513cc7e504bc674f76afa70a3442_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\beta_0\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"17\" style=\"vertical-align: -4px;\"><\/p>\n<p> jest sta\u0142\u0105 r\u00f3wnania regresji.<\/li>\n<li style=\"margin-bottom:5px\">\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/statorials.org\/wp-content\/ql-cache\/quicklatex.com-dff50ab66b848b910ea781069cba1094_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\beta_1\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"16\" style=\"vertical-align: -4px;\"><\/p>\n<p> jest nachyleniem r\u00f3wnania regresji.<\/li>\n<li style=\"margin-bottom:5px\">\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/statorials.org\/wp-content\/ql-cache\/quicklatex.com-dad27a9703483183e1afd245f5232b83_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x_i\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"15\" style=\"vertical-align: -3px;\"><\/p>\n<p> jest warto\u015bci\u0105 zmiennej niezale\u017cnej X danych i.<\/li>\n<li style=\"margin-bottom:5px\">\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/statorials.org\/wp-content\/ql-cache\/quicklatex.com-761333a1d61654bd1cb5c7224b0d1994_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y_i\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: -4px;\"><\/p>\n<p> jest warto\u015bci\u0105 zmiennej zale\u017cnej Y danych i.<\/li>\n<li style=\"margin-bottom:5px\">\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/statorials.org\/wp-content\/ql-cache\/quicklatex.com-a39858a792fb4fe9a3173e004701f2a7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\overline{x}\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"11\" style=\"vertical-align: 0px;\"><\/p>\n<p> jest \u015bredni\u0105 warto\u015bci zmiennej niezale\u017cnej<\/li>\n<li style=\"margin-bottom:5px\">\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/statorials.org\/wp-content\/ql-cache\/quicklatex.com-ed5becac4ccb36fec040f449ba9fa52d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\overline{y}\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"10\" style=\"vertical-align: -4px;\"><\/p>\n<p> jest \u015bredni\u0105 warto\u015bci zmiennej zale\u017cnej Y. <\/li>\n<\/ul>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"ejemplo-del-calculo-de-la-ecuacion-de-regresion\"><\/span> Przyk\u0142ad obliczenia r\u00f3wnania regresji<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<ul>\n<li> Po zdaniu egzaminu statystycznego pi\u0119ciu student\u00f3w zosta\u0142o zapytanych, ile godzin nauki sp\u0119dzili na egzaminie, dane przedstawia poni\u017csza tabela. Na podstawie zebranych danych statystycznych oblicz r\u00f3wnanie regresji, aby liniowo powi\u0105za\u0107 liczb\u0119 godzin nauki z uzyskan\u0105 ocen\u0105. Nast\u0119pnie ustal, jak\u0105 ocen\u0119 otrzyma ucze\u0144, kt\u00f3ry uczy\u0142 si\u0119 8 godzin. <\/li>\n<\/ul>\n<figure class=\"wp-block-image aligncenter size-full is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/statorials.org\/wp-content\/uploads\/2023\/08\/donnees-de-regression-lineaire-simple.png\" alt=\"\" class=\"wp-image-6503\" width=\"213\" height=\"220\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<p> Aby znale\u017a\u0107 r\u00f3wnanie regresji dla przyk\u0142adowych danych, musimy wyznaczy\u0107 wsp\u00f3\u0142czynniki b <sub>0<\/sub> i b <sub>1<\/sub> r\u00f3wnania i w tym celu skorzysta\u0107 ze wzor\u00f3w przedstawionych w powy\u017cszej sekcji.