{"id":19009,"date":"2016-12-06T19:48:58","date_gmt":"2016-12-06T19:48:58","guid":{"rendered":"http:\/\/www.oracletutoring.ca\/blog\/?p=19009"},"modified":"2016-12-06T19:48:58","modified_gmt":"2016-12-06T19:48:58","slug":"linear-algebra-best-fit-line-method-of-least-squares-by-hand","status":"publish","type":"post","link":"https:\/\/www.oracletutoring.ca\/blog\/linear-algebra-best-fit-line-method-of-least-squares-by-hand\/","title":{"rendered":"Linear algebra: best fit line: method of least squares by hand"},"content":{"rendered":"

The tutor shows a nice trick to obtain a least squares line using matrix algebra.<\/h1>\n

In my Nov 25<\/a> post I mentioned how, with Excel or LibreOffice Calc, to get a best-fit line for<\/p>\n\n\n\n\n\n\n
x<\/td>\ny<\/td>\n<\/tr>\n
0<\/td>\n3<\/td>\n<\/tr>\n
2<\/td>\n5<\/td>\n<\/tr>\n
6<\/td>\n10<\/td>\n<\/tr>\n
8<\/td>\n12<\/td>\n<\/tr>\n<\/table>\n

The same line can be gotten by hand, with the following ideas:<\/p>\n

    \n
  1. \nImagining the best fit line to be of form y=mx+b, the table above can be related as the matrix equation
    \n<\/li>\n
  2. We multiply each side by the transpose matrix:
    \n<\/li>\n
  3. Performing the multiplication on each side yields
    \n<\/li>\n

    which means<\/p>\n

    <\/li>\n

    Solving, we get m=1.15, b=2.9: our best-fit line is y=1.15x + 2.9, same with the result from November 25<\/a> using a spreadsheet.<\/p>\n

    Neat, eh?<\/p>\n

    Source:<\/p>\n

    Johnson\/Riess\/Arnold. Introduction to Linear Algebra<\/u>, Second Ed. Don Mills: Addison-Wesley, 1989.<\/p>\n

    Jack of Oracle Tutoring by Jack and Diane,<\/a> Campbell River, BC.<\/p>\n","protected":false},"excerpt":{"rendered":"

    The tutor shows a nice trick to obtain a least squares line using matrix algebra. In my Nov 25 post I mentioned how, with Excel or LibreOffice Calc, to get a best-fit line for x y 0 3 2 5 …<\/p>\n

    Linear algebra: best fit line: method of least squares by hand<\/span> Read More »<\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[11,302,3,7],"tags":[2047,2049,2048,2058,2057],"class_list":["post-19009","post","type-post","status-publish","format-standard","hentry","category-chemistry","category-linear-algebra","category-math","category-physics","tag-best-fit-line","tag-data-fitting","tag-least-squares-line","tag-least-squares-line-by-hand","tag-method-of-least-squares"],"_links":{"self":[{"href":"https:\/\/www.oracletutoring.ca\/blog\/wp-json\/wp\/v2\/posts\/19009","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.oracletutoring.ca\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.oracletutoring.ca\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.oracletutoring.ca\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.oracletutoring.ca\/blog\/wp-json\/wp\/v2\/comments?post=19009"}],"version-history":[{"count":14,"href":"https:\/\/www.oracletutoring.ca\/blog\/wp-json\/wp\/v2\/posts\/19009\/revisions"}],"predecessor-version":[{"id":19023,"href":"https:\/\/www.oracletutoring.ca\/blog\/wp-json\/wp\/v2\/posts\/19009\/revisions\/19023"}],"wp:attachment":[{"href":"https:\/\/www.oracletutoring.ca\/blog\/wp-json\/wp\/v2\/media?parent=19009"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.oracletutoring.ca\/blog\/wp-json\/wp\/v2\/categories?post=19009"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.oracletutoring.ca\/blog\/wp-json\/wp\/v2\/tags?post=19009"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}