29:null 2a:["$","title",null,{"children":"Split-Normal distribution | Mattias Villani | Observable"}] 2f:[] 2d:{"id":"1d97c9ee6e70964c","slug":"split-normal-distribution","title":"Split-Normal distribution","thumbnail":"4131fad98e6dafc64b954b876a5881e392c42992e7274db6d7871da9ec650a3b","access_level":"public","create_time":"$D2023-06-01T17:44:10.992Z","update_time":"$D2023-12-30T20:26:07.088Z","view_time":null,"first_public_time":"$D2023-06-01T17:44:10.992Z","has_unpublished_edits":null,"fork":{"id":"0520b079c9ae0892","slug":"split-t-distribution","title":"Split-t distribution","owner":{"id":"7592a7ab121f74a0","login":"mattiasvillani","name":"Mattias Villani"}},"stars":1,"starred":false,"owner":{"type":"team","id":"7592a7ab121f74a0","login":"mattiasvillani","name":"Mattias Villani","avatar_url":"https://avatars-next.observableusercontent.com/a440aaf39a3abefb86cbcc758fd7a5ab42555ae68d6825b52ec006f385a94d1d","tier":"education_2026"},"version":1167,"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,"lapsed":null,"subscribed":null,"forks":1,"collection":{"id":"40c8ef5af1a51aff","slug":"distributions","title":"Statistical distributions","owner":{"id":"7592a7ab121f74a0","login":"mattiasvillani","name":"Mattias Villani"}},"type":"notebook","body":{"title":"Split-Normal distribution","cells":[{"id":38,"value":"# Split-Normal distribution","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":629,"value":"

${tex`X\\sim \\mathrm{Split}\\text{-}\\mathrm{Normal}(\\mu, \\sigma_1, \\sigma_2)`}, for ${tex`-\\infty \\mu\n\\end{cases}\n`}\n\nwhere ${tex`c = \\sqrt{\\frac{2}{\\pi}}(\\sigma_1+\\sigma_2)^{-1}`} and\n\n${tex`\\kappa(x \\vert \\mu, \\sigma) = \\exp\\Big(-\\frac{1}{2\\sigma^2}(x-\\mu)^2 \\Big)`}\n\nis the kernel (unnormalized version) of the Normal density ${tex`\\mathrm{N}(\\mu,\\sigma)`}.\n\nThe Split-Normal is the limiting special case of the [Split-*t* distribution](https://observablehq.com/@mattiasvillani/split-t-distribution) ${tex`\\mathrm{Split}{\\text -}t(\\mu, \\tau, \\lambda, \\nu)`}, where ${tex`\\lambda = \\sigma_2/\\sigma_1`}, when the degrees of freedom ${tex`\\nu\\rightarrow \\infty`}, .\n","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":118,"value":"viewof params = Inputs.form([\n Inputs.range([-10, 10], {value: 0, step: 0.1, label: tex`\\text{location, }\\mu`}),\n Inputs.range([0, 10], {value: 1, step: 0.1, label: tex`\\text{left scale, }\\sigma_1`}),\n Inputs.range([0, 10], {value: 1.5, step: 0.01, label: tex`\\text{right scale, }\\sigma_2`}),\n Inputs.range([-10, 10], {value: -1, step: 0.01, label: \"quantile:\"})\n ])","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":461,"value":"dist_quantile = tex`\\text{If } X \\sim \\mathrm{Split}{\\text -}\\mathrm{Normal}(${mu},${(sigma1).toPrecision(2)},${(sigma2).toPrecision(2)}) \\text{ then } `","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":555,"value":"moments = {\n \n return tex`\n \\begin{aligned}\n &E( X) = \\sqrt{2/\\pi}(\\sigma_2- \\sigma_1) = ${(mean).toPrecision(3)} \\\\[1em]\n &\\mathrm{Stdev}( X) = \\sqrt{ (1-2/\\pi)(\\sigma_2-\\sigma_1)^2 + \\sigma_1\\sigma_2 } = \n ${(Math.sqrt(variance)).toPrecision(3)}\\\\[1em]\n &\\mathrm{Skewness}( X) = ${(skewness).toPrecision(3)} \\\\[1em]\n &P( X\\leq ${quantile}) =${cdf.toPrecision(4)} \\\\\n \\end{aligned}\n`\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":305,"value":"plt_pdf = Plot.plot({\n color: {\n legend: true\n },\n x: {\n label: \"x\",\n axis: true,\n domain: [-10,10]\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.line(pdfdata, {x: \"x\", y: \"pdf\", stroke : \"orange\", strokeWidth: 2}),\n Plot.areaY(pdfdata, {filter: d => d.x <= quantile, x: \"x\", y: \"pdf\", fill: \"orange\", opacity: 0.2})\n ]\n })","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"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","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":221,"value":"#### Imports","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":26,"value":"jstat = require('jstat')","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":750,"value":"mathjs = require('mathjs')","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1011,"value":"beta = require( 'https://cdn.jsdelivr.net/gh/stdlib-js/math-base-special-beta@umd/browser.js' )","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1031,"value":"// betainc(x,a,b) return the usual stats use of the incomplete beta function\nbetainc = require( 'https://cdn.jsdelivr.net/gh/stdlib-js/math-base-special-betainc@umd/browser.js' )","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":223,"value":"#### Data","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":654,"value":"mu = params[0]","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":656,"value":"sigma1 = params[1]","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":802,"value":"sigma2 = params[2]","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":662,"value":"quantile = params[3]","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":706,"value":"mean = mu + Math.sqrt(2/Math.PI)*(sigma2-sigma1)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":766,"value":"variance = (1-2/Math.PI)*(sigma2-sigma1)**2 + sigma1*sigma2","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":989,"value":"skewness = Math.sqrt(2/Math.PI)*(sigma2-sigma1)*( (4/Math.PI - 1)*(sigma2-sigma1)**2 +sigma1*sigma2 )","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":830,"value":"function splitnpdf(x, mu, sigma1, sigma2){\n const c = Math.sqrt(2/Math.PI)/(sigma1 + sigma2)\n if (x <= mu){\n return c*Math.exp( -(0.5/sigma1**2)*(x-mu)**2 )\n }\n else{\n return c*Math.exp( -(0.5/sigma2**2)*(x-mu)**2 )\n } \n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":845,"value":"function splitncdf(x, mu, sigma1, sigma2){\n const c = Math.sqrt(2/Math.PI)/(sigma1 + sigma2)\n if (x <= mu){\n return c*Math.sqrt(2*Math.PI)*sigma1*jstat.normal.cdf((x-mu)/sigma1,0,1)\n }\n else{\n return c*Math.sqrt(2*Math.PI)*sigma1*0.5 + (c*Math.sqrt(2*Math.PI)*sigma2*jstat.normal.cdf((x-mu)/sigma2,0,1) - 0.5)\n }\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":17,"value":"pdfdata = {\n const x = d3.range(-10, 10, 0.01); \n const pdf = x.map(x => ({x: x, pdf: splitnpdf(x, mu, sigma1, sigma2 )}));\n return pdf\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":19,"value":"cdf = splitncdf(quantile, mu, sigma1, sigma2)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":229,"value":"#### Style","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":202,"value":"","pinned":false,"hidden":false,"mode":"html","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":58,"value":"viewof md_colour = Inputs.color({label: \"Highlight color cells\", value: \"#F5F5DC\"})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"}],"files":"$W2f","stdlib":"1","titling":"auto"}} 2e:"$2d:body"