{"id":4971,"date":"2023-06-27T11:19:11","date_gmt":"2023-06-27T09:19:11","guid":{"rendered":"https:\/\/softinery.com\/pl\/?page_id=4971"},"modified":"2024-08-06T13:47:25","modified_gmt":"2024-08-06T11:47:25","slug":"python-regresja-liniowa","status":"publish","type":"page","link":"https:\/\/softinery.com\/pl\/blog\/python-regresja-liniowa\/","title":{"rendered":"Python &#8211; regresja liniowa"},"content":{"rendered":"<style>.kb-row-layout-id4971_8f1aac-fb > .kt-row-column-wrap{align-content:start;}:where(.kb-row-layout-id4971_8f1aac-fb > .kt-row-column-wrap) > .wp-block-kadence-column{justify-content:start;}.kb-row-layout-id4971_8f1aac-fb > .kt-row-column-wrap{column-gap:var(--global-kb-gap-md, 2rem);row-gap:var(--global-kb-gap-md, 2rem);padding-top:var(--global-kb-spacing-sm, 1.5rem);padding-bottom:var(--global-kb-spacing-sm, 1.5rem);grid-template-columns:minmax(0, 2fr) minmax(0, 1fr);}.kb-row-layout-id4971_8f1aac-fb > .kt-row-layout-overlay{opacity:0.30;}@media all and (max-width: 1024px){.kb-row-layout-id4971_8f1aac-fb > .kt-row-column-wrap{grid-template-columns:minmax(0, 2fr) minmax(0, 1fr);}}@media all and (max-width: 767px){.kb-row-layout-id4971_8f1aac-fb > .kt-row-column-wrap{grid-template-columns:minmax(0, 1fr);}}<\/style><div class=\"kb-row-layout-wrap kb-row-layout-id4971_8f1aac-fb alignnone wp-block-kadence-rowlayout\"><div class=\"kt-row-column-wrap kt-has-2-columns kt-row-layout-left-golden kt-tab-layout-inherit kt-mobile-layout-row kt-row-valign-top\">\n<style>.kadence-column4971_697f88-ce > .kt-inside-inner-col{padding-right:var(--global-kb-spacing-xs, 1rem);padding-left:var(--global-kb-spacing-xs, 1rem);}.kadence-column4971_697f88-ce > .kt-inside-inner-col,.kadence-column4971_697f88-ce > .kt-inside-inner-col:before{border-top-left-radius:0px;border-top-right-radius:0px;border-bottom-right-radius:0px;border-bottom-left-radius:0px;}.kadence-column4971_697f88-ce > .kt-inside-inner-col{column-gap:var(--global-kb-gap-sm, 1rem);}.kadence-column4971_697f88-ce > .kt-inside-inner-col{flex-direction:column;}.kadence-column4971_697f88-ce > .kt-inside-inner-col > .aligncenter{width:100%;}.kadence-column4971_697f88-ce > .kt-inside-inner-col:before{opacity:0.3;}.kadence-column4971_697f88-ce{position:relative;}@media all and (max-width: 1024px){.kadence-column4971_697f88-ce > .kt-inside-inner-col{flex-direction:column;justify-content:center;}}@media all and (max-width: 767px){.kadence-column4971_697f88-ce > .kt-inside-inner-col{flex-direction:column;justify-content:center;}}<\/style>\n<div class=\"wp-block-kadence-column kadence-column4971_697f88-ce\"><div class=\"kt-inside-inner-col\"><style>.wp-block-kadence-advancedheading.kt-adv-heading4971_18e3b5-a1, .wp-block-kadence-advancedheading.kt-adv-heading4971_18e3b5-a1[data-kb-block=\"kb-adv-heading4971_18e3b5-a1\"]{font-style:normal;}.wp-block-kadence-advancedheading.kt-adv-heading4971_18e3b5-a1 mark.kt-highlight, .wp-block-kadence-advancedheading.kt-adv-heading4971_18e3b5-a1[data-kb-block=\"kb-adv-heading4971_18e3b5-a1\"] mark.kt-highlight{font-style:normal;color:#f76a0c;-webkit-box-decoration-break:clone;box-decoration-break:clone;padding-top:0px;padding-right:0px;padding-bottom:0px;padding-left:0px;}.wp-block-kadence-advancedheading.kt-adv-heading4971_18e3b5-a1 img.kb-inline-image, .wp-block-kadence-advancedheading.kt-adv-heading4971_18e3b5-a1[data-kb-block=\"kb-adv-heading4971_18e3b5-a1\"] img.kb-inline-image{width:150px;vertical-align:baseline;}<\/style>\n<h2 class=\"kt-adv-heading4971_18e3b5-a1 wp-block-kadence-advancedheading\" data-kb-block=\"kb-adv-heading4971_18e3b5-a1\">Wprowadzenie<\/h2>\n\n\n\n<p class=\"has-24292-e-color has-text-color\">Regresja liniowa to bardzo prosta i u\u017cyteczna metoda matematyczna, stosowana w analizie wszelkich danych. M\u00f3wi\u0105c najpro\u015bciej celem regresji liniowej jest znalezienie r\u00f3wnania linii prostej, kt\u00f3ra najlepiej &#8220;pasuje&#8221; do zbioru danych x, y. Opis teoretyczny mo\u017cna znale\u017a\u0107 na wielu r\u00f3\u017cnych stronach, mi\u0119dzy innymi na <a href=\"https:\/\/pl.wikipedia.org\/wiki\/Regresja_liniowa\">Wikipedii<\/a>. R\u00f3wnie\u017c w tym artykule om\u00f3wione s\u0105 teoretyczne aspekty tej metody. Aby si\u0119 z nimi zapozna\u0107 przejd\u017a do sekcji Regresja liniowa: teoria.<\/p>\n\n\n<style>.wp-block-kadence-advancedheading.kt-adv-heading4971_7c0b18-8c, .wp-block-kadence-advancedheading.kt-adv-heading4971_7c0b18-8c[data-kb-block=\"kb-adv-heading4971_7c0b18-8c\"]{font-style:normal;}.wp-block-kadence-advancedheading.kt-adv-heading4971_7c0b18-8c mark.kt-highlight, .wp-block-kadence-advancedheading.kt-adv-heading4971_7c0b18-8c[data-kb-block=\"kb-adv-heading4971_7c0b18-8c\"] mark.kt-highlight{font-style:normal;color:#f76a0c;-webkit-box-decoration-break:clone;box-decoration-break:clone;padding-top:0px;padding-right:0px;padding-bottom:0px;padding-left:0px;}.wp-block-kadence-advancedheading.kt-adv-heading4971_7c0b18-8c img.kb-inline-image, .wp-block-kadence-advancedheading.kt-adv-heading4971_7c0b18-8c[data-kb-block=\"kb-adv-heading4971_7c0b18-8c\"] img.kb-inline-image{width:150px;vertical-align:baseline;}<\/style>\n<h2 class=\"kt-adv-heading4971_7c0b18-8c wp-block-kadence-advancedheading\" data-kb-block=\"kb-adv-heading4971_7c0b18-8c\"><strong>Zastosowanie regresji liniowej<\/strong><\/h2>\n\n\n\n<p class=\"has-24292-e-color has-text-color\">Regresja liniowa jest szeroko stosowana w r\u00f3\u017cnych dziedzinach, takich jak ekonomia, nauki spo\u0142eczne, nauki przyrodnicze, in\u017cynieria itp. O tym jak wa\u017cna jest ta metoda \u015bwiadczy mi\u0119dzy innymi to, \u017ce mo\u017cemy znale\u017a\u0107 dziesi\u0105tki ksi\u0105\u017cek po\u015bwi\u0119conych wy\u0142\u0105cznie temu zagadnieniu. Regresja liniowa pojawia si\u0119 niemal w ka\u017cdym podr\u0119czniku do ekonometrii<sup>1<\/sup>. R\u00f3wnie\u017c w badaniach naukowych jest ona wykorzystywana bardzo cz\u0119sto. Naukowcy staraj\u0105 si\u0119 znale\u017a\u0107 zale\u017cno\u015bci pomi\u0119dzy zmiennymi, dzi\u0119ki czemu mog\u0105 zrozumie\u0107 badane zagadnienie. W ksi\u0105\u017cce Linear Regression Analysis: Theory and Computing<sup>2<\/sup> opisano eksperyment, w kt\u00f3rym badano wp\u0142yw palenia papieros\u00f3w na \u015bmiertelno\u015b\u0107. Mo\u017cemy r\u00f3wnie\u017c znale\u017a\u0107 inne ciekawe badania naukowe dotycz\u0105ce zastosowania regresji liniowej. Przyk\u0142adem mo\u017ce by\u0107 praca naukowa<sup>3<\/sup>, w kt\u00f3rej wykorzystano t\u0119 metod\u0119 do okre\u015blania wieku autora na podstawie pisanych przez niego tekst\u00f3w. <\/p>\n\n\n<style>.wp-block-kadence-advancedheading.kt-adv-heading4971_9600f1-ba, .wp-block-kadence-advancedheading.kt-adv-heading4971_9600f1-ba[data-kb-block=\"kb-adv-heading4971_9600f1-ba\"]{font-style:normal;}.wp-block-kadence-advancedheading.kt-adv-heading4971_9600f1-ba mark.kt-highlight, .wp-block-kadence-advancedheading.kt-adv-heading4971_9600f1-ba[data-kb-block=\"kb-adv-heading4971_9600f1-ba\"] mark.kt-highlight{font-style:normal;color:#f76a0c;-webkit-box-decoration-break:clone;box-decoration-break:clone;padding-top:0px;padding-right:0px;padding-bottom:0px;padding-left:0px;}.wp-block-kadence-advancedheading.kt-adv-heading4971_9600f1-ba img.kb-inline-image, .wp-block-kadence-advancedheading.kt-adv-heading4971_9600f1-ba[data-kb-block=\"kb-adv-heading4971_9600f1-ba\"] img.kb-inline-image{width:150px;vertical-align:baseline;}<\/style>\n<h2 class=\"kt-adv-heading4971_9600f1-ba wp-block-kadence-advancedheading\" data-kb-block=\"kb-adv-heading4971_9600f1-ba\">Biblioteki Pythona do regresji liniowej<\/h2>\n\n\n\n<p class=\"has-24292-e-color has-text-color\">W j\u0119zyku Python mamy dwie g\u0142\u00f3wne <a href=\"https:\/\/softinery.com\/pl\/2023\/04\/28\/python-dla-inzynierow\/\">biblioteki<\/a>, kt\u00f3re mog\u0105 by\u0107 wykorzystane do regresji liniowej: scikit-learn oraz statsmodels. Druga z wymienionych jest du\u017co bardziej rozbudowana i nie b\u0119dziemy mieli potrzeby korzysta\u0107 z jej funkcjonalno\u015bci. Biblioteka scikit-learn zawiera wszystkie kluczowe funkcje na potrzeby wi\u0119kszo\u015bci analiz.<br>Wykonanie regresji liniowej w Pythonie z u\u017cyciem bilbioteki scikitlearn jest bardzo proste. Opr\u00f3cz tej biblioteki pos\u0142u\u017cymy si\u0119 r\u00f3wnie\u017c:<\/p>\n\n\n\n<ul class=\"has-24292-e-color has-text-color wp-block-list\">\n<li>numpy &#8211; biblioteka do oblicze\u0144, w szczeg\u00f3lno\u015bci na macierzach,<\/li>\n\n\n\n<li>matplotlib &#8211; biblioteka do tworzenia wykres\u00f3w.<\/li>\n<\/ul>\n\n\n<style>.wp-block-kadence-advancedheading.kt-adv-heading4971_690fd4-55, .wp-block-kadence-advancedheading.kt-adv-heading4971_690fd4-55[data-kb-block=\"kb-adv-heading4971_690fd4-55\"]{font-style:normal;}.wp-block-kadence-advancedheading.kt-adv-heading4971_690fd4-55 mark.kt-highlight, .wp-block-kadence-advancedheading.kt-adv-heading4971_690fd4-55[data-kb-block=\"kb-adv-heading4971_690fd4-55\"] mark.kt-highlight{font-style:normal;color:#f76a0c;-webkit-box-decoration-break:clone;box-decoration-break:clone;padding-top:0px;padding-right:0px;padding-bottom:0px;padding-left:0px;}.wp-block-kadence-advancedheading.kt-adv-heading4971_690fd4-55 img.kb-inline-image, .wp-block-kadence-advancedheading.kt-adv-heading4971_690fd4-55[data-kb-block=\"kb-adv-heading4971_690fd4-55\"] img.kb-inline-image{width:150px;vertical-align:baseline;}<\/style>\n<h2 class=\"kt-adv-heading4971_690fd4-55 wp-block-kadence-advancedheading\" data-kb-block=\"kb-adv-heading4971_690fd4-55\">Program do regresji liniowej w j\u0119zyku Python<\/h2>\n\n\n\n<p class=\"has-24292-e-color has-text-color\">Najpierw zaimportujemy wspomniane biblioteki:<\/p>\n\n\n\n<div class=\"wp-block-kevinbatdorf-code-block-pro cbp-has-line-numbers cbp-highlight-hover\" data-code-block-pro-font-family=\"Code-Pro-JetBrains-Mono\" style=\"font-size:.875rem;font-family:Code-Pro-JetBrains-Mono,ui-monospace,SFMono-Regular,Menlo,Monaco,Consolas,monospace;--cbp-line-number-color:#24292e;--cbp-line-number-width:calc(1 * 0.6 * .875rem);--cbp-line-highlight-color:rgba(16, 41, 67, 0.2);line-height:1.25rem;--cbp-tab-width:2;tab-size:var(--cbp-tab-width, 2)\"><span style=\"display:flex;align-items:center;padding:16px 0 0 16px;width:100%;text-align:left;background-color:#ffffff\"><span style=\"background:#2f363c;padding:0.3rem 0.5rem 0.2rem;border-radius:1rem;font-size:0.8em;line-height:1;height:1.25rem;text-align:center;display:inline-flex;align-items:center;justify-content:center;color:#ffffff\">Python<\/span><\/span><span role=\"button\" tabindex=\"0\" data-code=\"import numpy as np\nimport matplotlib.pyplot as plt\nfrom sklearn.linear_model