The Keeling Spiral<\/title><meta name=\"generator\" content=\"MATLAB 8.5\"><meta name=\"DC.date\" content=\"2015-03-31\"><meta name=\"DC.source\" content=\"keelingspiral.m\"><style type=\"text\/css\">\r\nhtml,body,div,span,applet,object,iframe,h1,h2,h3,h4,h5,h6,p,blockquote,pre,a,abbr,acronym,address,big,cite,code,del,dfn,em,font,img,ins,kbd,q,s,samp,small,strike,strong,sub,sup,tt,var,b,u,i,center,dl,dt,dd,ol,ul,li,fieldset,form,label,legend,table,caption,tbody,tfoot,thead,tr,th,td{margin:0;padding:0;border:0;outline:0;font-size:100%;vertical-align:baseline;background:transparent}body{line-height:1}ol,ul{list-style:none}blockquote,q{quotes:none}blockquote:before,blockquote:after,q:before,q:after{content:'';content:none}:focus{outine:0}ins{text-decoration:none}del{text-decoration:line-through}table{border-collapse:collapse;border-spacing:0}\r\n\r\nhtml { min-height:100%; margin-bottom:1px; }\r\nhtml body { height:100%; margin:0px; font-family:Arial, Helvetica, sans-serif; font-size:10px; color:#000; line-height:140%; background:#fff none; overflow-y:scroll; }\r\nhtml body td { vertical-align:top; text-align:left; }\r\n\r\nh1 { padding:0px; margin:0px 0px 25px; font-family:Arial, Helvetica, sans-serif; font-size:1.5em; color:#d55000; line-height:100%; font-weight:normal; }\r\nh2 { padding:0px; margin:0px 0px 8px; font-family:Arial, Helvetica, sans-serif; font-size:1.2em; color:#000; font-weight:bold; line-height:140%; border-bottom:1px solid #d6d4d4; display:block; }\r\nh3 { padding:0px; margin:0px 0px 5px; font-family:Arial, Helvetica, sans-serif; font-size:1.1em; color:#000; font-weight:bold; line-height:140%; }\r\n\r\na { color:#005fce; text-decoration:none; }\r\na:hover { color:#005fce; text-decoration:underline; }\r\na:visited { color:#004aa0; text-decoration:none; }\r\n\r\np { padding:0px; margin:0px 0px 20px; }\r\nimg { padding:0px; margin:0px 0px 20px; border:none; }\r\np img, pre img, tt img, li img, h1 img, h2 img { margin-bottom:0px; } \r\n\r\nul { padding:0px; margin:0px 0px 20px 23px; list-style:square; }\r\nul li { padding:0px; margin:0px 0px 7px 0px; }\r\nul li ul { padding:5px 0px 0px; margin:0px 0px 7px 23px; }\r\nul li ol li { list-style:decimal; }\r\nol { padding:0px; margin:0px 0px 20px 0px; list-style:decimal; }\r\nol li { padding:0px; margin:0px 0px 7px 23px; list-style-type:decimal; }\r\nol li ol { padding:5px 0px 0px; margin:0px 0px 7px 0px; }\r\nol li ol li { list-style-type:lower-alpha; }\r\nol li ul { padding-top:7px; }\r\nol li ul li { list-style:square; }\r\n\r\n.content { font-size:1.2em; line-height:140%; padding: 20px; }\r\n\r\npre, code { font-size:12px; }\r\ntt { font-size: 1.2em; }\r\npre { margin:0px 0px 20px; }\r\npre.codeinput { padding:10px; border:1px solid #d3d3d3; background:#f7f7f7; }\r\npre.codeoutput { padding:10px 11px; margin:0px 0px 20px; color:#4c4c4c; }\r\npre.error { color:red; }\r\n\r\n@media print { pre.codeinput, pre.codeoutput { word-wrap:break-word; width:100%; } }\r\n\r\nspan.keyword { color:#0000FF }\r\nspan.comment { color:#228B22 }\r\nspan.string { color:#A020F0 }\r\nspan.untermstring { color:#B20000 }\r\nspan.syscmd { color:#B28C00 }\r\n\r\n.footer { width:auto; padding:10px 0px; margin:25px 0px 0px; border-top:1px dotted #878787; font-size:0.8em; line-height:140%; font-style:italic; color:#878787; text-align:left; float:none; }\r\n.footer p { margin:0px; }\r\n.footer a { color:#878787; }\r\n.footer a:hover { color:#878787; text-decoration:underline; }\r\n.footer a:visited { color:#878787; }\r\n\r\ntable th { padding:7px 5px; text-align:left; vertical-align:middle; border: 1px solid #d6d4d4; font-weight:bold; }\r\ntable td { padding:7px 5px; text-align:left; vertical-align:top; border:1px solid #d6d4d4; }\r\n\r\n\r\n\r\n\r\n\r\n <\/style><\/head><body><div class=\"content\"><h1>The Keeling Spiral<\/h1>Today we're going to take a look at what is probably the most famous dataset in atmospheric science, the <a href=\"https:\/\/scripps.ucsd.edu\/programs\/keelingcurve\/\">Keeling Curve<\/a>. This is a timeseries dataset of the concentration of carbon dioxide at an observatory on Mauna Loa. <a href=\"http:\/\/en.wikipedia.org\/wiki\/Charles_David_Keeling\">Charles Keeling<\/a> started collecting this data in 1974. It's available online at the following URL.<\/p><pre class=\"codeinput\">s = urlread('ftp:\/\/aftp.cmdl.noaa.gov\/products\/trends\/co2\/co2_weekly_mlo.txt'<\/span>);\r\nc = strsplit(s,'\\n'<\/span>);\r\n<\/pre>The file is a pretty simple text format. We just need to break it into individual lines and pull out the fields we want. We want the date (which is stored in the first 3 columns) and the \"CO2 molfrac\" which is stored in the 5th column.<\/p><pre class=\"codeinput\">nl = length(c);\r\ndates = datetime.empty;\r\nmolfrac = [];\r\nfor<\/span> i=1:nl\r\n % Comment lines start with #<\/span>\r\n if<\/span> isempty(c{i}) || c{i}(1) == '#'<\/span>\r\n continue<\/span>\r\n end<\/span>\r\n % Get all the fields<\/span>\r\n lab = strsplit(c{i});\r\n value_str = lab{6};\r\n dates(end+1) = datetime(str2double(lab{2}), ...