2f:T430,chart = { const d3 = await require("d3@5"); const radius = 2; const n = randomNumbers.length; const height = 300; const dodge = dodger(radius * 2 + 1); const margin = { top: 0, right: 10, bottom: 20, left: 10 }; const values = randomNumbers; const minMax = [dataProfile[0].min, dataProfile[0].max]; const x = d3 .scaleLinear(d3.extent(values), [margin.left, width - margin.right]) .nice(); const svg = d3.select(DOM.svg(width, height)).style("overflow", "visible"); svg .append("g") .attr("transform", `translate(0,${height - margin.bottom})`) .call(d3.axisBottom(x)); for (let i = 0; i < n; ++i) { if (i % 5 === 0) yield svg.node(); const cx = x(values[i]); const cy = height - margin.bottom - dodge(cx) - radius - 1; svg .append("circle") .attr("cx", cx) .attr("cy", -400) .attr("r", radius) .attr("minMax", values[i] === minMax[0] || values[i] === minMax[1]) .transition() .duration(750) .ease(d3.easeBounce) .attr("cy", cy); } yield svg.node(); }30:T629,## Extra: Euler's Formula * [${tex`{e^{i\pi} + 1= 0}`}](https://en.wikipedia.org/wiki/Euler%27s_formula) * See discussion of 'something' out of 'nothing' in [The Thoughtful Universe: The Brain vs. Quantum Vacuums](https://observablehq.com/@mariodelgadosr/perspective-extra-the-thoughtful-universe?collection=@mariodelgadosr/musings#cell-30) and the essay [What is a ?](https://observablehq.com/@mariodelgadosr/what-is-a?collection=@mariodelgadosr/musings): > ...${tex`0`} (nothing) is theorized by some cosmologist to be the origin of the 'something' we experience: >>...What produced the energy [of the Universe] before inflation? This is perhaps the ***ultimate question***. As crazy as it might seem, the energy may have come out of nothing! The meaning of "nothing" is somewhat ambiguous here. It might be the vacuum in some pre-existing space and time, or it could be nothing at all – that is, all concepts of space and time were created with the universe itself.... >> -- [***The Cosmos: Astronomy in the New Millennium***](https://www.cambridge.org/us/academic/subjects/physics/astronomy-general/cosmos-astronomy-new-millennium-5th-edition?format=PB&isbn=9781108431385) * [Demo Link](https://www.desmos.com/calculator/shrmvda7r4) 31:T7c0,linear = { const d3 = await require("d3-array@3", "d3-axis@3", "d3-regression@1", "d3-scale@4", "d3-shape@3", "d3-selection@3"); const { margin, base, innerWidth, innerHeight } = dimensions; const xScaleLinear = d3.scaleLinear().domain([-4, 16]).range([0, innerWidth]); const yScaleLinear = d3 .scaleLinear() .domain([-4, 16]) .range([innerHeight, 0]); const xAxisLinear = d3.axisBottom(xScaleLinear).tickSize(innerHeight + 10); const yAxisLinear = d3.axisLeft(yScaleLinear).tickSize(innerWidth + 10); const svg = d3.select( DOM.svg( innerWidth + margin.left + margin.right, innerHeight + margin.top + margin.bottom ) ); const dataLinear = [ { x: 8, y: 3 }, { x: 2, y: 10 }, { x: 11, y: 3 }, { x: 6, y: 6 }, { x: 5, y: 8 }, { x: 4, y: 12 }, { x: 12, y: 1 }, { x: 9, y: 4 }, { x: 6, y: 9 }, { x: 1, y: 14 } ]; const linearRegression = d3 .regressionLinear() .x((d) => d.x) .y((d) => d.y) .domain([-1.7, 16]); const g = svg .append("g") .attr("transform", `translate(${margin.left}, ${margin.top})`); g.append("g") .attr("class", "axis") .call(xAxisLinear) .selectAll(".tick") .classed("baseline", (d) => d == 0); g.append("g") .attr("class", "axis") .attr("transform", `translate(${innerWidth})`) .call(yAxisLinear) .selectAll(".tick") .classed("baseline", (d) => d == 0); g.selectAll("circle") .data(dataLinear) .enter() .append("circle") .attr("r", 3) .attr("cx", (d) => xScaleLinear(d.x)) .attr("cy", (d) => yScaleLinear(d.y)); g.append("line") .attr("class", "regression") .datum(linearRegression(dataLinear)) .attr("x1", (d) => xScaleLinear(d[0][0])) .attr("x2", (d) => xScaleLinear(d[1][0])) .attr("y1", (d) => yScaleLinear(d[0][1])) .attr("y2", (d) => yScaleLinear(d[1][1])); //return svg.node(); return { svg, dataLinear, linearRegression }; }32:T7e6,polynomial = { const d3 = await require("d3-array@3", "d3-axis@3", "d3-regression@1", "d3-scale@4", "d3-shape@3", "d3-selection@3"); const { margin, base, innerWidth, innerHeight } = dimensions; const polynomialRegression = d3 .regressionPoly() .x((d) => d.x) .y((d) => d.y) .order(3); const dataPolynomial = [ { x: 0, y: 140 }, { x: 1, y: 149 }, { x: 2, y: 159.6 }, { x: 3, y: 159 }, { x: 4, y: 155.9 }, { x: 5, y: 169 }, { x: 6, y: 162.9 }, { x: 7, y: 169 }, { x: 8, y: 180 } ]; const xScalePolynomial = d3 .scaleLinear() .domain([0, 8]) .range([0, innerWidth]); const yScalePolynomial = d3 .scaleLinear() .domain([140, 180]) .range([innerHeight, 0]); const xAxisPolynomial = d3 .axisBottom(xScalePolynomial) .tickSize(innerHeight + 10); const yAxisPolynomial = d3 .axisLeft(yScalePolynomial) .tickSize(innerWidth + 10); const linePolynomial = d3 .line() .x((d) => xScalePolynomial(d[0])) .y((d) => yScalePolynomial(d[1])); const svg = d3.select( DOM.svg( innerWidth + margin.left + margin.right, innerHeight + margin.top + margin.bottom ) ); const g = svg .append("g") .attr("transform", `translate(${margin.left}, ${margin.top})`); g.append("g") .attr("class", "axis") .call(xAxisPolynomial) .selectAll(".tick") .classed("baseline", (d) => d === 0); g.append("g") .attr("class", "axis") .attr("transform", `translate(${innerWidth})`) .call(yAxisPolynomial) .selectAll(".tick") .classed("baseline", (d) => d === 0); g.selectAll("circle") .data(dataPolynomial) .enter() .append("circle") .attr("r", 3) .attr("cx", (d) => xScalePolynomial(d.x)) .attr("cy", (d) => yScalePolynomial(d.y)); g.append("path") .attr("class", "regression") .datum(polynomialRegression(dataPolynomial)) .attr("d", linePolynomial); return { svg, dataPolynomial, polynomialRegression }; }33:T7b7,exponential = { const d3 = await require("d3-array@3", "d3-axis@3", "d3-regression@1", "d3-scale@4", "d3-shape@3", "d3-selection@3"); const { margin, base, innerWidth, innerHeight } = dimensions; const exponentialRegression = d3 .regressionExp() .x((d) => d.x) .y((d) => d.y) .domain([-2, 5.06]); const dataExponential = [ { x: 0, y: 3 }, { x: 1, y: 7 }, { x: 2, y: 10 }, { x: 3, y: 24 }, { x: 4, y: 50 }, { x: 5, y: 95 } ]; const xScaleExponential = d3 .scaleLinear() .domain([-2, 10]) .range([0, innerWidth]); const yScaleExponential = d3 .scaleLinear() .domain([-10, 100]) .range([innerHeight, 0]); const xAxisExponential = d3 .axisBottom(xScaleExponential) .tickSize(innerHeight + 10); const