28:null 29:["$","title",null,{"children":"Mixture of Normals distribution | Mattias Villani | Observable"}] 2e:T8c7,viewof params = { switch (K) { case 1: return Inputs.form([ Inputs.range([-5, 5], {value: 0, step: 0.01, label: tex`\mu_1`}), Inputs.range([0, 5], {value: 1, step: 0.01, label: tex`\sigma_1`}) ]) break; case 2: return Inputs.form([ Inputs.range([0, 1], {value: 1/2, step: 0.01, label: tex`\omega_1`}), Inputs.range([-5, 5], {value: 0, step: 0.01, label: tex`\mu_1`}), Inputs.range([-5, 5], {value: 3, step: 0.01, label: tex`\mu_2`}), Inputs.range([0, 5], {value: 1, step: 0.01, label: tex`\sigma_1`}), Inputs.range([0, 5], {value: 2, step: 0.01, label: tex`\sigma_2`}) ]) break; case 3: return Inputs.form([ Inputs.range([0, 1], {value: 1/3, step: 0.01, label: tex`\omega_1`}), Inputs.range([0, 1], {value: 1/3, step: 0.01, label: tex`\omega_2`}), Inputs.range([-5, 5], {value: 0, step: 0.01, label: tex`\mu_1`}), Inputs.range([-5, 5], {value: 1, step: 0.01, label: tex`\mu_2`}), Inputs.range([-5, 5], {value: 4, step: 0.01, label: tex`\mu_3`}), Inputs.range([0, 5], {value: 1, step: 0.01, label: tex`\sigma_1`}), Inputs.range([0, 5], {value: 2, step: 0.01, label: tex`\sigma_2`}), Inputs.range([0, 5], {value: 1, step: 0.01, label: tex`\sigma_3`}) ]) break; case 4: return Inputs.form([ Inputs.range([0, 1], {value: 1/4, step: 0.01, label: tex`\omega_1`}), Inputs.range([0, 1], {value: 1/4, step: 0.01, label: tex`\omega_2`}), Inputs.range([0, 1], {value: 1/4, step: 0.01, label: tex`\omega_3`}), Inputs.range([-5, 5], {value: 0, step: 0.01, label: tex`\mu_1`}), Inputs.range([-5, 5], {value: 1, step: 0.01, label: tex`\mu_2`}), Inputs.range([-5, 5], {value: 2, step: 0.01, label: tex`\mu_3`}), Inputs.range([-5, 5], {value: 3, step: 0.01, label: tex`\mu_4`}), Inputs.range([0, 5], {value: 1, step: 0.01, label: tex`\sigma_1`}), Inputs.range([0, 5], {value: 2, step: 0.01, label: tex`\sigma_2`}), Inputs.range([0, 5], {value: 3, step: 0.01, label: tex`\sigma_3`}), Inputs.range([0, 5], {value: 4, step: 0.01, label: tex`\sigma_4`}) ]) break; } } 2f:T464,plt = { if (pdfdata.pdfs == 0){ return md`😰 Oops! ${tex`\sum_{k=1}^K\omega_k >1`}. Please change weights.
Note that ${tex`\omega_${K}`} is implicitly given as ${tex`\omega_${K}=1-\sum_{k=1}^${K-1}\omega_k`}.` } else{ return Plot.plot({ color: { legend: true, domain: legendarray, scheme: d3.Paired }, x: { label: "x", axis: true }, y: { label: "density", axis: false, //domain: [0,1.05*jstat.normal.pdf(myinputs[0], myinputs[0], myinputs[1])] }, marks: [ Plot.ruleY([0]), //Plot.ruleX([0]), Plot.areaY(pdfdata.pdfs, {filter: d => d.comp == "mixture", x: "x", y: "dens", fill: "comp", opacity: 0.1}), Plot.areaY(pdfdata.pdfs, {filter: d => showquantile && d.comp == "mixture" && d.x <= quantile, x: "x", y: "dens", fill: "comp", opacity: 0.5}), Plot.line(pdfdata.pdfs, {filter: d => d.comp == "mixture", x: "x", y: "dens", stroke: "comp", strokeWidth: 4}), Plot.line(pdfdata.pdfs, {filter: d => d.comp != "mixture", x: "x", y: "dens", stroke: "comp", strokeWidth: 2, opacity: 0.5}) ] }) } }30:T4b2,pdfdata = { var mus = new Array(K); var sigmas = new Array(K); var omegas = new Array(K); for (let k = 0; k <= K-2; k++){ omegas[k] = params[k] mus[k] = params[(K-1)+k] sigmas[k] = params[(K-1) + K + k] } if (d3.sum(omegas.slice(0,K-1))>1){ return 0; } omegas[K-1] = 1 - d3.sum(omegas.slice(0,K-1)) mus[K-1] = params[(K-2)+K] sigmas[K-1] = params[(K-2) + 2*K] var pdfs; var mixturepdf; const xgrid = d3.range(-10, 10, 0.01); pdfs = xgrid.map(x => ({x: x, dens: jstat.normal.pdf(x, mus[0], sigmas[0]), comp: "comp "+1})); mixturepdf = pdfs.filter(d => d.comp == ("comp "+1)).map(d => d.dens).map(d => d*omegas[0]) var tmp; for (let k = 1; k <= K-1 ; k++){ tmp = xgrid.map(x => ({x: x, dens: jstat.normal.pdf(x, mus[k], sigmas[k]), comp: "comp "+ (k+1)})) pdfs = pdfs.concat(tmp) const tmp2 = pdfs.filter(d => d.comp == ("comp "+ (k+1))).map(d => d.dens).map(d => d*omegas[k]) mixturepdf = mixturepdf.map((x,idx) => x + tmp2[idx]) } const mixturepdf_obj = mixturepdf.map((d, idx) => ({x: xgrid[idx], dens: d, comp: "mixture"})) pdfs = pdfs.concat(mixturepdf_obj) return {pdfs: pdfs, mus: mus, sigmas: sigmas, omegas: omegas} }31:[] 2c:{"id":"c3caeeba3214c8ef","slug":"normal-mixture","title":"Mixture of Normals distribution","thumbnail":"abf4ce3c5d68c6ebcf957a6c3216bf1b89c8e952851f6011178661c35dce2be4","access_level":"public","create_time":"$D2023-04-07T11:09:31.659Z","update_time":"$D2024-05-29T17:01:02.466Z","view_time":null,"first_public_time":"$D2023-04-07T11:09:31.659Z","has_unpublished_edits":null,"fork":{"id":"112b4245ebf148d9","slug":"ar-processes","title":"Autoregressive processes - simulation and priors","owner":{"id":"7592a7ab121f74a0","login":"mattiasvillani","name":"Mattias