2b:null 2c:["$","title",null,{"children":"Klein Four Monte Carlo | Lao | Observable"}] 31:T1ea2,betaRandom = { let numCurves = 3 let population = set.N let samples = [set.n1, set.n2, set.n3] let numBins = set.bin const w = width; const h = 400; const pad = {left:25, right:25, top:45, bottom:35} const color = ['#B93CC9', '#2CB1B3', '#55D15D'] // ["#e15759","#59a14f","#4e79a7"] // display options let range = [8, 0] let domain = [-1, 1] // linear scales let xScale = d3.scaleLinear(domain, [pad.left, w - pad.right]); let yScale = d3.scaleLinear(range, [pad.top, h - pad.bottom]); let xScaleInverse = d3.scaleLinear([pad.left, w - pad.right], domain); let yScaleInverse = d3.scaleLinear([pad.top, h - pad.bottom], range); // make data let handles = [] for(let i=0; i { sim = [ new Array(population*samples[0]).fill().map(() => randb(handles[0].alpha, handles[0].beta)*2-1), new Array(population*samples[1]).fill().map(() => randb(handles[1].alpha, handles[1].beta)*2-1), new Array(population*samples[2]).fill().map(() => randb(handles[2].alpha, handles[2].beta)*2-1) ] mutable simData = sim binData = [] histogramData = [] for( let i=0; i0) ) for( let j=0; j xScale(d[0])) .y((d) => yScale(d[1])) let histogram = svg.selectAll("histogram") // graphs the probability density functions .data(histogramData) // one graph per handles .enter() .append("path") .attr("stroke", 'none' ) .attr("fill", (d,i) => color[i]) .attr("fill-opacity", .3) .style("mix-blend-mode", "darken") .attr("d", line) let pdf = svg.selectAll("curve") // graphs the probability density functions .data(handles) // one graph per handles .enter() .append("path") .attr("stroke", (d) => d.color) .attr("stroke-width", 3) .attr("fill", (d) => d.color) .attr("fill-opacity", 0) .datum((d,i) => graphCurve(xScaleInverse(handles[i].x), yScaleInverse(handles[i].y))) // input parameter for curve .attr("d", line) // TEXT LABELS let textOffset = -13 // how much the alpha and beta coordinates are offset from point when mouseover let shape // for the values of alpha and beta at that circle's mean and gamma var text = svg.selectAll("text") .data(handles) .enter() .append("text") .text((d) => { shape = map(xScaleInverse(d.x), yScaleInverse(d.y)) return shape[0].toFixed(1) + " , " +shape[1].toFixed(1) }) .attr("x", (d) => d.x ) .attr("y", (d) => d.y + textOffset) .style("text-anchor", "middle") .attr("font-family","calibri") .attr("font-weight","bold") .attr("font-size","12pt") .attr("fill", (d) => d.color) .attr("pointer-events", "none") .style("opacity", 1e-6); // Moveable point var points = svg .append("g") .attr("cursor", "move") .selectAll("circle") .data(handles) .enter() .append("circle") .attr("fill", (d) => d.color) .attr("stroke", "white") .attr("stroke-width", 2) .attr("cx", (d) => d.x) .attr("cy", (d) => d.y) .attr("r", (d) => d.r) .call(d3.drag().on("drag", dragged)); let labelData = [ {x:xScale(0), y:35, text:'Simulation Parameters and Random Data', size:"16px", anchor:"middle"}, {x:xScale(-1), y:yScale(0)+35, text:'False Positive', size:"12px", anchor:"start"}, {x:xScale(0), y:yScale(0)+35, text:'Inconclusive', size:"12px", anchor:"middle"}, {x:xScale(1), y:yScale(0)+35, text:'True Positive', size:"12px", anchor:"end"} ] const label = svg.append('g') .selectAll("label") .data(labelData) .enter() .append("text") .attr("x", d => d.x) .attr("y", d => d.y ) .attr("text-anchor", d => d.anchor) .style("font-size", d => d.size) .attr("font-family", "Arial") .attr("font-weight",700) .text(d => d.text); // INTERACTION ////////////////////////////////////////////////////////////////// points .on('mouseover', function(event, d) { d3.select(this) .attr("r", (d) => d.r+2) // grow point by 2 when mouse over text .style("opacity", e => e === d ? 1 : 1e-6) }) .on('mouseout', function(event, d) { d3.select(this) .attr("r", (d) => d.r) // shrink point to default when mouse moves away text.style("opacity", 1e-6 ) }) let id function dragged(event, d) { d3.select(this) .attr("cx", () => { d.x = Math.min(w-pad.right-1, Math.max(pad.left+1, event.x)); // moves x position of circle return d.x+1; }) .attr("cy", () => { d.y = Math.min(yScale(0)-1, Math.max(pad.top, event.y)); // moves y position of circle d.alpha = map(xScaleInverse(d.x), yScaleInverse(d.y))[0] d.beta = map(xScaleInverse(d.x), yScaleInverse(d.y))[1] return d.y; }) .raise(); //brings point to front text .data(handles) .style("fill-opacity", e => e === d ? 