yAxisExponential = d3 .axisLeft(yScaleExponential) .tickSize(innerWidth + 10); const lineExponential = d3 .line() .x((d) => xScaleExponential(d[0])) .y((d) => yScaleExponential(d[1])); const svg = d3.select( DOM.svg( innerWidth + margin.left + margin.right, innerHeight + margin.top + margin.bottom ) ); const g = svg .append("g") .attr("transform", `translate(${margin.left}, ${margin.top})`); g.append("g") .attr("class", "axis") .call(xAxisExponential) .selectAll(".tick") .classed("baseline", (d) => d === 0); g.append("g") .attr("class", "axis") .attr("transform", `translate(${innerWidth})`) .call(yAxisExponential) .selectAll(".tick") .classed("baseline", (d) => d === 0); g.selectAll("circle") .data(dataExponential) .enter() .append("circle") .attr("r", 3) .attr("cx", (d) => xScaleExponential(d.x)) .attr("cy", (d) => yScaleExponential(d.y)); g.append("path") .attr("class", "regression") .datum(exponentialRegression(dataExponential)) .attr("d", lineExponential); return { svg, dataExponential, exponentialRegression }; }34:Tcff,logistic = { const numHoursStudied = [ 0.5, 0.75, 1.0, 1.25, 1.5, 1.75, 1.75, 2.0, 2.25, 2.5, 2.75, 3.0, 3.25, 3.5, 4.0, 4.25, 4.5, 4.75, 5.0, 5.5 ]; const studentPassedExam = [ 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 1, 1, 1 ]; // range of hours where both 0 and 1 studentPassedExam values can be found; no exclusive const failPassRange = [ studentPassedExam.findIndex((d) => d == 1), studentPassedExam.findLastIndex((d) => d == 0) ]; const dataLogistic = numHoursStudied.map((h, i) => ({ "Number of Hours Studied": h, "Student Passed Exam": studentPassedExam[i] })); const x1 = d3 .range(...[0, 6], 0.1) // https://github.com/d3/d3-array/blob/v3.2.3/README.md#range .concat(6); // https://github.com/d3/d3-array/issues/90 const y = x1.map((x) => 1 / (1 + Math.pow(Math.E, -(B0 + B1 * x)))); const w = width * 0.6; const h = w * 0.5; const plot = Plot.plot({ width: w, height: h, x: { domain: [0, 6], label: "x", grid: true }, y: { domain: [0, 1], label: "y", grid: true }, caption: tex`\Large Pr(y) = \frac{1}{1+e^{-(${B0} + ${B1}\cdot x)}}`, marks: [ Plot.ruleX([0]), // y axis Plot.ruleY([0]), // x axis showFailPass ? Plot.line( [ [numHoursStudied[failPassRange[0]], 0], [numHoursStudied[failPassRange[1]], 0], [numHoursStudied[failPassRange[1]], 1], [numHoursStudied[failPassRange[0]], 1], [numHoursStudied[failPassRange[0]], 0] ], { fill: "lightgray" } ) : [], showTestScores ? Plot.dot(dataLogistic, { x: "Number of Hours Studied", y: "Student Passed Exam", r: 7, fill: "purple", title: (d) => `Hours Studied: ${d["Number of Hours Studied"]}, ${ !d["Student Passed Exam"] ? "Failed" : "Passed" }` }) : [], showRegression ? Plot.line(x1, { x: x1, y: y, stroke: "blue" }) : [], // y = f(x) arrays plotted , showRegression ? Plot.dot([[-B0 / B1, 0.5]], { fill: "red", r: 5, title: (d) => `x: ${d3.format(".3")(d[0])}, y: ${d[1]}` }) : [], // location parameter https://en.wikipedia.org/wiki/Location_parameter , showRegression ? Plot.text([[-B0 / B1, 0.5]], { text: () => `LP`, dy: -5, dx: 15 }) : [], showRegression ? Plot.line( [ [0, 0.5], [-B0 / B1, 0.5] ], { stroke: "red", strokeDasharray: [5, 5] } ) : [], showRegression ? Plot.line( [ [-B0 / B1, 