<\/p>\n<p> Aby jednak zastosowa\u0107 wzory na r\u00f3wnanie regresji liniowej, musimy najpierw obliczy\u0107 \u015bredni\u0105 zmiennej niezale\u017cnej i \u015bredni\u0105 zmiennej zale\u017cnej:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/statorials.org\/wp-content\/ql-cache\/quicklatex.com-7a7aa6f1f20fa4ff0d61a2ad0dd2ea1f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{array}{c}\\overline{x}=\\cfrac{11+5+10+12+7}{5}=9\\\\[4ex]\\overline{y}=\\cfrac{7+4+5+8+6}{5}=6\\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"105\" width=\"226\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Teraz, gdy znamy \u015brednie zmiennych, obliczamy wsp\u00f3\u0142czynnik \u03b2 <sub>1<\/sub> modelu, korzystaj\u0105c z odpowiedniego wzoru:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/statorials.org\/wp-content\/ql-cache\/quicklatex.com-d9407478d6d590c0d970ac41963f6fbe_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{array}{c}\\beta_1=\\cfrac{\\displaystyle \\sum_{i=1}^n (x_i-\\overline{x})(y_i-\\overline{y})}{\\displaystyle \\sum_{i=1}^n (x_i-\\overline{x})^2}\\\\[10ex] \\beta_1=\\cfrac{\\begin{array}{c}(11-9)(7-6)+(5-9)(4-6)+(10-9)(5-6)+\\\\+(12-9)(8-6)+(7-9)(6-6)\\end{array}}{(11-9)^2+(5-9)^2+(10-9)^2+(12-9)^2+(7-9)^2}\\\\[6ex]\\beta_1=0,4412\\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"264\" width=\"461\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Na koniec obliczamy wsp\u00f3\u0142czynnik \u03b2 <sub>0<\/sub> modelu, korzystaj\u0105c z odpowiedniego wzoru:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/statorials.org\/wp-content\/ql-cache\/quicklatex.com-a8872fb5b9904b106d02b504ae36bb92_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{array}{l}\\beta_0=\\overline{y}-\\beta_1\\overline{x}\\\\[3ex]\\beta_0=6-0,4412\\cdot 9 \\\\[3ex]\\beta_0=2,0294\\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"107\" width=\"146\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Kr\u00f3tko m\u00f3wi\u0105c, r\u00f3wnanie linii regresji liniowej problemu wygl\u0105da nast\u0119puj\u0105co:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/statorials.org\/wp-content\/ql-cache\/quicklatex.com-7097952b105295743a390d0332e7a2b3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y=2,0294+0,4412x\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"170\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p> Poni\u017cej mo\u017cesz zobaczy\u0107 graficzn\u0105 reprezentacj\u0119 przyk\u0142adowych danych wraz z prostym r\u00f3wnaniem modelu regresji liniowej: <\/p>\n<figure class=\"wp-block-image aligncenter size-full is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/statorials.org\/wp-content\/uploads\/2023\/08\/exemple-de-regression-lineaire-simple.png\" alt=\"przyk\u0142ad linii regresji liniowej\" class=\"wp-image-6517\" width=\"454\" height=\"304\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<p> Po obliczeniu r\u00f3wnania regresji, aby przewidzie\u0107 ocen\u0119, jak\u0105 uzyska student, kt\u00f3ry uczy\u0142 si\u0119 8 godzin, wystarczy podstawi\u0107 t\u0119 warto\u015b\u0107 do otrzymanego r\u00f3wnania regresji:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/statorials.org\/wp-content\/ql-cache\/quicklatex.com-d1ef36c064448262d7c58b7f51b215c4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y=2,0294+0,4412\\cdot 8=5,56\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"239\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p> Zatem zgodnie z przeprowadzonym modelem regresji liniowej, je\u015bli student uczy\u0142 si\u0119 osiem godzin, uzyska na egzaminie not\u0119 5,56. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"calculadora-de-la-ecuacion-de-regresion\"><\/span> Kalkulator r\u00f3wna\u0144 regresji<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Pod\u0142\u0105cz przyk\u0142adowe dane do poni\u017cszego kalkulatora, aby obliczy\u0107 r\u00f3wnanie regresji. Nale\u017cy rozdzieli\u0107 pary danych tak, aby w pierwszym polu znajdowa\u0142y si\u0119 tylko warto\u015bci zmiennej niezale\u017cnej X, a w drugim polu znajdowa\u0142y si\u0119 wy\u0142\u0105cznie warto\u015bci zmiennej zale\u017cnej Y.<\/p>\n<p> Dane nale\u017cy oddzieli\u0107 spacj\u0105 i wprowadzi\u0107 z u\u017cyciem kropki jako separatora dziesi\u0119tnego.<\/p>\n<form action=\"\" method=\"post\">\n<ul>\n<li> Niezale\u017cna zmienna <\/li>\n<\/ul>\n<p><textarea name=\"datosX\" style=\"border:1.5px solid #4FC3F7; border-radius:15px;\" placeholder=\"1 4 8 5 7.2 9 ...