{"id":168210,"date":"2023-05-19T15:56:12","date_gmt":"2023-05-19T14:56:12","guid":{"rendered":"https:\/\/datascientest.com\/en\/?p=168210"},"modified":"2023-10-30T19:05:59","modified_gmt":"2023-10-30T18:05:59","slug":"pca-principal-component-analysis-what-is-it","status":"publish","type":"post","link":"https:\/\/datascientest.com\/en\/pca-principal-component-analysis-what-is-it","title":{"rendered":"PCA (Principal Component Analysis): What is it?"},"content":{"rendered":"\t\t<div data-elementor-type=\"wp-post\" data-elementor-id=\"168210\" class=\"elementor elementor-168210\" data-elementor-post-type=\"post\">\n\t\t\t\t\t\t\t<div class=\"elementor-element elementor-element-2991667 e-flex e-con-boxed e-con\" data-id=\"2991667\" data-element_type=\"container\" data-settings=\"{&quot;content_width&quot;:&quot;boxed&quot;}\">\n\t\t\t\t\t<div class=\"e-con-inner\">\n\t\t\t\t<div class=\"elementor-element elementor-element-5df5dd3 elementor-widget elementor-widget-heading\" data-id=\"5df5dd3\" data-element_type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t<style>\/*! elementor - v3.15.0 - 02-08-2023 *\/\n.elementor-heading-title{padding:0;margin:0;line-height:1}.elementor-widget-heading .elementor-heading-title[class*=elementor-size-]>a{color:inherit;font-size:inherit;line-height:inherit}.elementor-widget-heading .elementor-heading-title.elementor-size-small{font-size:15px}.elementor-widget-heading .elementor-heading-title.elementor-size-medium{font-size:19px}.elementor-widget-heading .elementor-heading-title.elementor-size-large{font-size:29px}.elementor-widget-heading .elementor-heading-title.elementor-size-xl{font-size:39px}.elementor-widget-heading .elementor-heading-title.elementor-size-xxl{font-size:59px}<\/style><h2 class=\"elementor-heading-title elementor-size-default\">Do you know the PCA? A very useful method used in dimension reduction, discover how it works in this article.<\/h2>\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t<div class=\"elementor-element elementor-element-981eaad e-flex e-con-boxed e-con\" data-id=\"981eaad\" data-element_type=\"container\" data-settings=\"{&quot;content_width&quot;:&quot;boxed&quot;}\">\n\t\t\t\t\t<div class=\"e-con-inner\">\n\t\t\t\t<div class=\"elementor-element elementor-element-5e67d00 elementor-widget elementor-widget-heading\" data-id=\"5e67d00\" data-element_type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t<h3 class=\"elementor-heading-title elementor-size-default\">What is the Principal Component Analysis?<\/h3>\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-136e53a elementor-widget elementor-widget-text-editor\" data-id=\"136e53a\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t<style>\/*! elementor - v3.15.0 - 02-08-2023 *\/\n.elementor-widget-text-editor.elementor-drop-cap-view-stacked .elementor-drop-cap{background-color:#69727d;color:#fff}.elementor-widget-text-editor.elementor-drop-cap-view-framed .elementor-drop-cap{color:#69727d;border:3px solid;background-color:transparent}.elementor-widget-text-editor:not(.elementor-drop-cap-view-default) .elementor-drop-cap{margin-top:8px}.elementor-widget-text-editor:not(.elementor-drop-cap-view-default) .elementor-drop-cap-letter{width:1em;height:1em}.elementor-widget-text-editor .elementor-drop-cap{float:left;text-align:center;line-height:1;font-size:50px}.elementor-widget-text-editor .elementor-drop-cap-letter{display:inline-block}<\/style>\t\t\t\t<p><span style=\"font-weight: 400;\">Who has never had in his hands a dataset containing a very large number of variables without knowing which are the most important?\u00a0 How to reduce this dataset to represent it simply on 2 or 3 axes? Here is the PCA!<\/span><\/p><p><span style=\"font-weight: 400;\">Principal Component Analysis answers these questions. PCA is a well-known <\/span><b>method of dimension reduction<\/b><span style=\"font-weight: 400;\"> that will allow the transformation of <\/span><b>highly correlated variables<\/b><span style=\"font-weight: 400;\"> into new variables that are decorrelated from each other.\u00a0<\/span><\/p><p><span style=\"font-weight: 400;\">The principle is simple: It is a matter of <\/span><b>summarizing the information <\/b><span style=\"font-weight: 400;\">contained in a large <\/span><a href=\"https:\/\/datascientest.com\/en\/database-what-is-it\"><span style=\"font-weight: 400;\">database<\/span><\/a><span style=\"font-weight: 400;\"> into a certain number of synthetic variables called: Principal Components.