0], [-B0 / B1, 0.5] ], { stroke: "red", strokeDasharray: [5, 5] } ) : [] /* Optional technique to show formula in the plot area ,showRegression ? () => svg` ${MathJax.tex2svg( String.raw`Pr(y) = \frac{1}{1+e^{-(${B0} + ${B1}\cdot x)}}` ).querySelector("svg")} ` : [] */ ] }); return { dataLogistic, plot }; }35:T5be,prospect = { // https://en.wikipedia.org/wiki/Prospect_theory#/media/File:Loss_Aversion.png // https://observablehq.com/@mcmcclur/plot-for-mathematicians const prospectData = [ [-0.2, -48], [-0.1, -45], [-0.05, -40], [-0.025, -30], [0, 0], [0.025, 13], [0.05, 20], [0.1, 23], [0.2, 26] ].map(([lossGain, value]) => ({ lossGain, value })); const w = width * 0.6; const h = w * 0.75; const plot = Plot.plot({ width: w, height: h, x: { domain: [-0.2, 0.2] }, y: { domain: [-60, 60] }, marks: [ Plot.axisX({ y: 0 }), Plot.axisY({ x: 0 }), Plot.ruleX([0]), // y axis Plot.ruleY([0]), // x axis Plot.line( [ [-0.05, 0], [-0.05, -40] ], { stroke: "blue", strokeDasharray: [5, 5] } ), Plot.line( [ [-0.05, -40], [0, -40] ], { stroke: "blue", strokeDasharray: [5, 5] } ), Plot.line( [ [0, 20], [0.05, 20] ], { stroke: "blue", strokeDasharray: [5, 5] } ), Plot.line( [ [0.05, 20], [0.05, 0] ], { stroke: "blue", strokeDasharray: [5, 5] } ), Plot.line(prospectData, { x: "lossGain", y: "value", curve: "catmull-rom" // https://observablehq.com/plot/features/curves }) ] }); return { prospectData, plot }; }36:[{"id":"8e177756c34dc29842dad3fbc29a5d576444e0541c23fa946d5d62fdf7dec3e101927ae4032119df70c49cb2add3b51c4778fa711a9f1ac647381daadabc681f","href":"https://static.observableusercontent.com/files/8e177756c34dc29842dad3fbc29a5d576444e0541c23fa946d5d62fdf7dec3e101927ae4032119df70c49cb2add3b51c4778fa711a9f1ac647381daadabc681f","name":"image.png","type":"image/png","lastModified":1682950278343.684,"size":65379}] 2d:{"title":"Just Enough Statistics","cells":[{"id":0,"value":"# Just Enough Statistics","pinned":false,"hidden":false,"mode":"md"},{"id":9,"value":"toc(\"h2,h3,h4\")","pinned":false,"hidden":false,"mode":"ojs"},{"id":12,"value":"## [Metadata](https://en.wikipedia.org/wiki/Metadata): Data About Data","pinned":false,"hidden":false,"mode":"md"},{"id":15,"value":"viewof randomFunction = {\n const d3 = await require(\"d3@5\");\n const randomFunctions = [\n { format: \"Uniform\", fn: d3.randomUniform(0, 1000) },\n { format: \"Normal\", fn: d3.randomNormal() },\n { format: \"Exponetial\", fn: d3.randomExponential(2) }\n ];\n\n return Inputs.radio(randomFunctions, {\n label: \"If you generated 1,000 random numbers that were charateristically\",\n format: (d) => d.format,\n value: randomFunctions[0]\n });\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":20,"value":"randomNumbers = Array.from({ length: 1000 }, randomFunction.fn)","pinned":false,"hidden":false,"mode":"ojs"},{"id":18,"value":"### Qualitative\nWith a [visualization](https://en.wikipedia.org/wiki/Data_and_information_visualization), [perceptual psychology](https://www.interaction-design.org/literature/book/the-encyclopedia-of-human-computer-interaction-2nd-ed/data-visualization-for-human-perception) reveals [qualifications](https://www.simplypsychology.org/qualitative-quantitative.html) of one-dimensional