<\/span>\r\n str2double(lab{3}), ...<\/span>\r\n str2double(lab{4}));\r\n % -999.99 means missing data<\/span>\r\n if<\/span> strcmp(value_str,'-999.99'<\/span>)\r\n molfrac(end+1) = nan;\r\n else<\/span>\r\n molfrac(end+1) = str2double(value_str);\r\n end<\/span>\r\nend<\/span>\r\n<\/pre>At this point, we have two arrays. * dates is an array of datetime values when the measurements were taken. * molfrac is a numeric array with the measurements in parts per million.<\/p>Lets start out with a simple plot.<\/p><pre class=\"codeinput\">plot(dates,molfrac)\r\nxlabel('Date'<\/span>)\r\nylabel('parts per million'<\/span>)\r\ntitle('Atmospheric CO_2 at Mauna Loa'<\/span>);\r\n<\/pre><img decoding=\"async\" vspace=\"5\" hspace=\"5\" src=\"https:\/\/blogs.mathworks.com\/images\/graphics\/2015\/keelingspiral_01.png\" alt=\"\"> As you can see, the CO2 level keeps going up every year, but it also bounces up and down quite a bit. That's because there's a seasonal variation in the level of CO2 that's related to things like the rate at which plants are growing. We can see this if we zoom in on a single year.<\/p><pre class=\"codeinput\">mask = year(dates)==1987;\r\nplot(dates(mask),molfrac(mask))\r\ntitle('Atmospheric CO_2 at Mauna Loa, 1987'<\/span>);\r\n<\/pre><img decoding=\"async\" vspace=\"5\" hspace=\"5\" src=\"https:\/\/blogs.mathworks.com\/images\/graphics\/2015\/keelingspiral_02.png\" alt=\"\"> One simple way to deal with this seasonality is to break the dataset into individual years, and plot them all together. We do this by looping over the years, pulling out the values where \"year(dates)\" matches our loop index, and then computing the number of days since January 1st of that year.<\/p><pre class=\"codeinput\">cla reset<\/span>\r\nhold on<\/span>\r\nfor<\/span> yidx=1974:2015\r\n mask = year(dates)==yidx;\r\n plot(days(dates(mask)-datetime(yidx,1,1)),molfrac(mask));\r\nend<\/span>\r\nxlim([0 365])\r\nxlabel('Day of Year'<\/span>)\r\nylabel('parts per million'<\/span>)\r\ntitle('Atmospheric CO_2 at Mauna Loa'<\/span>);\r\n<\/pre><img decoding=\"async\" vspace=\"5\" hspace=\"5\" src=\"https:\/\/blogs.mathworks.com\/images\/graphics\/2015\/keelingspiral_03.png\" alt=\"\"> But we really need to improve the tick labels with this view. We can use the <a href=\"https:\/\/www.mathworks.com\/help\/matlab\/ref\/datestr.html\">datestr<\/a> function to get a nicer label for the days.<\/p><pre class=\"codeinput\">xlabel(''<\/span>)\r\nax = gca;\r\nx = ax.XTick;\r\nlab = cell(size(x));\r\nfor<\/span> i=1:length(x)\r\n lab{i} = datestr(datetime(2015,1,1) + days(x(i)), 'mmm dd'<\/span>);\r\nend<\/span>\r\nax.XTickLabel = lab;\r\nax.XTickLabelRotation = 45;\r\n<\/pre><img decoding=\"async\" vspace=\"5\" hspace=\"5\" src=\"https:\/\/blogs.mathworks.com\/images\/graphics\/2015\/keelingspiral_04.png\" alt=\"\"> Stacking 2D lines shows us how the CO2 level moves up each year, but what would it look like if we plotted it in 3D? That's pretty easily done by creating an array of years and an array of \"day of the year\".<\/p><pre class=\"codeinput\">yr = zeros(size(molfrac));\r\ndofy = zeros(size(molfrac));\r\nfor<\/span> yidx=1974:2015\r\n mask = year(dates) == yidx;\r\n yr(mask) = yidx;\r\n dofy(mask) = days(dates(mask) - datetime(yidx,1,1));\r\nend<\/span>\r\n\r\nhold off<\/span>\r\ncla reset<\/span>\r\nfor<\/span> yidx=1974:2015\r\n mask = year(dates) == yidx;\r\n line(dofy(mask), yr(mask), molfrac(mask));\r\nend<\/span>\r\nbox on<\/span>\r\ngrid on<\/span>\r\nview(3)\r\nxlim([0 365])\r\nylim([1974 2015])\r\n<\/pre><img decoding=\"async\" vspace=\"5\" hspace=\"5\" src=\"https:\/\/blogs.mathworks.com\/images\/graphics\/2015\/keelingspiral_05.png\" alt=\"\"> And again, we'll want to pretty up the labels.<\/p><pre class=\"codeinput\">ax = gca;\r\nx = ax.XTick;\r\nlab = cell(size(x));\r\nfor<\/span> i=1:length(x)\r\n lab{i} = datestr(datetime(2015,1,1) + days(x(i)), 'mmm dd'<\/span>);\r\nend<\/span>\r\nax.XTickLabel = lab;\r\nxlabel('Day'<\/span>)\r\nylabel('Year'<\/span>)\r\nzlabel('parts per million'<\/span>)\r\ntitle('Atmospheric CO_2 at Mauna Loa'<\/span>);\r\n<\/pre><img decoding=\"async\" vspace=\"5\" hspace=\"5\" src=\"https:\/\/blogs.mathworks.com\/images\/graphics\/2015\/keelingspiral_06.png\" alt=\"\"> We could also use <a href=\"https:\/\/www.mathworks.com\/help\/matlab\/ref\/scatter3.html\">scatter3<\/a> to plot molfrac against the yr and dofy arrays. That gives us something that looks a little like a surface.<\/p><pre class=\"codeinput\">scatter3(dofy,yr,molfrac)\r\nxlim([0 365])\r\nylim([1974 2015])\r\n<\/pre><img decoding=\"async\" vspace=\"5\" hspace=\"5\" src=\"https:\/\/blogs.mathworks.com\/images\/graphics\/2015\/keelingspiral_07.png\" alt=\"\"> But it'd be nice if that really was a surface plot. We can't use the surface command because this isn't a regular grid. We could pad the first year and last year out to 52 weeks by adding nans, but there are also years that have 53 weeks, such as 1978.