2d:null 2e:["$","title",null,{"children":"Super-generalised Normal Distribution | andrew-primer-e | Observable"}] 33:[] 31:{"id":"febdf72cfd6b4325","slug":"super-generalised-normal-distribution","title":"Super-generalised Normal Distribution","thumbnail":"6e78380070759a1af0d963d54db27070f48ddc375aed828e895fcf51700d8c7a","access_level":"public","create_time":"$D2021-06-18T02:16:35.445Z","update_time":"$D2021-08-02T23:10:36.345Z","view_time":null,"first_public_time":"$D2022-12-23T02:14:26.635Z","has_unpublished_edits":null,"fork":true,"stars":2,"starred":false,"owner":{"type":"team","id":"88d93d5b2c1ed632","login":"andrew-primer-e","name":"","avatar_url":"https://avatars-next.observableusercontent.com/89ec4302ee0f1449de085937ca4510ed79534223a290deb2d18acb6d106c4507","tier":"starter_2024"},"version":836,"publish_version":null,"role":null,"authored":false,"authors":[{"type":"individual","id":"7e6b31ff7a0ad7a8","login":"andrew-primer-e","name":"","avatar_url":"https://avatars-next.observableusercontent.com/89ec4302ee0f1449de085937ca4510ed79534223a290deb2d18acb6d106c4507","tier":null}],"members":[{"type":"individual","id":"7e6b31ff7a0ad7a8","login":"andrew-primer-e","name":"","avatar_url":"https://avatars-next.observableusercontent.com/89ec4302ee0f1449de085937ca4510ed79534223a290deb2d18acb6d106c4507","tier":null,"role":"editor"}],"comments":[],"custom_thumbnail":false,"trashed":false,"lapsed":null,"subscribed":null,"forks":2,"collection":null,"type":"notebook","body":{"title":"Super-generalised Normal Distribution","cells":[{"id":0,"value":"# Super-generalised Normal Distribution","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":765,"value":"This is a combination of the two generalised normal distributions that are listed on [Wikipedia](https://en.wikipedia.org/wiki/Generalized_normal_distribution). It allows for modification of both kertosis and skewness.","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":8,"value":"chart = svg``","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":130,"value":"viewof xi = Inputs.range([-3, 3], {\n value: 0,\n step: 0.1,\n label: tex`\\xi \\text{ (location)}`\n})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":128,"value":"viewof alpha = Inputs.range([0, 3], {\n value: Math.sqrt(2),\n step: 0.1,\n label: tex`\\alpha \\text{ (scale)}`\n})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":127,"value":"viewof beta = Inputs.range([1.1, 8.1], {\n value: 2,\n step: 0.01,\n label: tex`\\beta \\text{ (kertosis)}`\n})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":327,"value":"viewof kappa = Inputs.range([-1, 1], {\n value: 0,\n step: 0.01,\n label: tex`\\kappa \\text{ (skew)}`\n})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":615,"value":"## Parameters\n\n - ${tex`\\xi \\in \\R`}\n - ${tex`\\alpha \\in \\R_{>0}`}\n - ${tex`\\beta \\in (1, \\infty)`}\n - ${tex`\\kappa \\in (-1, 1)`}\n","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":672,"value":"## Support\n${tex`\nx \\in \\begin{cases} \n (-\\infty, \\xi + \\alpha / \\kappa) & \\kappa > 0 \\\\\n (-\\infty, +\\infty) & \\kappa = 0 \\\\\n (\\xi + \\alpha / \\kappa, +\\infty) & \\kappa < 0 \n \\end{cases}\n`}","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":685,"value":"## PDF\n\n${tex`P(X = x) = \\frac{\\phi'(y)}{\\alpha - \\kappa (x - \\xi)}`}, where","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":754,"value":"$${tex`\n y = \\begin{cases}\n - \\frac{1}{\\kappa} \\log{(1 - \\frac{\\kappa (x - \\xi)}{\\alpha})} & \\kappa \\neq 0 \\\\\n \\frac{x - \\xi}{\\alpha} & \\kappa = 0\n \\end{cases}\n`}","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":758,"value":"$${tex`\\phi'(x) = \\frac{\\beta}{2 \\Gamma(1 / \\beta)} e^{|x|^\\beta}`}","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":597,"value":"## Notes","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":592,"value":"$${tex`\\beta`} and ${tex`\\kappa`} should both have larger domains.\nHowever, if the domains are extended then problematic peaks can sometimes occur away from the center of the distribution.\nI think these are caused by the leptokurtic normal distribution decaying more slowly than the biasing function and leading to blow-up.\nIf the biasing function can be tweaked, so that it always decays slowly enough, then it should be possible to extend the domain of ${tex`\\beta`} and ${tex`\\kappa`}.","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":807,"value":"It seems that removing the denominator improves the function in many ways.\nI'm not sure why it's included at all.\nMaybe it maintains a certain relationship between the skew and kertosis.\nMaybe it is there to maintain the total probability (area under the curve) as one.","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":811,"value":"This bears a lot of similarity to the skewed normal GLM that I came up with ages ago.