data:","pinned":false,"hidden":false,"mode":"md"},{"id":77,"value":"$2f","pinned":false,"hidden":false,"mode":"ojs"},{"id":84,"value":"### Quantitative\nThe discipline of [Statistics](https://en.wikipedia.org/wiki/Statistics) uses [mathematical rigor](https://en.wikipedia.org/wiki/Rigour#Mathematics) to compute [quantifications](https://en.wikipedia.org/wiki/Quantitative_analysis). ","pinned":false,"hidden":false,"mode":"md"},{"id":92,"value":"// https://github.com/vega/datalib/wiki/Statistics#dl_summary\n// q1: 25th percentile\n// q3: 75th percentile\ndataProfile = dl.summary(randomNumbers.map((d) => ({ number: d })))","pinned":false,"hidden":false,"mode":"ojs"},{"id":99,"value":"Inputs.table(dataProfile)","pinned":false,"hidden":false,"mode":"ojs"},{"id":131,"value":"{\n if (!showMinMax) return md``;\n const d3 = await require(\"d3@5\");\n d3.select(chart)\n .selectAll(\"circle[minMax=true]\")\n .attr(\"fill\", \"red\")\n .attr(\"r\", 5);\n\n return md`* Minimum: ${dataProfile[0].min} \n* Maximum: ${dataProfile[0].max}`;\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":127,"value":"// https://observablehq.com/@mbostock/wait-until-button\nviewof showMinMax = {\n randomFunction;\n return Inputs.button(\"Isolate Minimum and Maximum\", { value: null });\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":578,"value":"Related Notebook: [Standard error of the mean](https://observablehq.com/@bluprince13/standard-error-of-the-mean). ","pinned":false,"hidden":false,"mode":"md"},{"id":105,"value":"## [Regression Analyses](https://en.wikipedia.org/wiki/Regression_analysis) Can Quantify the Dynamics in Two-Dimensional Data\n\nThe following visualizations are from [Introducing d3-regression](https://observablehq.com/@harrystevens/introducing-d3-regression):","pinned":false,"hidden":false,"mode":"md"},{"id":164,"value":"### Linear\n\nLinear regressions take the form ${tex.options({ trust: true, displayMode: true\n})`\\href{https://en.wikipedia.org/wiki/Linear_equation#Slope%E2%80%93intercept_form_or_Gradient-intercept_form}{y = mx + y_0}\\\\`}\n\n\nWhere: \n${tex.options({ trust: true, \n})`\\bullet \\hspace{.5cm} \\href{https://en.wikipedia.org/wiki/Slope_(mathematics%29}{m} = \\text{the line's slope}\\\\`}\n${tex.options({ trust: true, \n})`\\bullet \\hspace{.5cm} \\href{https://en.wikipedia.org/wiki/Y-intercept}{y_0} = \\text{the line's y-intercept}`}\n\nAssuming the following data:","pinned":false,"hidden":false,"mode":"md"},{"id":227,"value":"Inputs.table(linear.dataLinear, { width: 150 })","pinned":false,"hidden":false,"mode":"ojs"},{"id":233,"value":"A [linear regression](https://en.wikipedia.org/wiki/Linear_regression) analysis determines a relationship described by the equation: \n${tex.options({\n displayMode: true\n})` y = ${linear.linearRegression(linear.dataLinear).a.toFixed(3)}x + ${linear.linearRegression(linear.dataLinear).b.toFixed(3)}`}\n","pinned":false,"hidden":false,"mode":"md"},{"id":222,"value":"linear.svg.node()","pinned":false,"hidden":false,"mode":"ojs"},{"id":281,"value":"### Polynomial\n[Polynomial regressions](https://en.wikipedia.org/wiki/Polynomial_regression) take the form ${tex.options({ trust: true, displayMode: true\n})`\\href{https://en.wikipedia.org/wiki/Polynomial}{y = a_nx^n + … + a_1x + a_0}`}\nAssuming the following data:","pinned":false,"hidden":false,"mode":"md"},{"id":292,"value":"Inputs.table(polynomial.dataPolynomial, { width: 150 })","pinned":false,"hidden":false,"mode":"ojs"},{"id":314,"value":"A polynomial regression analysis determines a relationship described by the equation: \n${tex.options({\n displayMode: true\n})`y = ${polynomial.polynomialRegression(polynomial.dataPolynomial).coefficients.reverse().map((d, i) => {\n const deg = Math.abs(polynomial.polynomialRegression(polynomial.dataPolynomial).coefficients.length - i - 1);\n return d.toFixed(2) + (deg > 0 ? \"x\" : \"\") + (deg > 1 ? \"^\" + deg : \"\")\n}).join(\"+\")}`}\n","pinned":false,"hidden":false,"mode":"md"},{"id":290,"value":"polynomial.svg.node()","pinned":false,"hidden":false,"mode":"ojs"},{"id":333,"value":"### Exponential\n[Exponential regressions](https://en.wikipedia.org/wiki/Nonlinear_regression) take the form ${tex.options({ trust: true, displayMode: true\n})`\\href{https://en.wikipedia.org/wiki/Polynomial}{y = ae^{bx}}`}\n\nWhere ${tex`e`} is [Euler's number](https://en.wikipedia.org/wiki/Eulers_number):\n\n${tex.block `e \\approx \\sum_{n=0}^{${max}}\\frac{1}{n!} = ${e_to_x_approximation(1, max)}`}\n\n","pinned":false,"hidden":false,"mode":"md"},{"id":360,"value":"viewof max = Inputs.range([0, 100], {\n label: \"Uppler Limit\",\n step: 1,\n value: 6\n})","pinned":false,"hidden":false,"mode":"ojs"},{"id":355,"value":"Assuming the following data:","pinned":false,"hidden":false,"mode":"md"},{"id":346,"value":"Inputs.table(exponential.dataExponential, { width: 150 })","pinned":false,"hidden":false,"mode":"ojs"},{"id":432,"value":"The regression is described by the equation ${tex.options({ displayMode: true\n})`y = ${exponential.exponentialRegression(exponential.dataExponential).a.toFixed(3)}e^{${exponential.exponentialRegression(exponential.dataExponential).b.toFixed(3)}x}`}.","pinned":false,"hidden":false,"mode":"md"},{"id":349,"value":"exponential.svg.node()","pinned":false,"hidden":false,"mode":"ojs"},{"id":399,"value":"### Logistic\n[Logistic regressions](https://en.wikipedia.org/wiki/Logistic_regression) take the form of a [sigmoid function](https://en.wikipedia.org/wiki/Sigmoid_function): ${tex.options({ trust: true, displayMode: true\n})`\\href{https://en.wikipedia.org/wiki/Polynomial}{Pr(y) = \\frac{1}{1+e^{-(\\beta_0 + \\beta_1\\cdot x)}}}`}\nWhere ${tex`Pr(y)`} is the predicted probability dependent on ${tex`x`}.","pinned":false,"hidden":false,"mode":"md"},{"id":408,"value":"Assuming the following [data](https://en.wikipedia.org/wiki/Logistic_regression#Example):","pinned":false,"hidden":false,"mode":"md"},{"id":411,"value":"Inputs.table(logistic.dataLogistic, { width: 400 })","pinned":false,"hidden":false,"mode":"ojs"},{"id":443,"value":"The regression analyses determine the values of the coefficients ${tex`B_0`} and ${tex`B_1`}. \n\nThe [location paramater](https://en.wikipedia.org/wiki/Location_parameter) (LP) is the midpoint of the curve where: \n\n${tex.block`\n \\begin{aligned}\n Pr(y) &= \\frac{1}{2} \\\\ \n \\text{for } x &\\approx \\frac{-B_0}{B_1} \\\\ \n &\\approx \\frac{${-B0}}{${B1}} \\\\ \n &\\approx ${B1 != 0 ? d3.format(\".3\")(-B0/B1) : `-∞`} \\text{ hours} \n \\end{aligned}`}","pinned":false,"hidden":false,"mode":"md"},{"id":429,"value":"logistic.plot","pinned":false,"hidden":false,"mode":"ojs"},{"id":582,"value":"viewof showTestScores = Inputs.toggle({\n label: htl.html`