<\/p>We can skip the gridding problem by using <a href=\"https:\/\/www.mathworks.com\/help\/matlab\/ref\/trisurf.html\">trisurf<\/a>, but that means we'll need to triangulate the data. Lets try using <a href=\"https:\/\/www.mathworks.com\/help\/matlab\/ref\/delaunay.html\">Delaunay triangulation<\/a>. BTW, the divide by 7 is required because Delaunay triangulation works best when the X & Y scaling is close to uniform.<\/p><pre class=\"codeinput\">tri = delaunay(yr,dofy\/7);\r\nh = trisurf(tri,dofy,yr,molfrac);\r\nh.EdgeColor = 'none'<\/span>;\r\nh.SpecularStrength = .25;\r\nylim([1974 2015])\r\ncamlight\r\n<\/pre><img decoding=\"async\" vspace=\"5\" hspace=\"5\" src=\"https:\/\/blogs.mathworks.com\/images\/graphics\/2015\/keelingspiral_08.png\" alt=\"\"> That looks pretty interesting. We can now rotate it around and see some details about how the different years relate to each other. But as you can see there is something funny going on at the edges. If we look straight down and zoom in, we can figure out what's happening:<\/p><pre class=\"codeinput\">xlim([0 200])\r\nylim([1970 1980])\r\nh.FaceLighting = 'none'<\/span>;\r\nh.EdgeColor = 'black'<\/span>;\r\nview(2)\r\n<\/pre><img decoding=\"async\" vspace=\"5\" hspace=\"5\" src=\"https:\/\/blogs.mathworks.com\/images\/graphics\/2015\/keelingspiral_09.png\" alt=\"\"> The call to delaunay created triangles between the first day (19-May-1974) and lots of different days in 1975. The Delaunay triangulation includes these triangles because they're part of the convex hull of the points, but they're not really meaningful for our purposes. We can get rid of those by walking through the array of triangles and throwing away any with long edges.<\/p><pre class=\"codeinput\">max_x = 1;\r\nmax_y = 15;\r\nlong_x = (abs(yr(tri(:,1))-yr(tri(:,2))) > max_x) ...<\/span>\r\n | (abs(yr(tri(:,2))-yr(tri(:,3))) > max_x) ...<\/span>\r\n | (abs(yr(tri(:,3))-yr(tri(:,1))) > max_x);\r\nlong_y = (abs(dofy(tri(:,1))-dofy(tri(:,2))) > max_y) ...<\/span>\r\n | (abs(dofy(tri(:,2))-dofy(tri(:,3))) > max_y) ...<\/span>\r\n | (abs(dofy(tri(:,3))-dofy(tri(:,1))) > max_y);\r\ntri = tri(~(long_x | long_y),:);\r\nh = trisurf(tri,dofy,yr,molfrac);\r\nh.EdgeColor = 'none'<\/span>;\r\nh.SpecularStrength = .25;\r\ncamlight\r\nview(2)\r\n<\/pre><img decoding=\"async\" vspace=\"5\" hspace=\"5\" src=\"https:\/\/blogs.mathworks.com\/images\/graphics\/2015\/keelingspiral_10.png\" alt=\"\"> Now we'll go back to our 3D view.<\/p><pre class=\"codeinput\">view(3)\r\nxlim([0 365])\r\nylim([1974 2015])\r\nzlim([-inf inf])\r\n\r\nax = gca;\r\nt = ax.XTick;\r\nlab = cell(size(t));\r\nfor<\/span> i=1:length(t)\r\n lab{i} = datestr(datetime(2015,1,1) + days(t(i)), 'mmm dd'<\/span>);\r\nend<\/span>\r\nax.XTickLabel = lab;\r\nxlabel('Day'<\/span>)\r\nylabel('Year'<\/span>)\r\nzlabel('parts per million'<\/span>)\r\nbox on<\/span>\r\n<\/pre><img decoding=\"async\" vspace=\"5\" hspace=\"5\" src=\"https:\/\/blogs.mathworks.com\/images\/graphics\/2015\/keelingspiral_11.png\" alt=\"\"> But there's a problem with this visualization. The last day of one year is actually adjacent to the first day of the next year, but in this visualization they're on opposite sides of the chart. What if we rolled this around a cylinder? We'll make a cylindrical chart where the angle is the day of the year, the Z is the year, and the radius is the measurement of the CO2 level.<\/p><pre class=\"codeinput\">[x,y,z] = pol2cart(2*pi*dofy\/365, molfrac, yr+dofy\/365);\r\nh = trisurf(tri,x,y,z);\r\nh.FaceVertexCData = molfrac';\r\nh.FaceColor = 'interp'<\/span>;\r\nh.SpecularStrength = .25;\r\nline([0 0],[0 0],[1970 2020],'Color'<\/span>,[.75 .75 .75],'LineWidth'<\/span>,3)\r\ncamlight\r\nxlim([-405 405])\r\nylim([-405 405])\r\nzlim([1974 2015])\r\n<\/pre><img decoding=\"async\" vspace=\"5\" hspace=\"5\" src=\"https:\/\/blogs.mathworks.com\/images\/graphics\/2015\/keelingspiral_12.png\" alt=\"\"> That's kind of interesting, but we can see that there's a rip down the back. That's because the Delaunay triangulation didn't know that those samples were near each other. It was working in a 2D plane which had a cut rather than in the cyclic surface of the cylinder. We'll need to do our own triangulation. This can be kind of tedious, so I'll just pull out one I prepared earlier like <a href=\"http:\/\/en.wikipedia.org\/wiki\/Julia_Child\">Julia Child<\/a> used to do on her TV show. You can <a href=\"https:\/\/blogs.mathworks.com\/images\/graphics\/2015\/stripyears.m\">download it here<\/a>.<\/p><pre class=\"codeinput\">tri = stripyears(yr,dofy);\r\n[x,y,z] = pol2cart(2*pi*dofy\/365, molfrac, yr+dofy\/365);\r\nh = trisurf(tri,x,y,z);\r\nh.FaceVertexCData = molfrac';\r\nh.FaceColor = 'interp'<\/span>;\r\nh.SpecularStrength = .25;\r\nline([0 0],[0 0],[1970 2020],'Color'<\/span>,[.75 .75 .75],'LineWidth'<\/span>,3)\r\nlight('Position'<\/span>,[-1 -.5 -.25]);\r\nlight('Position'<\/span>,[-1.5 .5 .25]);\r\nxlim([-405 405])\r\nylim([-405 405])\r\nzlim([1974 2015])\r\n<\/pre><img decoding=\"async\" vspace=\"5\" hspace=\"5\" src=\"https:\/\/blogs.mathworks.com\/images\/graphics\/2015\/keelingspiral_13.png\" alt=\"\"> This results in a fairly unusual visualization that I saw used a few times back in the 80's, but I don't think that it's been used much since. That was really the dawn of 3D in data visualization, so we were probably just getting carried away with our new toy. It certainly has some limitations. But lets see if there's anything we can do to make it more useful.