2b:null 2c:["$","title",null,{"children":"Covariance matrix's semi definite | Chuncheng | Observable"}] 31:T73b,{ const { mat, svd, S } = exampleData; let plt1, plt2, plt3, plt4, width = 400; { const array = mat2array(mat); plt1 = Plot.plot({ title: "Covariance matrix", x: { nice: true }, y: { nice: true }, width, color: { nice: true, scheme: "BuRd" }, aspectRatio: 1.0, marks: [ Plot.cell(array, { x: "i", y: "j", fill: "v" }), Plot.text(array, { x: "i", y: "j", text: (d) => d.v.toFixed(2), fill: "white" }) ] }); } { const array = mat2array(svd.U); plt2 = Plot.plot({ title: "U vectors", x: { nice: true }, y: { nice: true }, width: width / 2, color: { nice: true, scheme: "BuRd" }, aspectRatio: 1.0, marks: [Plot.cell(array, { x: "i", y: "j", fill: "v" })] }); } { const array = mat2array(svd.V); plt3 = Plot.plot({ title: "V vectors", x: { nice: true }, y: { nice: true }, width: width / 2, color: { nice: true, scheme: "BuRd" }, aspectRatio: 1.0, marks: [Plot.cell(array, { x: "i", y: "j", fill: "v" })] }); } { const array = mat2array(ML.Matrix.diag(S)); plt4 = Plot.plot({ title: "Eigenvalues", x: { nice: true }, y: { nice: true }, width, color: { nice: true, scheme: "BuRd" }, aspectRatio: 1.0, marks: [ Plot.cell(array, { x: "i", y: "j", fill: "v" }), Plot.text(array, { x: "i", y: "j", text: (d) => d.v.toFixed(2), fill: (d) => (d.v < -0.0000001 ? "black" : "white"), fontSize: (d) => (d.v === 0 ? undefined : width / 20) }) ] }); } return htl.html`

${plt1} ${plt4}

${plt2} ${plt3}

`; }32:T4d9,{ const { eigenvalueArray } = mutipleSimulation; const ratio = eigenvalueArray.filter((d) => d.v < -0.0001).length / eigenvalueArray.length; let plt1, plt2; { const ratios = []; let array; for (let j = 0; j < n; ++j) { array = eigenvalueArray.filter((d) => d.j === j); ratios.push({ j, v: array.filter((d) => d.v < -0.0001).length / array.length }); } plt2 = Plot.plot({ title: "Neg. ratio at eigenvalues idx", x: { nice: true }, y: { nice: true, reverse: true }, color: { nice: true, scheme: "Blues", legend: true }, marks: [ Plot.barX(ratios, { y: "j", x: "v", fill: "v" }), Plot.text(ratios, { y: "j", x: (d) => d.v * 0.9, text: "v", fill: "white", fontSize: 20 }) ] }); } { plt1 = Plot.plot({ title: `Eigen values (Neg. ratio = ${ratio.toFixed(2)})`, x: { nice: true }, y: { nice: true }, width, aspectRatio: 1.0, color: { nice: true, scheme: "BuRd", legend: true }, marks: [Plot.dot(eigenvalueArray, { x: "i", y: "j", fill: "v" })] }); } return htl.html`