\" required=\"\" oninvalid=\"this.setCustomValidity('Introduce los datos de la variable explicativa aqu\u00ed')\" oninput=\"this.setCustomValidity('')\"><\/textarea><\/p>\n<ul style=\"margin-top:25px\">\n<li> Zmienna zale\u017cna Y: <\/li>\n<\/ul>\n<p><textarea name=\"datosY\" style=\"border:1.5px solid #4FC3F7; border-radius:15px;\" placeholder=\"2 5 7 3 2 1 ...\" required=\"\" oninvalid=\"this.setCustomValidity('Introduce los datos de la variable respuesta aqu\u00ed')\" oninput=\"this.setCustomValidity('')\"><\/textarea><\/p>\n<div style=\"text-align:center\"><input align=\"center\" style=\"border-radius:30px; margin: 20px\" type=\"submit\" name=\"submit\" value=\"Oblicz\"><\/div>\n<\/form>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"ecuacion-de-regresion-lineal-multiple\"><\/span> R\u00f3wnanie wielokrotnej regresji liniowej<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> W\u0142a\u015bnie widzieli\u015bmy, czym jest proste r\u00f3wnanie regresji liniowej, jednak model regresji mo\u017ce by\u0107 r\u00f3wnie\u017c modelem wielokrotnej regresji liniowej, kt\u00f3ry obejmuje dwie lub wi\u0119cej niezale\u017cnych zmiennych. Zatem wielokrotna regresja liniowa umo\u017cliwia liniowe powi\u0105zanie kilku zmiennych obja\u015bniaj\u0105cych ze zmienn\u0105 odpowiedzi.<\/p>\n<p> <strong>R\u00f3wnanie modelu wielokrotnej regresji liniowej<\/strong> wygl\u0105da nast\u0119puj\u0105co:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/statorials.org\/wp-content\/ql-cache\/quicklatex.com-fd3ba8386b5954b654ca555774108ac0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y=\\beta_0+\\beta_1 x_1+\\beta_2 x_2+\\dots+\\beta_m x_m+\\varepsilon\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"305\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p style=\"margin-bottom:5px\"> Z\u0142oto: <\/p>\n<ul style=\"color:#FF8A05; font-weight: bold;\">\n<li style=\"margin-bottom:5px\">\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/statorials.org\/wp-content\/ql-cache\/quicklatex.com-38461fc041e953482219abf5d4cce1cb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: -4px;\"><\/p>\n<p> jest zmienn\u0105 zale\u017cn\u0105.<\/li>\n<li style=\"margin-bottom:5px\">\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/statorials.org\/wp-content\/ql-cache\/quicklatex.com-dad27a9703483183e1afd245f5232b83_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x_i\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"15\" style=\"vertical-align: -3px;\"><\/p>\n<p> jest zmienn\u0105 niezale\u017cn\u0105 i.<\/li>\n<li style=\"margin-bottom:5px\">\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/statorials.org\/wp-content\/ql-cache\/quicklatex.com-c5ba513cc7e504bc674f76afa70a3442_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\beta_0\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"17\" style=\"vertical-align: -4px;\"><\/p>\n<p> jest sta\u0142\u0105 r\u00f3wnania regresji liniowej.<\/li>\n<li style=\"margin-bottom:5px\">\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/statorials.org\/wp-content\/ql-cache\/quicklatex.com-ff540c55c6ee8f10a1dab8e2422947ab_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\beta_i\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"15\" style=\"vertical-align: -4px;\"><\/p>\n<p> jest wsp\u00f3\u0142czynnikiem regresji powi\u0105zanym ze zmienn\u0105<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/statorials.org\/wp-content\/ql-cache\/quicklatex.com-dad27a9703483183e1afd245f5232b83_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x_i\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"15\" style=\"vertical-align: -3px;\"><\/p>\n<p> .<\/li>\n<li style=\"margin-bottom:5px\">\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/statorials.org\/wp-content\/ql-cache\/quicklatex.com-29b8f7fac5f2df4b101dff63e95516c5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{\\varepsilon}\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"8\" style=\"vertical-align: 0px;\"><\/p>\n<p> jest b\u0142\u0119dem lub reszt\u0105, to znaczy r\u00f3\u017cnic\u0105 mi\u0119dzy warto\u015bci\u0105 obserwowan\u0105 a warto\u015bci\u0105 oszacowan\u0105 przez model.