{"id":427,"date":"2024-02-29T14:35:59","date_gmt":"2024-02-29T19:35:59","guid":{"rendered":"https:\/\/pressbooks.bccampus.ca\/businessanalytics\/chapter\/distances\/"},"modified":"2024-02-29T14:38:50","modified_gmt":"2024-02-29T19:38:50","slug":"distances","status":"publish","type":"chapter","link":"https:\/\/pressbooks.bccampus.ca\/businessanalytics\/chapter\/distances\/","title":{"raw":"Distances","rendered":"Distances"},"content":{"raw":"
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---<\/span>\r\ntitle<\/span>:<\/span> Distances<\/span>,<\/span> Norms<\/span> and<\/span> Metrics<\/span>\r\nauthor<\/span>:<\/span> Amy<\/span> Goldlist<\/span>\r\nformat<\/span>:<\/span>\r\n    html<\/span>:<\/span>\r\n        embed<\/span>-<\/span>resources<\/span>:<\/span> true<\/span>\r\n        code<\/span>-<\/span>fold<\/span>:<\/span> false<\/span>\r\n<\/pre>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n
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Distance (or L2 Norm \/ Metric)\u00b6<\/a><\/h2>\r\nThe distance between two points [latex](x_1, y_1)[\/latex] and [latex](x_2, y_2)[\/latex] is given by:\r\n\r\n$$\\|(x_2, y_2)-(x_1, y_1)\\|= d = \\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$$\r\n\r\nWe use the bars $\\|\\cdot\\|$ to mean \"distance\".\r\n\r\nWe can extend this to as many dimensions as we want:\r\n\r\n$$\\|(x_2, y_2, z_2, w_2)-(x_1, y_1, z_1, w_1)\\|= d = \\sqrt{(x_2-x_1)^2+(y_2-y1)^2+(z_2-z_1)^2+(w_2-w_1)^2}$$\r\n\r\nFor example:\r\n\r\n$$\\|(4, 3)-(3,-1)\\|= d = \\sqrt{(4-1)^2+(3+1)^2}= \\sqrt{25} = 5$$\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n
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import<\/span> numpy<\/span> as<\/span> np<\/span>\r\nfrom<\/span> numpy.linalg<\/span> import<\/span> *<\/span>\r\n\r\n## In Python:<\/span>\r\n## Let's define two points: <\/span>\r\n\r\na<\/span>=<\/span>np<\/span>.<\/span>array<\/span>([<\/span>4<\/span>,<\/span>3<\/span>])<\/span>\r\nb<\/span>=<\/span>np<\/span>.<\/span>array<\/span>([<\/span>1<\/span>,<\/span>-<\/span>1<\/span>])<\/span>\r\n\r\n##and find the distance between them:<\/span>\r\nprint<\/span>(<\/span>\"a = \"<\/span>,<\/span>a<\/span>,<\/span> \"b= \"<\/span>,<\/span> b<\/span>)<\/span>\r\nnorm<\/span>(<\/span>a<\/span>-<\/span>b<\/span>)<\/span>\r\n<\/pre>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n
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a =  [4 3] b=  [ 1 -1]\r\n<\/pre>\r\n<\/div>\r\n<\/div>\r\n
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5.0<\/pre>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n
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## And let's graph it:<\/span>\r\n\r\nimport<\/span> matplotlib.pyplot<\/span> as<\/span> plt<\/span> \r\n\r\nplt<\/span>.<\/span>plot<\/span>([<\/span>a<\/span>[<\/span>0<\/span>],<\/span>b<\/span>[<\/span>0<\/span>]],<\/span> [<\/span>a<\/span>[<\/span>1<\/span>],<\/span>b<\/span>[<\/span>1<\/span>]],<\/span> 'red'<\/span>,<\/span> linestyle<\/span>=<\/span>'-'<\/span>,<\/span> marker<\/span>=<\/span>'*'<\/span>)<\/span>\r\n\r\n\r\nplt<\/span>.<\/span>axis<\/span>(<\/span>'equal'<\/span>)<\/span>  ## This is important to get the display correct<\/span>\r\nplt<\/span>.<\/span>title<\/span>(<\/span>'L2 Distance between 2 points'<\/span>)<\/span> \r\n\r\nplt<\/span>.<\/span>xticks<\/span>(<\/span>range<\/span>(<\/span>-<\/span>1<\/span>,<\/span>8<\/span>))<\/span>\r\nplt<\/span>.<\/span>yticks<\/span>(<\/span>range<\/span>(<\/span>-<\/span>5<\/span>,<\/span>8<\/span>))<\/span>\r\n\r\n\r\n# Show plot with grid <\/span>\r\nplt<\/span>.<\/span>grid<\/span>()<\/span> \r\n\r\n##always important to actually display!<\/span>\r\nplt<\/span>.<\/span>show<\/span>()<\/span> \r\n<\/pre>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n
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Wait, isn't that a vector?\u00b6<\/a><\/h2>\r\nYes. Yes it is. Hopefully, you can see that the distance between 2 points is the magnitude of the vector found by subtracting one from the other. Phew!\r\n\r\nBut what this (should) lead to is... Are there different ways of finding distance? What if you lived in Manhattan, and needed to drive - then calculating a diagonal distance doesn't make much sense.\r\n\r\nI'm not going through a formal definition, but we can develop math with the basic idea of \"distance\", then sub in any way of calculating distance that we choose.\r\n
Definition of \"Distance\"\u00b6<\/a><\/h3>\r\n