${plt1} ${plt2}

`; }33:[] 2f:{"id":"091ca8dc1c46d8bb","slug":"covariance-matrixs-semi-definite","title":"Covariance matrix's semi definite","thumbnail":"4e4bd566881c5ba84f85a6b5f6adb80c860bc2cc67f4f5485289e0124c4b276a","access_level":"public","create_time":"$D2023-12-06T05:54:41.491Z","update_time":"$D2023-12-16T12:16:48.892Z","view_time":null,"first_public_time":"$D2023-12-06T05:54:41.491Z","fork":false,"stars":0,"starred":false,"owner":{"type":"team","id":"bcd65cef1cb4d2ed","login":"listenzcc","name":"Chuncheng","avatar_url":"https://avatars-next.observableusercontent.com/bb3c68e77a708a587ffc2d663bb83421f20d67991dbe6c1623c6e5d2fb73d15e","tier":"starter_2024"},"version":535,"publish_version":null,"role":null,"authored":false,"authors":[{"type":"individual","id":"16328058352682a8","login":"listenzcc","name":"Chuncheng","avatar_url":"https://avatars-next.observableusercontent.com/bb3c68e77a708a587ffc2d663bb83421f20d67991dbe6c1623c6e5d2fb73d15e","tier":null}],"members":[{"type":"individual","id":"16328058352682a8","login":"listenzcc","name":"Chuncheng","avatar_url":"https://avatars-next.observableusercontent.com/bb3c68e77a708a587ffc2d663bb83421f20d67991dbe6c1623c6e5d2fb73d15e","tier":null,"role":"editor"}],"comments":[],"custom_thumbnail":false,"trashed":false,"subscribed":null,"forks":0,"collection":{"id":"3aa143c61490363d","slug":"math","title":"Math","owner":{"id":"bcd65cef1cb4d2ed","login":"listenzcc","name":"Chuncheng"}},"type":"notebook","body":{"title":"Covariance matrix's semi definite","cells":[{"id":0,"value":"# Covariance matrix's semi definite","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":14,"value":"viewof n = Inputs.range([3, 15], { label: \"N\", step: 1 })","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":265,"value":"viewof k = Inputs.range([0, 20], { label: \"K\", step: 1, value: 2 })","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":387,"value":"viewof repeat = Inputs.range([100, 2000], {\n label: \"Repeat\",\n step: 1,\n value: 1000\n})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":57,"value":"viewof fixDiag = Inputs.toggle({ label: \"Fix diag.\", value: true })","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":510,"value":"

\n ${plotlyGraph}\n

","pinned":false,"hidden":false,"mode":"html","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":418,"value":"plotlyGraph = {\n const { vsArray } = mutipleSimulation;\n\n const layout = {\n title: \"Dim: \" + k,\n scene: { camera: { eye: { x: 1.87, y: 0.88, z: -0.64 } } },\n autosize: false,\n width: 600,\n height: 600,\n margin: {\n l: 65,\n r: 50,\n b: 65,\n t: 90\n }\n };\n\n const cs = d3.scaleLinear().domain([-1, 1]).range([0, 255]);\n\n const trace1 = {\n x: vsArray.map((d) => d.v1),\n y: vsArray.map((d) => d.v2),\n z: vsArray.map((d) => d.v3),\n type: \"scatter3d\",\n mode: \"markers\",\n // opacity: 0.5,\n marker: {\n size: 1.5,\n color: vsArray.map((d) => d3.rgb(cs(d.v1), cs(d.v2), cs(d.v3)).hex())\n }\n };\n\n const div = htl.html`