import LinearRegression\nfrom sklearn.metrics import r2_score\" style=\"color:#24292e;display:none\" aria-label=\"Copy\" class=\"code-block-pro-copy-button\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" style=\"width:24px;height:24px\" fill=\"none\" viewBox=\"0 0 24 24\" stroke=\"currentColor\" stroke-width=\"2\"><path class=\"with-check\" stroke-linecap=\"round\" stroke-linejoin=\"round\" d=\"M9 5H7a2 2 0 00-2 2v12a2 2 0 002 2h10a2 2 0 002-2V7a2 2 0 00-2-2h-2M9 5a2 2 0 002 2h2a2 2 0 002-2M9 5a2 2 0 012-2h2a2 2 0 012 2m-6 9l2 2 4-4\"><\/path><path class=\"without-check\" stroke-linecap=\"round\" stroke-linejoin=\"round\" d=\"M9 5H7a2 2 0 00-2 2v12a2 2 0 002 2h10a2 2 0 002-2V7a2 2 0 00-2-2h-2M9 5a2 2 0 002 2h2a2 2 0 002-2M9 5a2 2 0 012-2h2a2 2 0 012 2\"><\/path><\/svg><\/span><pre class=\"shiki github-light\" style=\"background-color: #fff\" tabindex=\"0\"><code><span class=\"line\"><span style=\"color: #D73A49\">import<\/span><span style=\"color: #24292E\"> numpy <\/span><span style=\"color: #D73A49\">as<\/span><span style=\"color: #24292E\"> np<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D73A49\">import<\/span><span style=\"color: #24292E\"> matplotlib.pyplot <\/span><span style=\"color: #D73A49\">as<\/span><span style=\"color: #24292E\"> plt<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D73A49\">from<\/span><span style=\"color: #24292E\"> sklearn.linear_model <\/span><span style=\"color: #D73A49\">import<\/span><span style=\"color: #24292E\"> LinearRegression<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D73A49\">from<\/span><span style=\"color: #24292E\"> sklearn.metrics <\/span><span style=\"color: #D73A49\">import<\/span><span style=\"color: #24292E\"> r2_score<\/span><\/span><\/code><\/pre><\/div>\n\n\n\n<p class=\"has-24292-e-color has-text-color\">Do regresji wykorzystamy dane wygenerowane przy u\u017cyciu biblioteki numpy. Wektor X b\u0119dzie zawiera\u0142 losowe warto\u015bci z przedzia\u0142u od 0 do 1, natomiast y to warto\u015bci obliczone z r\u00f3wnania linii prostej, do kt\u00f3rych dodany b\u0119dzie losowy &#8220;b\u0142\u0105d&#8221;.<\/p>\n\n\n\n<div class=\"wp-block-kevinbatdorf-code-block-pro cbp-has-line-numbers cbp-highlight-hover\" data-code-block-pro-font-family=\"Code-Pro-JetBrains-Mono\" style=\"font-size:.875rem;font-family:Code-Pro-JetBrains-Mono,ui-monospace,SFMono-Regular,Menlo,Monaco,Consolas,monospace;--cbp-line-number-color:#24292e;--cbp-line-number-width:calc(1 * 0.6 * .875rem);--cbp-line-highlight-color:rgba(16, 41, 67, 0.2);line-height:1.25rem;--cbp-tab-width:2;tab-size:var(--cbp-tab-width, 2)\"><span style=\"display:flex;align-items:center;padding:16px 0 0 16px;width:100%;text-align:left;background-color:#ffffff\"><span style=\"background:#2f363c;padding:0.3rem 0.5rem 0.2rem;border-radius:1rem;font-size:0.8em;line-height:1;height:1.25rem;text-align:center;display:inline-flex;align-items:center;justify-content:center;color:#ffffff\">Python<\/span><\/span><span role=\"button\" tabindex=\"0\" data-code=\"np.random.seed(0)\nX = np.random.rand(100, 1)  # Zmienna niezale\u017cna\ny = 3 * X + 2 + np.random.randn(100, 1)\/10  # Zmienna zale\u017cna + &quot;b\u0142\u0105d&quot;\" style=\"color:#24292e;display:none\" aria-label=\"Copy\" class=\"code-block-pro-copy-button\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" style=\"width:24px;height:24px\" fill=\"none\" viewBox=\"0 0 24 24\" stroke=\"currentColor\" stroke-width=\"2\"><path class=\"with-check\" stroke-linecap=\"round\" stroke-linejoin=\"round\" d=\"M9 5H7a2 2 0 00-2 2v12a2 2 0 002 2h10a2 2 0 002-2V7a2 2 0 00-2-2h-2M9 5a2 2 0 002 2h2a2 2 0 002-2M9 5a2 2 0 012-2h2a2 2 0 012 2m-6 9l2 2 4-4\"><\/path><path class=\"without-check\" stroke-linecap=\"round\" stroke-linejoin=\"round\" d=\"M9 5H7a2 2 0 00-2 2v12a2 2 0 002 2h10a2 2 0 002-2V7a2 2 0 00-2-2h-2M9 5a2 2 0 002 2h2a2 2 0 002-2M9 5a2 2 0 012-2h2a2 2 0 012 2\"><\/path><\/svg><\/span><pre class=\"shiki github-light\" style=\"background-color: #fff\" tabindex=\"0\"><code><span class=\"line\"><span style=\"color: #24292E\">np.random.seed(<\/span><span style=\"color: #005CC5\">0<\/span><span style=\"color: #24292E\">)<\/span><\/span>\n<span class=\"line\"><span style=\"color: #24292E\">X <\/span><span style=\"color: #D73A49\">=<\/span><span style=\"color: #24292E\"> np.random.rand(<\/span><span style=\"color: #005CC5\">100<\/span><span style=\"color: #24292E\">, <\/span><span style=\"color: #005CC5\">1<\/span><span style=\"color: #24292E\">)  <\/span><span style=\"color: #6A737D\"># Zmienna niezale\u017cna<\/span><\/span>\n<span class=\"line\"><span style=\"color: #24292E\">y <\/span><span style=\"color: #D73A49\">=<\/span><span style=\"color: #24292E\"> <\/span><span style=\"color: #005CC5\">3<\/span><span style=\"color: #24292E\"> <\/span><span style=\"color: #D73A49\">*<\/span><span style=\"color: #24292E\"> X <\/span><span style=\"color: #D73A49\">+<\/span><span style=\"color: #24292E\"> <\/span><span style=\"color: #005CC5\">2<\/span><span style=\"color: #24292E\"> <\/span><span style=\"color: #D73A49\">+<\/span><span style=\"color: #24292E\"> np.random.randn(<\/span><span style=\"color: #005CC5\">100<\/span><span style=\"color: #24292E\">, <\/span><span style=\"color: #005CC5\">1<\/span><span style=\"color: #24292E\">)<\/span><span style=\"color: #D73A49\">\/<\/span><span style=\"color: #005CC5\">10<\/span><span style=\"color: #24292E\">  <\/span><span style=\"color: #6A737D\"># Zmienna zale\u017cna + &quot;b\u0142\u0105d&quot;<\/span><\/span><\/code><\/pre><\/div>\n\n\n\n<p class=\"has-24292-e-color has-text-color\">W nast\u0119pnym kroku utworzymy model regresji liniowej. Metoda <em>LinearRegression<\/em>() zosta\u0142a wcze\u015bniej zaimportowana z biblioteki sklearn. Aby dopasowa\u0107 lini\u0119 prost\u0105 do danych X, y, wywo\u0142ujemy metod\u0119 <em>fit<\/em>(). <\/p>\n\n\n\n<div class=\"wp-block-kevinbatdorf-code-block-pro cbp-has-line-numbers cbp-highlight-hover\" data-code-block-pro-font-family=\"Code-Pro-JetBrains-Mono\" style=\"font-size:.875rem;font-family:Code-Pro-JetBrains-Mono,ui-monospace,SFMono-Regular,Menlo,Monaco,Consolas,monospace;--cbp-line-number-color:#24292e;--cbp-line-number-width:calc(1 * 0.6 * .875rem);--cbp-line-highlight-color:rgba(16, 41, 67, 0.2);line-height:1.25rem;--cbp-tab-width:2;tab-size:var(--cbp-tab-width, 2)\"><span style=\"display:flex;align-items:center;padding:16px 0 0 16px;width:100%;text-align:left;background-color:#ffffff\"><span style=\"background:#2f363c;padding:0.3rem 0.5rem 0.2rem;border-radius:1rem;font-size:0.8em;line-height:1;height:1.25rem;text-align:center;display:inline-flex;align-items:center;justify-content:center;color:#ffffff\">Python<\/span><\/span><span role=\"button\" tabindex=\"0\" data-code=\"# Utworzenie modelu regresji liniowej i dopasowanie\nmodel = LinearRegression()\nmodel.fit(X, y)\" style=\"color:#24292e;display:none\" aria-label=\"Copy\" class=\"code-block-pro-copy-button\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" style=\"width:24px;height:24px\" fill=\"none\" viewBox=\"0 0 24 24\" stroke=\"currentColor\" stroke-width=\"2\"><path class=\"with-check\" stroke-linecap=\"round\" stroke-linejoin=\"round\" d=\"M9 5H7a2 2 0 00-2 2v12a2 2 0 002 2h10a2 2 0 002-2V7a2 2 0 00-2-2h-2M9 5a2 2 0 002 2h2a2 2 0 002-2M9 5a2 2 0 012-2h2a2 2 0 012 2m-6 9l2 2 4-4\"><\/path><path class=\"without-check\" stroke-linecap=\"round\" stroke-linejoin=\"round\" d=\"M9 5H7a2 2 0 00-2 2v12a2 2 0 002 2h10a2 2 0 002-2V7a2 2 0 00-2-2h-2M9 5a2 2 0 002 2h2a2 2 0 002-2M9 5a2 2 0 012-2h2a2 2 0 012 2\"><\/path><\/svg><\/span><pre class=\"shiki github-light\" style=\"background-color: #fff\" tabindex=\"0\"><code><span class=\"line\"><span style=\"color: #6A737D\"># Utworzenie modelu regresji liniowej i dopasowanie<\/span><\/span>\n<span class=\"line\"><span style=\"color: #24292E\">model <\/span><span style=\"color: #D73A49\">=<\/span><span style=\"color: #24292E\"> LinearRegression()<\/span><\/span>\n<span class=\"line\"><span style=\"color: #24292E\">model.fit(X, y)<\/span><\/span><\/code><\/pre><\/div>\n\n\n\n<p class=\"has-24292-e-color has-text-color\">Do oceny dopasania prostej pos\u0142u\u017cymy si\u0119 wsp\u00f3\u0142czynnikiem R<sup>2<\/sup>. Z teorii wiemy, \u017ce jego warto\u015b\u0107 powinna by\u0107 mo\u017cliwie bliska jedno\u015bci, co \u015bwiadczy o dobrym dopasowaniu prostej do danych. Je\u017celi warto\u015b\u0107 jest znacznie mniejsza od jedno\u015bci (&#8220;znacznie&#8221; zale\u017cy od konkretnego problemu), to pomi\u0119dzy danymi X i y nie zachodzi zale\u017cno\u015b\u0107 liniowa. Aby wyznaczy\u0107 R<sup>2<\/sup> najpierw obliczamy warto\u015bci y przewidywane przez stworzony <a href=\"https:\/\/softinery.com\/pl\/oferta\/\">model<\/a> regresji liniowej (metoda <em>predict<\/em>) dla istniej\u0105cych warto\u015bci X. Nast\u0119pnie otrzymane warto\u015bci wykorzystujemy do wywo\u0142ania metody <em>r2score<\/em>.<\/p>\n\n\n\n<div class=\"wp-block-kevinbatdorf-code-block-pro cbp-has-line-numbers cbp-highlight-hover\" data-code-block-pro-font-family=\"Code-Pro-JetBrains-Mono\" style=\"font-size:.875rem;font-family:Code-Pro-JetBrains-Mono,ui-monospace,SFMono-Regular,Menlo,Monaco,Consolas,monospace;--cbp-line-number-color:#24292e;--cbp-line-number-width:calc(1 * 0.6 * .875rem);--cbp-line-highlight-color:rgba(16, 41, 67, 0.2);line-height:1.25rem;--cbp-tab-width:2;tab-size:var(--cbp-tab-width, 2)\"><span style=\"display:flex;align-items:center;padding:16px 0 0 16px;width:100%;text-align:left;background-color:#ffffff\"><span style=\"background:#2f363c;padding:0.3rem 0.5rem 0.2rem;border-radius:1rem;font-size:0.8em;line-height:1;height:1.25rem;text-align:center;display:inline-flex;align-items:center;justify-content:center;color:#ffffff\">Python<\/span><\/span><span role=\"button\" tabindex=\"0\" data-code=\"# Warto\u015bci przewidywane przez model regresji liniowej\ny_pred = model.predict(X)\n\n# Obliczenie R-kwadrat.\nr2 = r2_score(y, y_pred)\n\nprint(&quot;R kwadrat:&quot;, r2)\" style=\"color:#24292e;display:none\" aria-label=\"Copy\" class=\"code-block-pro-copy-button\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" style=\"width:24px;height:24px\" fill=\"none\" viewBox=\"0 0 24 24\" stroke=\"currentColor\" stroke-width=\"2\"><path class=\"with-check\" stroke-linecap=\"round\" stroke-linejoin=\"round\" d=\"M9 5H7a2 2 0 00-2 2v12a2 2 0 002 2h10a2 2 0 002-2V7a2 2 0 00-2-2h-2M9 5a2 2 0 002 2h2a2 2 0 002-2M9 5a2 2 0 012-2h2a2 2 0 012 2m-6 9l2 2 4-4\"><\/path><path class=\"without-check\" stroke-linecap=\"round\" stroke-linejoin=\"round\" d=\"M9 5H7a2 2 0 00-2 2v12a2 2 0 002 2h10a2 2 0 002-2V7a2 2 0 00-2-2h-2M9 5a2 2 0 002 2h2a2 2 0 002-2M9 5a2 2 0 012-2h2a2 2 0 012 2\"><\/path><\/svg><\/span><pre class=\"shiki github-light\" style=\"background-color: #fff\" tabindex=\"0\"><code><span class=\"line\"><span style=\"color: #6A737D\"># Warto\u015bci przewidywane przez model regresji liniowej<\/span><\/span>\n<span class=\"line\"><span style=\"color: #24292E\">y_pred <\/span><span style=\"color: #D73A49\">=<\/span><span style=\"color: #24292E\"> model.predict(X)<\/span><\/span>\n<span class=\"line\"><\/span>\n<span class=\"line\"><span style=\"color: #6A737D\"># Obliczenie R-kwadrat.