{"id":4480,"date":"2026-07-29T16:56:13","date_gmt":"2026-07-29T14:56:13","guid":{"rendered":"https:\/\/ellistat.com\/?post_type=guide-dutilisateur&#038;p=4480"},"modified":"2026-07-30T10:31:31","modified_gmt":"2026-07-30T08:31:31","slug":"principal-component-analysis-pca","status":"publish","type":"guide-dutilisateur","link":"https:\/\/ellistat.com\/en\/guide-dutilisateur\/analyse-en-composantes-principales-acp\/","title":{"rendered":"Principal Component Analysis (PCA)"},"content":{"rendered":"<p class=\"wp-block-paragraph\">PCA summarizes several correlated quantitative variables into a small number of dimensions called&nbsp;<strong>main components<\/strong>, labeled CP1, CP2, and CP3. Each axis accounts for a portion of the total variability; the first two account for the bulk of it.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">3.1 The Summary Tab<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Zone 1<\/strong>&nbsp;: display settings.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u21d2&nbsp;<strong>Show individuals<\/strong>&nbsp;: Overlays the scatter plot of individuals on the correlation circle.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u21d2&nbsp;<strong>Show labels<\/strong>&nbsp;: Write the name of each person next to their dot.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u21d2&nbsp;<strong>Unsupervised Classification (K-Means)<\/strong>&nbsp;: Color the individuals by family and mark the centers of each family.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u21d2&nbsp;<strong>Number of groups<\/strong>&nbsp;: the number of families being sought.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u21d2&nbsp;<strong>Chart Type<\/strong>&nbsp;: 2D or 3D. In 3D, the CP3 axis is added.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u21d2&nbsp;<strong>Label Positions<\/strong>&nbsp;: Appears only after you've moved a label, and lets you put everything back in place.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Zone 2<\/strong>&nbsp;: the graph. The circle with radius 1 is the&nbsp;<strong>circle of correlations<\/strong>.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Zone 3<\/strong>&nbsp;: Below the graph, the table of eigenvectors lists the coordinates of each variable on the axes.<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"644\" height=\"480\" src=\"https:\/\/ellistat.com\/wp-content\/uploads\/userguide3-1.png\" alt=\"\" class=\"wp-image-4481\" srcset=\"https:\/\/ellistat.com\/wp-content\/uploads\/userguide3-1.png 644w, https:\/\/ellistat.com\/wp-content\/uploads\/userguide3-1-300x224.png 300w, https:\/\/ellistat.com\/wp-content\/uploads\/userguide3-1-16x12.png 16w\" sizes=\"auto, (max-width: 644px) 100vw, 644px\" \/><\/figure>\n\n\n\n<p class=\"wp-block-paragraph\">\ud83d\udca1&nbsp;<strong>Labels can be moved.<\/strong>&nbsp;In 2D, click and drag a variable or category name with the mouse: this is the easiest way to separate overlapping labels. The positions are saved and will be included with the graph if you save it.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">3.2 Interpreting the Correlation Circle<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Each variable is an arrow extending from the origin. Its position is determined by two criteria.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Length<\/strong>&nbsp;measures the quality of representation. An arrow that almost touches the circle is well represented on the displayed plane: it can be interpreted. A short arrow, close to the center, is poorly represented on this plane \u2014&nbsp;<strong>It should not be interpreted here<\/strong>, you need to consider the following points.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>The angle between two arrows<\/strong>&nbsp;measures their correlation, provided that both are long:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u21d2 two arrows side by side: positively correlated variables; ;<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u21d2 two opposing arrows: negatively correlated variables; ;<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u21d2 two arrows at 90\u00b0: independent variables.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">The coordinate of a variable on an axis&nbsp;<strong>is<\/strong>&nbsp;its correlation with that axis. It is therefore always between \u22121 and 1, which explains the circle.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">3.3 Reading the Cloud of Individuals<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Two individuals that are close to each other on the plot are similar in terms of the variables analyzed. The center of the cloud is the origin: an individual far from the center is atypical.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u26a0\ufe0f The cloud is&nbsp;<strong>scaled to the size of the circle<\/strong>&nbsp;to be superimposable. The distances between individuals and their directions are accurate, but their scale is not the same as that of the correlations: we see a&nbsp;<strong>management<\/strong>&nbsp;(\u00abThis individual is moving in the direction of the pressure\u00bb), not a value.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">3.4 The Pareto Tab<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">For each axis, it provides the&nbsp;<strong>equity<\/strong>, the&nbsp;<strong>explained variance<\/strong>&nbsp;and the&nbsp;<strong>cumulative variance<\/strong>, along with the corresponding scree.