\u00a0<\/span><\/p><p><span style=\"font-weight: 400;\">The idea is then to be able to project these data on the <\/span><b>nearest hyperplane<\/b><span style=\"font-weight: 400;\"> to have a simple representation of our data.<\/span><\/p><p><span style=\"font-weight: 400;\">Of course, <\/span><b>dimension reduction means a loss of information<\/b><span style=\"font-weight: 400;\">. This is the challenge of a Principal Component Analysis. We must be able to reduce the dimension of our data while keeping a maximum of information.<\/span><\/p>\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t<div class=\"elementor-element elementor-element-81baa19 e-flex e-con-boxed e-con\" data-id=\"81baa19\" data-element_type=\"container\" data-settings=\"{&quot;content_width&quot;:&quot;boxed&quot;}\">\n\t\t\t\t\t<div class=\"e-con-inner\">\n\t\t\t\t<div class=\"elementor-element elementor-element-3d12cb0 elementor-widget elementor-widget-heading\" data-id=\"3d12cb0\" data-element_type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t<h3 class=\"elementor-heading-title elementor-size-default\">How does a Principal Component Analysis work?<\/h3>\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-35c71b1 elementor-widget elementor-widget-text-editor\" data-id=\"35c71b1\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<p><span style=\"font-weight: 400;\">To illustrate the principle of PCA, we will<\/span><b> take for example a dataset <\/b><span style=\"font-weight: 400;\">named &#8216;country_data&#8217; which as its name indicates gathers several pieces of information (GDP; Average income; Life expectancy; Birth\/mortality rate, etc&#8230;) on different countries.<\/span><\/p><p><span style=\"font-weight: 400;\">Here are the first 5 lines:<\/span><\/p>\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-b57a4f2 elementor-widget__width-initial elementor-widget elementor-widget-image\" data-id=\"b57a4f2\" data-element_type=\"widget\" data-widget_type=\"image.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t<style>\/*! elementor - v3.15.0 - 02-08-2023 *\/\n.elementor-widget-image{text-align:center}.elementor-widget-image a{display:inline-block}.elementor-widget-image a img[src$=\".svg\"]{width:48px}.elementor-widget-image img{vertical-align:middle;display:inline-block}<\/style>\t\t\t\t\t\t\t\t\t\t\t\t<img decoding=\"async\" width=\"800\" height=\"246\" src=\"https:\/\/datascientest.com\/en\/files\/2023\/05\/ACP_1.webp\" class=\"attachment-large size-large wp-image-168216\" alt=\"ACP_1\" srcset=\"https:\/\/datascientest.com\/en\/files\/2023\/05\/ACP_1.webp 1024w, https:\/\/datascientest.com\/en\/files\/2023\/05\/ACP_1-300x92.webp 300w, https:\/\/datascientest.com\/en\/files\/2023\/05\/ACP_1-768x236.webp 768w\" sizes=\"(max-width: 800px) 100vw, 800px\" \/>\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-48e285f elementor-widget elementor-widget-text-editor\" data-id=\"48e285f\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<p><span style=\"font-weight: 400;\">Thereafter it is important to <\/span><b>center and reduce our variables<\/b><span style=\"font-weight: 400;\"> to mitigate the scale effect because they are not calculated on the same basis.<\/span><\/p><p><span style=\"font-weight: 400;\">Once this step has been completed, we must see our data as a <\/span><b>matrix<\/b><span style=\"font-weight: 400;\"> from which we will calculate from which we will <\/span><b>calculate eigenvalues<\/b><span style=\"font-weight: 400;\"> and <\/span><b>eigenvectors<\/b><span style=\"font-weight: 400;\">.<\/span><\/p><p><span style=\"font-weight: 400;\">In linear algebra, the notion of eigenvector corresponds to the <\/span><b>study of privileged axes<\/b><span style=\"font-weight: 400;\">, according to which an application of a space in itself behaves like a dilation, multiplying the vectors by a constant called an eigenvalue. The vectors to which it applies are called eigenvectors,<\/span><b> combined in an eigenspace.<\/b><\/p><p><span style=\"font-weight: 400;\">After importing the PCA module from sklearn.decomposition, the eigenvalues returned are\u00a0<\/span><\/p><p><span style=\"font-weight: 400;\">The eigenvalues are: [3.48753851 1.47902877 1.15061758 0.93557048 0.65529084 0.15140052]<\/span><\/p><p><span style=\"font-weight: 400;\">These eigenvalues will allow us to <\/span><b>determine the optimal number of factors\/principal components<\/b><span style=\"font-weight: 400;\"> for our PCA. For example, if the optimal number of components is 2, then our data will be represented on<\/span><b> two axes<\/b><span style=\"font-weight: 400;\">, and so on.