<\/p>The first thing we should do is fix the datatips so that they display the date and value from the dataset, rather than the X, Y, & Z coordinates. That's actually pretty easy. We just need this simple function:<\/p><pre class=\"codeinput\">type keeling_datatip.m<\/span>\r\ndcm = datacursormode(gcf);\r\ndcm.Enable = 'on'<\/span>;\r\ndcm.UpdateFcn = @keeling_datatip;\r\n<\/pre><pre class=\"codeoutput\">\r\nfunction txt = keeling_datatip(~,evd)\r\n [th,r,z] = cart2pol(evd.Position(1),evd.Position(2),evd.Position(3));\r\n th = mod(th,2*pi);\r\n d = datetime(floor(z),1,1) + days(365*th\/(2*pi));\r\n txt = {['Date: ', datestr(d)], ...\r\n ['Value: ', num2str(r,4)]};\r\nend\r\n<\/pre><img decoding=\"async\" vspace=\"5\" hspace=\"5\" src=\"https:\/\/blogs.mathworks.com\/images\/graphics\/2015\/datatip.png\" alt=\"\"> Another thing we can do is to exagerate the radius. That will make the variations in the radius more visible. We could also get rid of the edges and switch to Gouraud lighting.<\/p><pre class=\"codeinput\">roffset = 300;\r\n[x,y,z] = pol2cart(2*pi*dofy\/365, molfrac-roffset, yr+dofy\/365);\r\nh = trisurf(tri,x,y,z);\r\nh.FaceVertexCData = molfrac';\r\nh.FaceColor = 'interp'<\/span>;\r\nh.EdgeColor = 'none'<\/span>;\r\nh.FaceLighting = 'gouraud'<\/span>;\r\nh.SpecularStrength = .25;\r\nline([0 0],[0 0],[1970 2020],'Color'<\/span>,[.75 .75 .75],'LineWidth'<\/span>,3)\r\nl1 = light('Position'<\/span>,[-1 -.5 -.25]);\r\nl2 = light('Position'<\/span>,[-1.5 .5 .25]);\r\nxlim([-105 105])\r\nylim([-105 105])\r\nzlim([1974 2015])\r\n<\/pre><img decoding=\"async\" vspace=\"5\" hspace=\"5\" src=\"https:\/\/blogs.mathworks.com\/images\/graphics\/2015\/keelingspiral_15.png\" alt=\"\"> But when with the new radius scale, we need to fix the X & Y ticks to account for the offset.<\/p><pre class=\"codeinput\">ax = gca;\r\nt = ax.XTick;\r\nlab = cell(size(t));\r\nfor<\/span> i=1:length(t)\r\n lab{i} = num2str(abs(t(i))+roffset);\r\nend<\/span>\r\nax.XTickLabel = lab;\r\nax.YTick = t;\r\nax.YTickLabel = lab;\r\nxlabel('parts per million'<\/span>)\r\nylabel('parts per million'<\/span>)\r\nzlabel('Year'<\/span>)\r\ntitle('Atmospheric CO_2 at Mauna Loa'<\/span>);\r\n<\/pre><img decoding=\"async\" vspace=\"5\" hspace=\"5\" src=\"https:\/\/blogs.mathworks.com\/images\/graphics\/2015\/keelingspiral_16.png\" alt=\"\"> What do you think? Can you think of cases where this visualization might be useful? Can you think of any additional tweaks which might make it more useful?<\/p> <a href=\"https:\/\/www.mathworks.com\/products\/matlab\/\">Published with MATLAB® R2015a<\/a> <\/p><\/div><\/body><\/html>","protected":false},"excerpt":{"rendered":"<div class=\"overview-image\"><img src=\"https:\/\/blogs.mathworks.com\/graphics\/files\/feature_image\/keelingthumbnail.png\" class=\"img-responsive attachment-post-thumbnail size-post-thumbnail wp-post-image\" alt=\"\" decoding=\"async\" loading=\"lazy\" \/><\/div>\r\n\r\n \r\n The Keeling... <a class=\"read-more\" href=\"https:\/\/blogs.mathworks.com\/graphics\/2015\/03\/31\/keeling-spiral\/\">read more >><\/a><\/p>","protected":false},"author":89,"featured_media":225,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":[],"categories":[7],"tags":[],"_links":{"self":[{"href":"https:\/\/blogs.mathworks.com\/graphics\/wp-json\/wp\/v2\/posts\/222"}],"collection":[{"href":"https:\/\/blogs.mathworks.com\/graphics\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/blogs.mathworks.com\/graphics\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/blogs.mathworks.com\/graphics\/wp-json\/wp\/v2\/users\/89"}],"replies":[{"embeddable":true,"href":"https:\/\/blogs.mathworks.com\/graphics\/wp-json\/wp\/v2\/comments?post=222"}],"version-history":[{"count":5,"href":"https:\/\/blogs.mathworks.com\/graphics\/wp-json\/wp\/v2\/posts\/222\/revisions"}],"predecessor-version":[{"id":831,"href":"https:\/\/blogs.mathworks.com\/graphics\/wp-json\/wp\/v2\/posts\/222\/revisions\/831"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/blogs.mathworks.com\/graphics\/wp-json\/wp\/v2\/media\/225"}],"wp:attachment":[{"href":"https:\/\/blogs.mathworks.com\/graphics\/wp-json\/wp\/v2\/media?parent=222"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blogs.mathworks.com\/graphics\/wp-json\/wp\/v2\/categories?post=222"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blogs.mathworks.com\/graphics\/wp-json\/wp\/v2\/tags?post=222"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}

