2a:null 2b:["$","title",null,{"children":"Scaled inverse chi2 distribution | Mattias Villani | Observable"}] 31:[] 2e:{"id":"d668b1efe2f23f3c","slug":"scaled-inverse-chi-2-distribution","title":"Scaled inverse chi2 distribution","thumbnail":"6ab4cc39231e26f18a573ad3933bce150fe68c94ef1016ae489b84fa2f533df7","access_level":"public","create_time":"$D2023-02-11T21:34:08.833Z","update_time":"$D2023-09-04T09:18:50.780Z","view_time":null,"first_public_time":"$D2023-02-11T21:34:08.833Z","fork":{"id":"0ce89a02b7e0a2f4","slug":"beta-distribution","title":"Beta 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":823,"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":2,"collection":{"id":"40c8ef5af1a51aff","slug":"distributions","title":"Statistical distributions","owner":{"id":"7592a7ab121f74a0","login":"mattiasvillani","name":"Mattias Villani"}},"type":"notebook","body":{"title":"Scaled inverse chi2 distribution","cells":[{"id":38,"value":"# Scaled inverse chi2 distribution","pinned":false,"hidden":false,"mode":"md"},{"id":662,"value":"disttext = md`

${tex`X\\sim \\operatorname{Inv-\\chi^2}(\\nu,\\tau^2)`}, for ${tex`X \\in (0,\\infty)`}`","pinned":false,"hidden":false,"mode":"ojs"},{"id":496,"value":"## Density function and moments","pinned":false,"hidden":false,"mode":"md"},{"id":118,"value":"viewof params = Inputs.form([\n Inputs.range([0.1, 100], {value: 3, step: 0.1, label: tex`\\nu`}),\n Inputs.range([0.1, 5], {value: 1, step: 0.1, label: tex`\\tau^2`}),\n Inputs.range([0, 5], {value: 1, step: 0.01, label: \"Quantile:\"})\n ])","pinned":false,"hidden":false,"mode":"ojs"},{"id":461,"value":"dist_quantile = tex`\\text{If } X \\sim \\operatorname{Inv-\\chi^2}(${params[0]},${params[1]}) \\text{ then } `","pinned":false,"hidden":false,"mode":"ojs"},{"id":555,"value":"moments = tex`\n\\begin{aligned}\n&E( X) =\\frac{\\tau^2\\nu}{\\nu - 2} = ${densmean.toPrecision(2)} \\text{ for }\\nu>2 \\\\[1em]\n&Var( X) = \\frac{2\\tau^4}{(\\nu-2)^2(\\nu-4)} = ${densvar.toPrecision(2)} \\text{ for }\\nu>4 \\\\[1em]\n&P( X\\leq ${params[2]}) =${scaledinvchi2cdf.toPrecision(4)}\n\\end{aligned}\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, scaledinvchi2pdf((params[0]*params[1])/(params[0] + 2) , params[0], params[1]) ] \n },\n marks: [\n Plot.ruleY([0]),\n Plot.ruleX([0]),\n Plot.line(scaledinvchi2pdfvals, {x: \"x\", y: \"pdf\", stroke : \"red\", strokeWidth: 2}),\n Plot.areaY(scaledinvchi2pdfvals, {filter: d => d.x <= params[2], x: \"x\", y: \"pdf\", fill: \"red\", opacity: 0.2})\n ]\n })","pinned":false,"hidden":false,"mode":"ojs"},{"id":785,"value":"## Relations to other distributions","pinned":false,"hidden":false,"mode":"md"},{"id":787,"value":"

If ${tex`X \\sim \\chi^2(\\nu)`} then\n

${tex`\\frac{\\nu\\tau^2}{X} \\sim \\mathrm{Inv-}\\chi^2(\\nu,\\tau^2)`}

\n ","pinned":false,"hidden":false,"mode":"html"},{"id":724,"value":"

\n\n\n## Implementation from InvGamma distribution in jStat","pinned":false,"hidden":false,"mode":"md"},{"id":726,"value":"function scaledinvchi2pdf(x, nu, tau2){\n // ScaledInverseChiSq(ν,τ²) = InverseGamma(ν/2,ν*τ²/2)\n return jstat.invgamma.pdf(x, nu/2, nu*tau2/2)\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":731,"value":"function randscaledinvchi2(nu, tau2, n = 1){\n if (n==1){\n return jstat.invgamma.sample(nu/2, nu*tau2/2)\n }else{\n return d3.range(n).map(i => jstat.invgamma.sample(nu/2, nu*tau2/2))\n }\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":109,"value":"
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\n\n---\n## 🙈 Appendix 🙈\n\nJust some code and details that you probably want to ignore.","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":223,"value":"#### Data","pinned":false,"hidden":false,"mode":"md"},{"id":17,"value":"scaledinvchi2pdfvals = {\n const x = d3.range(0, 10, 0.01);\n const pdf = x.map(x => ({x: x, pdf: scaledinvchi2pdf(x, params[0], params[1])}));\n return pdf\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":19,"value":"scaledinvchi2cdf = jstat.invgamma.cdf(params[2], params[0]/2, params[0]*params[1]/2)","pinned":false,"hidden":false,"mode":"ojs"},{"id":808,"value":"nu = params[0]","pinned":true,"hidden":false,"mode":"ojs"},{"id":811,"value":"tau2 = params[1]","pinned":true,"hidden":false,"mode":"ojs"},{"id":767,"value":"densmean = nu*tau2/(nu-2)","pinned":true,"hidden":false,"mode":"ojs"},{"id":769,"value":"densvar = 2*tau2**2/((nu-2)**2*(nu-4))","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"