<\/span><\/span>\n<span class=\"line\"><span style=\"color: #24292E\">r2 <\/span><span style=\"color: #D73A49\">=<\/span><span style=\"color: #24292E\"> r2_score(y, y_pred)<\/span><\/span>\n<span class=\"line\"><\/span>\n<span class=\"line\"><span style=\"color: #005CC5\">print<\/span><span style=\"color: #24292E\">(<\/span><span style=\"color: #032F62\">&quot;R kwadrat:&quot;<\/span><span style=\"color: #24292E\">, r2)<\/span><\/span><\/code><\/pre><\/div>\n\n\n\n<p class=\"has-24292-e-color has-text-color\">Parametry prostej a*x + b otrzymujemy z atrybut\u00f3w obiektu model. coef_ to parametr a (nachylenie), a intercept_ to b (przeci\u0119cie z osi\u0105 OY).<\/p>\n\n\n\n<div class=\"wp-block-kevinbatdorf-code-block-pro cbp-has-line-numbers cbp-highlight-hover\" data-code-block-pro-font-family=\"Code-Pro-JetBrains-Mono\" style=\"font-size:.875rem;font-family:Code-Pro-JetBrains-Mono,ui-monospace,SFMono-Regular,Menlo,Monaco,Consolas,monospace;--cbp-line-number-color:#24292e;--cbp-line-number-width:calc(1 * 0.6 * .875rem);--cbp-line-highlight-color:rgba(16, 41, 67, 0.2);line-height:1.25rem;--cbp-tab-width:2;tab-size:var(--cbp-tab-width, 2)\"><span style=\"display:flex;align-items:center;padding:16px 0 0 16px;width:100%;text-align:left;background-color:#ffffff\"><span style=\"background:#2f363c;padding:0.3rem 0.5rem 0.2rem;border-radius:1rem;font-size:0.8em;line-height:1;height:1.25rem;text-align:center;display:inline-flex;align-items:center;justify-content:center;color:#ffffff\">Python<\/span><\/span><span role=\"button\" tabindex=\"0\" data-code=\"# Wsp\u00f3\u0142czynnik a w r\u00f3wnaniu y = a*x + b\na = model.coef_\nprint(&quot;Wsp\u00f3\u0142czynnik a:&quot;, a)\n\n# Wsp\u00f3\u0142czynnik b w r\u00f3wnaniu y = a*x + b\nb = model.intercept_\nprint(&quot;Wsp\u00f3\u0142czynnik b:&quot;, b)\" style=\"color:#24292e;display:none\" aria-label=\"Copy\" class=\"code-block-pro-copy-button\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" style=\"width:24px;height:24px\" fill=\"none\" viewBox=\"0 0 24 24\" stroke=\"currentColor\" stroke-width=\"2\"><path class=\"with-check\" stroke-linecap=\"round\" stroke-linejoin=\"round\" d=\"M9 5H7a2 2 0 00-2 2v12a2 2 0 002 2h10a2 2 0 002-2V7a2 2 0 00-2-2h-2M9 5a2 2 0 002 2h2a2 2 0 002-2M9 5a2 2 0 012-2h2a2 2 0 012 2m-6 9l2 2 4-4\"><\/path><path class=\"without-check\" stroke-linecap=\"round\" stroke-linejoin=\"round\" d=\"M9 5H7a2 2 0 00-2 2v12a2 2 0 002 2h10a2 2 0 002-2V7a2 2 0 00-2-2h-2M9 5a2 2 0 002 2h2a2 2 0 002-2M9 5a2 2 0 012-2h2a2 2 0 012 2\"><\/path><\/svg><\/span><pre class=\"shiki github-light\" style=\"background-color: #fff\" tabindex=\"0\"><code><span class=\"line\"><span style=\"color: #6A737D\"># Wsp\u00f3\u0142czynnik a w r\u00f3wnaniu y = a*x + b<\/span><\/span>\n<span class=\"line\"><span style=\"color: #24292E\">a <\/span><span style=\"color: #D73A49\">=<\/span><span style=\"color: #24292E\"> model.coef_<\/span><\/span>\n<span class=\"line\"><span style=\"color: #005CC5\">print<\/span><span style=\"color: #24292E\">(<\/span><span style=\"color: #032F62\">&quot;Wsp\u00f3\u0142czynnik a:&quot;<\/span><span style=\"color: #24292E\">, a)<\/span><\/span>\n<span class=\"line\"><\/span>\n<span class=\"line\"><span style=\"color: #6A737D\"># Wsp\u00f3\u0142czynnik b w r\u00f3wnaniu y = a*x + b<\/span><\/span>\n<span class=\"line\"><span style=\"color: #24292E\">b <\/span><span style=\"color: #D73A49\">=<\/span><span style=\"color: #24292E\"> model.intercept_<\/span><\/span>\n<span class=\"line\"><span style=\"color: #005CC5\">print<\/span><span style=\"color: #24292E\">(<\/span><span style=\"color: #032F62\">&quot;Wsp\u00f3\u0142czynnik b:&quot;<\/span><span style=\"color: #24292E\">, b)<\/span><\/span><\/code><\/pre><\/div>\n\n\n\n<p class=\"has-24292-e-color has-text-color\">W ostatnim kroku <a href=\"https:\/\/softinery.com\/pl\/katalog-szkolen\/python-data-science-kurs\/\">zwizualizujemy dane i wyniki<\/a>. Do tego celu pos\u0142u\u017cy nam <a href=\"https:\/\/softinery.com\/pl\/blog\/wykresy-python\/\">biblioteka Matplotlib<\/a>. Utworzymy wykres punktowy przy u\u017cyciu metody scatter. Metody xlabel i ylabel s\u0142u\u017c\u0105 do opisania osi, <em>legend<\/em>() do utworzenia legendy, na kt\u00f3rej wy\u015bwietlone zostan\u0105 wcze\u015bniej nadane etykiety (label). Wykres wy\u015bwietlamy przy u\u017cyciu <a href=\"https:\/\/softinery.com\/pl\/blog\/kurs-python-funkcje\/\">funkcji<\/a> <em>show<\/em>().<\/p>\n\n\n\n<div class=\"wp-block-kevinbatdorf-code-block-pro cbp-has-line-numbers cbp-highlight-hover\" data-code-block-pro-font-family=\"Code-Pro-JetBrains-Mono\" style=\"font-size:.875rem;font-family:Code-Pro-JetBrains-Mono,ui-monospace,SFMono-Regular,Menlo,Monaco,Consolas,monospace;--cbp-line-number-color:#24292e;--cbp-line-number-width:calc(1 * 0.6 * .875rem);--cbp-line-highlight-color:rgba(16, 41, 67, 0.2);line-height:1.25rem;--cbp-tab-width:2;tab-size:var(--cbp-tab-width, 2)\"><span style=\"display:flex;align-items:center;padding:16px 0 0 16px;width:100%;text-align:left;background-color:#ffffff\"><span style=\"background:#2f363c;padding:0.3rem 0.5rem 0.2rem;border-radius:1rem;font-size:0.8em;line-height:1;height:1.25rem;text-align:center;display:inline-flex;align-items:center;justify-content:center;color:#ffffff\">Python<\/span><\/span><span role=\"button\" tabindex=\"0\" data-code=\"plt.scatter(X, y, color='blue', label='Dane')\nplt.plot(X, y_pred, color='red', linewidth=2, label='Regresja liniowa')\nplt.xlabel('X')\nplt.ylabel('y')\nplt.legend()\nplt.show()\" style=\"color:#24292e;display:none\" aria-label=\"Copy\" class=\"code-block-pro-copy-button\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" style=\"width:24px;height:24px\" fill=\"none\" viewBox=\"0 0 24 24\" stroke=\"currentColor\" stroke-width=\"2\"><path class=\"with-check\" stroke-linecap=\"round\" stroke-linejoin=\"round\" d=\"M9 5H7a2 2 0 00-2 2v12a2 2 0 002 2h10a2 2 0 002-2V7a2 2 0 00-2-2h-2M9 5a2 2 0 002 2h2a2 2 0 002-2M9 5a2 2 0 012-2h2a2 2 0 012 2m-6 9l2 2 4-4\"><\/path><path class=\"without-check\" stroke-linecap=\"round\" stroke-linejoin=\"round\" d=\"M9 5H7a2 2 0 00-2 2v12a2 2 0 002 2h10a2 2 0 002-2V7a2 2 0 00-2-2h-2M9 5a2 2 0 002 2h2a2 2 0 002-2M9 5a2 2 0 012-2h2a2 2 0 012 2\"><\/path><\/svg><\/span><pre class=\"shiki github-light\" style=\"background-color: #fff\" tabindex=\"0\"><code><span class=\"line\"><span style=\"color: #24292E\">plt.scatter(X, y, <\/span><span style=\"color: #E36209\">color<\/span><span style=\"color: #D73A49\">=<\/span><span style=\"color: #032F62\">&#39;blue&#39;<\/span><span style=\"color: #24292E\">, <\/span><span style=\"color: #E36209\">label<\/span><span style=\"color: #D73A49\">=<\/span><span style=\"color: #032F62\">&#39;Dane&#39;<\/span><span style=\"color: #24292E\">)<\/span><\/span>\n<span class=\"line\"><span style=\"color: #24292E\">plt.plot(X, y_pred, <\/span><span style=\"color: #E36209\">color<\/span><span style=\"color: #D73A49\">=<\/span><span style=\"color: #032F62\">&#39;red&#39;<\/span><span style=\"color: #24292E\">, <\/span><span style=\"color: #E36209\">linewidth<\/span><span style=\"color: #D73A49\">=<\/span><span style=\"color: #005CC5\">2<\/span><span style=\"color: #24292E\">, <\/span><span style=\"color: #E36209\">label<\/span><span style=\"color: #D73A49\">=<\/span><span style=\"color: #032F62\">&#39;Regresja liniowa&#39;<\/span><span style=\"color: #24292E\">)<\/span><\/span>\n<span class=\"line\"><span style=\"color: #24292E\">plt.xlabel(<\/span><span style=\"color: #032F62\">&#39;X&#39;<\/span><span style=\"color: #24292E\">)<\/span><\/span>\n<span class=\"line\"><span style=\"color: #24292E\">plt.ylabel(<\/span><span style=\"color: #032F62\">&#39;y&#39;<\/span><span style=\"color: #24292E\">)<\/span><\/span>\n<span class=\"line\"><span style=\"color: #24292E\">plt.legend()<\/span><\/span>\n<span class=\"line\"><span style=\"color: #24292E\">plt.show()<\/span><\/span><\/code><\/pre><\/div>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"567\" height=\"432\" src=\"https:\/\/softinery.com\/wp-content\/uploads\/sites\/5\/2023\/06\/regresja-liniowa.webp\" alt=\"Python regresa liniowa - rysunek w Matplotlib\" class=\"wp-image-5010\" style=\"width:425px;height:324px\" title=\"Python regresa liniowa - rysunek w Matplotlib\" srcset=\"https:\/\/softinery.com\/pl\/wp-content\/uploads\/sites\/5\/2023\/06\/regresja-liniowa.webp 567w, https:\/\/softinery.com\/pl\/wp-content\/uploads\/sites\/5\/2023\/06\/regresja-liniowa-300x229.webp 300w\" sizes=\"auto, (max-width: 567px) 100vw, 567px\" \/><figcaption class=\"wp-element-caption\">Regresja liniowa (R2 = 0,99)<\/figcaption><\/figure>\n<\/div>\n\n\n<p class=\"has-24292-e-color has-text-color\">#data science #machine learning #statystyka #analiza danych #python regresja liniowa<\/p>\n\n\n\n<div style=\"height:67px\" aria-hidden=\"true\" class=\"wp-block-spacer\"><\/div>\n\n\n\n<h2 class=\"wp-block-heading has-text-align-center has-24292-e-color has-text-color\">Regresj\u0119 mo\u017cesz wykona\u0107 przy u\u017cyciu naszego narz\u0119dzia online<\/h2>\n\n\n<style>.kadence-column4971_c680a0-cd > .kt-inside-inner-col,.kadence-column4971_c680a0-cd > .kt-inside-inner-col:before{border-top-left-radius:0px;border-top-right-radius:0px;border-bottom-right-radius:0px;border-bottom-left-radius:0px;}.kadence-column4971_c680a0-cd > .kt-inside-inner-col{column-gap:var(--global-kb-gap-sm, 1rem);}.kadence-column4971_c680a0-cd > .kt-inside-inner-col{flex-direction:column;}.kadence-column4971_c680a0-cd > .kt-inside-inner-col > .aligncenter{width:100%;}.kadence-column4971_c680a0-cd > .kt-inside-inner-col:before{opacity:0.3;}.kadence-column4971_c680a0-cd, .kadence-column4971_c680a0-cd h1, .kadence-column4971_c680a0-cd h2, .kadence-column4971_c680a0-cd h3, .kadence-column4971_c680a0-cd h4, .kadence-column4971_c680a0-cd h5, .kadence-column4971_c680a0-cd h6{color:#24292e;}.kadence-column4971_c680a0-cd{position:relative;}@media all and (max-width: 1024px){.kadence-column4971_c680a0-cd > .kt-inside-inner-col{flex-direction:column;justify-content:center;}}@media all