<\/span><\/p>\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-b3e16c1 elementor-widget__width-initial elementor-widget elementor-widget-image\" data-id=\"b3e16c1\" data-element_type=\"widget\" data-widget_type=\"image.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<img decoding=\"async\" width=\"800\" height=\"572\" src=\"https:\/\/datascientest.com\/en\/files\/2023\/05\/ACP_3-3-1024x732.jpg\" class=\"attachment-large size-large wp-image-168221\" alt=\"\" srcset=\"https:\/\/datascientest.com\/en\/files\/2023\/05\/ACP_3-3-1024x732.jpg 1024w, https:\/\/datascientest.com\/en\/files\/2023\/05\/ACP_3-3-300x215.jpg 300w, https:\/\/datascientest.com\/en\/files\/2023\/05\/ACP_3-3-768x549.jpg 768w, https:\/\/datascientest.com\/en\/files\/2023\/05\/ACP_3-3.jpg 1344w\" sizes=\"(max-width: 800px) 100vw, 800px\" \/>\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-f60580b elementor-widget elementor-widget-text-editor\" data-id=\"f60580b\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<p><span style=\"font-weight: 400;\">On this graph which represents the number of factors to choose from according to the eigenvalues, we indicate that the <\/span><b>optimal choice of factor <\/b><span style=\"font-weight: 400;\">is 2 (thanks to the elbow method). Thus, we will go from dimension 9 to dimension 2 which considerably<\/span><b> reduces the basic dimension<\/b><span style=\"font-weight: 400;\">. As said before, there will necessarily be a<\/span><b> loss of information<\/b><span style=\"font-weight: 400;\"> following this reduction. However, we still keep a rate of information of almost 70% which will allow us to have a representation close to my representation in 9 dimensions.<\/span><\/p><p><span style=\"font-weight: 400;\">Once the PCA module has <\/span><b>calculated the coordinates of our data<\/b><span style=\"font-weight: 400;\">, all that remains to be done is to represent them, but before doing so, we are going to<\/span><b> take an interest in a tool<\/b><span style=\"font-weight: 400;\"> that is very often used when performing a Principal Component Analysis, namely the <\/span><b>correlation circle<\/b><span style=\"font-weight: 400;\">.<\/span><\/p>\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-2d47979 elementor-widget__width-initial elementor-widget elementor-widget-image\" data-id=\"2d47979\" data-element_type=\"widget\" data-widget_type=\"image.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<img decoding=\"async\" width=\"512\" height=\"474\" src=\"https:\/\/datascientest.com\/en\/files\/2023\/05\/ACP_4.jpg\" class=\"attachment-large size-large wp-image-168223\" alt=\"ACP_4\" srcset=\"https:\/\/datascientest.com\/en\/files\/2023\/05\/ACP_4.jpg 512w, https:\/\/datascientest.com\/en\/files\/2023\/05\/ACP_4-300x278.jpg 300w\" sizes=\"(max-width: 512px) 100vw, 512px\" \/>\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-4becc23 elementor-widget elementor-widget-text-editor\" data-id=\"4becc23\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<p><span style=\"font-weight: 400;\">As our representation is done on 2 axes, the correlation circle is a<\/span><b> practical tool <\/b><span style=\"font-weight: 400;\">that allows us to<\/span><b> visualize the importance of each explanatory<\/b><span style=\"font-weight: 400;\"> variable for each representation axis. The direction of each arrow indicates the axis explained by the variable and the direction indicates whether the correlation is positive or negative. <\/span><\/p><p><span style=\"font-weight: 400;\">We notice that <\/span><b>variables<\/b><span style=\"font-weight: 400;\"> such as &#8216;income&#8217;, &#8216;gdpp&#8217;, and &#8216;health&#8217; are positively correlated to the first axis, while &#8216;child_mort&#8217; or &#8216;total_fer&#8217; are also <\/span><b>positively correlated<\/b><span style=\"font-weight: 400;\"> but negatively. We can then look at the representation of countries in the two axes chosen by the PCA and see the influence of the variable &#8216;life_expec&#8217; on their representations.<\/span><\/p>\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-ea56081 elementor-widget__width-initial elementor-widget elementor-widget-image\" data-id=\"ea56081\" data-element_type=\"widget\" data-widget_type=\"image.