1 : 1e-6 ) .attr("x", d => d.x ) // moves x position of text .attr("y", d => d.y + textOffset) // moves y position of text .attr("fill", (d,i) => { return handles[i].color }) .text((d) => { shape = map(xScaleInverse(d.x), yScaleInverse(d.y)) return d.alpha.toFixed(1) + " , " +d.beta.toFixed(1) // updates the alpha and beta of label }) pdf .datum((d,i) => graphCurve(xScaleInverse(handles[i].x), yScaleInverse(handles[i].y)) ) // recalcs all PDFs .attr("d", line) .attr("fill", (d,i) => { return handles[i].color }) .attr("stroke",(d,i)=> { return handles[i].color }) calcHistogram(sim) histogram .data(histogramData) .attr("d", line) } let xAxis1 = d3.axisBottom(xScale).ticks(21); svg.append("g").attr("transform", `translate(0,${yScale(0)})`).call(xAxis1); return svg.node(); }32:Tb32,coherenceTest = { const w = width const h = 300 const pad = {left:25, right:25, top:35, bottom:35} let color = ['red', 'green'] const numBins = set.bin let meanSim = d3.mean(simScore) let meanPerm = d3.mean(permutation) let devSim = d3.deviation(simScore) let domain = [-Math.min(1, Math.abs(meanSim)+3*devSim), Math.min(1,Math.abs(meanSim)+3*devSim) ] let modelData = [permutation, simScore] let scale = [ numBins/set.N, numBins/set.N ] let histogramData, bin, binData let calcHistogram = () => { binData = [] histogramData = [] for( let i=0; i<2; i++){ binData.push( new Array(numBins).fill().map(()=>0) ) for( let j=0; j xScale(d[0])) .y((d) => yScale(d[1])) let histogram = svg.selectAll("g") // graphs the probability density functions .data(histogramData) // one graph per handles .enter() .append("path") .attr("stroke", 'none' ) .attr("opacity", .75 ) .attr("fill", (d,i) => color[i]) .style("mix-blend-mode", "hard-light") .attr("d", line ) let labelData = [ {x:xScale(0), y:35, text:'Normalized Coherence Score', size:"16px", anchor:"middle"}, {x:pad.left, y:yScale(0)+35, text:'Anticoherent', size:"12px", anchor:"start"}, {x:xScale(0), y:yScale(0)+35, text:'Incoherent', size:"12px", anchor:"middle"}, {x:w-pad.right, y:yScale(0)+35, text:'Coherent', size:"12px", anchor:"end"} ] const label = svg.append('g') .selectAll("label") .data(labelData) .enter() .append("text") .attr("x", d => d.x) .attr("y", d => d.y ) .attr("text-anchor", d => d.anchor) .style("font-size", d => d.size) .attr("font-family", "Arial") .attr("font-weight",700) .text(d => d.text); let xAxis1 = d3.axisBottom(xScale); svg.append("g").attr("transform", `translate(0,${yScale(0)})`).call(xAxis1); return svg.node() }33:Taa3,viewof set = { let max = 200 const form = html` `; form.oninput = () => { form.out.value = 'Number of products = ' + form.n1.value + ' × ' + form.n2.value + ' × ' + form.n3.value + ' = ' + form.n1.valueAsNumber * form.n2.valueAsNumber * form.n3.valueAsNumber return form.value = ({ N: form.N.valueAsNumber, n1: form.n1.valueAsNumber, n2: form.n2.valueAsNumber, n3: form.n3.valueAsNumber, bin: form.bin.valueAsNumber }); } form.oninput(); return form; } 34:T7e0,The [Klein four dichotomy correlation notebook](https://observablehq.com/@laotzunami/klein-four-dichotomy-correlation) presented how a Klein four group relation could be tested by taking the product of three random variables. When at least one dichotomy mean was zero, the null and alternative distribution were the same, but when all dichotomy means were far from zero, at least half of the two distributions overlapped. This is the beginning of an asymmetry that can be tested, but a single set of observations does not move the distributions far enough to significantly distinguish between the null and alternative hypothesis. However, adding sets of observations separates the two distributions further apart. Continuous distributions can be added together using [convolution](https://en.wikipedia.org/wiki/Convolution), which due to the [convolution theorm](https://en.wikipedia.org/wiki/Convolution_theorem#Functions_of_a_continuous_variable), can be implemented with the [fast Fourier transform](https://en.wikipedia.org/wiki/Fast_Fourier_transform). This is hard, so in the mean time, I made this notebook with a [monte carlo](https://en.wikipedia.org/wiki/Monte_Carlo_method) simulation. Instead of convolution in the continuous case, multiplying and adding random values is as simple as multiplying and adding the random data from the simulation. The variables n1, n2 and n3 are the number of samples from each dichotomy. When sampling each dichotomy only once, it is the same case in the [Klein four dichotomy correlation](https://observablehq.com/@laotzunami/klein-four-dichotomy-correlation), where at least half of the distributions must overlap. The coherence score is the sum of products. Summing random variables expands their support. The sum of two random variables with a support of -1 to 1 will expand to