<\/li>\n<li style=\"margin-bottom:5px\">\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/statorials.org\/wp-content\/ql-cache\/quicklatex.com-fdc40b8ad1cdad0aab9d632215459d28_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"m\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"15\" style=\"vertical-align: 0px;\"><\/p>\n<p> jest ca\u0142kowit\u0105 liczb\u0105 zmiennych w modelu.<\/li>\n<\/ul>\n<p> Je\u015bli wi\u0119c mamy pr\u00f3bk\u0119 o \u0142\u0105cznej warto\u015bci<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/statorials.org\/wp-content\/ql-cache\/quicklatex.com-ec4217f4fa5fcd92a9edceba0e708cf7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"n\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"11\" style=\"vertical-align: 0px;\"><\/p>\n<p> obserwacji mo\u017cemy przedstawi\u0107 model wielokrotnej regresji liniowej w postaci macierzowej:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/statorials.org\/wp-content\/ql-cache\/quicklatex.com-5848686c8ed0857f16e7e24e2a31024e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{pmatrix}y_1\\\\y_2\\\\\\vdots\\\\y_n\\end{pmatrix}=\\begin{pmatrix}1&amp;x_{11}&amp;\\dots&amp;x_{1m}\\\\1&amp;x_{21}&amp;\\dots&amp;x_{2m}\\\\ \\vdots&amp;\\vdots&amp;\\ddots&amp;\\vdots\\\\1&amp;x_{n1}&amp;\\dots&amp;x_{nm}\\end{pmatrix}\\cdot\\begin{pmatrix}\\beta_0\\\\\\beta_1\\\\\\vdots\\\\\\beta_m\\end{pmatrix}+\\begin{pmatrix}\\varepsilon_1\\\\\\varepsilon_2\\\\\\vdots\\\\\\varepsilon_n\\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"96\" width=\"370\" style=\"vertical-align: -43px;\"><\/p>\n<\/p>\n<p> Powy\u017csze wyra\u017cenie macierzowe mo\u017cna przepisa\u0107, przypisuj\u0105c liter\u0119 do ka\u017cdej macierzy:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/statorials.org\/wp-content\/ql-cache\/quicklatex.com-7614ddbb78ced2e2b8b6c7642d9969c3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"Y=X\\beta+\\varepsilon\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"95\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p> Zatem, stosuj\u0105c kryterium najmniejszych kwadrat\u00f3w, mo\u017cemy otrzyma\u0107 <strong>wz\u00f3r na oszacowanie wsp\u00f3\u0142czynnik\u00f3w r\u00f3wnania regresji wielokrotnej liniowej<\/strong> :<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/statorials.org\/wp-content\/ql-cache\/quicklatex.com-b6ef097cee722e7355fa4eb77b7ea3e5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\widehat{\\beta}=\\left(X^tX\\right)^{-1}X^tY\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"146\" style=\"vertical-align: -7px;\"><\/p>\n<\/p>\n<p> Stosowanie tej formu\u0142y jest jednak bardzo pracoch\u0142onne i czasoch\u0142onne, dlatego w praktyce zaleca si\u0119 stosowanie program\u00f3w komputerowych (takich jak Minitab czy Excel), kt\u00f3re pozwalaj\u0105 znacznie szybciej stworzy\u0107 model regresji wielokrotnej. <\/p>\n<div style=\"background-color:#FFFDE7; padding-top: 10px; padding-bottom: 10px; padding-right: 20px; padding-left: 30px; border: 2.5px dashed #FFB74D; border-radius:20px;\"> <span style=\"color:#ff951b\">\u27a4<\/span> <strong>Zobacz:<\/strong> <a href=\"https:\/\/statorials.org\/pl\/wielokrotna-regresja-liniowa-1\/\">Co to jest wielokrotna regresja liniowa?<\/a><\/div>\n","protected":false},"excerpt":{"rendered":"<p>W tym artykule wyja\u015bniono, czym jest r\u00f3wnanie regresji i do czego si\u0119 go stosuje. Podobnie dowiesz si\u0119, jak znale\u017a\u0107 r\u00f3wnanie regresji, rozwi\u0105zane \u0107wiczenie i wreszcie kalkulator online umo\u017cliwiaj\u0105cy obliczenie r\u00f3wnania regresji dla dowolnego zbioru danych. Co to jest r\u00f3wnanie regresji? R\u00f3wnanie regresji to r\u00f3wnanie, kt\u00f3re najlepiej pasuje do wykresu punktowego, co oznacza, \u017ce r\u00f3wnanie regresji [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[6],"tags":[],"class_list":["post-318","post","type-post","status-publish","format-standard","hentry","category-statystyka"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v21.5 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>\u25b7 R\u00f3wnanie regresji<\/title>\n<meta name=\"description\" content=\"Tutaj znajdziesz czym jest r\u00f3wnanie regresji, jak jest obliczane (wz\u00f3r), rozwi\u0105zane \u0107wiczenie oraz kalkulator r\u00f3wnania regresji.