{"id":222,"date":"2015-03-31T14:16:21","date_gmt":"2015-03-31T18:16:21","guid":{"rendered":"https:\/\/blogs.mathworks.com\/graphics\/?p=222"},"modified":"2019-06-16T14:15:38","modified_gmt":"2019-06-16T18:15:38","slug":"keeling-spiral","status":"publish","type":"post","link":"https:\/\/blogs.mathworks.com\/graphics\/2015\/03\/31\/keeling-spiral\/","title":{"rendered":"Keeling Spiral"},"content":{"rendered":"\r\n\r\n \r\n The Keeling Spiral<\/title><meta name=\"generator\" content=\"MATLAB 8.5\"><meta name=\"DC.date\" content=\"2015-03-31\"><meta name=\"DC.source\" content=\"keelingspiral.m\"><style type=\"text\/css\">\r\nhtml,body,div,span,applet,object,iframe,h1,h2,h3,h4,h5,h6,p,blockquote,pre,a,abbr,acronym,address,big,cite,code,del,dfn,em,font,img,ins,kbd,q,s,samp,small,strike,strong,sub,sup,tt,var,b,u,i,center,dl,dt,dd,ol,ul,li,fieldset,form,label,legend,table,caption,tbody,tfoot,thead,tr,th,td{margin:0;padding:0;border:0;outline:0;font-size:100%;vertical-align:baseline;background:transparent}body{line-height:1}ol,ul{list-style:none}blockquote,q{quotes:none}blockquote:before,blockquote:after,q:before,q:after{content:'';content:none}:focus{outine:0}ins{text-decoration:none}del{text-decoration:line-through}table{border-collapse:collapse;border-spacing:0}\r\n\r\nhtml { min-height:100%; margin-bottom:1px; }\r\nhtml body { height:100%; margin:0px; font-family:Arial, Helvetica, sans-serif; font-size:10px; color:#000; line-height:140%; background:#fff none; overflow-y:scroll; }\r\nhtml body td { vertical-align:top; text-align:left; }\r\n\r\nh1 { padding:0px; margin:0px 0px 25px; font-family:Arial, Helvetica, sans-serif; font-size:1.5em; color:#d55000; line-height:100%; font-weight:normal; }\r\nh2 { padding:0px; margin:0px 0px 8px; font-family:Arial, Helvetica, sans-serif; font-size:1.2em; color:#000; font-weight:bold; line-height:140%; border-bottom:1px solid #d6d4d4; display:block; }\r\nh3 { padding:0px; margin:0px 0px 5px; font-family:Arial, Helvetica, sans-serif; font-size:1.1em; color:#000; font-weight:bold; line-height:140%; }\r\n\r\na { color:#005fce; text-decoration:none; }\r\na:hover { color:#005fce; text-decoration:underline; }\r\na:visited { color:#004aa0; text-decoration:none; }\r\n\r\np { padding:0px; margin:0px 0px 20px; }\r\nimg { padding:0px; margin:0px 0px 20px; border:none; }\r\np img, pre img, tt img, li img, h1 img, h2 img { margin-bottom:0px; } \r\n\r\nul { padding:0px; margin:0px 0px 20px 23px; list-style:square; }\r\nul li { padding:0px; margin:0px 0px 7px 0px; }\r\nul li ul { padding:5px 0px 0px; margin:0px 0px 7px 23px; }\r\nul li ol li { list-style:decimal; }\r\nol { padding:0px; margin:0px 0px 20px 0px; list-style:decimal; }\r\nol li { padding:0px; margin:0px 0px 7px 23px; list-style-type:decimal; }\r\nol li ol { padding:5px 0px 0px; margin:0px 0px 7px 0px; }\r\nol li ol li { list-style-type:lower-alpha; }\r\nol li ul { padding-top:7px; }\r\nol li ul li { list-style:square; }\r\n\r\n.content { font-size:1.2em; line-height:140%; padding: 20px; }\r\n\r\npre, code { font-size:12px; }\r\ntt { font-size: 1.2em; }\r\npre { margin:0px 0px 20px; }\r\npre.codeinput { padding:10px; border:1px solid #d3d3d3; background:#f7f7f7; }\r\npre.codeoutput { padding:10px 11px; margin:0px 0px 20px; color:#4c4c4c; }\r\npre.error { color:red; }\r\n\r\n@media print { pre.codeinput, pre.codeoutput { word-wrap:break-word; width:100%; } }\r\n\r\nspan.keyword { color:#0000FF }\r\nspan.comment { color:#228B22 }\r\nspan.string { color:#A020F0 }\r\nspan.untermstring { color:#B20000 }\r\nspan.syscmd { color:#B28C00 }\r\n\r\n.footer { width:auto; padding:10px 0px; margin:25px 0px 0px; border-top:1px dotted #878787; font-size:0.8em; line-height:140%; font-style:italic; color:#878787; text-align:left; float:none; }\r\n.footer p { margin:0px; }\r\n.footer a { color:#878787; }\r\n.footer a:hover { color:#878787; text-decoration:underline; }\r\n.footer a:visited { color:#878787; }\r\n\r\ntable th { padding:7px 5px; text-align:left; vertical-align:middle; border: 1px solid #d6d4d4; font-weight:bold; }\r\ntable td { padding:7px 5px; text-align:left; vertical-align:top; border:1px solid #d6d4d4; }\r\n\r\n\r\n\r\n\r\n\r\n <\/style><\/head><body><div class=\"content\"><h1>The Keeling Spiral<\/h1>Today we're going to take a look at what is probably the most famous dataset in atmospheric science, the <a href=\"https:\/\/scripps.ucsd.edu\/programs\/keelingcurve\/\">Keeling Curve<\/a>. This is a timeseries dataset of the concentration of carbon dioxide at an observatory on Mauna Loa. <a href=\"http:\/\/en.wikipedia.org\/wiki\/Charles_David_Keeling\">Charles Keeling<\/a> started collecting this data in 1974. It's available online at the following URL.<\/p><pre class=\"codeinput\">s = urlread('ftp:\/\/aftp.cmdl.noaa.gov\/products\/trends\/co2\/co2_weekly_mlo.txt'<\/span>);\r\nc = strsplit(s,'\\n'<\/span>);\r\n<\/pre>The file is a pretty simple text format. We just need to break it into individual lines and pull out the fields we want. We want the date (which is stored in the first 3 columns) and the \"CO2 molfrac\" which is stored in the 5th column.