Test Scores`,\n value: true\n})","pinned":false,"hidden":false,"mode":"ojs"},{"id":656,"value":"viewof showFailPass = Inputs.toggle({\n label: htl.html`

Indeterminate Fail/Pass Range`,\n value: showTestScores\n})","pinned":false,"hidden":false,"mode":"ojs"},{"id":629,"value":"viewof showRegression = Inputs.toggle({\n label: htl.html`

Logistic Regression`,\n value: true\n})","pinned":false,"hidden":false,"mode":"ojs"},{"id":418,"value":"viewof B0 = Inputs.range([-10, 0], {\n step: 0.1,\n label: tex`\\LARGE \\beta_0`,\n value: -4.1\n})","pinned":false,"hidden":false,"mode":"ojs"},{"id":421,"value":"viewof B1 = Inputs.range([0, 4], {\n step: 0.1,\n label: tex`\\LARGE \\beta_1`,\n value: 1.5\n})","pinned":false,"hidden":false,"mode":"ojs"},{"id":445,"value":"## Food for Thought: [Prospect Theory](https://en.wikipedia.org/wiki/Prospect_theory)/[Behavioral Economics](https://en.wikipedia.org/wiki/Behavioral_economics)/[***Thinking Fast and Slow***](https://en.wikipedia.org/wiki/Thinking,_Fast_and_Slow)","pinned":false,"hidden":false,"mode":"md"},{"id":519,"value":"![image.png](${await FileAttachment(\"image.png\").url()})","pinned":false,"hidden":false,"mode":"md"},{"id":523,"value":"Humans (and [other species](https://stanford.edu/~knutson/bad/tom07.pdf)) have quantified, non-linear responses to every day decisions. Accurate Artificial/Co-Intelligence systems will have to incorporate these non-linear dynamics.\n\nThe following data illustrates that contrary to the [expected utility theory](https://en.wikipedia.org/wiki/Expected_utility_hypothesis)—which models the decision that perfectly rational agents would make—[Prospect Theory](https://en.wikipedia.org/wiki/Prospect_theory) seeks to describe observed behaviors:","pinned":false,"hidden":false,"mode":"md"},{"id":525,"value":"Inputs.table(prospect.prospectData, { width: 250 })","pinned":false,"hidden":false,"mode":"ojs"},{"id":458,"value":"prospect.plot","pinned":false,"hidden":false,"mode":"ojs"},{"id":608,"value":"Another dynamic to factor into Artificial/[Co-Intelligence](https://en.wikipedia.org/wiki/Collective_intelligence) systems: \n> [WYSIATI](https://en.wikipedia.org/wiki/Illusion_of_validity) is a concept developed by Daniel Kahneman, a professor of psychology and a Nobel Prize laureate in economic sciences in 2002, in his book [Thinking, Fast and Slow](https://en.wikipedia.org/wiki/Thinking,_Fast_and_Slow)... \n-- [Kahneman’s word: WYSIATI](https://narrativeweliveby.wordpress.com/2020/03/19/kahnmans-word-wysiati/)\n\n\nMore: [Perspective](https://observablehq.com/@mariodelgadosr/perspective?collection=@mariodelgadosr/observable-for-active-retirement-communities)","pinned":false,"hidden":false,"mode":"md"},{"id":375,"value":"$30","pinned":false,"hidden":false,"mode":"md"},{"id":686,"value":"### The [Golden Ratio](https://en.wikipedia.org/wiki/Golden_ratio) in terms of Euler's Number with a [Continued