2f:T59f,## Overview This notebook estimates the [${tex`B_0`} and ${tex`B_1`}](https://en.wikipedia.org/wiki/Logistic_regression#Parameter_estimation) coefficients for the [sigmoid function](https://en.wikipedia.org/wiki/Sigmoid_function) that represents the derived [fit](https://en.wikipedia.org/wiki/Logistic_regression#Fit) for the example detailed in the Wikipedia article on [Logistic Regression](https://en.wikipedia.org/wiki/Logistic_regression). The regression is represented by the following relationship: ${tex.options({ displayMode: true })`Pr(y) = \frac{1}{1+e^{-(\beta_0 + \beta_1\cdot x)}}`} with a [location paramater](https://en.wikipedia.org/wiki/Location_parameter) (LP) at the midpoint of the curve where ${tex.block` \begin{aligned} Pr(y) &= \frac{1}{2} \space \\ for \space LP_x &= \frac{-B_0}{B_1} \end{aligned} `} The Wikipedia example discusses the estimation of ${tex`B_0`} and ${tex`B_1`} by ["numerical methods"](https://en.wikipedia.org/wiki/Logistic_regression#Parameter_estimation) and asserts that they "are found to be": ${tex.block` \begin{aligned} \beta_0 &\approx ${-4.1} \\ \beta_1 &\approx ${1.5} \\ LP_x &\approx 2.73 \end{aligned} `} This notebook's [optimization](https://www.tensorflow.org/js/guide/train_models#optimizer_loss_and_metric) utilizes TensorFlow.js numerical methods to implement the [cross entropy](https://en.wikipedia.org/wiki/Cross_entropy) detailed in the article. 30:T546,// https://talk.observablehq.com/t/input-form-template-option-modification-and-examples/7762/2 viewof inputs = { // https://observablehq.com/@tophtucker/interactive-plot-legend const styles = { width: "1em", height: "1em", display: "inline-block", verticalAlign: "middle" }; const selections = [ { value: "Logistic Regression", color: "blue" }, { value: "Indeterminate Fail/Pass Range", color: "lightgray" }, { value: "Test Scores", color: "purple" } ]; return Inputs.form( { plotType: Inputs.radio(["2D", "3D"], { value: "2D", format: (d) => `${d} Plot` }), estimateParameters: Inputs.button( "Visualize Estimated Parameters in 2D", { value: entropy ? 0 : 0 } ), selections: Inputs.checkbox( new Map( selections.map( (d, i) => [d.value, d.value] /*[ htl.html`

${ d.value }`, d.value ]*/ ) ), { value: selections.map((d) => d.value) } ) }, { template: (options) => htl.html`

${Object.values( options )}

` } ); }31:T912,fit = { if (inputs.estimateParameters == 0 || inputs.plotType == "3D") { yield []; return; } let B0, B1, predict, loss, loss_value, optimizer; // The parameters to optimize for // variable: https://js.tensorflow.org/api/latest/#variable // tidy: https://js.tensorflow.org/api/latest/#tidy // Feature weights and bias - the parameters we want to learn B0 = tf.tidy(() => tf.variable(tf.randomNormal([1], 0, 1.0), true)); B1 = tf.tidy(() => tf.variable(tf.randomNormal([1, 1], 0, 1.0), true)); const variables = [B0, B1]; // Tensor.js built-in optimizer using stochiastic gradient descent // train: https://js.tensorflow.org/api/latest/#Training-Optimizers optimizer = tf.train.sgd(1); // Tensor.js operation using the parameters that will be minimized // sigmoid: https://js.tensorflow.org/api/latest/#sigmoid // Logistic regression model: P(y=1 | x) = Bernoulli(y=1 | sigm(w.T * x + b)) // where * is matrix multiplication, // w, x ∈ ℝ^D, b ∈ ℝ, and y ∈ {0,1} predict = (x) => x.matMul(B1).add(B0).sigmoid().as1D(); // Also minimize the loss for the cross entropy (see: https://en.wikipedia.org/wiki/Logistic_regression#Fit) //https://js.tensorflow.org/api/latest/#metrics.binaryCrossentropy loss = (pred, truth) => tf.metrics.binaryCrossentropy(truth, pred); const estimatedParameters = []; // Optimization loop: https://deepai.org/machine-learning-glossary-and-terms/epoch for (let epoch = 1; epoch <= maxEpoch; epoch++) { for (let i = 0; i < xk.shape[0]; i++) { // https://js.tensorflow.org/api/latest/#tidy // https://www.tensorflow.org/js/guide/train_models#optimizer_loss_and_metric loss_value = tf.tidy(() => optimizer.minimize(() => loss(predict(xk), yk), true, variables) ); } // for i const estimatedParameter = { epoch, B0: B0.dataSync()[0], B1: B1.dataSync()[0], loss_value: loss_value.dataSync()[0] }; estimatedParameters.push(estimatedParameter); yield estimatedParameters; } // for epoch // https://observablehq.com/@observablehq/invalidation try { yield invalidation; } finally { //https://js.tensorflow.org/api/latest/#disposeVariables //tf.disposeVariables(); B0.dispose(); B1.dispose(); loss_value.dispose(); } }32:Tc20,// https://developer.mozilla.org/en-US/docs/Web/JavaScript/Reference/Global_Objects/JSON // Saved output of JSON.stringify(fit) to a cell. // Then used JSON.parse on the string to create a permanent dataset of a previously run estimation sequence. // Used for quick display of 3D plot fit3d = JSON.parse( `[{"epoch":1,"B0":-1.7202575206756592,"B1":0.6984502077102661,"loss_value":0.4706369638442993},{"epoch":2,"B0":-2.565659523010254,"B1":1.0011142492294312,"loss_value":0.42557820677757263},{"epoch":3,"B0":-3.050428867340088,"B1":1.1605863571166992,"loss_value":0.4117744266986847},{"epoch":4,"B0":-3.356598377227783,"B1":1.2623101472854614,"loss_value":0.4063202142715454},{"epoch":5,"B0":-3.5615057945251465,"B1":1.330794095993042,"loss_value":0.4038894772529602},{"epoch":6,"B0":-3.7035019397735596,"B1":1.3784326314926147,"loss_value":0.40272560715675354},{"epoch":7,"B0":-3.8041272163391113,"B1":1.4122779369354248,"loss_value":0.4021422564983368},{"epoch":8,"B0":-3.876514196395874,"B1":1.4366683959960938,"loss_value":0.4018407166004181},{"epoch":9,"B0":-3.929131269454956,"B1":1.454419732093811,"loss_value":0.4016815423965454},{"epoch":10,"B0":-3.967658519744873,"B1":1.4674293994903564,"loss_value":0.4015962481498718},{"epoch":11,"B0":-3.996018648147583,"B1":1.477012038230896,"loss_value":0.4015501141548157},{"epoch":12,"B0":-4.016973972320557,"B1":1.4840961694717407,"loss_value":0.4015248417854309},{"epoch":13,"B0":-4.032501697540283,"B1":1.489347219467163,"loss_value":0.401511013507843},{"epoch":14,"B0":-4.044031143188477,"B1":1.493247151374817,"loss_value":0.40150341391563416},{"epoch":15,"B0":-4.052604675292969,"B1":1.496147871017456,"loss_value":0.4014992117881775},{"epoch":16,"B0":-4.058987140655518,"B1":1.4983073472976685,"loss_value":0.40149688720703125},{"epoch":17,"B0":-4.063742160797119,"B1":1.4999165534973145,"loss_value":0.4014955461025238},{"epoch":18,"B0":-4.067287921905518,"B1":1.5011166334152222,"loss_value":0.4014948010444641},{"epoch":19,"B0":-4.069930553436279,"B1":1.5020110607147217,"loss_value":0.40149441361427307},{"epoch":20,"B0":-4.071903228759766,"B1":1.5026787519454956,"loss_value":0.40149417519569397},{"epoch":21,"B0":-4.073375701904297,"B1":1.5031770467758179,"loss_value":0.4014940857887268},{"epoch":22,"B0":-4.074474334716797,"B1":1.5035489797592163,"loss_value":0.4014940559864044},{"epoch":23,"B0":-4