Villani"}},"stars":2,"starred":false,"owner":{"type":"team","id":"7592a7ab121f74a0","login":"mattiasvillani","name":"Mattias Villani","avatar_url":"https://avatars-next.observableusercontent.com/a440aaf39a3abefb86cbcc758fd7a5ab42555ae68d6825b52ec006f385a94d1d","tier":"education_2026"},"version":1205,"publish_version":null,"role":null,"authored":false,"authors":[{"type":"individual","id":"56499be8cf43f0ef","login":"mattiasvillani","name":"Mattias Villani","avatar_url":"https://avatars-next.observableusercontent.com/a440aaf39a3abefb86cbcc758fd7a5ab42555ae68d6825b52ec006f385a94d1d","tier":null}],"members":[{"type":"individual","id":"56499be8cf43f0ef","login":"mattiasvillani","name":"Mattias Villani","avatar_url":"https://avatars-next.observableusercontent.com/a440aaf39a3abefb86cbcc758fd7a5ab42555ae68d6825b52ec006f385a94d1d","tier":null,"role":"editor"}],"comments":[],"custom_thumbnail":false,"trashed":false,"lapsed":null,"subscribed":null,"forks":4,"collection":{"id":"40c8ef5af1a51aff","slug":"distributions","title":"Statistical distributions","owner":{"id":"7592a7ab121f74a0","login":"mattiasvillani","name":"Mattias Villani"}},"type":"notebook","body":{"title":"Mixture of Normals distribution","cells":[{"id":38,"value":"# Mixture of Normals distribution","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":1079,"value":"

${tex`X\\sim \\mathrm{MoN}(\\boldsymbol{\\mu}, \\boldsymbol{\\sigma},\\boldsymbol{\\omega})`}, for ${tex`-\\infty \\leq X \\leq \\infty`}","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":406,"value":"The density for a normal mixture is of the form","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":408,"value":"model_text = tex`\nf(x) = \\sum_{k=1}^K \\omega_k \\, N(x \\mid \\mu_k, \\sigma_k)\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":769,"value":"where ${tex`N(x\\mid\\mu,\\sigma)`} is the density of a normal variable ${tex`X\\sim N(\\mu,\\sigma)`} evaluated at ${tex`x`} and the mixture weights satisfy ${tex`\\sum_{k=1}^K \\omega_k`}.","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":666,"value":"viewof K = Inputs.range([1, 4], {value: 2, step: 1, label: \"number of components, K\"})\n ","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":672,"value":"$2e","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":694,"value":"$2f","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1186,"value":"viewof showquantile = Inputs.toggle({label: \"show quantile\", value: false})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1162,"value":"viewof quantile = Inputs.range([-10,10],{label: \"quantile\", value: -4, step: 0.1})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1160,"value":"quantiletext = {\n return tex`\n \\begin{aligned}\n &P( X\\leq ${quantile}) =${mixturecdf.toPrecision(4)}\n \\end{aligned}\n`\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":109,"value":"
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\n\n---\n## 🙈 Appendix 🙈\n\nHere is only code and details that you probably don't want to read.","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":221,"value":"#### Imports","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":26,"value":"jstat = require('jstat')","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":747,"value":"mathjs = require('mathjs')","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":971,"value":"import {Legend, Swatches} from \"@d3/color-legend\"","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":620,"value":"### Functions","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":223,"value":"#### Data","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":17,"value":"$30","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1059,"value":"mixturecdf = {\n var cdf = 0; \n for (let k = 0; k < K; k++){\n cdf += pdfdata.omegas[k]*jstat.normal.cdf(quantile, pdfdata.mus[k], pdfdata.sigmas[k]);\n }\n return cdf\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":947,"value":"legendarray = {\n var tmp = d3.range(K).map((k) => `comp ${k + 1}`);\n return [\"mixture\"].concat(tmp)\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":993,"value":"colorscheme = {\n var tmp = d3.schemeCategory10.slice(0,K);\n tmp.push(\"#383838\")\n return tmp\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":229,"value":"#### Style","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":202,"value":"","pinned":false,"hidden":false,"mode":"html","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":58,"value":"viewof md_colour = Inputs.color({label: \"Highlight color cells\", value: \"#F5F5DC\"})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"}],"files":"$W31","stdlib":"1","titling":"auto"}} 2d:"$2c:body"