2a:null 2b:["$","title",null,{"children":"Generalized extreme value distribution | Mattias Villani | Observable"}] 31:[] 2e:{"id":"0527b03232426322","slug":"generalized-extreme-value-distribution","title":"Generalized extreme value distribution","thumbnail":"930188b48dda004a941b1d0c8afba00be3eba593901d2ee64d1fe32604d19d84","access_level":"public","create_time":"$D2024-01-01T22:07:59.742Z","update_time":"$D2025-01-24T09:26:51.503Z","view_time":null,"first_public_time":"$D2024-01-01T23:21:30.249Z","fork":{"id":"80b67ff4e974308e","slug":"logit-normal-distribution","title":"LogitNormal 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":973,"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":"Generalized extreme value distribution","cells":[{"id":38,"value":"# Generalized extreme value distribution","pinned":false,"hidden":false,"mode":"md"},{"id":629,"value":"

${tex`X\\sim \\mathrm{GEV}(\\mu,\\sigma,\\xi)`}","pinned":false,"hidden":false,"mode":"md"},{"id":496,"value":"The density function of a ${tex`X\\sim \\mathrm{GEV}(\\mu,\\sigma,\\xi)`} random variable is\n\n${tex`f(x)=\\frac{1}{\\sigma}t(x)^{\\xi+1}e^{-t(x)}`}\n\nwhere ${tex`\nt(x) = \n\\begin{cases}\n\\big( 1 + \\xi (\\frac{x-\\mu}{\\sigma}) \\big)^{-1/\\xi} & \\text{ if } \\xi \\neq 0 \\\\\n\\exp \\big(-\\big(\\frac{x-\\mu}{\\sigma}\\big) \\big) & \\text{ if } \\xi = 0\n\\end{cases}\n`}","pinned":false,"hidden":false,"mode":"md"},{"id":118,"value":"viewof inputs = Inputs.form({\n mu: Inputs.range([-5, 5], {value: -2, step: 0.1, label: tex`\\mu`}),\n sigma: Inputs.range([0, 5], {value: 0.5, step: 0.01, label: tex`\\sigma`}),\n xi: Inputs.range([-5, 5], {value: 0, step: 0.01, label: tex`\\xi`}),\n quantile: Inputs.range([-5, 5], {value: -1, step: 0.01, label: \"quantile:\"})\n})","pinned":false,"hidden":false,"mode":"ojs"},{"id":461,"value":"dist_quantile = tex`\\text{If } X \\sim \\mathrm{GEV}(${inputs.mu},${(inputs.sigma).toPrecision(2)}, ${inputs.xi}) \\text{ then } `","pinned":false,"hidden":false,"mode":"ojs"},{"id":555,"value":"moments = {\n \n return tex`\n \\begin{aligned}\n &\\mathrm{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 domain: support(inputs.mu, inputs.sigma, inputs.xi)\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([-5]),\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":false,"hidden":false,"mode":"ojs"},{"id":899,"value":"digamma = require( 'https://cdn.jsdelivr.net/gh/stdlib-js/math-base-special-digamma@umd/browser.js' )","pinned":false,"hidden":false,"mode":"ojs"},{"id":828,"value":"#### Functions","pinned":false,"hidden":false,"mode":"md"},{"id":909,"value":"function t(x){\n if (inputs.xi != 0){\n return (1 + inputs.xi*((x-inputs.mu)/inputs.sigma))**(-1/inputs.xi)\n }else{\n return Math.exp(-(x-inputs.mu)/inputs.sigma)\n }\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":903,"value":"function pdfgev(x, mu, sigma, xi){\n return (1/sigma)*(t(x)**(xi+1))*Math.exp(-t(x))\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":907,"value":"function cdfgev(x, mu, sigma, xi){\n return Math.exp(-t(x))\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":939,"value":"function support(mu, sigma, xi){\n if(inputs.xi == 0){\n return [xmin, xmax] \n }else{\n if (inputs.xi > 0){\n return [(mu-sigma)/xi, xmax]\n }else{\n return [xmin, (mu-sigma)/xi]\n }\n }\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":223,"value":"#### Data","pinned":false,"hidden":false,"mode":"md"},{"id":942,"value":"xmin = -5","pinned":false,"hidden":false,"mode":"ojs"},{"id":944,"value":"xmax = 5","pinned":false,"hidden":false,"mode":"ojs"},{"id":896,"value":"γ = -digamma(1) // Euler's number","pinned":false,"hidden":false,"mode":"ojs"},{"id":850,"value":"mean = {\n if(inputs.xi == 0){\n return inputs.mu + inputs.sigma*γ\n }else{\n if (inputs.xi < 1){\n const g1 = mathjs.gamma(1-inputs.xi)\n return inputs.mu + inputs.sigma*(g1 - 1)/inputs.xi\n }else{\n return Infinity\n }\n }\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":766,"value":"variance = {\n if(inputs.xi == 0){\n return ((Math.PI**2)/6)*inputs.sigma**2\n }else{\n if (inputs.xi < 1/2){\n const g1 = mathjs.gamma(1-inputs.xi)\n const g2 = mathjs.gamma(1-2*inputs.xi)\n return (inputs.sigma**2)*(g2 - g1**2)/(inputs.xi**2)\n }else{\n return Infinity\n }\n }\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":17,"value":"pdfdata = {\n const supp = support(inputs.mu, inputs.sigma, inputs.xi);\n const x = d3.range(supp[0], supp[1], 0.01); \n const pdf = x.map(x => ({x: x, pdf: pdfgev(x, inputs.mu, inputs.sigma, inputs.xi)}));\n return pdf\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":19,"value":"cdf = cdfgev(inputs.quantile, inputs.mu, inputs.sigma, inputs.xi);","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"