.0752949714660645,"B1":1.5038267374038696,"loss_value":0.40149396657943726},{"epoch":24,"B0":-4.0759077072143555,"B1":1.504034161567688,"loss_value":0.40149393677711487},{"epoch":25,"B0":-4.076364994049072,"B1":1.504188895225525,"loss_value":0.40149393677711487},{"epoch":26,"B0":-4.0767059326171875,"B1":1.5043044090270996,"loss_value":0.40149393677711487},{"epoch":27,"B0":-4.076961040496826,"B1":1.5043907165527344,"loss_value":0.4014939069747925},{"epoch":28,"B0":-4.077151775360107,"B1":1.5044554471969604,"loss_value":0.40149393677711487},{"epoch":29,"B0":-4.077293395996094,"B1":1.5045032501220703,"loss_value":0.40149393677711487},{"epoch":30,"B0":-4.077399730682373,"B1":1.5045392513275146,"loss_value":0.40149393677711487}]` )33:Te1d,function showPlot2D({ epoch, B0, B1, loss_value } = { B0: 1, B1: 0 }) { const { x, y } = sigmoid(B0, B1); const w = width * 0.6; const h = w * 0.5; // range of hours where both 0 and 1 studentPassedExam values can be found; no exclusive const failPassRange = [ dataForEstimate.findIndex((d) => d["Student Passed Exam"] == 1), dataForEstimate.findLastIndex((d) => d["Student Passed Exam"] == 0) ]; const selections = [ { value: "Logistic Regression", color: "blue" }, { value: "Indeterminate Fail/Pass Range", color: "lightgray" }, { value: "Test Scores", color: "purple" } ]; 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: inputs.estimateParameters == 0 ? `` : tex`\Large Epoch: ${epoch} \\ Loss = ${loss_value} \\ {B_0} \approx ${B0} \\ {B_1} \approx ${B1} \\ Location Parameter (LP) \approx ${-B0 / B1} \\ Pr(y) = \frac{1}{1+e^{-(${d3.format(".5")(B0)} + ${d3.format(".5")( B1 )}\cdot x)}}`, */ marks: [ Plot.ruleX([0]), // y axis Plot.ruleY([0]), // x axis //inputs.showFailPass inputs.selections.includes("Indeterminate Fail/Pass Range") ? Plot.line( [ [dataForEstimate[failPassRange[0]]["Number of Hours Studied"], 0], [dataForEstimate[failPassRange[1]]["Number of Hours Studied"], 0], [dataForEstimate[failPassRange[1]]["Number of Hours Studied"], 1], [dataForEstimate[failPassRange[0]]["Number of Hours Studied"], 1], [dataForEstimate[failPassRange[0]]["Number of Hours Studied"], 0] ], { fill: "lightgray" } ) : [], inputs.selections.includes("Test Scores") ? Plot.dot(dataForEstimate, { 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" }` }) : [], inputs.selections.includes("Logistic Regression") && inputs.estimateParameters != 0 ? Plot.line(x, { x: x, y: y, stroke: "blue" }) : [], // y = f(x) arrays plotted , inputs.selections.includes("Logistic Regression") && inputs.estimateParameters != 0 ? 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 , inputs.selections.includes("Logistic Regression") && inputs.estimateParameters != 0 ? Plot.text([[-B0 / B1, 0.5]], { text: () => `LP`, dy: -5, dx: 15 }) : [], inputs.selections.includes("Logistic Regression") && inputs.estimateParameters != 0 ? Plot.line( [ [0, 0.5], [-B0 / B1, 0.5] ], { stroke: "red", strokeDasharray: [5, 5] } ) : [], inputs.selections.includes("Logistic Regression") && inputs.estimateParameters != 0 ? Plot.line( [ [-B0 / B1, 0], [-B0 / B1, 0.5] ], { stroke: "red", strokeDasharray: [5, 5] } ) : [] ] }); return plot; }34:T711,// Displays previously generated data function* showPlot3D(fit) { const lastEpoch = fit[fit.length - 1].epoch; // Spinner yield html`