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">This is where you decide how many axes to keep. The cumulative variance is displayed in red as long as it is less than 80 %, and in green when it exceeds that value: keep enough axes to reach the green level.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\ud83d\udca1 Another classic guideline: keep only the axes with an eigenvalue greater than 1; an axis that explains less than a single variable does not provide a summary.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">3.5 The Variables Tab<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">One line per variable, and four columns for each axis:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u21d2&nbsp;<strong>C1, C2, C3<\/strong>&nbsp;: the correlation of the variable with the axis; ;<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u21d2&nbsp;<strong>Contribution<\/strong>&nbsp;: the weight of the variable in the construction of the axis, expressed as a percentage. The contributions of an axis add up to 100; ;<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u21d2&nbsp;<strong>Cos\u00b2<\/strong>&nbsp;: the quality of the variable's representation on the axis; ;<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u21d2&nbsp;<strong>p<\/strong>&nbsp;: the probability that the observed correlation is due to chance. The value is color-coded from highly significant to not significant.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\ud83d\udca1 To label an axis, identify the variables whose&nbsp;<strong>contribution<\/strong>&nbsp;is significantly higher than the average (100 \/ number of variables) and whose&nbsp;<strong>Cos\u00b2<\/strong>&nbsp;is high. They are what give the axis its meaning.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">3.6 The Individuals Tab<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Same logic, one line per individual: distance from the center, coordinates, contributions, and Cos\u00b2.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">An individual whose contribution far exceeds that of others&nbsp;<strong>pulls the axle all by itself<\/strong>&nbsp;: Make sure it isn't a data entry error before drawing a conclusion.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">An individual's Cos\u00b2 values add up to 1 across all axes. On the axes shown here, their sum therefore indicates what portion of their position is actually visible.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">3.7 Additional Variables and Labels<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Zone 2<\/strong>&nbsp;The top of the screen (above the tabs) features two selectors.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Additional Variables<\/strong>&nbsp;\u2014 also known as illustrative. They are&nbsp;<strong>projected<\/strong>&nbsp;on the axes but do not contribute to their construction. This is a good way to compare a result with an external variable without that variable influencing the analysis.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u21d2 An additional variable&nbsp;<strong>quantitative<\/strong>&nbsp;appears as an additional arrow in the circle, and in the \u00abAdditional Quantitative Variables\u00bb tab along with its correlation with each axis.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u21d2 An additional variable&nbsp;<strong>qualitative<\/strong>&nbsp;place the&nbsp;<strong>center of mass<\/strong>&nbsp;for each category on the map: the midpoint of the individuals representing that category. The size of the symbol is proportional to the number of individuals in that category.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">The \u00abAdditional Qualitative Variables\u00bb tab displays, for each modality, its distance from the center, its coordinates, its Cos\u00b2, and a&nbsp;<strong>p-value by axis<\/strong>. This p-value answers the question: Does this variable distinguish between individuals along this axis? A low p-value indicates that the categories occupy distinctly different positions on the axis.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Label<\/strong>&nbsp;Identifies individuals: their marker turns the color of the selected mode, and their name is displayed if \u00abShow Labels\u00bb is checked. This is the most direct way to see if a family pattern is emerging.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\ud83d\udca1 Do not set a variable as both active and additional: it would construct the axis and then be evaluated on that axis, which proves nothing.<\/p>","protected":false},"featured_media":0,"menu_order":0,"template":"","meta":{"_acf_changed":true},"menu-guide-dutilisateur":[40],"class_list":["post-4480","guide-dutilisateur","type-guide-dutilisateur","status-publish","hentry","menu-guide-dutilisateur-6-statistiques-multivariees"],"acf":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v28.3 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Analyse en composantes principales (ACP) - Ellistat<\/title>\n<meta name=\"description\" content=\"D\u00e9couvrez comment le module Data Analysis d&#039;Ellistat identifie l&#039;origine des d\u00e9rives qualit\u00e9 gr\u00e2ce \u00e0 l&#039;analyse statistique multivari\u00e9e.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/ellistat.com\/en\/users-guide\/principal-component-analysis-pca\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Analyse en composantes principales (ACP) - 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