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/statorials.org\/pl\/rownanie-regresji\/\" \/>\n<meta property=\"og:locale\" content=\"pl_PL\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"\u25b7 R\u00f3wnanie regresji\" \/>\n<meta property=\"og:description\" content=\"Tutaj znajdziesz czym jest r\u00f3wnanie regresji, jak jest obliczane (wz\u00f3r), rozwi\u0105zane \u0107wiczenie oraz kalkulator r\u00f3wnania regresji.\" \/>\n<meta property=\"og:url\" content=\"https:\/\/statorials.org\/pl\/rownanie-regresji\/\" \/>\n<meta property=\"og:site_name\" content=\"Statorials\" \/>\n<meta property=\"article:published_time\" content=\"2023-08-02T13:13:50+00:00\" \/>\n<meta property=\"og:image\" content=\"https:\/\/statorials.org\/wp-content\/ql-cache\/quicklatex.com-83e43f626a469e9de3d5ecfed9a216ac_l3.png\" \/>\n<meta name=\"author\" content=\"Benjamin Anderson\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:label1\" content=\"Napisane przez\" \/>\n\t<meta name=\"twitter:data1\" content=\"Benjamin Anderson\" \/>\n\t<meta name=\"twitter:label2\" content=\"Szacowany czas czytania\" \/>\n\t<meta name=\"twitter:data2\" content=\"4 minuty\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\/\/schema.org\",\"@graph\":[{\"@type\":\"WebPage\",\"@id\":\"https:\/\/statorials.org\/pl\/rownanie-regresji\/\",\"url\":\"https:\/\/statorials.org\/pl\/rownanie-regresji\/\",\"name\":\"\u25b7 R\u00f3wnanie regresji\",\"isPartOf\":{\"@id\":\"https:\/\/statorials.org\/pl\/#website\"},\"datePublished\":\"2023-08-02T13:13:50+00:00\",\"dateModified\":\"2023-08-02T13:13:50+00:00\",\"author\":{\"@id\":\"https:\/\/statorials.org\/pl\/#\/schema\/person\/6484727a4612df3e69f016c3129c6965\"},\"description\":\"Tutaj znajdziesz czym jest r\u00f3wnanie regresji, jak jest obliczane (wz\u00f3r), rozwi\u0105zane \u0107wiczenie oraz kalkulator r\u00f3wnania regresji.\",\"breadcrumb\":{\"@id\":\"https:\/\/statorials.org\/pl\/rownanie-regresji\/#breadcrumb\"},\"inLanguage\":\"pl-PL\",\"potentialAction\":[{\"@type\":\"ReadAction\",\"target\":[\"https:\/\/statorials.org\/pl\/rownanie-regresji\/\"]}]},{\"@type\":\"BreadcrumbList\",\"@id\":\"https:\/\/statorials.org\/pl\/rownanie-regresji\/#breadcrumb\",\"itemListElement\":[{\"@type\":\"ListItem\",\"position\":1,\"name\":\"Dom\",\"item\":\"https:\/\/statorials.org\/pl\/\"},{\"@type\":\"ListItem\",\"position\":2,\"name\":\"R\u00f3wnanie regresji\"}]},{\"@type\":\"WebSite\",\"@id\":\"https:\/\/statorials.org\/pl\/#website\",\"url\":\"https:\/\/statorials.org\/pl\/\",\"name\":\"Statorials\",\"description\":\"Tw\u00f3j przewodnik po kompetencjach statystycznych!\",\"potentialAction\":[{\"@type\":\"SearchAction\",\"target\":{\"@type\":\"EntryPoint\",\"urlTemplate\":\"https:\/\/statorials.org\/pl\/?s={search_term_string}\"},\"query-input\":\"required name=search_term_string\"}],\"inLanguage\":\"pl-PL\"},{\"@type\":\"Person\",\"@id\":\"https:\/\/statorials.org\/pl\/#\/schema\/person\/6484727a4612df3e69f016c3129c6965\",\"name\":\"Benjamin Anderson\",\"image\":{\"@type\":\"ImageObject\",\"inLanguage\":\"pl-PL\",\"@id\":\"https:\/\/statorials.org\/pl\/#\/schema\/person\/image\/\",\"url\":\"https:\/\/statorials.org\/pl\/wp-content\/uploads\/2023\/11\/Benjamin-Anderson-96x96.jpg\",\"contentUrl\":\"https:\/\/statorials.org\/pl\/wp-content\/uploads\/2023\/11\/Benjamin-Anderson-96x96.jpg\",\"caption\":\"Benjamin Anderson\"},\"description\":\"Cze\u015b\u0107, jestem Benjamin i jestem emerytowanym profesorem statystyki, kt\u00f3ry zosta\u0142 oddanym nauczycielem Statorials. 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