`; const traces = [ { x: hours.map((d) => lastEpoch), // Epoch y: hours, // Hours z: pass, // Probability mode: "markers", name: "Test Scores", text: pass.map((d) => (d == 1 ? "Pass" : "Fail")), // https://plotly.com/javascript/text-and-annotations/ hovertemplate: `Hours: %{y}
%{text}`, marker: { size: 5, color: "purple" }, type: "scatter3d" } ]; const epochColor = (epoch) => epoch == maxEpoch ? "rgb(0,0,255)" : d3.interpolateRgb( "lightgray", "blue" )( d3 .scaleLinear() .domain([1, maxEpoch - 1]) .range([0, 0.4])(epoch) ); fit.forEach((eParm) => { const { epoch, B0, B1, loss_value } = eParm; const { x, y } = sigmoid(B0, B1); traces.push({ x: x.map((d) => epoch), // Epoch y: x, // Hours z: y, // Probability mode: "lines", name: `Epoch ${epoch}`, hovertemplate: `Epoch: ${epoch}
B0: ${B0}
B1: ${B1}
`, line: { color: epochColor(epoch), width: 10 }, fill: "tonexty", type: "scatter3d" }); }); // forEach var layout = { scene: { xaxis: { title: "Epoch", tickvals: [0, 30], range: [0, 30] }, yaxis: { title: "Hours" }, zaxis: { title: "Probability" } } }; const div = DOM.element("div"); yield Plotly.newPlot(div, traces, layout); }35:Tfcc,fitTesting = { /* if (inputs.estimateParameters == 0) { yield {}; return; } let B0, B1, predict, loss, loss_value, optimizer; switch (entropy) { case "Binary Cross Entropy": // The parameters to optimize for // variable: https://js.tensorflow.org/api/latest/#variable // tidy: https://js.tensorflow.org/api/latest/#tidy // Feature weights and bias - the parameters we want to learn B0 = tf.tidy(() => tf.variable(tf.randomNormal([1], 0, 1.0), true)); B1 = tf.tidy(() => tf.variable(tf.randomNormal([1, 1], 0, 1.0), true)); const variables = [B0, B1]; // Tensor.js built-in optimizer using stochiastic gradient descent // train: https://js.tensorflow.org/api/latest/#Training-Optimizers optimizer = tf.train.sgd(1); // Tensor.js operation using the parameters that will be minimized // sigmoid: https://js.tensorflow.org/api/latest/#sigmoid // Logistic regression model: P(y=1 | x) = Bernoulli(y=1 | sigm(w.T * x + b)) // where * is matrix multiplication, // w, x ∈ ℝ^D, b ∈ ℝ, and y ∈ {0,1} predict = (x) => x.matMul(B1).add(B0).sigmoid().as1D(); // Also minimize the loss for the cross entropy (see: https://en.wikipedia.org/wiki/Logistic_regression#Fit) //https://js.tensorflow.org/api/latest/#metrics.binaryCrossentropy loss = (pred, truth) => tf.metrics.binaryCrossentropy(truth, pred); // Optimization loop: https://deepai.org/machine-learning-glossary-and-terms/epoch for (let epoch = 0; epoch < 30; epoch++) { for (let i = 0; i < xk.shape[0]; i++) { // https://js.tensorflow.org/api/latest/#tidy // https://www.tensorflow.org/js/guide/train_models#optimizer_loss_and_metric loss_value = tf.tidy(() => optimizer.minimize(() => loss(predict(xk), yk), true, variables) ); } // for i yield { epoch, B0: B0.dataSync()[0], B1: B1.dataSync()[0], loss_value: loss_value.dataSync()[0] }; } // for epoch break; case "Sigmoid Cross Entropy": // The parameters to optimize for // variable: https://js.tensorflow.org/api/latest/#variable // tidy: https://js.tensorflow.org/api/latest/#tidy B0 = tf.tidy(() => tf.variable(tf.scalar(0))); B1 = tf.tidy(() => tf.variable(tf.scalar(0))); // Tensor.js built-in optimizer using stochiastic gradient descent // train: https://js.tensorflow.org/api/latest/#Training-Optimizers optimizer = tf.train.sgd(0.1); // Tensor.js operation using the parameters that will be minimized // sigmoid: https://js.tensorflow.org/api/latest/#sigmoid predict = (x) => { return