2b:null 2c:["$","title",null,{"children":"Beta distribution (generalized four-parameter) | Mattias Villani | Observable"}] 31:[] 2f:{"id":"bf4527573922da02","slug":"four-parameter-beta-distribution","title":"Beta distribution (generalized four-parameter)","thumbnail":"1338ef2c46d24cb96530aac3cf1d255847d3e31fad4d6d51b0f562b451bdfaf5","access_level":"public","create_time":"$D2023-12-29T15:05:09.315Z","update_time":"$D2023-12-29T20:43:02.112Z","view_time":null,"first_public_time":"$D2023-12-29T18:32:17.975Z","fork":{"id":"a580de9aa23cdd70","slug":"three-parameter-beta-distribution","title":"Beta distribution (generalized three-parameter)","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":860,"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":0,"collection":{"id":"40c8ef5af1a51aff","slug":"distributions","title":"Statistical distributions","owner":{"id":"7592a7ab121f74a0","login":"mattiasvillani","name":"Mattias Villani"}},"type":"notebook","body":{"title":"Beta distribution (generalized four-parameter)","cells":[{"id":38,"value":"# Beta distribution (generalized four-parameter)","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":662,"value":"disttext = md`\n

${tex`X\\sim \\operatorname{Beta}(\\alpha,\\beta, \\gamma, \\delta)`}
\n*Domain*: ${tex`X \\in (0,\\delta) \\text{ for } \\gamma>0 \\text{ and } X \\in (\\delta,\\infty) \\text{ for } \\gamma<0`}
\n*Parameters*: ${tex`\\alpha>0, \\beta > 0, \\gamma \\neq 0, \\delta >0`}\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":496,"value":"The density function of a ${tex`X\\sim \\operatorname{Beta}(\\alpha,\\beta,\\gamma,\\delta)`} random variable is \n\n${tex`f(x)=\\frac{1}{\\Beta(\\alpha,\\beta)} \\frac{1}{\\delta} \\vert \\gamma \\vert (x/\\delta)^{\\alpha\\gamma-1}(1-(x/\\delta)^\\gamma)^{\\beta-1}`}\n\nThe distribution arises from ${tex`X = \\delta Y^{1/\\gamma}`} where ${tex`Y\\sim\\mathrm{Beta}(\\alpha,\\beta)`}.","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":118,"value":"viewof inputs = Inputs.form({\n alpha: Inputs.range([0.1, 20], {value: 3, step: 0.1, label: tex`\\alpha`}),\n beta: Inputs.range([0.1, 20], {value: 3, step: 0.1, label: tex`\\beta`}),\n gamma: Inputs.range([-0.55, 10], {value: 1, step: 0.1, label: tex`\\gamma`}),\n delta: Inputs.range([0.1, 20], {value: 1, step: 0.1, label: tex`\\delta`}),\n quantile: Inputs.range([0, 10], {value: 0.2, step: 0.01, label: \"Quantile:\"}) \n })","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":461,"value":"dist_quantile = tex`\\text{If } X \\sim \\operatorname{Beta}(${inputs.alpha},${inputs.beta},${inputs.gamma},${inputs.delta}) \\text{ then } `","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":555,"value":"moments = tex`\n\\begin{aligned}\n&E( X) =${mean.toPrecision(4)} \\\\[0.5em]\n&\\mathrm{Stdev}( X) =${Math.sqrt(variance).toPrecision(4)} \\\\[0.5em]\n&P( X\\leq ${inputs.quantile}) =${betacdf.toPrecision(4)}\n\\end{aligned}\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"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.beta.pdf((params[0]-1)/(params[0]+params[1]-2), params[0], params[1])]\n },\n marks: [\n Plot.ruleY([0]),\n Plot.ruleX([0]),\n Plot.line(betapdf, {x: \"x\", y: \"pdf\", stroke : \"orange\", strokeWidth: 2}),\n Plot.areaY(betapdf, {filter: d => d.x <= inputs.quantile, x: \"x\", y: \"pdf\", fill: \"orange\", opacity: 0.2})\n ]\n })","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"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","author":"$undefined","summary":"$undefined"},{"id":221,"value":"#### Imports","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":26,"value":"jstat = require('jstat')","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":744,"value":"math = require(\"mathjs\");","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":223,"value":"#### Data","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":737,"value":"function beta4pdf(x, alpha, beta, gamma, delta){\n return (Math.abs(gamma)*(x/delta)**(alpha*gamma - 1)*(1 - (x/delta)**gamma)**(beta - 1)) / (delta*betafunction(alpha, beta))\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":754,"value":"function beta4cdf(quantile, alpha, beta, gamma, delta){\nif (gamma > 0){\n return jstat.beta.cdf(quantile**gamma, alpha, beta);\n }else{\n return 1 - jstat.beta.cdf((delta/quantile)**Math.abs(gamma), alpha, beta);\n }\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":17,"value":"betapdf = {\n var x;\n if (inputs.gamma > 0){\n x = d3.range(0.001, inputs.delta + 0.001, 0.001)\n }else{\n x = d3.range(inputs.delta + 0.001, 10, 0.001)\n }\n const pdf = x.map(x => ({x: x, pdf: beta4pdf(x, inputs.alpha, inputs.beta, inputs.gamma, inputs.delta)}));\n return pdf\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":19,"value":"betacdf = beta4cdf(inputs.quantile, inputs.alpha, inputs.beta, inputs.gamma, inputs.delta);","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":739,"value":"function betafunction(a, b){\n return Math.exp(math.lgamma(a) + math.lgamma(b) - math.lgamma(a+b) )\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":788,"value":"mean = inputs.delta*betafunction(inputs.alpha + 1/inputs.gamma, inputs.beta)/betafunction(inputs.alpha, inputs.beta)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":800,"value":"variance = inputs.delta**2*betafunction(inputs.alpha + 2/inputs.gamma, inputs.beta)/betafunction(inputs.alpha, inputs.beta) - mean**2","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":229,"value":"#### Style","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":202,"value":"","pinned":false,"hidden":false,"mode":"html","author":"$undefined","summary":"$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"}],"files":"$W31","stdlib":"1","titling":"auto"}} 30:"$2f:body"