and (max-width: 767px){.kadence-column4971_c680a0-cd > .kt-inside-inner-col{flex-direction:column;justify-content:center;}}<\/style>\n<div class=\"wp-block-kadence-column kadence-column4971_c680a0-cd\"><div class=\"kt-inside-inner-col\"><style>.kadence-column4971_5f937c-94{max-width:356px;margin-left:auto;margin-right:auto;}.wp-block-kadence-column.kb-section-dir-horizontal:not(.kb-section-md-dir-vertical)>.kt-inside-inner-col>.kadence-column4971_5f937c-94{-webkit-flex:0 1 356px;flex:0 1 356px;max-width:unset;margin-left:unset;margin-right:unset;}.kadence-column4971_5f937c-94 > .kt-inside-inner-col,.kadence-column4971_5f937c-94 > .kt-inside-inner-col:before{border-top-left-radius:0px;border-top-right-radius:0px;border-bottom-right-radius:0px;border-bottom-left-radius:0px;}.kadence-column4971_5f937c-94 > .kt-inside-inner-col{column-gap:var(--global-kb-gap-sm, 1rem);}.kadence-column4971_5f937c-94 > .kt-inside-inner-col{flex-direction:column;}.kadence-column4971_5f937c-94 > .kt-inside-inner-col > .aligncenter{width:100%;}.kadence-column4971_5f937c-94 > .kt-inside-inner-col:before{opacity:0.3;}.kadence-column4971_5f937c-94{position:relative;}@media all and (min-width: 1025px){.wp-block-kadence-column.kb-section-dir-horizontal>.kt-inside-inner-col>.kadence-column4971_5f937c-94{-webkit-flex:0 1 356px;flex:0 1 356px;max-width:unset;margin-left:unset;margin-right:unset;}}@media all and (max-width: 1024px){.kadence-column4971_5f937c-94 > .kt-inside-inner-col{flex-direction:column;justify-content:center;}}@media all and (max-width: 767px){.wp-block-kadence-column.kb-section-sm-dir-vertical:not(.kb-section-sm-dir-horizontal):not(.kb-section-sm-dir-specificity)>.kt-inside-inner-col>.kadence-column4971_5f937c-94{max-width:356px;-webkit-flex:1;flex:1;margin-left:auto;margin-right:auto;}.kadence-column4971_5f937c-94 > .kt-inside-inner-col{flex-direction:column;justify-content:center;}}<\/style>\n<div class=\"wp-block-kadence-column kadence-column4971_5f937c-94\"><div class=\"kt-inside-inner-col\"><style>.kadence-column4971_3bd2ec-52 > .kt-inside-inner-col,.kadence-column4971_3bd2ec-52 > .kt-inside-inner-col:before{border-top-left-radius:0px;border-top-right-radius:0px;border-bottom-right-radius:0px;border-bottom-left-radius:0px;}.kadence-column4971_3bd2ec-52 > .kt-inside-inner-col{column-gap:var(--global-kb-gap-sm, 1rem);}.kadence-column4971_3bd2ec-52 > .kt-inside-inner-col{flex-direction:column;}.kadence-column4971_3bd2ec-52 > .kt-inside-inner-col > .aligncenter{width:100%;}.kadence-column4971_3bd2ec-52 > .kt-inside-inner-col:before{opacity:0.3;}.kadence-column4971_3bd2ec-52 a{color:var(--global-palette3, #1A202C);}.kadence-column4971_3bd2ec-52 a:hover{color:var(--global-palette4, #2D3748);}.kadence-column4971_3bd2ec-52{position:relative;}@media all and (max-width: 1024px){.kadence-column4971_3bd2ec-52 > .kt-inside-inner-col{flex-direction:column;justify-content:center;}}@media all and (max-width: 767px){.kadence-column4971_3bd2ec-52 > .kt-inside-inner-col{flex-direction:column;justify-content:center;}}<\/style>\n<div class=\"wp-block-kadence-column kadence-column4971_3bd2ec-52\"><div class=\"kt-inside-inner-col\"><style>.wp-block-kadence-image.kb-image4971_93cdd2-3a:not(.kb-specificity-added):not(.kb-extra-specificity-added){margin-bottom:var(--global-kb-spacing-lg, 3rem);}.kb-image4971_93cdd2-3a:not(.kb-image-is-ratio-size) .kb-img, .kb-image4971_93cdd2-3a.kb-image-is-ratio-size{padding-top:0px;padding-right:0px;padding-left:0px;}.kb-image4971_93cdd2-3a .kb-image-has-overlay:after{opacity:0.3;}<\/style>\n<div class=\"wp-block-kadence-image kb-image4971_93cdd2-3a blog_link_image grey-box-with-shadow\"><figure class=\"aligncenter size-full kb-image-is-ratio-size\"><a href=\"https:\/\/tools.softinery.com\/CurveFitter\/\" class=\"kb-advanced-image-link\"><div class=\"kb-is-ratio-image kb-image-ratio-square\"><img decoding=\"async\" src=\"https:\/\/softinery.com\/wp-content\/uploads\/2023\/09\/CurveFitterThumbnail.jpg\" alt=\"Online Curve Fitting Tool\" class=\"kb-img wp-image-5019\" title=\"Online Curve Fitting Tool\"\/><\/div><\/a><\/figure><\/div>\n\n\n<style>.wp-block-kadence-advancedheading.kt-adv-heading4971_1a1007-9d, .wp-block-kadence-advancedheading.kt-adv-heading4971_1a1007-9d[data-kb-block=\"kb-adv-heading4971_1a1007-9d\"]{margin-top:0px;margin-bottom:var(--global-kb-spacing-sm, 1.5rem);text-align:center;font-size:var(--global-kb-font-size-md, 1.25rem);font-style:normal;color:var(--global-palette3, #1A202C);}.wp-block-kadence-advancedheading.kt-adv-heading4971_1a1007-9d mark.kt-highlight, .wp-block-kadence-advancedheading.kt-adv-heading4971_1a1007-9d[data-kb-block=\"kb-adv-heading4971_1a1007-9d\"] mark.kt-highlight{font-style:normal;color:#f76a0c;-webkit-box-decoration-break:clone;box-decoration-break:clone;padding-top:0px;padding-right:0px;padding-bottom:0px;padding-left:0px;}.wp-block-kadence-advancedheading.kt-adv-heading4971_1a1007-9d img.kb-inline-image, .wp-block-kadence-advancedheading.kt-adv-heading4971_1a1007-9d[data-kb-block=\"kb-adv-heading4971_1a1007-9d\"] img.kb-inline-image{width:150px;vertical-align:baseline;}<\/style>\n<h3 class=\"kt-adv-heading4971_1a1007-9d wp-block-kadence-advancedheading has-theme-palette-3-color has-text-color\" data-kb-block=\"kb-adv-heading4971_1a1007-9d\"><a href=\"https:\/\/tools.softinery.com\/CurveFitter\/\">Online Curve Fitting Tool<\/a><\/h3>\n<\/div><\/div>\n<\/div><\/div>\n<\/div><\/div>\n\n\n\n<div style=\"height:67px\" aria-hidden=\"true\" class=\"wp-block-spacer\"><\/div>\n\n\n<style>.wp-block-kadence-advancedheading.kt-adv-heading4971_d3025d-3b, .wp-block-kadence-advancedheading.kt-adv-heading4971_d3025d-3b[data-kb-block=\"kb-adv-heading4971_d3025d-3b\"]{padding-top:var(--global-kb-spacing-xxs, 0.5rem);padding-bottom:var(--global-kb-spacing-xxs, 0.5rem);padding-left:var(--global-kb-spacing-xxs, 0.5rem);font-style:normal;}.wp-block-kadence-advancedheading.kt-adv-heading4971_d3025d-3b mark.kt-highlight, .wp-block-kadence-advancedheading.kt-adv-heading4971_d3025d-3b[data-kb-block=\"kb-adv-heading4971_d3025d-3b\"] mark.kt-highlight{font-style:normal;color:#f76a0c;-webkit-box-decoration-break:clone;box-decoration-break:clone;padding-top:0px;padding-right:0px;padding-bottom:0px;padding-left:0px;}.wp-block-kadence-advancedheading.kt-adv-heading4971_d3025d-3b img.kb-inline-image, .wp-block-kadence-advancedheading.kt-adv-heading4971_d3025d-3b[data-kb-block=\"kb-adv-heading4971_d3025d-3b\"] img.kb-inline-image{width:150px;vertical-align:baseline;}<\/style>\n<h2 class=\"kt-adv-heading4971_d3025d-3b wp-block-kadence-advancedheading\" data-kb-block=\"kb-adv-heading4971_d3025d-3b\">Wi\u0119cej na temat oblicze\u0144 z u\u017cyciem Pythona dowiesz si\u0119 z naszych <a href=\"https:\/\/softinery.com\/pl\/katalog-szkolen\/\">kurs\u00f3w programowania<\/a>.<\/h2>\n\n\n<style>.wp-block-kadence-advancedheading.kt-adv-heading4971_73419c-e7, .wp-block-kadence-advancedheading.kt-adv-heading4971_73419c-e7[data-kb-block=\"kb-adv-heading4971_73419c-e7\"]{font-style:normal;}.wp-block-kadence-advancedheading.kt-adv-heading4971_73419c-e7 mark.kt-highlight, .wp-block-kadence-advancedheading.kt-adv-heading4971_73419c-e7[data-kb-block=\"kb-adv-heading4971_73419c-e7\"] mark.kt-highlight{font-style:normal;color:#f76a0c;-webkit-box-decoration-break:clone;box-decoration-break:clone;padding-top:0px;padding-right:0px;padding-bottom:0px;padding-left:0px;}.wp-block-kadence-advancedheading.kt-adv-heading4971_73419c-e7 img.kb-inline-image, .wp-block-kadence-advancedheading.kt-adv-heading4971_73419c-e7[data-kb-block=\"kb-adv-heading4971_73419c-e7\"] img.kb-inline-image{width:150px;vertical-align:baseline;}<\/style>\n<h2 class=\"kt-adv-heading4971_73419c-e7 wp-block-kadence-advancedheading\" data-kb-block=\"kb-adv-heading4971_73419c-e7\">Regresja liniowa: teoria<\/h2>\n\n\n\n<p>Regresja liniowa jest metod\u0105 statystyczn\u0105 s\u0142u\u017c\u0105c\u0105 do okre\u015blenia zale\u017cno\u015bci mi\u0119dzy jedn\u0105 zmienn\u0105 a drug\u0105 (regresja liniowa prosta) lub kilkoma zmiennymi niezale\u017cnymi a zmienn\u0105 zale\u017cn\u0105 (regresja liniowa wielokrotna). Metoda ta znajduje szerokie zastosowanie w r\u00f3\u017cnych dziedzinach, np. w naukach przyrodniczych, ekonomii, medycynie czy <a href=\"https:\/\/softinery.com\/pl\/katalog-szkolen\/wprowadzenie-do-obliczen-naukowych\/\">naukach technicznych<\/a>. Obecnie regresj\u0119 liniow\u0105 uwa\u017ca si\u0119 za jeden z algorytm\u00f3w nadzorowanych <a href=\"https:\/\/softinery.com\/pl\/model-predykcyjny-czym-jest-i-wykorzystac-go-w-przemysle\/\">uczenia maszynowego<\/a>.<\/p>\n\n\n\n<h3 class=\"wp-block-heading has--font-size\"><strong>Regresja liniowa prosta<\/strong><\/h3>\n\n\n\n<p>Regresja liniowa prosta jest najprostszym przypadkiem regresji liniowej, gdzie analizowany jest zwi\u0105zek mi\u0119dzy jedn\u0105 zmienn\u0105 niezale\u017cn\u0105 (<em>X<\/em>) a jedn\u0105 zmienn\u0105 zale\u017cn\u0105 (<em>Y<\/em>).<\/p>\n\n\n\n<h4 class=\"wp-block-heading\">Model regresji prostej<\/h4>\n\n\n\n<p class=\"has-text-align-center\"><em>Y<\/em> = <em>\u03b2<\/em><sub>0<\/sub> + <em>\u03b2<\/em><sub>1<\/sub><em>X<\/em><\/p>\n\n\n\n<p>gdzie:<\/p>\n\n\n\n<p><em>Y<\/em> &#8211; zmienna zale\u017cna,<\/p>\n\n\n\n<p><em>X<\/em> &#8211; zmienna niezale\u017cna,<\/p>\n\n\n\n<p><em>\u03b2<\/em><sub>0\u200b <\/sub>&#8211; wyraz wolny<\/p>\n\n\n\n<p><em>\u03b2<\/em><sub>1\u200b<\/sub> &#8211; wsp\u00f3\u0142czynnik regresji, okre\u015blaj\u0105cy nachylenie linii regresji,<\/p>\n\n\n\n<p><strong>Interpretacja wsp\u00f3\u0142czynnik\u00f3w<\/strong><\/p>\n\n\n\n<p><em>\u03b2<\/em><sub>0\u200b <\/sub> to warto\u015b\u0107 <em>Y<\/em>, gdy <em>X<\/em>=0, czyli punkt przeci\u0119cia linii regresji z osi\u0105 <em>Y<\/em>.<\/p>\n\n\n\n<p><em>\u03b2<\/em><sub>1\u200b<\/sub> to wsp\u00f3\u0142czynnik nachylenia, m\u00f3wi\u0105cy o zmianie Y w odpowiedzi na jednostkow\u0105 zmian\u0119 <em>X<\/em>.<\/p>\n\n\n\n<h4 class=\"wp-block-heading\">Przyk\u0142ady praktycznego zastosowania<\/h4>\n\n\n\n<p>O tym jak wa\u017cnym zagadnieniem jest regresja liniowa (w tym prosta) \u015bwiadczy\u0107 mo\u017ce fakt powstawania na ten temat podr\u0119cznik\u00f3w &#8211; np. <a href=\"https:\/\/books.google.pl\/books?hl=pl&amp;lr=&amp;id=xd0tNdFOOjcC&amp;oi=fnd&amp;pg=PR7&amp;dq=linear+regression&amp;ots=dW1wHqCyHS&amp;sig=YMzXa_Ab_mXKVo_edQGavU1dXs0&amp;redir_esc=y#v=onepage&amp;q=linear%20regression&amp;f=false\">&#8220;Applied linear regression&#8221;<\/a>. Do dziedzin, w kt\u00f3rych wykorzystywana jest regresja liniowa prosta zaliczamy:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Chemia &#8211; analiza wielu zjawisk chemicznych wykorzystuje regresj\u0119 liniow\u0105. Przyk\u0142adem mo\u017ce by\u0107 okre\u015blanie <a href=\"https:\/\/tools.softinery.com\">szybko\u015bci reakcji chemicznych<\/a>.