{"id":42,"date":"2010-03-04T11:48:35","date_gmt":"2010-03-04T11:48:35","guid":{"rendered":"https:\/\/blog.informationgeometry.org\/?p=42"},"modified":"2021-07-31T11:50:00","modified_gmt":"2021-07-31T11:50:00","slug":"the-geometric-median","status":"publish","type":"post","link":"https:\/\/blog.informationgeometry.org\/the-geometric-median\/","title":{"rendered":"The geometric median"},"content":{"rendered":"<p>The center of mass (=centroid) is defined as the center point minimizing the squared of the Euclidean distances (=variance). If one of the source point is an outlier corrupting your dataset, and if that outlier goes to infinity, then your centroid follows it can is clearly not a robust centerpoint. The median on the contrary is defined as the center point minimising the sum of Euclidean distances. It is robust as it breaks only if n\/2 outlier points go to infinity. However, there is no closed form solutions.<\/p>\n<p>Doing some survey, I found that since Fermat (who allegedly first ask the question for 3 points), it has been studied and rediscovered in many communities. The current labeling of this point should be Fermat-Fagnano-Weber-Torricelli-Steiner point, and I forgot many names&#8230;<\/p>\n<p>History is full of insights!<br \/>\nFrank.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>The center of mass (=centroid) is defined as the center point minimizing the squared of the Euclidean distances (=variance). If one of the source point is an outlier corrupting your dataset, and if that outlier goes to infinity, then your<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[2],"tags":[],"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/blog.informationgeometry.org\/wp-json\/wp\/v2\/posts\/42"}],"collection":[{"href":"https:\/\/blog.informationgeometry.org\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/blog.informationgeometry.org\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/blog.informationgeometry.org\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/blog.informationgeometry.org\/wp-json\/wp\/v2\/comments?post=42"}],"version-history":[{"count":1,"href":"https:\/\/blog.informationgeometry.org\/wp-json\/wp\/v2\/posts\/42\/revisions"}],"predecessor-version":[{"id":43,"href":"https:\/\/blog.informationgeometry.org\/wp-json\/wp\/v2\/posts\/42\/revisions\/43"}],"wp:attachment":[{"href":"https:\/\/blog.informationgeometry.org\/wp-json\/wp\/v2\/media?parent=42"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blog.informationgeometry.org\/wp-json\/wp\/v2\/categories?post=42"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blog.informationgeometry.org\/wp-json\/wp\/v2\/tags?post=42"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}