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<img decoding=\"async\" width=\"512\" height=\"493\" src=\"https:\/\/datascientest.com\/en\/files\/2023\/05\/ACP_5.jpg\" class=\"attachment-large size-large wp-image-168222\" alt=\"ACP_5\" srcset=\"https:\/\/datascientest.com\/en\/files\/2023\/05\/ACP_5.jpg 512w, https:\/\/datascientest.com\/en\/files\/2023\/05\/ACP_5-300x289.jpg 300w\" sizes=\"(max-width: 512px) 100vw, 512px\" \/>\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-cbfcd57 elementor-widget elementor-widget-text-editor\" data-id=\"cbfcd57\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<p><span style=\"font-weight: 400;\">Here is a <\/span><b>representation of each country<\/b><span style=\"font-weight: 400;\"> (167) on 2 axes. To judge the quality of our representation, we decided to color each country according to the life expectancy of each one in 3 groups, we can observe a certain trend. We can then notice that the countries with a high life expectancy are concentrated in the <\/span><b>lower right part<\/b><span style=\"font-weight: 400;\"> of the graph. According to the <\/span><b>correlation circle<\/b><span style=\"font-weight: 400;\">, the individuals in this part are partly explained by the variables &#8216;health&#8217;, &#8216;income&#8217;, or &#8216;gdb&#8217;. It can be concluded that <\/span><b>countries spending more on health have a higher life expectancy<\/b><span style=\"font-weight: 400;\">. The same is true for the countries in the upper left part of the graph. According to the correlation circle, this part is mostly explained by the variables &#8216;child_death&#8217; or &#8216;total_iron&#8217;.<\/span><\/p><p><span style=\"font-weight: 400;\">If you want to learn more about <\/span><b>Principal Component Analysis<\/b><span style=\"font-weight: 400;\"> or other <\/span><b>dimension reduction methods<\/b><span style=\"font-weight: 400;\">, there are several modules dedicated to it in <\/span><a href=\"https:\/\/datascientest.com\/en\/data-analyst-course\"><span style=\"font-weight: 400;\">our Data Analyst training<\/span><\/a><span style=\"font-weight: 400;\">.<\/span><\/p>\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-90fade4 elementor-align-center button-principal elementor-widget elementor-widget-button\" data-id=\"90fade4\" data-element_type=\"widget\" data-widget_type=\"button.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<div class=\"elementor-button-wrapper\">\n\t\t\t<a class=\"elementor-button elementor-button-link elementor-size-md\" href=\"https:\/\/datascientest.com\/en\/data-courses\">\n\t\t\t\t\t\t<span class=\"elementor-button-content-wrapper\">\n\t\t\t\t\t\t<span class=\"elementor-button-text\">Discover our Data Science courses<\/span>\n\t\t<\/span>\n\t\t\t\t\t<\/a>\n\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t<div class=\"elementor-element elementor-element-befcc6b e-flex e-con-boxed e-con\" data-id=\"befcc6b\" data-element_type=\"container\" data-settings=\"{&quot;content_width&quot;:&quot;boxed&quot;}\">\n\t\t\t\t\t<div class=\"e-con-inner\">\n\t\t\t\t<div class=\"elementor-element elementor-element-66eb91f elementor-widget elementor-widget-html\" data-id=\"66eb91f\" data-element_type=\"widget\" data-widget_type=\"html.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t<script type=\"application\/ld+json\">\r\n{\r\n  \"@context\": \"https:\/\/schema.org\",\r\n  \"@type\": \"FAQPage\",\r\n  \"mainEntity\": [{\r\n    \"@type\": \"Question\",\r\n    \"name\": \"What is the Principal Component Analysis ?\",\r\n    \"acceptedAnswer\": {\r\n      \"@type\": \"Answer\",\r\n      \"text\": \"Who has never had in his hands a dataset containing a very large number of variables without knowing which are the most important?  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A very useful method used in dimension reduction, discover how it works in this article. What is the Principal Component Analysis? Who has never had in his hands a dataset containing a very large number of variables without knowing which are the most important?&nbsp; How to reduce this dataset to [&hellip;]<\/p>\n","protected":false},"author":45,"featured_media":168212,"comment_status":"open","ping_status":"open","sticky":false,"template":"elementor_theme","format":"standard","meta":{"content-type":"","inline_featured_image":false},"categories":[2425],"tags":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO Premium plugin v21.7 (Yoast SEO v21.7) - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>PCA (Principal Component Analysis): What is it? | DataScientest<\/title>\n<meta name=\"description\" content=\"Do you know the PCA? 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