<\/li>\n\n\n\n<li>Ekonomia &#8211; przyk\u0142adem zastosowania regresji liniowej prostej jest <a href=\"https:\/\/dl.acm.org\/doi\/abs\/10.1145\/3465631.3465778\">badanie wp\u0142ywu Covid-19 na PKB<\/a>.<\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\">Regresja liniowa wielokrotna<\/h3>\n\n\n\n<p>Regresja liniowa wielokrotna s\u0142u\u017cy do okre\u015blenia zale\u017cno\u015bci mi\u0119dzy jedn\u0105 zmienn\u0105 zale\u017cn\u0105 (<em>Y<\/em>) a dwiema lub wi\u0119cej zmiennymi niezale\u017cnymi <em>X<sub>1<\/sub><\/em>, <em>X<sub>2<\/sub><\/em>, &#8230;, <em>Xn<\/em>.<\/p>\n\n\n\n<h4 class=\"wp-block-heading\">Model regresji wielokrotnej<\/h4>\n\n\n\n<p class=\"has-text-align-center\"><em>Y<\/em> = <em>\u03b2<\/em><sub>0<\/sub> + <em>\u03b2<\/em><sub>1<\/sub><em>X<\/em><sub>1<\/sub> + \u03b2<sub>1<\/sub><em>X<\/em><sub>2<\/sub> + &#8230; + <em>\u03b2<\/em><sub><em>n<\/em><\/sub><em>X<\/em><sub><em>n<\/em><\/sub><\/p>\n\n\n\n<p>gdzie:<\/p>\n\n\n\n<p><em>Y<\/em> &#8211; zmienna zale\u017cna,<\/p>\n\n\n\n<p><em>X<\/em><sub>1<\/sub>, <em>X<\/em><sub>2<\/sub>, &#8230;, <em>Xn<\/em> &#8211; zmienne niezale\u017cne,<\/p>\n\n\n\n<p><em>\u03b2<\/em><sub>0\u200b <\/sub>&#8211; wyraz wolny<\/p>\n\n\n\n<p><em>\u03b2<\/em><sub>1<\/sub>, <em>\u03b2<\/em><sub>2<\/sub>, &#8230;, <em>\u03b2n<\/em> &#8211; wsp\u00f3\u0142czynniki regresji, okre\u015blaj\u0105ce wp\u0142yw poszczeg\u00f3lnych zmiennych na Y,<\/p>\n\n\n\n<p><strong>Interpretacja wsp\u00f3\u0142czynnik\u00f3w<\/strong><\/p>\n\n\n\n<p><em>\u03b2<\/em><sub>0\u200b <\/sub> to warto\u015b\u0107 <em>Y<\/em>, gdy wszystkie zmienne niezale\u017cne <em>X<sub>i<\/sub><\/em>\u200b s\u0105 r\u00f3wne zero.<\/p>\n\n\n\n<p><em>\u03b2<\/em><sub>1<\/sub>, <em>\u03b2<\/em><sub>2<\/sub>, &#8230;, <em>\u03b2<sub>n<\/sub><\/em> okre\u015blaj\u0105 jak zmienia si\u0119 <em>Y<\/em> w odpowiedzi na jednostkow\u0105 zmian\u0119 poszczeg\u00f3lnych <em>X<sub>i<\/sub><\/em>, rzy za\u0142o\u017ceniu, \u017ce pozosta\u0142e zmienne niezale\u017cne pozostaj\u0105 sta\u0142e.<\/p>\n\n\n\n<p>przy za\u0142o\u017ceniu, \u017ce pozosta\u0142e zmienne niezale\u017cne pozostaj\u0105 sta\u0142e<\/p>\n\n\n\n<h5 class=\"wp-block-heading\">Przyk\u0142ady praktycznego zastosowania<\/h5>\n\n\n\n<p>Mo\u017cemy znale\u017a\u0107 wiele przyk\u0142ad\u00f3w zastosowania regresji liniowej wielokrotnej. Poni\u017cej przedstawiamy wybrane:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Biologia &#8211; prognozowanie plonu j\u0119czmienia.<\/li>\n\n\n\n<li>Sport &#8211; identyfikacja czynnik\u00f3w wp\u0142ywaj\u0105cych na wyniki dru\u017cyny pi\u0142karskiej.<\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\">Dopasowanie wsp\u00f3\u0142czynnik\u00f3w w regresji liniowej<\/h3>\n\n\n\n<p>Dopasowanie wsp\u00f3\u0142czynnik\u00f3w jest kluczowym procesem w regresji liniowej, kt\u00f3ry polega na znalezieniu takich warto\u015bci, kt\u00f3re najlepiej odwzorowuj\u0105 zwi\u0105zek mi\u0119dzy zmiennymi niezale\u017cnymi a zmienn\u0105 zale\u017cn\u0105. Jedn\u0105 z metod s\u0142u\u017c\u0105cych do dopasowania wsp\u00f3\u0142czynnik\u00f3w jest metoda najmniejszych kwadrat\u00f3w.<\/p>\n\n\n\n<h4 class=\"wp-block-heading\">Metoda najmniejszych kwadrat\u00f3w<\/h4>\n\n\n\n<p>Metoda najmniejszych kwadrat\u00f3w polega na minimalizacji sumy kwadrat\u00f3w r\u00f3\u017cnic mi\u0119dzy rzeczywistymi warto\u015bciami zmiennej zale\u017cnej a warto\u015bciami przewidywanymi przez model, czyli sum\u0119 b\u0142\u0119du kwadratowego. W przypadku regresji liniowej prostej, minimalizujemy wyra\u017cenie:<\/p>\n\n\n<p class=\"ql-center-displayed-equation\" style=\"line-height: 49px;\"><span class=\"ql-right-eqno\"> &nbsp; <\/span><span class=\"ql-left-eqno\"> &nbsp; <\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/softinery.com\/pl\/wp-content\/ql-cache\/quicklatex.com-4905f65e73f3477dce3ecb834b795e09_l3.png\" height=\"49\" width=\"239\" class=\"ql-img-displayed-equation quicklatex-auto-format\" alt=\"&#92;&#91;&#83;&#83;&#69;&#61;&#92;&#115;&#117;&#109;&#95;&#123;&#105;&#61;&#49;&#125;&#94;&#123;&#109;&#125;&#40;&#89;&#95;&#105;&#45;&#40;&#92;&#98;&#101;&#116;&#97;&#95;&#48;&#43;&#92;&#98;&#101;&#116;&#97;&#95;&#49;&#88;&#95;&#49;&#32;&#41;&#41;&#94;&#50;&#92;&#93;\" title=\"Rendered by QuickLaTeX.com\"\/><\/p>\n\n\n\n<p>Dla regresji liniowej wielokrotnej minimalizujemy analogiczne wyra\u017cenie:<\/p>\n\n\n<p class=\"ql-center-displayed-equation\" style=\"line-height: 49px;\"><span class=\"ql-right-eqno\"> &nbsp; <\/span><span class=\"ql-left-eqno\"> &nbsp; <\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/softinery.com\/pl\/wp-content\/ql-cache\/quicklatex.com-9ee7d408c4b521986230a2ad1d5761bf_l3.png\" height=\"49\" width=\"401\" class=\"ql-img-displayed-equation quicklatex-auto-format\" alt=\"&#92;&#91;&#83;&#83;&#69;&#61;&#92;&#115;&#117;&#109;&#95;&#123;&#105;&#61;&#49;&#125;&#94;&#123;&#109;&#125;&#40;&#89;&#95;&#105;&#45;&#40;&#92;&#98;&#101;&#116;&#97;&#95;&#48;&#43;&#92;&#98;&#101;&#116;&#97;&#95;&#49;&#88;&#95;&#50;&#43;&#92;&#98;&#101;&#116;&#97;&#95;&#49;&#88;&#95;&#50;&#43;&#46;&#46;&#46;&#43;&#92;&#98;&#101;&#116;&#97;&#95;&#110;&#88;&#95;&#110;&#41;&#41;&#94;&#50;&#92;&#93;\" title=\"Rendered by QuickLaTeX.com\"\/><\/p>\n\n\n\n<p>gdzie <em>m<\/em> to liczba obserwacji.<\/p>\n\n\n\n<h4 class=\"wp-block-heading\"><strong>Ocena dopasowania w regresji liniowej<\/strong><\/h4>\n\n\n\n<p>Zale\u017cno\u015b\u0107 wyznaczona poprzez zastosowanie regresji liniowej powinna zosta\u0107 poddana ocenie, dzi\u0119ki kt\u00f3rej b\u0119dziemy wiedzie\u0107, jak dobrze model regresji liniowej pasuje do danych. Istnieje kilka popularnych miar s\u0142u\u017c\u0105cych do tego celu. Jedn\u0105 z nich jest wsp\u00f3\u0142czynnik determinacji R<sup>2<\/sup>.<\/p>\n\n\n\n<h5 class=\"wp-block-heading\">Wsp\u00f3\u0142czynnik determinacji R<sup>2<\/sup><\/h5>\n\n\n\n<p>Wsp\u00f3\u0142czynnik determinacji <em>R<\/em><sup>2<\/sup> jest definiowany jako stosunek wyja\u015bnianej zmienno\u015bci do ca\u0142kowitej zmienno\u015bci zmiennej zale\u017cnej:<\/p>\n\n\n<p class=\"ql-center-displayed-equation\" style=\"line-height: 46px;\"><span class=\"ql-right-eqno\"> &nbsp; <\/span><span class=\"ql-left-eqno\"> &nbsp; <\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/softinery.com\/pl\/wp-content\/ql-cache\/quicklatex.com-5cdbfe104432b8e688ecf8db6f205b51_l3.png\" height=\"46\" width=\"194\" class=\"ql-img-displayed-equation quicklatex-auto-format\" alt=\"&#92;&#91;&#82;&#94;&#50;&#61;&#49;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#115;&#117;&#109;&#95;&#123;&#105;&#61;&#49;&#125;&#94;&#123;&#109;&#125;&#40;&#89;&#95;&#105;&#45;&#92;&#104;&#97;&#116;&#123;&#89;&#95;&#105;&#125;&#41;&#94;&#50;&#125;&#123;&#92;&#115;&#117;&#109;&#95;&#123;&#105;&#61;&#49;&#125;&#94;&#123;&#109;&#125;&#40;&#89;&#95;&#105;&#45;&#92;&#98;&#97;&#114;&#123;&#89;&#125;&#41;&#94;&#50;&#125;&#92;&#93;\" title=\"Rendered by QuickLaTeX.com\"\/><\/p>\n\n\n\n<p>gdzie:<\/p>\n\n\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/softinery.com\/pl\/wp-content\/ql-cache\/quicklatex.com-1ce9a015a804a13965be499e5d05726c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#89;&#95;&#105;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"15\" style=\"vertical-align: -3px;\"\/> to obserwowane warto\u015bci zmiennej zale\u017cnej,<br \/>\n<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/softinery.com\/pl\/wp-content\/ql-cache\/quicklatex.com-adbe6d20249e700ea031c1cccc3e9e4c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#104;&#97;&#116;&#123;&#89;&#95;&#105;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"15\" style=\"vertical-align: -3px;\"\/> to przewidywane warto\u015bci zmiennej zale\u017cnej na podstawie modelu regresji,<br \/>\n<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/softinery.com\/pl\/wp-content\/ql-cache\/quicklatex.com-3228fab9e1be66d6be485f6b7cdebe61_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#97;&#114;&#123;&#89;&#95;&#105;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"15\" style=\"vertical-align: -3px;\"\/> to \u015brednia warto\u015b\u0107 zmiennej zale\u017cnej,<br \/>\n<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/softinery.com\/pl\/wp-content\/ql-cache\/quicklatex.com-6b41df788161942c6f98604d37de8098_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#109;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"15\" style=\"vertical-align: 0px;\"\/> &#8211; to liczba obserwacji.<\/p>\n\n\n\n<p>Warto\u015b\u0107 <em>R<\/em><sup>2<\/sup> mie\u015bci si\u0119 w przedziale od 0 do 1, gdzie warto\u015b\u0107 bli\u017cej 1 oznacza lepsze dopasowanie modelu do danych.<\/p>\n\n\n\n<h5 class=\"wp-block-heading\">\u015aredni b\u0142\u0105d kwadratowy<\/h5>\n\n\n\n<p>\u015aredni b\u0142\u0105d kwadratowy (MSE &#8211; mean square error) to \u015brednia kwadrat\u00f3w r\u00f3\u017cnic mi\u0119dzy warto\u015bci\u0105 zaobserwowan\u0105 w badaniu statystycznym a warto\u015bciami przewidywanymi na podstawie modelu. Por\u00f3wnuj\u0105c obserwacje z warto\u015bciami przewidywanymi, konieczne jest podniesienie r\u00f3\u017cnic do kwadratu, poniewa\u017c niekt\u00f3re warto\u015bci danych b\u0119d\u0105 wi\u0119ksze od przewidywa\u0144 (a wi\u0119c ich r\u00f3\u017cnice b\u0119d\u0105 dodatnie), a inne b\u0119d\u0105 mniejsze (a wi\u0119c ich r\u00f3\u017cnice b\u0119d\u0105 ujemne). \u015aredni b\u0142\u0105d kwadratowy obliczamy z zale\u017cno\u015bci:<\/p>\n\n\n<p class=\"ql-center-displayed-equation\" style=\"line-height: 49px;\"><span class=\"ql-right-eqno\"> &nbsp; <\/span><span class=\"ql-left-eqno\"> &nbsp; <\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/softinery.com\/pl\/wp-content\/ql-cache\/quicklatex.com-0fe5f58c7d47d3b1eedeea8c806d3fa5_l3.png\" height=\"49\" width=\"191\" class=\"ql-img-displayed-equation quicklatex-auto-format\" alt=\"&#92;&#91;&#77;&#83;&#69;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#109;&#125;&#92;&#115;&#117;&#109;&#95;&#123;&#105;&#61;&#49;&#125;&#94;&#123;&#109;&#125;&#40;&#89;&#95;&#105;&#45;&#92;&#104;&#97;&#116;&#123;&#89;&#95;&#105;&#125;&#41;&#94;&#50;&#92;&#93;\" title=\"Rendered by QuickLaTeX.com\"\/><\/p>\n\n\n\n<h2 class=\"wp-block-heading\">Podsumowanie<\/h2>\n\n\n\n<p class=\"has-24292-e-color has-text-color\">Z tego artyku\u0142u dowiedzia\u0142e\u015b si\u0119 czym jest regresja liniowa i jak napisa\u0107 program, kt\u00f3ry tworzy model regresji liniowej w Pythonie. W szczeg\u00f3lno\u015bci nauczy\u0142e\u015b si\u0119 jak:<\/p>\n\n\n\n<ul class=\"has-24292-e-color has-text-color wp-block-list\">\n<li>wykorzysta\u0107 bibliotek\u0119 scikit-learn do stworzenia modelu prostej regresji liniowej,<\/li>\n\n\n\n<li>wyznaczy\u0107 wsp\u00f3\u0142czynniki linii prostej oraz wsp\u00f3\u0142czynnik dopasowania R<sup>2<\/sup>,<\/li>\n\n\n\n<li><a href=\"https:\/\/softinery.com\/pl\/blog\/wykresy-python\/\">narysowa\u0107 wykres<\/a> z danymi oraz dopasowan\u0105 prost\u0105.