2a:null 2b:["$","title",null,{"children":"Zipf distribution | Mattias Villani | Observable"}] 31:[] 2e:{"id":"e6c40551335944ad","slug":"zipf-distribution","title":"Zipf distribution","thumbnail":"7d62a7510e7dd1f819f0d2c6068cf829abf6cea25391d8e78722dc35733afa31","access_level":"public","create_time":"$D2023-10-21T07:47:39.344Z","update_time":"$D2023-10-21T08:57:27.283Z","view_time":null,"first_public_time":"$D2023-10-21T07:47:50.714Z","fork":{"id":"ff603c873e322a52","slug":"skellam-distribution","title":"Skellam 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":1541,"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":"Zipf distribution","cells":[{"id":38,"value":"# Zipf distribution","pinned":false,"hidden":false,"mode":"md"},{"id":629,"value":"

${tex`X\\sim \\mathrm{Zipf}(N, s)\\hspace{0.3cm}`} for ${tex`X \\in \\{1,2,\\ldots,N\\}` }","pinned":false,"hidden":false,"mode":"md"},{"id":496,"value":"A random variable following the ${tex`\\mathrm{Zipf}(N,s)`} distribution has probability mass function\n\n${tex`P(X=k) = \\frac{1}{k^s}\\cdot\\frac{1}{H_{N,s}}`}\n\nwhere ${tex`H_{N,s}=\\sum_{k=1}^N \\frac{1}{k^s}`} is the [generalized Harmonic number](https://en.wikipedia.org/wiki/Harmonic_number#Generalized_harmonic_numbers).","pinned":false,"hidden":false,"mode":"md"},{"id":461,"value":"dist_quantile = tex`\\text{If } X \\sim \\mathrm{Zipf}(${N},${s}) \\text{ then } `","pinned":false,"hidden":false,"mode":"ojs"},{"id":555,"value":"moments = {\n return tex`\n \\begin{aligned}\n &E( X) = H(N,s-1)/H(N,s) = ${(mean).toPrecision(3)} \\\\[1em]\n &\\mathrm{Stdev}( X) = \\sqrt{ H(N,s-2)/H(N,s) - H(N,s-1)^2/H(N,s)^2} = \n ${(Math.sqrt(variance)).toPrecision(3)}\\\\[1em]\n &P( X\\leq ${quantile}) \\approx ${cdf.toPrecision(4)} \\\\\n \\end{aligned}\n`\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":118,"value":"viewof params = Inputs.form([\n Inputs.range([1, 50], {\n value: 10,\n step: 1,\n label: tex`N`\n }),\n Inputs.range([Number.EPSILON, 5], {\n value: 1,\n step: 0.1,\n label: tex`s`\n }),\n])","pinned":false,"hidden":false,"mode":"ojs"},{"id":1475,"value":"viewof quantile = Inputs.range([1, 50], {value: -2, step: 1, label: \"quantile:\"})","pinned":false,"hidden":false,"mode":"ojs"},{"id":305,"value":"plt_pdf = Plot.plot({\n color: { \n legend: false\n },\n x: {\n label: \"k\",\n axis: true,\n domain: d3.range(1, Math.floor(mean + 8*Math.sqrt(variance) ) )\n },\n y: {\n label: \"P(k)\",\n axis: true\n },\n tooltip: {\n fill: \"#673147\",\n stroke: \"#673147\",\n opacity: 1,\n },\n marks: [\n Plot.ruleY([0]),\n Plot.barY(pdfdata,{x: \"x\", y: \"pdf\", fill: \"purple\", strokeWidth: 0, opacity: 0.5,\n title: (d) => `P(X=${d.x}) = ${(d.pdf).toPrecision(4)}`}),\n Plot.barY(pdfdata, {filter: d => d.x <= quantile, x: \"x\", y: \"pdf\", fill: \"purple\", opacity: 0.7,\n title: (d) => `P(X=${d.x}) = ${(d.pdf).toPrecision(4)}`})\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":1375,"value":"import {Plot} from \"@mkfreeman/plot-tooltip\"","pinned":true,"hidden":false,"mode":"ojs"},{"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":1173,"value":"besselfunc = require(\"bessel@1.0.2/bessel.js\")","pinned":false,"hidden":false,"mode":"ojs"},{"id":223,"value":"#### Data","pinned":false,"hidden":false,"mode":"md"},{"id":654,"value":"N = params[0]","pinned":false,"hidden":false,"mode":"ojs"},{"id":656,"value":"s = params[1]","pinned":false,"hidden":false,"mode":"ojs"},{"id":1485,"value":"function H(N, s){\n let harmonicnumber = 0;\n for (let k = 1; k <= N; k++){\n harmonicnumber += 1/(k**s)\n }\n return harmonicnumber\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":706,"value":"mean = H(N, s-1)/H(N, s)","pinned":false,"hidden":false,"mode":"ojs"},{"id":766,"value":"variance = H(N, s-2)/H(N, s) - mean**2","pinned":false,"hidden":false,"mode":"ojs"},{"id":1199,"value":"gridsize = 0.01","pinned":false,"hidden":false,"mode":"ojs"},{"id":830,"value":"function zipfpdf(k, N, s) {\n return (1/k**s)*(1/H(N,s))\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":845,"value":"function zipfcdf(quantile, N, s){\n return H(quantile,s)/H(N,s)\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":17,"value":"pdfdata = {\n const x = d3.range(1, Math.floor(mean + 10*Math.sqrt(variance) ) )\n const pdf = x.map((x) => ({ x: x, pdf: zipfpdf(x, N, s) }));\n return pdf;\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":19,"value":"cdf = zipfcdf(quantile, N, s)","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"