<\/p><pre class=\"codeinput\">nl = length(c);\r\ndates = datetime.empty;\r\nmolfrac = [];\r\nfor<\/span> i=1:nl\r\n % Comment lines start with #<\/span>\r\n if<\/span> isempty(c{i}) || c{i}(1) == '#'<\/span>\r\n continue<\/span>\r\n end<\/span>\r\n % Get all the fields<\/span>\r\n lab = strsplit(c{i});\r\n value_str = lab{6};\r\n dates(end+1) = datetime(str2double(lab{2}), ...<\/span>\r\n str2double(lab{3}), ...<\/span>\r\n str2double(lab{4}));\r\n % -999.99 means missing data<\/span>\r\n if<\/span> strcmp(value_str,'-999.99'<\/span>)\r\n molfrac(end+1) = nan;\r\n else<\/span>\r\n molfrac(end+1) = str2double(value_str);\r\n end<\/span>\r\nend<\/span>\r\n<\/pre>At this point, we have two arrays. * dates is an array of datetime values when the measurements were taken. * molfrac is a numeric array with the measurements in parts per million.<\/p>Lets start out with a simple plot.<\/p><pre class=\"codeinput\">plot(dates,molfrac)\r\nxlabel('Date'<\/span>)\r\nylabel('parts per million'<\/span>)\r\ntitle('Atmospheric CO_2 at Mauna Loa'<\/span>);\r\n<\/pre><img decoding=\"async\" vspace=\"5\" hspace=\"5\" src=\"https:\/\/blogs.mathworks.com\/images\/graphics\/2015\/keelingspiral_01.png\" alt=\"\"> As you can see, the CO2 level keeps going up every year, but it also bounces up and down quite a bit. That's because there's a seasonal variation in the level of CO2 that's related to things like the rate at which plants are growing. We can see this if we zoom in on a single year.<\/p><pre class=\"codeinput\">mask = year(dates)==1987;\r\nplot(dates(mask),molfrac(mask))\r\ntitle('Atmospheric CO_2 at Mauna Loa, 1987'<\/span>);\r\n<\/pre><img decoding=\"async\" vspace=\"5\" hspace=\"5\" src=\"https:\/\/blogs.mathworks.com\/images\/graphics\/2015\/keelingspiral_02.png\" alt=\"\"> One simple way to deal with this seasonality is to break the dataset into individual years, and plot them all together. We do this by looping over the years, pulling out the values where \"year(dates)\" matches our loop index, and then computing the number of days since January 1st of that year.<\/p><pre class=\"codeinput\">cla reset<\/span>\r\nhold on<\/span>\r\nfor<\/span> yidx=1974:2015\r\n mask = year(dates)==yidx;\r\n plot(days(dates(mask)-datetime(yidx,1,1)),molfrac(mask));\r\nend<\/span>\r\nxlim([0 365])\r\nxlabel('Day of Year'<\/span>)\r\nylabel('parts per million'<\/span>)\r\ntitle('Atmospheric CO_2 at Mauna Loa'<\/span>);\r\n<\/pre><img decoding=\"async\" vspace=\"5\" hspace=\"5\" src=\"https:\/\/blogs.mathworks.com\/images\/graphics\/2015\/keelingspiral_03.png\" alt=\"\"> But we really need to improve the tick labels with this view. We can use the <a href=\"https:\/\/www.mathworks.com\/help\/matlab\/ref\/datestr.html\">datestr<\/a> function to get a nicer label for the days.<\/p><pre class=\"codeinput\">xlabel(''<\/span>)\r\nax = gca;\r\nx = ax.XTick;\r\nlab = cell(size(x));\r\nfor<\/span> i=1:length(x)\r\n lab{i} = datestr(datetime(2015,1,1) + days(x(i)), 'mmm dd'<\/span>);\r\nend<\/span>\r\nax.XTickLabel = lab;\r\nax.XTickLabelRotation = 45;\r\n<\/pre><img decoding=\"async\" vspace=\"5\" hspace=\"5\" src=\"https:\/\/blogs.mathworks.com\/images\/graphics\/2015\/keelingspiral_04.png\" alt=\"\"> Stacking 2D lines shows us how the CO2 level moves up each year, but what would it look like if we plotted it in 3D? That's pretty easily done by creating an array of years and an array of \"day of the year\".<\/p><pre class=\"codeinput\">yr = zeros(size(molfrac));\r\ndofy = zeros(size(molfrac));\r\nfor<\/span> yidx=1974:2015\r\n mask = year(dates) == yidx;\r\n yr(mask) = yidx;\r\n dofy(mask) = days(dates(mask) - datetime(yidx,1,1));\r\nend<\/span>\r\n\r\nhold off<\/span>\r\ncla reset<\/span>\r\nfor<\/span> yidx=1974:2015\r\n mask = year(dates) == yidx;\r\n line(dofy(mask), yr(mask), molfrac(mask));\r\nend<\/span>\r\nbox on<\/span>\r\ngrid on<\/span>\r\nview(3)\r\nxlim([0 365])\r\nylim([1974 2015])\r\n<\/pre><img decoding=\"async\" vspace=\"5\" hspace=\"5\" src=\"https:\/\/blogs.mathworks.com\/images\/graphics\/2015\/keelingspiral_05.png\" alt=\"\"> And again, we'll want to pretty up the labels.<\/p><pre class=\"codeinput\">ax = gca;\r\nx = ax.XTick;\r\nlab = cell(size(x));\r\nfor<\/span> i=1:length(x)\r\n lab{i} = datestr(datetime(2015,1,1) + days(x(i)), 'mmm dd'<\/span>);\r\nend<\/span>\r\nax.XTickLabel = lab;\r\nxlabel('Day'<\/span>)\r\nylabel('Year'<\/span>)\r\nzlabel('parts per million'<\/span>)\r\ntitle('Atmospheric CO_2 at Mauna Loa'<\/span>);\r\n<\/pre><img decoding=\"async\" vspace=\"5\" hspace=\"5\" src=\"https:\/\/blogs.mathworks.com\/images\/graphics\/2015\/keelingspiral_06.png\" alt=\"\"> We could also use <a href=\"https:\/\/www.mathworks.com\/help\/matlab\/ref\/scatter3.html\">scatter3<\/a> to plot molfrac against the yr and dofy arrays. That gives us something that looks a little like a surface.<\/p><pre class=\"codeinput\">scatter3(dofy,yr,molfrac)\r\nxlim([0 365])\r\nylim([1974 2015])\r\n<\/pre><img decoding=\"async\" vspace=\"5\" hspace=\"5\" src=\"https:\/\/blogs.mathworks.com\/images\/graphics\/2015\/keelingspiral_07.png\" alt=\"\"> But it'd be nice if that really was a surface plot. We can't use the surface command because this isn't a regular grid. We could pad the first year and last year out to 52 weeks by adding nans, but there are also years that have 53 weeks, such as 1978.