<\/li>\n<\/ul>\n\n\n\n<p>Om\u00f3wione zosta\u0142y teoretyczne podstawy regresji liniowej wielokrotnej i prostej.<\/p>\n\n\n\n<p class=\"has-24292-e-color has-text-color has-medium-font-size\"><strong>Odno\u015bniki<\/strong><\/p>\n\n\n\n<ol class=\"has-24292-e-color has-text-color wp-block-list\">\n<li><a href=\"https:\/\/www.tandfonline.com\/doi\/abs\/10.1080\/07350015.1996.10524677\">On Using Linear Regressions in Welfare Economics: Journal of Business &amp; Economic Statistics: Vol 14, No 4 (tandfonline.com)<\/a><\/li>\n\n\n\n<li><a href=\"https:\/\/dl.acm.org\/doi\/10.5555\/1717831\">Linear Regression Analysis: Theory and Computing | Guide books | ACM Digital Library<\/a><\/li>\n\n\n\n<li><a href=\"https:\/\/aclanthology.org\/W11-1515.pdf\">Nguyen, Dong, Noah A. Smith, and Carolyn Rose. &#8220;Author age prediction from text using linear regression.&#8221;&nbsp;<em>Proceedings of the 5th ACL-HLT workshop on language technology for cultural heritage, social sciences, and humanities<\/em>. 2011.<\/a><\/li>\n<\/ol>\n<\/div><\/div>\n\n\n<style>.kadence-column4971_9427cc-85 > .kt-inside-inner-col{min-height:1300px;}.kadence-column4971_9427cc-85 > .kt-inside-inner-col,.kadence-column4971_9427cc-85 > .kt-inside-inner-col:before{border-top-left-radius:0px;border-top-right-radius:0px;border-bottom-right-radius:0px;border-bottom-left-radius:0px;}.kadence-column4971_9427cc-85 > .kt-inside-inner-col{column-gap:var(--global-kb-gap-sm, 1rem);}.kadence-column4971_9427cc-85 > .kt-inside-inner-col{flex-direction:column;}.kadence-column4971_9427cc-85 > .kt-inside-inner-col > .aligncenter{width:100%;}.kadence-column4971_9427cc-85 > .kt-inside-inner-col:before{opacity:0.3;}.kadence-column4971_9427cc-85{position:relative;}@media all and (max-width: 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767px){.kadence-column8952_2cd610-bf > .kt-inside-inner-col{flex-direction:column;justify-content:center;}}.kadence-column8952_2cd610-bf {}<\/style>\n<div class=\"wp-block-kadence-column kadence-column8952_2cd610-bf\"><div class=\"kt-inside-inner-col\"><style>.kadence-column8952_9b934e-2c > .kt-inside-inner-col,.kadence-column8952_9b934e-2c > .kt-inside-inner-col:before{border-top-left-radius:0px;border-top-right-radius:0px;border-bottom-right-radius:0px;border-bottom-left-radius:0px;}.kadence-column8952_9b934e-2c > .kt-inside-inner-col{column-gap:var(--global-kb-gap-sm, 1rem);}.kadence-column8952_9b934e-2c > .kt-inside-inner-col{flex-direction:column;}.kadence-column8952_9b934e-2c > .kt-inside-inner-col > .aligncenter{width:100%;}.kadence-column8952_9b934e-2c > .kt-inside-inner-col:before{opacity:0.3;}.kadence-column8952_9b934e-2c{position:relative;}@media all and (max-width: 1024px){.kadence-column8952_9b934e-2c > .kt-inside-inner-col{flex-direction:column;justify-content:center;}}@media all and (max-width: 767px){.kadence-column8952_9b934e-2c > .kt-inside-inner-col{flex-direction:column;justify-content:center;}}<\/style>\n<div class=\"wp-block-kadence-column kadence-column8952_9b934e-2c\"><div class=\"kt-inside-inner-col\">\n<div class=\"wp-block-columns is-layout-flex wp-container-core-columns-is-layout-9d6595d7 wp-block-columns-is-layout-flex\">\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\" style=\"flex-basis:70%\">\n<div class=\"wp-block-columns is-layout-flex wp-container-core-columns-is-layout-9d6595d7 wp-block-columns-is-layout-flex\">\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\" style=\"flex-basis:40%\"><style>.kadence-column8952_448464-bd > .kt-inside-inner-col{display:flex;}.kadence-column8952_448464-bd > .kt-inside-inner-col,.kadence-column8952_448464-bd > .kt-inside-inner-col:before{border-top-left-radius:0px;border-top-right-radius:0px;border-bottom-right-radius:0px;border-bottom-left-radius:0px;}.kadence-column8952_448464-bd > .kt-inside-inner-col{column-gap:var(--global-kb-gap-sm, 1rem);}.kadence-column8952_448464-bd > .kt-inside-inner-col{flex-direction:column;justify-content:center;align-items:center;}.kadence-column8952_448464-bd > .kt-inside-inner-col > .kb-image-is-ratio-size{align-self:stretch;}.kadence-column8952_448464-bd > .kt-inside-inner-col > .wp-block-kadence-advancedgallery{align-self:stretch;}.kadence-column8952_448464-bd > .kt-inside-inner-col > .aligncenter{width:100%;}.kt-row-column-wrap > .kadence-column8952_448464-bd{align-self:center;}.kt-inner-column-height-full:not(.kt-has-1-columns) > .wp-block-kadence-column.kadence-column8952_448464-bd{align-self:auto;}.kt-inner-column-height-full:not(.kt-has-1-columns) > .wp-block-kadence-column.kadence-column8952_448464-bd > .kt-inside-inner-col{flex-direction:column;justify-content:center;}.kadence-column8952_448464-bd > .kt-inside-inner-col:before{opacity:0.3;}.kadence-column8952_448464-bd{position:relative;}@media all and (max-width: 1024px){.kt-row-column-wrap > .kadence-column8952_448464-bd{align-self:center;}}@media all and (max-width: 1024px){.kt-inner-column-height-full:not(.kt-has-1-columns) > .wp-block-kadence-column.kadence-column8952_448464-bd{align-self:auto;}}@media all and (max-width: 1024px){.kt-inner-column-height-full:not(.kt-has-1-columns) > .wp-block-kadence-column.kadence-column8952_448464-bd > .kt-inside-inner-col{flex-direction:column;justify-content:center;}}@media all and (max-width: 1024px){.kadence-column8952_448464-bd > .kt-inside-inner-col{flex-direction:column;justify-content:center;align-items:center;}}@media all and (max-width: 767px){.kt-row-column-wrap > .kadence-column8952_448464-bd{align-self:center;}.kt-inner-column-height-full:not(.kt-has-1-columns) > .wp-block-kadence-column.kadence-column8952_448464-bd{align-self:auto;}.kt-inner-column-height-full:not(.kt-has-1-columns) > .wp-block-kadence-column.kadence-column8952_448464-bd > .kt-inside-inner-col{flex-direction:column;justify-content:center;}.kadence-column8952_448464-bd > .kt-inside-inner-col{flex-direction:column;justify-content:center;align-items:center;}}<\/style>\n<div class=\"wp-block-kadence-column kadence-column8952_448464-bd\"><div class=\"kt-inside-inner-col\"><style>.kb-image8952_2b2db0-f3.kb-image-is-ratio-size, .kb-image8952_2b2db0-f3 .kb-image-is-ratio-size{max-width:50px;width:100%;}.wp-block-kadence-column > .kt-inside-inner-col > .kb-image8952_2b2db0-f3.kb-image-is-ratio-size, .wp-block-kadence-column > .kt-inside-inner-col > .kb-image8952_2b2db0-f3 .kb-image-is-ratio-size{align-self:unset;}.kb-image8952_2b2db0-f3{max-width:50px;}.image-is-svg.kb-image8952_2b2db0-f3{-webkit-flex:0 1 100%;flex:0 1 100%;}.image-is-svg.kb-image8952_2b2db0-f3 img{width:100%;}.kb-image8952_2b2db0-f3 .kb-image-has-overlay:after{opacity:0.3;}<\/style>\n<figure class=\"wp-block-kadence-image kb-image8952_2b2db0-f3 size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"288\" height=\"208\" src=\"https:\/\/softinery.com\/pl\/wp-content\/uploads\/sites\/5\/2024\/06\/computer-2.png\" alt=\"python szkolenia\" class=\"kb-img wp-image-8956\"\/><\/figure>\n<\/div><\/div>\n<\/div>\n\n\n\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\" style=\"flex-basis:60%\"><style>.kb-image8952_cc6008-a1.kb-image-is-ratio-size, .kb-image8952_cc6008-a1 .kb-image-is-ratio-size{max-width:100px;width:100%;}.wp-block-kadence-column > .kt-inside-inner-col > .kb-image8952_cc6008-a1.kb-image-is-ratio-size, .wp-block-kadence-column > .kt-inside-inner-col > .kb-image8952_cc6008-a1 .kb-image-is-ratio-size{align-self:unset;}.kb-image8952_cc6008-a1 figure{max-width:100px;}.kb-image8952_cc6008-a1 .image-is-svg, .kb-image8952_cc6008-a1 .image-is-svg img{width:100%;}.kb-image8952_cc6008-a1 .kb-image-has-overlay:after{opacity:0.3;}<\/style>\n<div class=\"wp-block-kadence-image kb-image8952_cc6008-a1\"><figure class=\"aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"276\" height=\"50\" src=\"https:\/\/softinery.com\/pl\/wp-content\/uploads\/sites\/5\/2024\/06\/level_beginner.png\" alt=\"pozaiom pocz\u0105tkuj\u0105cy\" class=\"kb-img wp-image-8958\"\/><\/figure><\/div>\n\n\n\n<p class=\"has-text-align-center margin_zero\" style=\"font-size:0.75em\">POCZ\u0104TKUJ\u0104CY<\/p>\n<\/div>\n<\/div>\n<\/div>\n\n\n\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\" style=\"flex-basis:30%\">\n<p class=\"has-text-align-center has-medium-font-size\">1600 z\u0142<\/p>\n<\/div>\n<\/div>\n<\/div><\/div>\n\n\n<style>.kadence-column8952_e8c9c1-24 > .kt-inside-inner-col,.kadence-column8952_e8c9c1-24 > .kt-inside-inner-col:before{border-top-left-radius:0px;border-top-right-radius:0px;border-bottom-right-radius:0px;border-bottom-left-radius:0px;}.kadence-column8952_e8c9c1-24 > .kt-inside-inner-col{column-gap:var(--global-kb-gap-sm, 1rem);}.kadence-column8952_e8c9c1-24 > .kt-inside-inner-col{flex-direction:column;}.kadence-column8952_e8c9c1-24 > .kt-inside-inner-col > .aligncenter{width:100%;}.kadence-column8952_e8c9c1-24 > .kt-inside-inner-col:before{opacity:0.3;}.kadence-column8952_e8c9c1-24{position:relative;}.kadence-column8952_e8c9c1-24, .kt-inside-inner-col > .kadence-column8952_e8c9c1-24:not(.specificity){margin-top:-30px;}@media all and (max-width: 1024px){.kadence-column8952_e8c9c1-24 > .kt-inside-inner-col{flex-direction:column;justify-content:center;}}@media all and (max-width: 767px){.kadence-column8952_e8c9c1-24 > .kt-inside-inner-col{flex-direction:column;justify-content:center;}}<\/style>\n<div class=\"wp-block-kadence-column kadence-column8952_e8c9c1-24\"><div class=\"kt-inside-inner-col\">\n<p class=\"has-medium-font-size\"><strong>Kurs programowania Python Online dla pocz\u0105tkuj\u0105cych<\/strong> (Zdalnie)<\/p>\n<\/div><\/div>\n\n\n<style>.kadence-column8952_e52106-1c > .kt-inside-inner-col,.kadence-column8952_e52106-1c > .kt-inside-inner-col:before{border-top-left-radius:0px;border-top-right-radius:0px;border-bottom-right-radius:0px;border-bottom-left-radius:0px;}.kadence-column8952_e52106-1c > .kt-inside-inner-col{column-gap:var(--global-kb-gap-sm, 1rem);}.kadence-column8952_e52106-1c > .kt-inside-inner-col{flex-direction:column;}.kadence-column8952_e52106-1c > .kt-inside-inner-col > .aligncenter{width:100%;}.kadence-column8952_e52106-1c > .kt-inside-inner-col:before{opacity:0.3;}.kadence-column8952_e52106-1c{position:relative;}@media