2d:null 2e:["$","title",null,{"children":"Local Warming - Construct Basis Matrix (with MATLAB) | Ginda Chen | Observable"}] 33:[] 31:{"id":"2d6225a2f8b1d304","slug":"local-warming","title":"Local Warming - Construct Basis Matrix (with MATLAB)","thumbnail":"7af6545d790a7d3308430628a5e93e258d730a5f6f715dcdff07a7ea3277a07b","access_level":"public","create_time":"$D2018-12-12T23:48:51.217Z","update_time":"$D2018-12-14T01:56:14.224Z","view_time":null,"first_public_time":"$D2018-12-13T02:35:37.640Z","has_unpublished_edits":null,"fork":false,"stars":0,"starred":false,"owner":{"type":"team","id":"88621bbaf496a8ae","login":"gindachen","name":"Ginda Chen","avatar_url":"https://avatars-next.observableusercontent.com/756c8cdb2a3d0fdf10196067ee393ad22da59d22d1944c0139df6b7fa0d712db","tier":"starter_2024"},"version":598,"publish_version":598,"role":null,"authored":false,"authors":[{"type":"individual","id":"5d9dfb3076bb6e74","login":"gindachen","name":"Ginda Chen","avatar_url":"https://avatars-next.observableusercontent.com/756c8cdb2a3d0fdf10196067ee393ad22da59d22d1944c0139df6b7fa0d712db","tier":null}],"members":[{"type":"individual","id":"5d9dfb3076bb6e74","login":"gindachen","name":"Ginda Chen","avatar_url":"https://avatars-next.observableusercontent.com/756c8cdb2a3d0fdf10196067ee393ad22da59d22d1944c0139df6b7fa0d712db","tier":null,"role":"editor"}],"comments":[],"custom_thumbnail":false,"trashed":false,"lapsed":null,"subscribed":null,"forks":1,"collection":{"id":"78b5b8dac5b57445","slug":"local-warming-or-global-warming","title":"Local Warming or Global Warming","owner":{"id":"88621bbaf496a8ae","login":"gindachen","name":"Ginda Chen"}},"type":"notebook","body":{"title":"Local Warming - Construct Basis Matrix (with MATLAB)","cells":[{"id":102,"value":"md `# Local Warming - Construct Basis Matrix (with MATLAB)\nAssistant page for our project about local warming.\n\n\\`@Author: Junda Chen, Doris Duan, Haoruo Zhao\\`\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":542,"value":"","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":105,"value":"md `# Summary: The Final Code`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":528,"value":"md `\n\\`\\`\\`MATLAB\n%% Construct a Basis Matrix \nn = 1000;\nx = (1:n)'; % X: the range\nT = [ 1 2 3 ]; % T: the cycle parameters \nones_col = ones(length(x), 1); % Represents the one columns\n\n%%%% 1. Calculate an outer product of x and T\nu = x ./ T * 2 * pi; \n%%%% 2. Concatenate each column vector component of the matrix X\nX = [ sin(u) cos(u) ones_col x ];\n\\`\\`\\`\n\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":192,"value":"md `## A Simple Basis Matrix`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":135,"value":"md `For example, you choose a basis function to represent the samples`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":142,"value":"tex `\\hat{y_i} = w_0 + w_1 x_i + w_2 sin(x_i) + w_3 cos(x_i)\\ \\forall \\ i = 1,2,...,n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":305,"value":"func1 = (w, x) => {return w[0] + w[1] * x + w[2] * Math.sin(x) + w[3] * Math.cos(x);}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":329,"value":"md `Edit the \\`w\\` so that you can control the function, and look at \\`value\\` to see how the actual values change accordingly`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":301,"value":"w = [ 1, 2 ,32, 4 ] // Edit me to change the values of 2","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":294,"value":"values = {\n let a = [];\n for(let i = 0; i < 100; i++){ a.push(Object({\n x: i, y: func1(w, i)\n }))}\n return a;\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":418,"value":"{\n let s = \"\";\n for(let i = 0; i < w.length; i++){\n s += \"\" + float_to_tex(w[i]);\n if(i < w2.length - 1){\n s += \"&\"\n }\n }\n return tex ` w = \\begin{bmatrix} ${s} \\end{bmatrix} ^T`;\n // return s\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":423,"value":"tex `\\hat{y_i} = ${w[0]} + ${w[1]} x_i + ${w[2]} sin(x_i) + ${w[3]} cos(x_i)\\ \\forall \\ i = 1,2,...,n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":290,"value":"vegalite({\n width: 800,\n height: 200,\n \"data\": {\n \"values\": values\n },\n \"layer\": [{\n \"mark\": \"line\",\n \"encoding\": {\n \"x\": {\"field\": \"x\", \"type\": \"quantitative\"},\n \"y\": {\"field\": \"y\", \"type\": \"quantitative\"}\n }\n }]\n})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":148,"value":"md `You may want to further express it in vector form`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":150,"value":"tex `\\hat{\\mathbf{y}} = \\mathbf{X} \\mathbf{w}, \n\n\n\\text{where} \\ \\hat{\\mathbf{y}} = \\begin{bmatrix}\ny_1\\\\\ny_2\\\\\n\\vdots \\\\\ny_n\n\\end{bmatrix},\n\n\\mathbf{w} = \\begin{bmatrix}\nw_0\\\\\nw_1\\\\\n\\vdots \\\\\nw_m\n\\end{bmatrix} \n\n\\text{, such that }\n\n(\\mathbf{X} \\mathbf{w})_i = w_0 + w_1 x_i + w_2 sin(x_i) + w_3 cos(x_i)\n\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":145,"value":"md `In this case, the basis matrix to construct looks like:`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":110,"value":"tex `\\mathbf{X} =\n\\begin{bmatrix} \n1 & x_1 & sin(x_1) & cos(x_1)\\\\\n1 & x_2 & sin(x_2) & cos(x_2)\\\\\n\\vdots & & \\ddots & \\vdots \\\\\n1 & x_n & sin(x_n) & cos(x_n)\\\\\n\\end{bmatrix}`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":109,"value":"md `or simply, since you know that \\`x_i = i\\`: `","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":186,"value":"tex `\n\\mathbf{X} =\n\\begin{bmatrix} \n1 & 1 & sin(1) & cos(1)\\\\\n1 & 2 & sin(2) & cos(2)\\\\\n\\vdots & & \\ddots & \\vdots \\\\\n1 & n & sin(n) & cos(n)\\\\\n\\end{bmatrix}`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":338,"value":"md `Here is the MATLAB code that helps you build this piece of matrix:`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":108,"value":"md`\n\\`\\`\\`MATLAB\n\nn = 1000; % Top range\nx = (1:n)'; % x-values (a column vector)\nones_column = ones(length(x),1);\n\n%% Represent X as the concatenated column vectors\nX = [ ones_column x sin(x) cos(x) ];\n\n\\`\\`\\` \n`\n// md `In matlab we could simply type:\n\n// \\`\\`\\` MATLAB\n// n = 1000; % Top range\n// x = (1:n)'; % x-values (a column vector)\n// ones_column = ones(length(x),1);\n\n// %% Represent X as the concatenated column vectors\n// X = [ ones_column x sin(x) cos(x) ];\n// \\`\\`\\` \n\n// // `","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":107,"value":"","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":106,"value":"md `## A More Complicated Basis Matrix`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":246,"value":"md `Now you want to represent the function as`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":248,"value":"tex `\n y_i = \\sum_{k=1}^{N} w_{k} sin(x_i \\cdot 2 \\pi / T_k) + \\sum_{k=1}^{N} w_{N+k} cos(x_i \\cdot 2 \\pi / T_k) + \nw_{2N+1} + w_{2N+2} \\cdot x_i\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":284,"value":"md `for example:`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":278,"value":"tex ` \ny_i = w_1 sin(x_i \\cdot 2 \\pi / T_1) + w_2 sin(x_i \\cdot 2 \\pi / T_2) +\nw_3 cos(x_i \\cdot 2 \\pi / T_1) + w_4 cos(x_i \\cdot 2 \\pi / T_2) + \nw_{5} + w_{6} \\cdot x_i\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":287,"value":"func2 = (w, T, x) => {\n let N = T.length;\n let a = 0;\n for(let i = 0; i < N; i++){\n a += w[ i] * Math.sin(x * 2 * Math.PI / T[i]) +\n w[N + i] * Math.cos(x * 2 * Math.PI / T[i]) ;\n }\n a = a + w[2 * N] * 1 + w[2 * N + 1] * x;\n return a\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":357,"value":"w2 = [2, 23, 22, 1, 1, 1]; // Edit me to see the difference","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":359,"value":"T2 = [20 * Math.PI , 10 * Math.PI]","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":350,"value":"values2 = {\n let a = [];\n let N = 300;\n for(let i = 0; i < N; i++){ a.push(Object({\n x: i, y: func2(w2, T2, i)\n }))}\n return a;\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":407,"value":"{\n let s = \"\";\n for(let i = 0; i < w2.length; i++){\n s += \"\" + float_to_tex(w2[i]);\n if(i < w2.length - 1){\n s += \"&\"\n }\n }\n return tex ` w2 = \\begin{bmatrix} ${s} \\end{bmatrix} ^T`;\n // return s\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":475,"value":"{\n let s = \"\";\n for(let i = 0; i < T2.length; i++){\n s += \"\" + T2[i].toFixed(1);\n if(i < T2.length - 1){\n s += \"&\"\n }\n }\n return tex ` T2 = \\begin{bmatrix} ${s} \\end{bmatrix} ^T`;\n // return s\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":432,"value":"{\n let s = \"\";\n let N = T2.length;\n for(let i = 0; i < N; i++){\n let r = T2[i].toFixed(1);\n s += \"(\" + w2[i] + \") \\\\cdot sin( x_i \\\\cdot 2 \\\\pi / \" + r + \") + \"\n }\n for(let i = 0; i < N; i++){\n let r = T2[i].toFixed(1);\n s += \"(\" + w2[N+i] + \") \\\\cdot cos( x_i \\\\cdot 2 \\\\pi / \" + r + \") + \"\n }\n s += \"(\" + w2[2*N] +\") + (\" + w2[2*N+1] +\" ) x_i\";\n \n return tex `y_i = ${s}`\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":347,"value":"vegalite({\n width: 800,\n height: 200,\n \"data\": {\n \"values\": values2\n },\n \"layer\": [{\n \"mark\": \"line\",\n \"encoding\": {\n \"x\": {\"field\": \"x\", \"type\": \"quantitative\"},\n \"y\": {\"field\": \"y\", \"type\": \"quantitative\"}\n }\n }]\n})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":261,"value":"md `so you want to construct something like`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":226,"value":"tex `\n\\mathbf{X} =\n\\begin{bmatrix} \nsin(1 \\cdot 2 \\pi / T_1) & sin(1 \\cdot 2 \\pi / T_2) & \\dots & cos(1 \\cdot 2 \\pi / T_1) & cos(1 \\cdot 2 \\pi / T_2) & 1 & 1 \\\\\nsin(2 \\cdot 2 \\pi / T_1) & sin(2 \\cdot 2 \\pi / T_2) & \\dots & cos(1 \\cdot 2 \\pi / T_1) & cos(1 \\cdot 2 \\pi / T_2) & 1 & 2\\\\\n\\vdots & & \\ddots&& & \\vdots \\\\\nsin(n \\cdot 2 \\pi / T_1) & sin(n \\cdot 2 \\pi / T_2) & \\dots & cos(n \\cdot 2 \\pi / T_1) & cos(n \\cdot 2 \\pi / T_2) & 1 & n \\\\\n\n\\end{bmatrix}`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":507,"value":"md `Therefore, here is the magic in matlab:\n\\`\\`\\`MATLAB\nn = 1000;\nx = (1:n)';\nT = [ 1 2 3 ]; % The cycle parameters \nones_col = ones(length(x), 1);\n\n%%%% 1. Calculate an outer product of x and T\nu = x ./ T * 2 * pi; \n%%%% 2. Concatenate each column vector component of the matrix X\nX = [ sin(u) cos(u) ones_col x ];\n\\`\\`\\`\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":521,"value":"","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":518,"value":"","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":497,"value":"md `# Reference`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":288,"value":"vegalite = require(\"@observablehq/vega-lite@0.1\")","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":316,"value":"import {abs, acos, cos, sin} from \"@fil/math\"","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":411,"value":"float_to_tex = x =>\n ('' + x).replace(/e(\\+|(\\-))([0-9]+)/, '\\\\times 10^{$2$3}')","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"}],"files":"$W33","stdlib":"1","titling":"auto"}} 32:"$31:body"