31:T60b,viewof params = { switch (K) { case 1: return Inputs.form([ Inputs.range([0.01, 5], {value: 0, step: 0.01, label: tex`\lambda_1`}) ]) break; case 2: return Inputs.form([ Inputs.range([0, 1], {value: 1/2, step: 0.01, label: tex`\omega_1`}), Inputs.range([0.01, 5], {value: 0, step: 0.01, label: tex`\lambda_1`}), Inputs.range([0.01, 5], {value: 3, step: 0.01, label: tex`\lambda_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([0.01, 5], {value: 0, step: 0.01, label: tex`\lambda_1`}), Inputs.range([0.01, 5], {value: 1, step: 0.01, label: tex`\lambda_2`}), Inputs.range([0.01, 5], {value: 4, step: 0.01, label: tex`\lambda_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([0.01, 5], {value: 0, step: 0.01, label: tex`\lambda_1`}), Inputs.range([0.01, 5], {value: 1, step: 0.01, label: tex`\lambda_2`}), Inputs.range([0.01, 5], {value: 2, step: 0.01, label: tex`\lambda_3`}), Inputs.range([0.01, 5], {value: 3, step: 0.01, label: tex`\lambda_4`}) ]) break; } } 32:T48b,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: true, //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, marker: "circle"}), Plot.line(pdfdata.pdfs, {filter: d => d.comp != "mixture", x: "x", y: "dens", stroke: "comp", strokeWidth: 2, opacity: 0.5, marker: "circle"}) ] }) } }33:T421,pdfdata = { var mus = 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] } 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] var pdfs; var mixturepdf; const xgrid = d3.range(0, 10, 1); pdfs = xgrid.map(x => ({x: x, dens: jstat.poisson.pdf(x, mus[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.poisson.pdf(x, mus[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, omegas: omegas} }34:[] 2f:{"title":"Mixture of Poissons","cells":[{"id":38,"value":"# Mixture of Poissons","pinned":false,"hidden":false,"mode":"md"},{"id":1079,"value":"

${tex`X\\sim \\mathrm{MoP}(\\boldsymbol{\\lambda}, \\boldsymbol{\\omega})`}, for ${tex`X \\in \\{0,1,\\ldots\\}`}","pinned":false,"hidden":false,"mode":"md"},{"id":406,"value":"The density for a mixture of Poissons is of the form","pinned":false,"hidden":false,"mode":"md"},{"id":408,"value":"model_text = tex`\nP(x) = \\sum_{k=1}^K \\omega_k \\, \\mathrm{Pois}(x \\mid \\lambda_k)\n`","pinned":false,"hidden":false,"mode":"ojs"},{"id":769,"value":"where ${tex`\\mathrm{Pois}(x \\mid \\lambda)`} is the density of a Pois variable ${tex`X \\sim \\mathrm{Pois}(\\lambda)`} evaluated at ${tex`x`}.","pinned":false,"hidden":false,"mode":"md"},{"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"},{"id":672,"value":"$31","pinned":false,"hidden":false,"mode":"ojs"},{"id":1162,"value":"viewof quantform = Inputs.form([\n Inputs.range([0,10],{label: \"quantile\", value: 1, step: 1}),\n Inputs.toggle({label: \"show quantile in plot\", value: false})\n])","pinned":false,"hidden":false,"mode":"ojs"},{"id":1226,"value":"moments = {\n return tex`\n \\begin{aligned}\n &E( X) = \\sum_{k=1}^K \\omega_k \\lambda_k = ${mean.toFixed(3)} \\\\[1em]\n &V( X) = \\sum_{k=1}^K \\omega_k \\lambda_k(1 + \\lambda_k) = ${(variance).toPrecision(3)} \\\\[1em]\n &P( X\\leq ${quantile}) =${mixturecdf.toPrecision(4)}\n \\end{aligned}\n `\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":694,"value":"$32","pinned":false,"hidden":false,"mode":"ojs"},{"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"},{"id":221,"value":"#### Imports","pinned":false,"hidden":false,"mode":"md"},{"id":26,"value":"jstat = require('jstat')","pinned":false,"hidden":false,"mode":"ojs"},{"id":747,"value":"mathjs = require('mathjs')","pinned":false,"hidden":false,"mode":"ojs"},{"id":971,"value":"import {Legend, Swatches} from \"@d3/color-legend\"","pinned":false,"hidden":false,"mode":"ojs"},{"id":620,"value":"### Functions","pinned":false,"hidden":false,"mode":"md"},{"id":223,"value":"#### Data","pinned":false,"hidden":false,"mode":"md"},{"id":1269,"value":"quantile = quantform[0]","pinned":true,"hidden":false,"mode":"ojs"},{"id":1273,"value":"showquantile = quantform[1]","pinned":true,"hidden":false,"mode":"ojs"},{"id":1222,"value":"mean = {\n var mean = 0;\n for (let i = 0; i `comp ${k + 1}`);\n return [\"mixture\"].concat(tmp)\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":993,"value":"colorscheme = {\n var tmp = d3.schemeCategory10.slice(0,K);\n tmp.push(\"#383838\")\n return tmp\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":229,"value":"#### Style","pinned":false,"hidden":false,"mode":"md"},{"id":202,"value":"","pinned":false,"hidden":false,"mode":"html"},{"id":58,"value":"viewof md_colour = Inputs.color({label: \"Highlight color cells\", value: \"#F5F5DC\"})","pinned":false,"hidden":false,"mode":"ojs"}],"files":"$W34","stdlib":"1","titling":"auto"}