a support of -2 to 2. The sum of coherences is normalized by dividing the sum by the number of products added together, so the support is from -1 to 1, the same as the random variables.35:T4ff,/* function from Mike Bostock's notebook Beta-Distributed Random Values https://observablehq.com/@mbostock/beta-distributed-random-values */ function randb(alpha, beta) { // Returns a normal deviate (mu=0, sigma=1). function randn() { let u, v, x, y, q; do { u = Math.random(); v = 1.7156 * (Math.random() - 0.5); x = u - 0.449871; y = Math.abs(v) + 0.386595; q = x * x + y * (0.19600 * y - 0.25472 * x); } while (q > 0.27597 && (q > 0.27846 || v * v > -4 * Math.log(u) * u * u)); return v / u; } // Returns a gamma deviate by the method of Marsaglia and Tsang. function randg(shape) { let oalph = shape, a1, a2, u, v, x, mat; if (!shape) shape = 1; if (shape < 1) shape += 1; a1 = shape - 1 / 3; a2 = 1 / Math.sqrt(9 * a1); do { do { x = randn(); v = 1 + a2 * x; } while (v <= 0); v = v * v * v; u = Math.random(); } while ( u > 1 - 0.331 * Math.pow(x, 4) && Math.log(u) > 0.5 * x * x + a1 * (1 - v + Math.log(v)) ); if (shape === oalph) return a1 * v; // alpha > 1 do u = Math.random(); while (u === 0); // alpha < 1 return Math.pow(u, 1 / oalph) * a1 * v; } const u = randg(alpha); return u / (u + randg(beta)); }36:[] 2f:{"id":"5b20ebb987cd992a","slug":"klein-four-monte-carlo","title":"Klein Four Monte Carlo","thumbnail":"646c721afdbf1f6933d03e37164273e0527d152229d58a8360a291f8245b6c9e","access_level":"public","create_time":"$D2022-05-27T03:57:06.139Z","update_time":"$D2024-07-18T13:59:13.234Z","view_time":null,"first_public_time":"$D2022-05-27T07:03:59.466Z","fork":false,"stars":9,"starred":false,"owner":{"type":"team","id":"cf45be0120eccaad","login":"laotzunami","name":"Lao","avatar_url":"https://avatars-next.observableusercontent.com/aeddc181df6ca80a347940ba62e17601ac6395e9d2c21142887eca5fd277326a","tier":"starter_2024"},"version":160,"publish_version":null,"role":null,"authored":false,"authors":[{"type":"individual","id":"5da1d0646b8496ac","login":"laotzunami","name":"Lao","avatar_url":"https://avatars-next.observableusercontent.com/9c14da488e1a68ca1868ed025eb7edcce3d300c8a9f3fbcdae66a3daa9ccca63","tier":null}],"members":[{"type":"individual","id":"5da1d0646b8496ac","login":"laotzunami","name":"Lao","avatar_url":"https://avatars-next.observableusercontent.com/9c14da488e1a68ca1868ed025eb7edcce3d300c8a9f3fbcdae66a3daa9ccca63","tier":null,"role":"editor"}],"comments":[],"custom_thumbnail":true,"trashed":false,"subscribed":null,"forks":7,"collection":null,"type":"notebook","body":{"title":"Klein Four Monte Carlo","cells":[{"id":0,"value":"# Klein Four Monte Carlo","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":8,"value":"$31","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":12,"value":"$32","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":15,"value":"$33","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":49,"value":"$34","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":18,"value":"### Variables","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":22,"value":"mutable simData = [[],[],[]]","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":25,"value":"simScore = {\n let scores = []\n \n let population = set.N\n let n01 = set.n1\n let n10 = set.n2\n let n11 = set.n3\n \n for(let i=0; i Math.round( Math.random() )*2-1),\n new Array(n10).fill().map(() => Math.round( Math.random() )*2-1),\n new Array(n11).fill().map(() => Math.round( Math.random() )*2-1)\n ]\n for(let j=0; j= critvalue){count++}\n }\n return Math.round(count *100 / simScore.length) \n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":34,"value":"critvalue = {\n let a = .05\n return permutation.sort()[Math.round(permutation.length*(1-a))-1]\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":82,"value":"### Functions","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":37,"value":"function map(mean, gamma) {\n mean = (mean + 1) / 2 // transforms range from [0,1] --> [-1,1]\n let a = 1.5 * mean * gamma ** 2 * (1 - 0.9 * (2 * mean - 1) ** 2);\n let b = (a - a * mean) / mean;\n let betaCoefficient, estimate;\n // refine alpha until curve intersects point\n for (let i = 0; i < 15; i++) {\n betaCoefficient = cephes.beta(a, b);\n estimate = betaPDF(mean, a, b, betaCoefficient);\n a = (a * gamma) / estimate;\n b = (a - a * mean) / mean;\n }\n return [a, b, betaCoefficient];\n }","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":40,"value":"function betaPDF(x, alpha, beta, B){\n if(x==0){x=.00000001}\n if(x==1){x=.99999999}\n if(0