`;\n\n const fig = await Plotly.newPlot(div, [trace1], layout);\n\n console.log(fig);\n\n return fig;\n}","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":310,"value":"{\n const { vsArray } = mutipleSimulation;\n\n return Plot.plot({\n aspectRatio: 1.0,\n width: 400,\n title: \"Dim: \" + k,\n x: { nice: true, domain: [-1, 1] },\n y: { nice: true, domain: [-1, 1] },\n color: { nice: true, scheme: \"Blues\", legend: true, range: [-0.1, 0.8] },\n marks: [\n Plot.density(vsArray, {\n x: \"v1\",\n y: \"v2\",\n bandwidth: 15,\n fill: \"density\",\n opacity: 0.8\n }),\n Plot.dot(vsArray, {\n x: \"v1\",\n y: \"v2\",\n fill: (d) => colormap(d.v3 + 0.5),\n r: 2,\n opacity: 0.5\n })\n // Plot.linearRegressionY(vsArray, { x: \"v1\", y: \"v2\", stroke: \"red\" })\n ]\n });\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":31,"value":"$31","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":170,"value":"$32","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":192,"value":"{\n const { eigenvalueArray } = mutipleSimulation;\n\n const ratios = [];\n\n let array;\n for (let j = 0; j < n; ++j) {\n array = eigenvalueArray.filter((d) => d.j === j);\n ratios.push({\n j,\n v: array.filter((d) => d.v < -0.0001).length / array.length\n });\n }\n\n return ratios;\n}","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":404,"value":"colormap = d3.interpolateReds","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":160,"value":"---\n## Multiple time simulation","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":162,"value":"mutipleSimulation = {\n const eigenvalueArray = [],\n vsArray = [];\n\n let data, v1, v2, v3;\n for (let i = 0; i < repeat; ++i) {\n data = mkExampleData();\n v1 = data.mat.data[0][1];\n v2 = data.mat.data[1][2];\n v3 = data.mat.data[0][2];\n\n vsArray.push({ v1, v2, v3, i });\n data.S.map((v, j) => eigenvalueArray.push({ v: v > 1 ? 1 : v, i, j }));\n }\n\n return { eigenvalueArray, vsArray };\n}","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":156,"value":"---\n\n## Covariance matrix computation","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":165,"value":"exampleData = mkExampleData()","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":26,"value":"/**\n * Make one data for example\n */\nmkExampleData = () => {\n const mat = mkCovMat(),\n array = mat2array(mat),\n svd = new ML.SVD(mat),\n S = reSignEigenvalues(svd);\n\n return { mat, array, svd, S };\n}","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":93,"value":"/**\n * Re-sign the eigenvalues\n */\nreSignEigenvalues = (svd) => {\n const { s, U, V } = svd,\n S = [];\n\n for (let i = 0; i < n; ++i) {\n S[i] = (s[i] * U.data[0][i]) / V.data[0][i];\n }\n\n return S;\n}","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":33,"value":"/**\n * Convert mat into array, it is an array of {v, i, j}\n */\nmat2array = (mat) => {\n let array = [];\n mat.data.map((vec, i) => vec.map((v, j) => array.push({ v, i, j })));\n return array;\n}","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":7,"value":"/**\n * Make random matrix for simulation of covariance matrix\n */\nmkCovMat = () => {\n let mat, v;\n\n if (k > 0) {\n let mat1 = normalizeRows(ML.Matrix.random(n, k).sub(0.5));\n\n mat = mat1.mmul(mat1.transpose());\n } else {\n mat = ML.Matrix.zeros(n, n);\n\n for (let i = 0; i < n; ++i) {\n for (let j = i; j < n; ++j) {\n v = i === j && fixDiag ? 1.0 : Math.random() - 0.5;\n mat.data[i][j] = v;\n mat.data[j][i] = v;\n }\n }\n }\n\n return mat;\n}","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":277,"value":"normalizeRows = (mat) => {\n const { rows } = mat;\n\n let s,\n diag = mat.mmul(mat.transpose()).diag();\n\n for (let j = 0; j < rows; ++j) {\n s = Math.sqrt(diag[j]);\n mat.data[j] = mat.data[j].map((d) => d / s);\n }\n\n return mat;\n}","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":263,"value":"mat2.mmul(mat2.transpose())","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":279,"value":"mat2 = normalizeRows(mat1)","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":291,"value":"mat1 = ML.Matrix.random(n, k)","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":5,"value":"ML = require(\"https://cdn.jsdelivr.net/npm/ml-matrix@6.10.5/matrix.umd.min.js\")","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":402,"value":"d3 = require(\"d3\")","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":416,"value":"Plotly = require(\"https://cdn.plot.ly/plotly-2.20.0.min.js\")","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"}],"files":"$W33","stdlib":"1","titling":"auto"}} 30:"$2f:body"