all and (max-width: 1024px){.kadence-column8952_e52106-1c > .kt-inside-inner-col{flex-direction:column;justify-content:center;}}@media all and (max-width: 767px){.kadence-column8952_e52106-1c > .kt-inside-inner-col{flex-direction:column;justify-content:center;}}<\/style>\n<div class=\"wp-block-kadence-column kadence-column8952_e52106-1c\"><div class=\"kt-inside-inner-col\"><style>.wp-block-kadence-advancedbtn.kb-btns8952_0b9057-c8{gap:var(--global-kb-gap-xs, 0.5rem );justify-content:center;align-items:center;}.kt-btns8952_0b9057-c8 .kt-button{font-weight:normal;font-style:normal;}.kt-btns8952_0b9057-c8 .kt-btn-wrap-0{margin-right:5px;}.wp-block-kadence-advancedbtn.kt-btns8952_0b9057-c8 .kt-btn-wrap-0 .kt-button{color:#555555;border-color:#555555;}.wp-block-kadence-advancedbtn.kt-btns8952_0b9057-c8 .kt-btn-wrap-0 .kt-button:hover, .wp-block-kadence-advancedbtn.kt-btns8952_0b9057-c8 .kt-btn-wrap-0 .kt-button:focus{color:#ffffff;border-color:#444444;}.wp-block-kadence-advancedbtn.kt-btns8952_0b9057-c8 .kt-btn-wrap-0 .kt-button::before{display:none;}.wp-block-kadence-advancedbtn.kt-btns8952_0b9057-c8 .kt-btn-wrap-0 .kt-button:hover, .wp-block-kadence-advancedbtn.kt-btns8952_0b9057-c8 .kt-btn-wrap-0 .kt-button:focus{background:#444444;}<\/style>\n<div class=\"wp-block-kadence-advancedbtn kb-buttons-wrap kb-btns8952_0b9057-c8\"><style>ul.menu .wp-block-kadence-advancedbtn .kb-btn8952_c5591b-79.kb-button{width:initial;}.wp-block-kadence-advancedbtn .kb-btn8952_c5591b-79.kb-button{color:var(--nv-site-bg);background:var(--nv-primary-accent);}<\/style><a class=\"kb-button kt-button button kb-btn8952_c5591b-79 kt-btn-size-standard kt-btn-width-type-auto kb-btn-global-fill  kt-btn-has-text-true kt-btn-has-svg-false  wp-block-kadence-singlebtn\" href=\"https:\/\/softinery.com\/pl\/katalog-szkolen\/kurs-python-dla-poczatkujacych\/\"><span class=\"kt-btn-inner-text\">Zobacz szczeg\u00f3\u0142y szkolenia<\/span><\/a><\/div>\n<\/div><\/div>\n<\/div><\/div>\n\n\n<style>.kadence-column8952_3cedb4-64 > .kt-inside-inner-col{padding-top:var(--global-kb-spacing-xs, 1rem);padding-right:var(--global-kb-spacing-xs, 1rem);padding-bottom:var(--global-kb-spacing-xs, 1rem);padding-left:var(--global-kb-spacing-xs, 1rem);}.kadence-column8952_3cedb4-64 > .kt-inside-inner-col{box-shadow:3px 3px 14px 0px var(--nv-c-2);}.kadence-column8952_3cedb4-64 > .kt-inside-inner-col,.kadence-column8952_3cedb4-64 > .kt-inside-inner-col:before{border-top-left-radius:0px;border-top-right-radius:0px;border-bottom-right-radius:0px;border-bottom-left-radius:0px;}.kadence-column8952_3cedb4-64 > .kt-inside-inner-col{column-gap:var(--global-kb-gap-sm, 1rem);}.kadence-column8952_3cedb4-64 > .kt-inside-inner-col{flex-direction:column;}.kadence-column8952_3cedb4-64 > .kt-inside-inner-col > .aligncenter{width:100%;}.kadence-column8952_3cedb4-64 > .kt-inside-inner-col:before{opacity:0.3;}.kadence-column8952_3cedb4-64{position:relative;}.kadence-column8952_3cedb4-64, .kt-inside-inner-col > .kadence-column8952_3cedb4-64:not(.specificity){margin-top:var(--global-kb-spacing-md, 2rem);margin-right:var(--global-kb-spacing-xxs, 0.5rem);margin-bottom:var(--global-kb-spacing-xs, 1rem);margin-left:var(--global-kb-spacing-xxs, 0.5rem);}@media all and (max-width: 1024px){.kadence-column8952_3cedb4-64 > .kt-inside-inner-col{flex-direction:column;justify-content:center;}}@media all and (max-width: 767px){.kadence-column8952_3cedb4-64 > .kt-inside-inner-col{flex-direction:column;justify-content:center;}}.kadence-column8952_3cedb4-64 {}<\/style>\n<div class=\"wp-block-kadence-column kadence-column8952_3cedb4-64\"><div class=\"kt-inside-inner-col\"><style>.kadence-column8952_44a3cc-33 > .kt-inside-inner-col,.kadence-column8952_44a3cc-33 > .kt-inside-inner-col:before{border-top-left-radius:0px;border-top-right-radius:0px;border-bottom-right-radius:0px;border-bottom-left-radius:0px;}.kadence-column8952_44a3cc-33 > .kt-inside-inner-col{column-gap:var(--global-kb-gap-sm, 1rem);}.kadence-column8952_44a3cc-33 > .kt-inside-inner-col{flex-direction:column;}.kadence-column8952_44a3cc-33 > .kt-inside-inner-col > .aligncenter{width:100%;}.kadence-column8952_44a3cc-33 > .kt-inside-inner-col:before{opacity:0.3;}.kadence-column8952_44a3cc-33{position:relative;}@media all and (max-width: 1024px){.kadence-column8952_44a3cc-33 > .kt-inside-inner-col{flex-direction:column;justify-content:center;}}@media all and (max-width: 767px){.kadence-column8952_44a3cc-33 > .kt-inside-inner-col{flex-direction:column;justify-content:center;}}<\/style>\n<div class=\"wp-block-kadence-column kadence-column8952_44a3cc-33\"><div class=\"kt-inside-inner-col\">\n<div class=\"wp-block-columns is-layout-flex wp-container-core-columns-is-layout-9d6595d7 wp-block-columns-is-layout-flex\">\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\" style=\"flex-basis:70%\">\n<div class=\"wp-block-columns is-layout-flex wp-container-core-columns-is-layout-9d6595d7 wp-block-columns-is-layout-flex\">\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\" style=\"flex-basis:40%\"><style>.kadence-column8952_bd53c7-a7 > .kt-inside-inner-col{display:flex;}.kadence-column8952_bd53c7-a7 > .kt-inside-inner-col,.kadence-column8952_bd53c7-a7 > .kt-inside-inner-col:before{border-top-left-radius:0px;border-top-right-radius:0px;border-bottom-right-radius:0px;border-bottom-left-radius:0px;}.kadence-column8952_bd53c7-a7 > .kt-inside-inner-col{column-gap:var(--global-kb-gap-sm, 1rem);}.kadence-column8952_bd53c7-a7 > .kt-inside-inner-col{flex-direction:column;justify-content:center;align-items:center;}.kadence-column8952_bd53c7-a7 > .kt-inside-inner-col > .kb-image-is-ratio-size{align-self:stretch;}.kadence-column8952_bd53c7-a7 > .kt-inside-inner-col > .wp-block-kadence-advancedgallery{align-self:stretch;}.kadence-column8952_bd53c7-a7 > .kt-inside-inner-col > .aligncenter{width:100%;}.kt-row-column-wrap > .kadence-column8952_bd53c7-a7{align-self:center;}.kt-inner-column-height-full:not(.kt-has-1-columns) > .wp-block-kadence-column.kadence-column8952_bd53c7-a7{align-self:auto;}.kt-inner-column-height-full:not(.kt-has-1-columns) > .wp-block-kadence-column.kadence-column8952_bd53c7-a7 > .kt-inside-inner-col{flex-direction:column;justify-content:center;}.kadence-column8952_bd53c7-a7 > .kt-inside-inner-col:before{opacity:0.3;}.kadence-column8952_bd53c7-a7{position:relative;}@media all and (max-width: 1024px){.kt-row-column-wrap > .kadence-column8952_bd53c7-a7{align-self:center;}}@media all and (max-width: 1024px){.kt-inner-column-height-full:not(.kt-has-1-columns) > .wp-block-kadence-column.kadence-column8952_bd53c7-a7{align-self:auto;}}@media all and (max-width: 1024px){.kt-inner-column-height-full:not(.kt-has-1-columns) > .wp-block-kadence-column.kadence-column8952_bd53c7-a7 > .kt-inside-inner-col{flex-direction:column;justify-content:center;}}@media all and (max-width: 1024px){.kadence-column8952_bd53c7-a7 > .kt-inside-inner-col{flex-direction:column;justify-content:center;align-items:center;}}@media all and (max-width: 767px){.kt-row-column-wrap > .kadence-column8952_bd53c7-a7{align-self:center;}.kt-inner-column-height-full:not(.kt-has-1-columns) > .wp-block-kadence-column.kadence-column8952_bd53c7-a7{align-self:auto;}.kt-inner-column-height-full:not(.kt-has-1-columns) > .wp-block-kadence-column.kadence-column8952_bd53c7-a7 > .kt-inside-inner-col{flex-direction:column;justify-content:center;}.kadence-column8952_bd53c7-a7 > .kt-inside-inner-col{flex-direction:column;justify-content:center;align-items:center;}}<\/style>\n<div class=\"wp-block-kadence-column kadence-column8952_bd53c7-a7\"><div class=\"kt-inside-inner-col\"><style>.kb-image8952_82f697-2d.kb-image-is-ratio-size, .kb-image8952_82f697-2d .kb-image-is-ratio-size{max-width:50px;width:100%;}.wp-block-kadence-column > .kt-inside-inner-col > .kb-image8952_82f697-2d.kb-image-is-ratio-size, .wp-block-kadence-column > .kt-inside-inner-col > .kb-image8952_82f697-2d .kb-image-is-ratio-size{align-self:unset;}.kb-image8952_82f697-2d{max-width:50px;}.image-is-svg.kb-image8952_82f697-2d{-webkit-flex:0 1 100%;flex:0 1 100%;}.image-is-svg.kb-image8952_82f697-2d img{width:100%;}.kb-image8952_82f697-2d .kb-image-has-overlay:after{opacity:0.3;}<\/style>\n<figure class=\"wp-block-kadence-image kb-image8952_82f697-2d size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"288\" height=\"208\" src=\"https:\/\/softinery.com\/pl\/wp-content\/uploads\/sites\/5\/2024\/06\/computer-2.png\" alt=\"python szkolenia\" class=\"kb-img wp-image-8956\"\/><\/figure>\n<\/div><\/div>\n<\/div>\n\n\n\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\" style=\"flex-basis:60%\"><style>.kb-image8952_815edb-63.kb-image-is-ratio-size, .kb-image8952_815edb-63 .kb-image-is-ratio-size{max-width:100px;width:100%;}.wp-block-kadence-column > .kt-inside-inner-col > .kb-image8952_815edb-63.kb-image-is-ratio-size, .wp-block-kadence-column > .kt-inside-inner-col > .kb-image8952_815edb-63 .kb-image-is-ratio-size{align-self:unset;}.kb-image8952_815edb-63 figure{max-width:100px;}.kb-image8952_815edb-63 .image-is-svg, .kb-image8952_815edb-63 .image-is-svg img{width:100%;}.kb-image8952_815edb-63 .kb-image-has-overlay:after{opacity:0.3;}<\/style>\n<div class=\"wp-block-kadence-image kb-image8952_815edb-63\"><figure class=\"aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"276\" height=\"50\" src=\"https:\/\/softinery.com\/pl\/wp-content\/uploads\/sites\/5\/2024\/06\/level_intermediate.png\" alt=\"\" class=\"kb-img wp-image-9027\"\/><\/figure><\/div>\n\n\n\n<p class=\"has-text-align-center margin_zero\" style=\"font-size:0.75em\">\u015aREDNIOZAAWANSOWANY<\/p>\n<\/div>\n<\/div>\n<\/div>\n\n\n\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\" style=\"flex-basis:30%\">\n<p class=\"has-text-align-center has-medium-font-size\">2200 z\u0142<\/p>\n<\/div>\n<\/div>\n<\/div><\/div>\n\n\n<style>.kadence-column8952_62efda-1e > .kt-inside-inner-col,.kadence-column8952_62efda-1e > .kt-inside-inner-col:before{border-top-left-radius:0px;border-top-right-radius:0px;border-bottom-right-radius:0px;border-bottom-left-radius:0px;}.kadence-column8952_62efda-1e > .kt-inside-inner-col{column-gap:var(--global-kb-gap-sm, 1rem);}.kadence-column8952_62efda-1e > .kt-inside-inner-col{flex-direction:column;}.kadence-column8952_62efda-1e > .kt-inside-inner-col > .aligncenter{width:100%;}.kadence-column8952_62efda-1e > .kt-inside-inner-col:before{opacity:0.3;}.kadence-column8952_62efda-1e{position:relative;}.kadence-column8952_62efda-1e, .kt-inside-inner-col > .kadence-column8952_62efda-1e:not(.specificity){margin-top:-30px;}@media all and (max-width: 1024px){.kadence-column8952_62efda-1e > .kt-inside-inner-col{flex-direction:column;justify-content:center;}}@media all and (max-width: 767px){.kadence-column8952_62efda-1e > .kt-inside-inner-col{flex-direction:column;justify-content:center;}}<\/style>\n<div class=\"wp-block-kadence-column kadence-column8952_62efda-1e\"><div class=\"kt-inside-inner-col\">\n<p class=\"has-medium-font-size\"><strong>Python Data Science<\/strong> (Zdalnie)<\/p>\n<\/div><\/div>\n\n\n<style>.kadence-column8952_1853fa-8a > .kt-inside-inner-col,.kadence-column8952_1853fa-8a > .kt-inside-inner-col:before{border-top-left-radius:0px;border-top-right-radius:0px;border-bottom-right-radius:0px;border-bottom-left-radius:0px;}.kadence-column8952_1853fa-8a > .kt-inside-inner-col{column-gap:var(--global-kb-gap-sm, 1rem);}.kadence-column8952_1853fa-8a > .kt-inside-inner-col{flex-direction:column;}.kadence-column8952_1853fa-8a > .kt-inside-inner-col > .aligncenter{width:100%;}.kadence-column8952_1853fa-8a > .kt-inside-inner-col:before{opacity:0.3;}.kadence-column8952_1853fa-8a{position:relative;}@media all and (max-width: 1024px){.kadence-column8952_1853fa-8a > .kt-inside-inner-col{flex-direction:column;justify-content:center;}}@media all and (max-width: 767px){.kadence-column8952_1853fa-8a > .kt-inside-inner-col{flex-direction:column;justify-content:center;}}<\/style>\n<div class=\"wp-block-kadence-column kadence-column8952_1853fa-8a\"><div class=\"kt-inside-inner-col\"><style>.wp-block-kadence-advancedbtn.kb-btns8952_a1a89d-47{gap:var(--global-kb-gap-xs, 0.5rem );justify-content:center;align-items:center;}.kt-btns8952_a1a89d-47 .kt-button{font-weight:normal;font-style:normal;}.kt-btns8952_a1a89d-47 .kt-btn-wrap-0{margin-right:5px;}.wp-block-kadence-advancedbtn.kt-btns8952_a1a89d-47 .kt-btn-wrap-0 .kt-button{color:#555555;border-color:#555555;}.wp-block-kadence-advancedbtn.kt-btns8952_a1a89d-47 .kt-btn-wrap-0 .kt-button:hover, .wp-block-kadence-advancedbtn.kt-btns8952_a1a89d-47 .kt-btn-wrap-0 .kt-button:focus{color:#ffffff;border-color:#444444;}.wp-block-kadence-advancedbtn.kt-btns8952_a1a89d-47 .kt-btn-wrap-0 .kt-button::before{display:none;}.wp-block-kadence-advancedbtn.kt-btns8952_a1a89d-47 .kt-btn-wrap-0 .kt-button:hover, .wp-block-kadence-advancedbtn.kt-btns8952_a1a89d-47 .kt-btn-wrap-0 .kt-button:focus{background:#444444;}<\/style>\n<div class=\"wp-block-kadence-advancedbtn kb-buttons-wrap kb-btns8952_a1a89d-47\"><style>ul.menu .wp-block-kadence-advancedbtn .kb-btn8952_d0a5ef-01.kb-button{width:initial;}.wp-block-kadence-advancedbtn .kb-btn8952_d0a5ef-01.kb-button{color:var(--nv-site-bg);background:var(--nv-primary-accent);}<\/style><a class=\"kb-button kt-button button kb-btn8952_d0a5ef-01 kt-btn-size-standard kt-btn-width-type-auto kb-btn-global-fill  kt-btn-has-text-true kt-btn-has-svg-false  wp-block-kadence-singlebtn\" href=\"https:\/\/softinery.com\/pl\/katalog-szkolen\/python-data-science-kurs\/\"><span class=\"kt-btn-inner-text\">Zobacz szczeg\u00f3\u0142y szkolenia<\/span><\/a><\/div>\n<\/div><\/div>\n<\/div><\/div>\n<\/div><\/div>\n<\/div><\/div>\n\n<\/div><\/div>\n\n<style>.kb-row-layout-wrap.wp-block-kadence-rowlayout.kb-row-layout-id9153_6dbf4b-95{margin-bottom:var(--global-kb-spacing-md, 2rem);}.kb-row-layout-id9153_6dbf4b-95 > .kt-row-column-wrap{align-content:start;}:where(.kb-row-layout-id9153_6dbf4b-95 > .kt-row-column-wrap) > .wp-block-kadence-column{justify-content:start;}.kb-row-layout-id9153_6dbf4b-95 > .kt-row-column-wrap{column-gap:var(--global-kb-gap-md, 2rem);row-gap:var(--global-kb-gap-md, 2rem);max-width:1200px;margin-left:auto;margin-right:auto;padding-top:var(--global-kb-spacing-xs, 1rem);grid-template-columns:minmax(0, calc(10% - ((var(--global-kb-gap-md, 2rem) * 1 )\/2)))minmax(0, calc(90% - ((var(--global-kb-gap-md, 2rem) * 1 )\/2)));}.kb-row-layout-id9153_6dbf4b-95 > .kt-row-layout-overlay{opacity:0.30;}@media all and (max-width: 1024px){.kb-row-layout-id9153_6dbf4b-95 > .kt-row-column-wrap{grid-template-columns:minmax(0, 1fr) minmax(0, 2fr);}}@media all and (max-width: 767px){.kb-row-layout-id9153_6dbf4b-95 > .kt-row-column-wrap{padding-left:var(--global-kb-spacing-xxs, 0.5rem);grid-template-columns:minmax(0, 1fr);}}<\/style><div class=\"kb-row-layout-wrap kb-row-layout-id9153_6dbf4b-95 alignnone wp-block-kadence-rowlayout\"><div class=\"kt-row-column-wrap kt-has-2-columns kt-row-layout-right-golden kt-tab-layout-inherit kt-mobile-layout-row kt-row-valign-top\">\n<style>.kadence-column9153_9cb95c-72 > .kt-inside-inner-col,.kadence-column9153_9cb95c-72 > .kt-inside-inner-col:before{border-top-left-radius:0px;border-top-right-radius:0px;border-bottom-right-radius:0px;border-bottom-left-radius:0px;}.kadence-column9153_9cb95c-72 > .kt-inside-inner-col{column-gap:var(--global-kb-gap-sm, 1rem);}.kadence-column9153_9cb95c-72 > .kt-inside-inner-col{flex-direction:column;}.kadence-column9153_9cb95c-72 > .kt-inside-inner-col > .aligncenter{width:100%;}.kadence-column9153_9cb95c-72 > .kt-inside-inner-col:before{opacity:0.3;}.kadence-column9153_9cb95c-72{position:relative;}@media all and (max-width: 1024px){.kadence-column9153_9cb95c-72 > .kt-inside-inner-col{flex-direction:column;justify-content:center;}}@media all and (max-width: 767px){.kadence-column9153_9cb95c-72 > .kt-inside-inner-col{flex-direction:column;justify-content:center;}}<\/style>\n<div class=\"wp-block-kadence-column kadence-column9153_9cb95c-72\"><div class=\"kt-inside-inner-col\"><style>.kb-image9153_a06f3f-1c .kb-image-has-overlay:after{opacity:0.3;}.kb-image9153_a06f3f-1c img.kb-img, .kb-image9153_a06f3f-1c .kb-img img{-webkit-mask-image:url(https:\/\/softinery.com\/pl\/wp-content\/plugins\/kadence-blocks\/includes\/assets\/images\/masks\/circle.svg);mask-image:url(https:\/\/softinery.com\/pl\/wp-content\/plugins\/kadence-blocks\/includes\/assets\/images\/masks\/circle.svg);-webkit-mask-size:auto;mask-size:auto;-webkit-mask-repeat:no-repeat;mask-repeat:no-repeat;-webkit-mask-position:center;mask-position:center;object-position:47% 9%;}<\/style>\n<div class=\"wp-block-kadence-image kb-image9153_a06f3f-1c is-style-rounded\"><figure class=\"aligncenter size-medium kb-image-is-ratio-size\"><div class=\"kb-is-ratio-image kb-image-ratio-land43\"><img loading=\"lazy\" decoding=\"async\" width=\"226\" height=\"300\" src=\"https:\/\/softinery.com\/pl\/wp-content\/uploads\/sites\/5\/2024\/06\/photo_v2d-226x300.webp\" alt=\"Szymon Skoneczny\" class=\"kb-img wp-image-9147\" srcset=\"https:\/\/softinery.com\/pl\/wp-content\/uploads\/sites\/5\/2024\/06\/photo_v2d-226x300.webp 226w, https:\/\/softinery.com\/pl\/wp-content\/uploads\/sites\/5\/2024\/06\/photo_v2d-770x1024.webp 770w, https:\/\/softinery.com\/pl\/wp-content\/uploads\/sites\/5\/2024\/06\/photo_v2d-768x1021.webp 768w, https:\/\/softinery.com\/pl\/wp-content\/uploads\/sites\/5\/2024\/06\/photo_v2d-1155x1536.webp 1155w, https:\/\/softinery.com\/pl\/wp-content\/uploads\/sites\/5\/2024\/06\/photo_v2d.webp 1328w\" sizes=\"auto, (max-width: 226px) 100vw, 226px\" \/><\/div><\/figure><\/div>\n<\/div><\/div>\n\n\n<style>.kadence-column9153_a7a3d2-8c > .kt-inside-inner-col{padding-left:var(--global-kb-spacing-xxs, 0.5rem);}.kadence-column9153_a7a3d2-8c > .kt-inside-inner-col,.kadence-column9153_a7a3d2-8c > .kt-inside-inner-col:before{border-top-left-radius:0px;border-top-right-radius:0px;border-bottom-right-radius:0px;border-bottom-left-radius:0px;}.kadence-column9153_a7a3d2-8c > .kt-inside-inner-col{column-gap:var(--global-kb-gap-sm, 1rem);}.kadence-column9153_a7a3d2-8c > .kt-inside-inner-col{flex-direction:column;}.kadence-column9153_a7a3d2-8c > .kt-inside-inner-col > .aligncenter{width:100%;}.kadence-column9153_a7a3d2-8c > .kt-inside-inner-col:before{opacity:0.3;}.kadence-column9153_a7a3d2-8c{position:relative;}.kadence-column9153_a7a3d2-8c, .kt-inside-inner-col > .kadence-column9153_a7a3d2-8c:not(.specificity){margin-top:0px;}@media all and (max-width: 1024px){.kadence-column9153_a7a3d2-8c > .kt-inside-inner-col{flex-direction:column;justify-content:center;}}@media all and (max-width: 767px){.kadence-column9153_a7a3d2-8c > .kt-inside-inner-col{flex-direction:column;justify-content:center;}}<\/style>\n<div class=\"wp-block-kadence-column kadence-column9153_a7a3d2-8c\"><div class=\"kt-inside-inner-col\">\n<p><br>Dr hab. in\u017c. Szymon Skoneczny ma kilkunastoletnie do\u015bwiadczenie w tworzeniu oprogramowania do cel\u00f3w naukowych i przemys\u0142owych. Pracowa\u0142 dla renomowanych firm, w tym Siemens, Electricite de France oraz ArcelorMittal. Jego obszar dzia\u0142alno\u015bci obejmuje r\u00f3wnie\u017c nauczanie i udzia\u0142 w projektach naukowych w instytucjach takich jak Akademia G\u00f3rniczo-Hutnicza w Krakowie oraz Politechnika Krakowska. Jest autorem ponad 40 publikacji naukowych dotycz\u0105cych symulacji komputerowych bioreaktor\u00f3w, kt\u00f3re ukaza\u0142y si\u0119 w cenionych czasopismach o mi\u0119dzynarodowym zasi\u0119gu. Ponadto wsp\u00f3\u0142pracowa\u0142 przy projektach naukowych finansowanych ze \u015brodk\u00f3w Unii Europejskiej, koncentruj\u0105cych si\u0119 na wykorzystaniu komputer\u00f3w du\u017cej mocy.<\/p>\n\n\n\n<p>Znajd\u017a mnie na <a href=\"https:\/\/www.linkedin.com\/in\/szymon-skoneczny-77213682\/\">Linkedin<\/a>.<\/p>\n<\/div><\/div>\n\n<\/div><\/div>","protected":false},"excerpt":{"rendered":"<p>Wprowadzenie Regresja liniowa to bardzo prosta i u\u017cyteczna metoda matematyczna, stosowana w analizie wszelkich danych. M\u00f3wi\u0105c najpro\u015bciej celem regresji liniowej jest znalezienie r\u00f3wnania linii prostej, kt\u00f3ra najlepiej &#8220;pasuje&#8221; do zbioru danych x, y. Opis teoretyczny mo\u017cna znale\u017a\u0107 na wielu r\u00f3\u017cnych stronach, mi\u0119dzy innymi na Wikipedii. R\u00f3wnie\u017c w tym artykule om\u00f3wione s\u0105 teoretyczne aspekty tej metody.&hellip;&nbsp;<a href=\"https:\/\/softinery.com\/pl\/blog\/python-regresja-liniowa\/\" rel=\"bookmark\">Dowiedz si\u0119 wi\u0119cej &raquo;<span class=\"screen-reader-text\">Python &#8211; regresja liniowa<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"parent":578,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"neve_meta_sidebar":"","neve_meta_container":"full-width","neve_meta_enable_content_width":"on","neve_meta_content_width":80,"neve_meta_title_alignment":"","neve_meta_author_avatar":"","neve_post_elements_order":"","neve_meta_disable_header":"","neve_meta_disable_footer":"","neve_meta_disable_title":"","footnotes":""},"categories":[],"tags":[],"class_list":["post-4971","page","type-page","status-publish","hentry"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.1.1 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Python - 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