2a:null 2b:["$","title",null,{"children":"LogitNormal distribution | Mattias Villani | Observable"}] 31:[] 2e:{"id":"80b67ff4e974308e","slug":"logit-normal-distribution","title":"LogitNormal distribution","thumbnail":"e32be32129932909ad224ceadcf8d913895282697302ac6ebf63560d70ff5ea3","access_level":"public","create_time":"$D2023-12-30T19:26:51.743Z","update_time":"$D2023-12-30T20:10:44.072Z","view_time":null,"first_public_time":"$D2023-12-30T20:05:00.701Z","fork":{"id":"b2776bb165853bef","slug":"kumaraswamy-distribution","title":"Kumaraswamy distribution","owner":{"id":"7592a7ab121f74a0","login":"mattiasvillani","name":"Mattias Villani"}},"stars":0,"starred":false,"owner":{"type":"team","id":"7592a7ab121f74a0","login":"mattiasvillani","name":"Mattias Villani","avatar_url":"https://avatars-next.observableusercontent.com/a440aaf39a3abefb86cbcc758fd7a5ab42555ae68d6825b52ec006f385a94d1d","tier":"pro_2024"},"version":877,"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,"subscribed":null,"forks":1,"collection":{"id":"40c8ef5af1a51aff","slug":"distributions","title":"Statistical distributions","owner":{"id":"7592a7ab121f74a0","login":"mattiasvillani","name":"Mattias Villani"}},"type":"notebook","body":{"title":"LogitNormal distribution","cells":[{"id":38,"value":"# LogitNormal distribution","pinned":false,"hidden":false,"mode":"md"},{"id":629,"value":"

${tex`X\\sim \\mathrm{LogitNormal}(\\mu,\\sigma^2)`}","pinned":false,"hidden":false,"mode":"md"},{"id":496,"value":"The density function of a ${tex`X\\sim \\mathrm{LogitNormal}(\\mu,\\sigma^2)`} random variable is\n\n${tex`f(x)=\\frac{1}{\\sqrt{2\\pi\\sigma^2}}e^{-\\frac{1}{2\\sigma^2}(\\mathrm{logit}(x)-\\mu)^2}\\frac{1}{x(1-x)}`}\n\nwhere ${tex`\\mathrm{logit}(x) = \\log \\big(\\frac{x}{1-x}\\big)`}.\n\nCharacterization: ${tex`X\\sim \\mathrm{LogitNormal}(\\mu,\\sigma^2) \\Leftrightarrow`} ${tex`\\mathrm{logit}(X)\\sim \\mathrm{Normal}(\\mu,\\sigma^2)`}","pinned":false,"hidden":false,"mode":"md"},{"id":118,"value":"viewof inputs = Inputs.form({\n mu: Inputs.range([-5, 5], {value: 0, step: 0.1, label: tex`\\mu`}),\n sigma: Inputs.range([0, 5], {value: 1, step: 0.01, label: tex`\\sigma`}),\n quantile: Inputs.range([0, 1], {value: 0.5, step: 0.01, label: \"quantile:\"})\n})","pinned":false,"hidden":false,"mode":"ojs"},{"id":461,"value":"dist_quantile = tex`\\text{If } X \\sim \\mathrm{LogitNormal}(${inputs.mu},${(inputs.sigma).toPrecision(2)}^2) \\text{ then } `","pinned":false,"hidden":false,"mode":"ojs"},{"id":555,"value":"moments = {\n \n return tex`\n \\begin{aligned}\n &E( X) = ${(mean).toPrecision(3)} \\\\[1em]\n &\\mathrm{Stdev( X)} =${(Math.sqrt(variance)).toPrecision(3)}\\\\[1em]\n &P( X\\leq ${inputs.quantile}) =${cdf.toPrecision(4)}\n \\end{aligned}\n `\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":305,"value":"plt_pdf = Plot.plot({\n color: {\n legend: true\n },\n x: {\n label: \"x\",\n axis: true\n },\n y: {\n label: \"f(x)\",\n //axis: false,\n //domain: [0,jstat.chisquare.pdf(mode, nu1, nu2)]\n },\n marks: [\n Plot.ruleY([0]),\n Plot.ruleX([0]),\n Plot.line(pdfdata, {x: \"x\", y: \"pdf\", stroke : \"purple\", strokeWidth: 2}),\n Plot.areaY(pdfdata, {filter: d => d.x <= inputs.quantile, x: \"x\", y: \"pdf\", fill: \"purple\", opacity: 0.2})\n ]\n })","pinned":false,"hidden":false,"mode":"ojs"},{"id":109,"value":"
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\n\n---\n## 🙈 Appendix 🙈\n\nHere is some code and data details that you probably do not 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":750,"value":"mathjs = require('mathjs')","pinned":true,"hidden":false,"mode":"ojs"},{"id":828,"value":"#### Functions","pinned":false,"hidden":false,"mode":"md"},{"id":830,"value":"function logit(x){\n return Math.log(x/(1-x))\n}","pinned":true,"hidden":false,"mode":"ojs"},{"id":841,"value":"function logistic(x){\n return 1/(1 + Math.exp(-x))\n}","pinned":true,"hidden":false,"mode":"ojs"},{"id":223,"value":"#### Data","pinned":false,"hidden":false,"mode":"md"},{"id":706,"value":"draws = d3.range(10000).map(d => logistic(jstat.normal.sample(inputs.mu, inputs.sigma)))","pinned":false,"hidden":false,"mode":"ojs"},{"id":850,"value":"mean = d3.mean(draws)","pinned":true,"hidden":false,"mode":"ojs"},{"id":766,"value":"variance = jstat.variance(draws)","pinned":false,"hidden":false,"mode":"ojs"},{"id":17,"value":"pdfdata = {\n const x = d3.range(0.00, 1, 0.001); \n const pdf = x.map(x => ({x: x, pdf: jstat.normal.pdf(logit(x), inputs.mu, inputs.sigma)/(x*(1-x))}));\n return pdf\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":19,"value":"cdf = jstat.normal.cdf(logit(inputs.quantile), inputs.mu, inputs.sigma);","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":"$W31","stdlib":"1","titling":"auto"}} 2f:"$2e:body"