<\/p>We can skip the gridding problem by using <a href=\"https:\/\/www.mathworks.com\/help\/matlab\/ref\/trisurf.html\">trisurf<\/a>, but that means we'll need to triangulate the data. Lets try using <a href=\"https:\/\/www.mathworks.com\/help\/matlab\/ref\/delaunay.html\">Delaunay triangulation<\/a>. BTW, the divide by 7 is required because Delaunay triangulation works best when the X & Y scaling is close to uniform.<\/p><pre class=\"codeinput\">tri = delaunay(yr,dofy\/7);\r\nh = trisurf(tri,dofy,yr,molfrac);\r\nh.EdgeColor = 'none'<\/span>;\r\nh.SpecularStrength = .25;\r\nylim([1974 2015])\r\ncamlight\r\n<\/pre><img decoding=\"async\" vspace=\"5\" hspace=\"5\" src=\"https:\/\/blogs.mathworks.com\/images\/graphics\/2015\/keelingspiral_08.png\" alt=\"\"> That looks pretty interesting. We can now rotate it around and see some details about how the different years relate to each other. But as you can see there is something funny going on at the edges. If we look straight down and zoom in, we can figure out what's happening:<\/p><pre class=\"codeinput\">xlim([0 200])\r\nylim([1970 1980])\r\nh.FaceLighting = 'none'<\/span>;\r\nh.EdgeColor = 'black'<\/span>;\r\nview(2)\r\n<\/pre><img decoding=\"async\" vspace=\"5\" hspace=\"5\" src=\"https:\/\/blogs.mathworks.com\/images\/graphics\/2015\/keelingspiral_09.png\" alt=\"\"> The call to delaunay created triangles between the first day (19-May-1974) and lots of different days in 1975. The Delaunay triangulation includes these triangles because they're part of the convex hull of the points, but they're not really meaningful for our purposes. We can get rid of those by walking through the array of triangles and throwing away any with long edges.<\/p><pre class=\"codeinput\">max_x = 1;\r\nmax_y = 15;\r\nlong_x = (abs(yr(tri(:,1))-yr(tri(:,2))) > max_x) ...<\/span>\r\n | (abs(yr(tri(:,2))-yr(tri(:,3))) > max_x) ...<\/span>\r\n | (abs(yr(tri(:,3))-yr(tri(:,1))) > max_x);\r\nlong_y = (abs(dofy(tri(:,1))-dofy(tri(:,2))) > max_y) ...<\/span>\r\n | (abs(dofy(tri(:,2))-dofy(tri(:,3))) > max_y) ...<\/span>\r\n | (abs(dofy(tri(:,3))-dofy(tri(:,1))) > max_y);\r\ntri = tri(~(long_x | long_y),:);\r\nh = trisurf(tri,dofy,yr,molfrac);\r\nh.EdgeColor = 'none'<\/span>;\r\nh.SpecularStrength = .25;\r\ncamlight\r\nview(2)\r\n<\/pre><img decoding=\"async\" vspace=\"5\" hspace=\"5\" src=\"https:\/\/blogs.mathworks.com\/images\/graphics\/2015\/keelingspiral_10.png\" alt=\"\"> Now we'll go back to our 3D view.<\/p><pre class=\"codeinput\">view(3)\r\nxlim([0 365])\r\nylim([1974 2015])\r\nzlim([-inf inf])\r\n\r\nax = gca;\r\nt = ax.XTick;\r\nlab = cell(size(t));\r\nfor<\/span> i=1:length(t)\r\n lab{i} = datestr(datetime(2015,1,1) + days(t(i)), 'mmm dd'<\/span>);\r\nend<\/span>\r\nax.XTickLabel = lab;\r\nxlabel('Day'<\/span>)\r\nylabel('Year'<\/span>)\r\nzlabel('parts per million'<\/span>)\r\nbox on<\/span>\r\n<\/pre><img decoding=\"async\" vspace=\"5\" hspace=\"5\" src=\"https:\/\/blogs.mathworks.com\/images\/graphics\/2015\/keelingspiral_11.png\" alt=\"\"> But there's a problem with this visualization. The last day of one year is actually adjacent to the first day of the next year, but in this visualization they're on opposite sides of the chart. What if we rolled this around a cylinder? We'll make a cylindrical chart where the angle is the day of the year, the Z is the year, and the radius is the measurement of the CO2 level.<\/p><pre class=\"codeinput\">[x,y,z] = pol2cart(2*pi*dofy\/365, molfrac, yr+dofy\/365);\r\nh = trisurf(tri,x,y,z);\r\nh.FaceVertexCData = molfrac';\r\nh.FaceColor = 'interp'<\/span>;\r\nh.SpecularStrength = .25;\r\nline([0 0],[0 0],[1970 2020],'Color'<\/span>,[.75 .75 .75],'LineWidth'<\/span>,3)\r\ncamlight\r\nxlim([-405 405])\r\nylim([-405 405])\r\nzlim([1974 2015])\r\n<\/pre><img decoding=\"async\" vspace=\"5\" hspace=\"5\" src=\"https:\/\/blogs.mathworks.com\/images\/graphics\/2015\/keelingspiral_12.png\" alt=\"\"> That's kind of interesting, but we can see that there's a rip down the back. That's because the Delaunay triangulation didn't know that those samples were near each other. It was working in a 2D plane which had a cut rather than in the cyclic surface of the cylinder. We'll need to do our own triangulation. This can be kind of tedious, so I'll just pull out one I prepared earlier like <a href=\"http:\/\/en.wikipedia.org\/wiki\/Julia_Child\">Julia Child<\/a> used to do on her TV show. You can <a href=\"https:\/\/blogs.mathworks.com\/images\/graphics\/2015\/stripyears.m\">download it here<\/a>.