tf.sigmoid(B0.add(B1.mul(x))); }; // Also minimize the loss for the cross entropy (see: https://en.wikipedia.org/wiki/Logistic_regression#Fit) loss = (y_true, y_predicted) => { return tf.losses.sigmoidCrossEntropy(y_true, y_predicted); }; // Optimization loop: https://deepai.org/machine-learning-glossary-and-terms/epoch for (let epoch = 0; epoch < 105; epoch++) { for (let i = 0; i < xk.shape[0]; i++) { // https://js.tensorflow.org/api/latest/#tidy tf.tidy(() => optimizer.minimize(() => { // https://www.tensorflow.org/js/guide/train_models#optimizer_loss_and_metric const y_predicted = predict(xk.slice([i], [1])); const l = loss(yk.slice([i], [1]), y_predicted); loss_value = l.dataSync()[0]; return l; }) ); } // for i yield { epoch, B0: B0.dataSync()[0], B1: B1.dataSync()[0], loss_value }; } // for epoch } //switch // https://observablehq.com/@observablehq/invalidation try { yield invalidation; } finally { //https://js.tensorflow.org/api/latest/#disposeVariables //tf.disposeVariables(); B0.dispose(); B1.dispose(); }*/ }36:[] 2d:{"title":"Estimating Logistic Regression Parameters","cells":[{"id":0,"value":"# Estimating Logistic Regression Parameters\n \n\n ","pinned":false,"hidden":false,"mode":"md"},{"id":406,"value":"toc(\"h2,h3,h4, h5\")","pinned":false,"hidden":false,"mode":"ojs"},{"id":410,"value":"$2f","pinned":false,"hidden":false,"mode":"md"},{"id":421,"value":"## Data\nThe following is the data from the Wikipedia article's [example](https://en.wikipedia.org/wiki/Logistic_regression#Example):","pinned":false,"hidden":false,"mode":"md"},{"id":212,"value":"viewof dataForEstimate = {\n const dataLogistic = hours.map((h, i) => ({\n \"Number of Hours Studied\": h,\n \"Student Passed Exam\": pass[i]\n }));\n return Inputs.table(dataLogistic, { width: 400, value: dataLogistic });\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":418,"value":"## Instructions\n\nBy default the parameters will be estimated for all the [data in the example](https://en.wikipedia.org/wiki/Logistic_regression#Example), but you can select a subset of the data in the table above to see how the regression progresses (2D Plot) as a function of the specific input data selected.\n\nClick on the ***Estimate Parameters*** button to start the parameters optimization. ","pinned":false,"hidden":false,"mode":"md"},{"id":423,"value":"## Visualization of Logistic Regression Parameter Estimation","pinned":false,"hidden":false,"mode":"md"},{"id":810,"value":"$30","pinned":false,"hidden":false,"mode":"ojs"},{"id":714,"value":"showCaption(\n inputs.plotType == \"2D\" ? fit[fit.length - 1] : fit3d[fit3d.length - 1]\n)","pinned":false,"hidden":false,"mode":"ojs"},{"id":92,"value":"plot = inputs.plotType == \"2D\"\n ? showPlot2D(fit[fit.length - 1])\n : showPlot3D(fit3d)","pinned":false,"hidden":false,"mode":"ojs"},{"id":134,"value":"## Notebook Objects","pinned":false,"hidden":false,"mode":"md"},{"id":5,"value":"$31","pinned":false,"hidden":false,"mode":"ojs"},{"id":732,"value":"$32","pinned":false,"hidden":false,"mode":"ojs"},{"id":51,"value":"$33","pinned":false,"hidden":false,"mode":"ojs"},{"id":537,"value":"$34","pinned":false,"hidden":false,"mode":"ojs"},{"id":711,"value":"function showCaption(\n { epoch, B0, B1, loss_value } = {\n epoch: 0,\n B0: 1,\n B1: 0,\n loss_value: undefined\n }\n) {\n const caption = tex`Epoch: ${epoch} \\\\ \n Loss = ${loss_value} \\\\\n {B_0} \\approx ${B0} \\\\ {B_1} \\approx ${B1} \\\\\n