Fraction](https://en.wikipedia.org/wiki/Golden_ratio#Continued_fraction_and_square_root)\n${tex.block`\n \\begin{aligned}\n e^{i\\pi} + 1 &= 0 \\\\\n e^{i\\pi} &= -1 \\\\\n -e^{i\\pi} &= 1 \\\\ \\\\\n \\phi &= \\frac{a + b}{a} \\\\\n &= \\frac{a}{b} \\approx 1.61803 \\\\\n\\phi &= 1 + \\cfrac{1}{1 + \\cfrac{1}{1 + \\cfrac{1}{1 + \\cfrac{1}{1 + \\cdots}}}} \\ \\ &= -e^{i\\pi} + \\cfrac{-e^{i\\pi}}{-e^{i\\pi} + \\cfrac{-e^{i\\pi}}{-e^{i\\pi} + \\cfrac{-e^{i\\pi}}{-e^{i\\pi} + \\cfrac{-e^{i\\pi}}{-e^{i\\pi} + \\cdots}}}}\n \\end{aligned}`}\n","pinned":false,"hidden":false,"mode":"md"},{"id":738,"value":"### The [Golden Ratio](https://en.wikipedia.org/wiki/Golden_ratio) in terms of Euler's Number with a [Continued Square Root](https://en.wikipedia.org/wiki/Golden_ratio#Continued_fraction_and_square_root): \n\n${tex.block`\n \\begin{aligned}\n e^{i\\pi} + 1 &= 0 \\\\\n e^{i\\pi} &= -1 \\\\\n -e^{i\\pi} &= 1 \\\\ \\\\\n \\phi &= \\frac{a + b}{a} \\\\\n &= \\frac{a}{b} \\approx 1.61803 \\\\\n\\phi &= \\sqrt{1+\\sqrt{1+\\sqrt{1+\\sqrt{1+\\cdots}}}}\n&= \\sqrt{-e^{i\\pi}+\\sqrt{-e^{i\\pi}+\\sqrt{-e^{i\\pi}+\\sqrt{-e^{i\\pi}+\\cdots}}}}\n \\end{aligned}`}","pinned":false,"hidden":false,"mode":"md"},{"id":162,"value":"## Notebook Objects","pinned":false,"hidden":false,"mode":"md"},{"id":171,"value":"dimensions = {\n const margin = { left: 29, right: 8, top: 13, bottom: 24 };\n const base = Math.min(width, 400);\n const innerWidth = base - margin.left - margin.right;\n const innerHeight = base - margin.top - margin.bottom;\n return { margin, base, innerWidth, innerHeight };\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":166,"value":"$31","pinned":false,"hidden":false,"mode":"ojs"},{"id":283,"value":"$32","pinned":false,"hidden":false,"mode":"ojs"},{"id":342,"value":"$33","pinned":false,"hidden":false,"mode":"ojs"},{"id":404,"value":"$34","pinned":false,"hidden":false,"mode":"ojs"},{"id":448,"value":"$35","pinned":false,"hidden":false,"mode":"ojs"},{"id":427,"value":"MathJax = require(\"https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-svg.js\").catch(\n () => window[\"MathJax\"]\n)","pinned":false,"hidden":false,"mode":"ojs"},{"id":207,"value":"html`\n\n`","pinned":false,"hidden":false,"mode":"ojs"},{"id":585,"value":"// https://observablehq.com/@tophtucker/interactive-plot-legend\nstyles = ({\n width: \"1em\",\n height: \"1em\",\n display: \"inline-block\",\n verticalAlign: \"middle\"\n})","pinned":false,"hidden":false,"mode":"ojs"},{"id":68,"value":"## Imports","pinned":false,"hidden":false,"mode":"md"},{"id":73,"value":"import { dodger } from \"@mbostock/pachinko-simulator\"","pinned":false,"hidden":false,"mode":"ojs"},{"id":5,"value":"import { toc } from \"@sugi2000/toc\"","pinned":false,"hidden":false,"mode":"ojs"},{"id":28,"value":"import { panchinko } from \"@d3/d3-random\"","pinned":false,"hidden":false,"mode":"ojs"},{"id":364,"value":"import {\n e_to_x_approximation,\n factorial\n} from \"@alekos92/eulers-number-approximation\"","pinned":false,"hidden":false,"mode":"ojs"},{"id":89,"value":"// https://github.com/vega/datalib\ndl = require(\"datalib\")","pinned":false,"hidden":false,"mode":"ojs"}],"files":"$W36","stdlib":"1","titling":"auto"}