<\/p><pre class=\"codeinput\">tri = stripyears(yr,dofy);\r\n[x,y,z] = pol2cart(2*pi*dofy\/365, molfrac, yr+dofy\/365);\r\nh = trisurf(tri,x,y,z);\r\nh.FaceVertexCData = molfrac';\r\nh.FaceColor = 'interp'<\/span>;\r\nh.SpecularStrength = .25;\r\nline([0 0],[0 0],[1970 2020],'Color'<\/span>,[.75 .75 .75],'LineWidth'<\/span>,3)\r\nlight('Position'<\/span>,[-1 -.5 -.25]);\r\nlight('Position'<\/span>,[-1.5 .5 .25]);\r\nxlim([-405 405])\r\nylim([-405 405])\r\nzlim([1974 2015])\r\n<\/pre><img decoding=\"async\" vspace=\"5\" hspace=\"5\" src=\"https:\/\/blogs.mathworks.com\/images\/graphics\/2015\/keelingspiral_13.png\" alt=\"\"> This results in a fairly unusual visualization that I saw used a few times back in the 80's, but I don't think that it's been used much since. That was really the dawn of 3D in data visualization, so we were probably just getting carried away with our new toy. It certainly has some limitations. But lets see if there's anything we can do to make it more useful.<\/p>The first thing we should do is fix the datatips so that they display the date and value from the dataset, rather than the X, Y, & Z coordinates. That's actually pretty easy. We just need this simple function:<\/p><pre class=\"codeinput\">type keeling_datatip.m<\/span>\r\ndcm = datacursormode(gcf);\r\ndcm.Enable = 'on'<\/span>;\r\ndcm.UpdateFcn = @keeling_datatip;\r\n<\/pre><pre class=\"codeoutput\">\r\nfunction txt = keeling_datatip(~,evd)\r\n [th,r,z] = cart2pol(evd.Position(1),evd.Position(2),evd.Position(3));\r\n th = mod(th,2*pi);\r\n d = datetime(floor(z),1,1) + days(365*th\/(2*pi));\r\n txt = {['Date: ', datestr(d)], ...\r\n ['Value: ', num2str(r,4)]};\r\nend\r\n<\/pre><img decoding=\"async\" vspace=\"5\" hspace=\"5\" src=\"https:\/\/blogs.mathworks.com\/images\/graphics\/2015\/datatip.png\" alt=\"\"> Another thing we can do is to exagerate the radius. That will make the variations in the radius more visible. We could also get rid of the edges and switch to Gouraud lighting.<\/p><pre class=\"codeinput\">roffset = 300;\r\n[x,y,z] = pol2cart(2*pi*dofy\/365, molfrac-roffset, yr+dofy\/365);\r\nh = trisurf(tri,x,y,z);\r\nh.FaceVertexCData = molfrac';\r\nh.FaceColor = 'interp'<\/span>;\r\nh.EdgeColor = 'none'<\/span>;\r\nh.FaceLighting = 'gouraud'<\/span>;\r\nh.SpecularStrength = .25;\r\nline([0 0],[0 0],[1970 2020],'Color'<\/span>,[.75 .75 .75],'LineWidth'<\/span>,3)\r\nl1 = light('Position'<\/span>,[-1 -.5 -.25]);\r\nl2 = light('Position'<\/span>,[-1.5 .5 .25]);\r\nxlim([-105 105])\r\nylim([-105 105])\r\nzlim([1974 2015])\r\n<\/pre><img decoding=\"async\" vspace=\"5\" hspace=\"5\" src=\"https:\/\/blogs.mathworks.com\/images\/graphics\/2015\/keelingspiral_15.png\" alt=\"\"> But when with the new radius scale, we need to fix the X & Y ticks to account for the offset.<\/p><pre class=\"codeinput\">ax = gca;\r\nt = ax.XTick;\r\nlab = cell(size(t));\r\nfor<\/span> i=1:length(t)\r\n lab{i} = num2str(abs(t(i))+roffset);\r\nend<\/span>\r\nax.XTickLabel = lab;\r\nax.YTick = t;\r\nax.YTickLabel = lab;\r\nxlabel('parts per million'<\/span>)\r\nylabel('parts per million'<\/span>)\r\nzlabel('Year'<\/span>)\r\ntitle('Atmospheric CO_2 at Mauna Loa'<\/span>);\r\n<\/pre><img decoding=\"async\" vspace=\"5\" hspace=\"5\" src=\"https:\/\/blogs.mathworks.com\/images\/graphics\/2015\/keelingspiral_16.png\" alt=\"\"> What do you think? Can you think of cases where this visualization might be useful? Can you think of any additional tweaks which might make it more useful?<\/p> <a href=\"https:\/\/www.mathworks.com\/products\/matlab\/\">Published with MATLAB® R2015a<\/a> <\/p><\/div><\/body><\/html>","protected":false},"excerpt":{"rendered":"<div class=\"overview-image\"><img src=\"https:\/\/blogs.mathworks.com\/graphics\/files\/feature_image\/keelingthumbnail.png\" class=\"img-responsive attachment-post-thumbnail size-post-thumbnail wp-post-image\" alt=\"\" decoding=\"async\" loading=\"lazy\" \/><\/div>\r\n\r\n \r\n The Keeling... <a class=\"read-more\" href=\"https:\/\/blogs.mathworks.com\/graphics\/2015\/03\/31\/keeling-spiral\/\">read more >><\/a><\/p>","protected":false},"author":89,"featured_media":225,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":[],"categories":[7],"tags":[],"_links":{"self":[{"href":"https:\/\/blogs.mathworks.com\/graphics\/wp-json\/wp\/v2\/posts\/222"}],"collection":[{"href":"https:\/\/blogs.mathworks.com\/graphics\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/blogs.mathworks.com\/graphics\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/blogs.mathworks.com\/graphics\/wp-json\/wp\/v2\/users\/89"}],"replies":[{"embeddable":true,"href":"https:\/\/blogs.mathworks.com\/graphics\/wp-json\/wp\/v2\/comments?post=222"}],"version-history":[{"count":5,"href":"https:\/\/blogs.mathworks.com\/graphics\/wp-json\/wp\/v2\/posts\/222\/revisions"}],"predecessor-version":[{"id":831,"href":"https:\/\/blogs.mathworks.com\/graphics\/wp-json\/wp\/v2\/posts\/222\/revisions\/831"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/blogs.mathworks.com\/graphics\/wp-json\/wp\/v2\/media\/225"}],"wp:attachment":[{"href":"https:\/\/blogs.mathworks.com\/graphics\/wp-json\/wp\/v2\/media?parent=222"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blogs.mathworks.com\/graphics\/wp-json\/wp\/v2\/categories?post=222"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blogs.mathworks.com\/graphics\/wp-json\/wp\/v2\/tags?post=222"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}