Location Parameter (LP) \\approx ${-B0 / B1} \\\\\n Pr(y) = \\frac{1}{1+e^{-(${d3.format(\".5\")(B0)} + ${d3.format(\".5\")(\n B1\n )}\\cdot x)}}`;\n\n return inputs.plotType == \"3D\"\n ? caption\n : inputs.estimateParameters == 0\n ? md``\n : caption;\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":678,"value":"maxEpoch = 30","pinned":false,"hidden":false,"mode":"ojs"},{"id":61,"value":"// https://en.wikipedia.org/wiki/Logistic_regression#Problem\nhours = [\n 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,\n 4.0, 4.25, 4.5, 4.75, 5.0, 5.5\n]","pinned":false,"hidden":false,"mode":"ojs"},{"id":63,"value":"// https://en.wikipedia.org/wiki/Logistic_regression#Problem\npass = [0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 1, 1, 1]","pinned":false,"hidden":false,"mode":"ojs"},{"id":32,"value":"xk = tf.tensor(\n dataForEstimate.map((d) => d[\"Number of Hours Studied\"]),\n [dataForEstimate.length, 1]\n)","pinned":false,"hidden":false,"mode":"ojs"},{"id":34,"value":"yk = tf.tensor(dataForEstimate.map((d) => d[\"Student Passed Exam\"]))","pinned":false,"hidden":false,"mode":"ojs"},{"id":803,"value":"","pinned":false,"hidden":false,"mode":"html"},{"id":625,"value":"function sigmoid(B0, B1) {\n const x = d3\n .range(...[0, 6], 0.1) // https://github.com/d3/d3-array/blob/v3.2.3/README.md#range\n .concat(6); // https://github.com/d3/d3-array/issues/90\n const y = x.map((x) => 1 / (1 + Math.pow(Math.E, -(B0 + B1 * x))));\n return { x, y };\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":132,"value":"## Imports","pinned":false,"hidden":false,"mode":"md"},{"id":7,"value":"tf = require(\"@tensorflow/tfjs@4.0.0/dist/tf.min.js\")","pinned":false,"hidden":false,"mode":"ojs"},{"id":400,"value":"import { toc } from \"@sugi2000/toc\"","pinned":false,"hidden":false,"mode":"ojs"},{"id":430,"value":"Plotly = require(\"https://cdn.plot.ly/plotly-latest.min.js\")","pinned":false,"hidden":false,"mode":"ojs"},{"id":340,"value":"## Code Used While Testing TensorFlow Sigmoid Cross Entropy vs Binary Cross Entropy\nSee [this Stack Overflow entry](https://stackoverflow.com/questions/47034888/how-to-choose-cross-entropy-loss-in-tensorflow) for a discussion of the various cross entropy methods. ","pinned":false,"hidden":false,"mode":"md"},{"id":268,"value":"viewof entropy = Inputs.radio(\n [\"Binary Cross Entropy\", \"Sigmoid Cross Entropy\"],\n {\n label: \"Entropy Optimizer\",\n value: \"Binary Cross Entropy\"\n }\n)","pinned":false,"hidden":false,"mode":"ojs"},{"id":336,"value":"xkTest = entropy == \"Binary Cross Entropy\"\n ? tf.tensor(\n dataForEstimate.map((d) => d[\"Number of Hours Studied\"]),\n [dataForEstimate.length, 1]\n )\n : tf.tensor1d(dataForEstimate.map((d) => d[\"Number of Hours Studied\"]))","pinned":false,"hidden":false,"mode":"ojs"},{"id":338,"value":"ykOld = entropy == \"Binary Cross Entropy\"\n ? tf.tensor(dataForEstimate.map((d) => d[\"Student Passed Exam\"]))\n : tf.tensor1d(dataForEstimate.map((d) => d[\"Student Passed Exam\"]))","pinned":false,"hidden":false,"mode":"ojs"},{"id":333,"value":"$35","pinned":false,"hidden":false,"mode":"ojs"},{"id":171,"value":"## Related\n* [Logistic Regression in TensorFlow.js](https://observablehq.com/@sorig/logistic-regression-in-tensorflow)\n* [Logistic regression](https://observablehq.com/@sorig/logistic-regression)","pinned":false,"hidden":false,"mode":"md"}],"files":"$W36","stdlib":"1","titling":"auto"}