\nThe advantage of that is that there should be an efficient way of finding the optimal parameters without performing a general numerical optimisation, as it is a true GLM.","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":188,"value":"## Appendix","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":770,"value":"This section includes the code that is used to produce the chart.","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":516,"value":"### Distribution Functions","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":285,"value":"//pdf = normalPdf(xi, alpha);\n//pdf = generalisedNormalV1Pdf(xi, alpha, beta);\n//pdf = generalisedNormalV2Pdf(xi, alpha, kappa);\npdf = generalisedNormalCombinedPdf(xi, alpha, beta, kappa);\n","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":277,"value":"normalPdf = function(xi, alpha) {\n return function(x) {\n return 1 / (alpha * Math.sqrt(2 * Math.PI)) * Math.exp(-0.5 * Math.pow((x - xi) / alpha, 2));\n }\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":138,"value":"generalisedNormalV1Pdf = function(xi, alpha, beta) {\n return function(x) {\n return beta / (2 * alpha * mathjs.gamma(1 / beta)) * Math.exp(- Math.pow(Math.abs(x - xi) / alpha, beta));\n }\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":300,"value":"generalisedNormalV2Pdf = function(xi, alpha, kappa) {\n return function(x) {\n if (kappa > 0 && (x >= xi + alpha / kappa)) {\n return 0;\n }\n\n if (kappa < 0 && (x <= xi + alpha / kappa)) {\n return 0;\n }\n \n let y = kappa == 0 ? (x - xi) / alpha : - 1 / kappa * Math.log(1 - kappa * (x - xi) / alpha);\n return normalPdf(0, 1)(y) / (alpha - kappa * (x - xi));\n }\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":319,"value":"generalisedNormalCombinedPdf = function(xi, alpha, beta, kappa) {\n return function(x) {\n if (kappa > 0 && (x >= xi + alpha / kappa)) {\n return 0;\n }\n\n if (kappa < 0 && (x <= xi + alpha / kappa)) {\n return 0;\n }\n \n let y = kappa == 0 ? (x - xi) / alpha : - 1 / kappa * Math.log(1 - kappa * (x - xi) / alpha);\n return generalisedNormalV1Pdf(0, 1, beta)(y) / (alpha - kappa * (x - xi));\n }\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":574,"value":"### Data","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":2,"value":"data = Array.from(Array(numXPoints), (_, i) => {let x = xRange[0] + (xRange[1] - xRange[0]) * i / (numXPoints - 1); return {x: x, y: pdf(x)}});","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":776,"value":"dataV1 = Array.from(Array(numXPoints), (_, i) => {\n let x = xRange[0] + (xRange[1] - xRange[0]) * i / (numXPoints - 1); \n let y = kappa == 0 ? (x - xi) / alpha : - 1 / kappa * Math.log(1 - kappa * (x - xi) / alpha);\n\n if (kappa > 0 && (x >= xi + alpha / kappa)) {\n return {x: x, y: 0}\n }\n\n if (kappa < 0 && (x <= xi + alpha / kappa)) {\n return {x: x, y: 0}\n }\n\n return {x: x, y: generalisedNormalV1Pdf(0, 1, beta)(y)};\n});","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":791,"value":"dataV2 = Array.from(Array(numXPoints), (_, i) => {\n let x = xRange[0] + (xRange[1] - xRange[0]) * i / (numXPoints - 1); \n\n return {x: x, y: 1 / (alpha - kappa * (x - xi))}\n});","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":522,"value":"### Layout","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":35,"value":"height = 400;","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":53,"value":"margin = ({top: 20, right: 30, bottom: 30, left: 40})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":539,"value":"### X Axis","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":449,"value":"viewof xMin = Inputs.range([-100, -1], {value: -3, step: 0.01, label: \"Min. X-Axis\"})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":451,"value":"viewof xMax = Inputs.range([1, 10], {value: 3, step: 0.01, label: \"Max. X-Axis\"})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":132,"value":"xRange = [xMin, xMax]","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":137,"value":"viewof numXPoints = Inputs.range([10, 10000], {value: 1000, step: 1, label: \"Num. X Points\"})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":22,"value":"xScale = d3.scaleLinear()\n .domain(d3.extent(data, d => d.x))\n .range([margin.left, width - margin.right]);","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":48,"value":"xAxis = g => g\n .attr(\"transform\", `translate(0,${height - margin.bottom})`)\n .call(d3.axisBottom(xScale).ticks(width / 80).tickSizeOuter(0))","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":542,"value":"### Y Axis","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":32,"value":"yScale = d3.scaleLinear()\n .domain(d3.extent(Array.from(data, d => d.y).concat([0, 1])))\n .range([height - margin.bottom, margin.top]);","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":59,"value":"yAxis = g => g\n .attr(\"transform\", `translate(${xScale(0)}, 0)`)\n .call(d3.axisLeft(yScale).ticks(width / 80).tickSizeOuter(0))","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":547,"value":"### Line","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":11,"value":"line = d3.line()\n .x(d => xScale(d.x))\n .y(d => yScale(d.y))","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":584,"value":"### Libraries","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":141,"value":"mathjs = import('https://cdn.skypack.dev/mathjs');","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"}],"files":"$W33","stdlib":"1","titling":"auto"}} 32:"$31:body"