28:null 29:["$","title",null,{"children":"Gaussian Process Playground | Herb Susmann | Observable"}] 2e:T109f,md`# Gaussian Process Playground Let ${tex`\{Y_x\}`} be a stochastic process indexed by ${tex`x\in\mathbb{R}`}. ${tex`{Y_x}`} is called a _Gaussian Process_ if any finite set of realizations ${tex`y`} of the process at points ${tex`x`} are multivariate normally distributed with some mean vector ${tex`\bm{\mu}`} and covariance matrix ${tex`\bm{\Sigma}`}: ${tex.block` y \sim MVN\left(\bm{\mu}, \bm{\Sigma} \right) `} Where do these mean and covariances come from? We define a _mean function_ ${tex`m(x)`} which gives the mean of the multivariate normal distribution at any point ${tex`x`}, and a _covariance kernel function_ ${tex`k(x)`} which gives a matrix defining the covariance between every point in ${tex`x`}. Then we write ${tex.block` y \sim \mathcal{GP}\left(m(x), k(x) \right) `} to indicate that ${tex`y`} is a realization from a Gaussian Process defined by a mean function and covariance kernel. For this notebook we set the mean function to zero: ${tex`m(x) = 0`}. ## Prediction Given observations ${tex`y`} at positions ${tex`x`}, we may want to predict the value ${tex`\tilde{y}`} of the GP at a new set of positions ${tex`\tilde{x}`}. Fortunately, the conditional distribution of a multivariate normal distribution is also multivariate normal with closed forms for its mean and covariance matrix. Assuming the mean function of the GP is zero, the conditional distribution is given by ${tex.block` \tilde{y} \mid \tilde{x}, y, x \sim MVN(K^\top \Sigma^{-1} y, \Omega - K^\top \Sigma^{-1} K ) `} where ${tex`\Sigma`} is the covariance between observed points ${tex`x`}, ${tex`\Omega`} is the covariance between new points ${tex`\tilde{x}`}, and ${tex`K`} is the covariance between observed points ${tex`x`} and new points ${tex`\tilde{x}`}. ## Measurement Error We can add an assumption of Gaussian measurement error to the observed measurement error by adding an extra variance term to the variance of the observed data. Then the covariance between the observed points ${tex`x`} is given by ${tex.block` \Sigma + \delta I `} where ${tex`\Sigma`} is the covariance between every point in ${tex`x`}, given by the kernel function, ${tex`\delta`} is the variance of the measurement error, and ${tex`I`} is the identity matrix. ## Covariance Kernels Several popular covariance kernels. See also [The Kernel Cookbook](https://www.cs.toronto.edu/~duvenaud/cookbook/) by David Duvenaud. ### Constant ${tex.block` k(x_1, x_2) = \sigma^2 `} Perhaps the simplest and the least practically useful kernel. Interpolates a flat line between the observed data. ### Squared Exponential ${tex.block` k_{SE}(x_1, x_2) = \sigma^2 \exp\left(-\frac{(x_1 - x_2)^2}{2\ell^2}\right) `} where ${tex`\ell`} is a length-scale parameter. The squared exponential kernel is infinitely differentiable. Also known as the Radial Basis Function kernel. ### Rational Quadratic ${tex.block` k_{RQ}(x_1, x_2) = \sigma^2 \left(1 + \frac{(x_1 - x_2)^2}{2\alpha\ell^2} \right)^{-\alpha} `} where ${tex`\ell`} is a length-scale parameter and ${tex`\alpha`} is a smoothness parameter. The kernel approaches the squared exponential kernel as ${tex`\alpha \rightarrow \infty`}. ### Periodic ${tex.block` k_{Per}(x_1, x_2) = \sigma^2 \exp\left(-\frac{2\sin^2(\pi|x_1 - x_2|/p)}{\ell^2} \right) `} where ${tex`\ell`} is a length-scale parameter and ${tex`p`} is the period at which the process repeats. ### AR(1) A first-order autoregressive (AR) process can be written conditionally as ${tex.block` y_t \sim N(\rho \cdot y_{t,1}, \sigma^2) `} where ${tex`t`} is an integer. The parameter ${tex`\rho`} controls the level of autocorrelation. We call it this a first-order AR process because the value of ${tex`y_t`} only depends on the value of ${tex`y_{t-1}`}. When ${tex`\rho=1`} then we retrieve a random walk. The AR process as given above is only defined at integer positions. A GP with a similar concept, but defined at any real position, is given by the following covariance kernel: ${tex.block` k_{AR}(x_1, x_2) = \sigma^2 \rho^{|x_1 - x_2|} `} where ${tex`\rho`} is an autoregressive parameter. ### Linear ${tex.block` k_{Linear}(x_1, x_2) = \sigma^2 + (x_1 - c)(x_2 - c) `} where ${tex`c`} is an offset parameter. Equivalent to Bayesian linear regression. `2f:Tc40,{ const svg = d3.create("svg") .attr("viewBox", [0, 0, width, height]); svg.append("g") .call(xAxis); svg.append("g") .call(yAxis); svg.on("click", (d, i) => { if (d3.event.defaultPrevented) return; // dragged mutable points = points.concat([{ x: d3.event.offsetX, y: d3.event.offsetY }]); render(); update(); }); function dragstarted(d) { d3.select(this).attr("stroke", "black"); } function dragged(d) { d3.select(this).raise().attr("cx", d.x = d3.event.x).attr("cy", d.y = d3.event.y); svg.node().dispatchEvent(new CustomEvent("input")); render(); update(); } function dragended(d) { d3.select(this).attr("stroke", null); } const drag = d3.drag() .on("start", dragstarted) .on("drag", dragged) .on("end", dragended); const line = d3.line() .curve(d3.curveBasis) .x(d => xscale(d.x)) .y(d => yscale(d.mean)) const area = d3.area() .curve(d3.curveBasis) .x(d => xscale(d.x)) .y0(d => yscale(d.lower)) .y1(d => yscale(d.upper)); const area2 = d3.area() .curve(d3.curveBasis) .x(d => xscale(d.x)) .y0(d => yscale(d.lower2)) .y1(d => yscale(d.upper2)); function render() { svg.selectAll(".error") .data(points) .join("line") .attr("class", "error") .attr("x1", d => d.x) .attr("x2", d => d.x) .attr("y1", d => xscale(xscale.invert(d.y) - 1.96 * Math.sqrt(delta))) .attr("y2", d => xscale(xscale.invert(d.y) + 1.96 * Math.sqrt(delta))) .attr("stroke", "rgba(0, 0, 0, 0.5)"); svg.selectAll("circle") .data(points) .join("circle") .attr("cx", d => d.x) .attr("cy", d => d.y) .attr("r", radius) .attr("fill", (d, i) => d3.schemeCategory10[i % 10]) .attr("cursor", "move") .call(drag) .on("click", clicked); } function update() { const dist = conditional_distribution(points.map((d) => xscale.invert(d.x)), points.map((d) => yscale.invert(d.y)), xtilde, kernel); svg.selectAll('.mean') .data([dist]) .join('path') .attr('class', 'mean') .attr('d', d => line(d)) .attr('stroke', 'black') .attr('fill', 'transparent'); svg.selectAll('.envelope') .data([dist]) .join('path') .attr('class', 'envelope') .attr('d', d => area(d)) .attr('stroke', 'transparent') .attr('fill', 'rgba(0, 0, 0, 0.05)'); svg.selectAll('.envelope2') .data([dist]) .join('path') .attr('class', 'envelope2') .attr('d', d => area2(d)) .attr('stroke', 'transparent') .attr('fill', 'rgba(0, 0, 0, 0.1)'); } function clicked(d, i) { if (d3.event.defaultPrevented) return; // dragged d3.event.stopPropagation(); points.splice(i, 1); mutable points = points.slice(); update(); render(); } update(); render(); return svg.node(); }30:T66f,optimize_and_update_inputs = () => { const result = maximum_likelihood(kernel_choice, delta); // Update sigma (viewof sigma)[0].value = Math.exp(result.argument[0]); (viewof sigma).dispatchEvent(new CustomEvent("input")); if(kernel_choice == "Squared Exponential" || kernel_choice == "Rational Quadratic" || kernel_choice == "Periodic") { // Update ell inputs[0][0][0].value = Math.exp(result.argument[1]); inputs[0][0].dispatchEvent(new CustomEvent("input", { bubbles: true })); } if(kernel_choice == "Rational Quadratic") { inputs[1][0][0].value = Math.exp(result.argument[2]); inputs[1][0].dispatchEvent(new CustomEvent("input", { bubbles: true })); } if(kernel_choice == "Periodic") { inputs[1][0][0].value = Math.exp(result.argument[2]); inputs[1][0].dispatchEvent(new CustomEvent("input", { bubbles: true })); } if(kernel_choice == "AR(1)") { inputs[0][0][0].value = Math.exp(result.argument[1]) / (Math.exp(result.argument[1]) + 1); inputs[0][0].dispatchEvent(new CustomEvent("input", { bubbles: true })); } if(kernel_choice == "Linear" || kernel_choice == "Linear × Linear") { inputs[0][0][0].value = Math.exp(result.argument[1]) / (Math.exp(result.argument[1]) + 1) * 10; inputs[0][0].dispatchEvent(new CustomEvent("input", { bubbles: true })); } if(kernel_choice == "Squared Exponential × Periodic") { for(let i = 0; i < 3; i++) { inputs[i][0][0].value = Math.exp(result.argument[i + 1]); inputs[i][0].dispatchEvent(new CustomEvent("input", { bubbles: true })); } } }31:T4ee,neg_log_dens_optim = (arr, kernel_choice, delta) => { const sigma_value = Math.exp(arr[0]); const values = {}; if(kernel_choice == "Squared Exponential" || kernel_choice == "Rational Quadratic" || kernel_choice == "Periodic") { values.ell = Math.exp(arr[1]); } if(kernel_choice == "Rational Quadratic") { values.alpha = Math.exp(arr[2]); } if(kernel_choice == "Periodic") { values.p = Math.exp(arr[2]); } if(kernel_choice == "AR(1)") { values.rho = Math.exp(arr[1]) / (Math.exp(arr[1]) + 1); } if(kernel_choice == "Linear" || kernel_choice == "Linear × Linear") { values.c = Math.exp(arr[1]) / (Math.exp(arr[1]) + 1) * 10 } if(kernel_choice == "Squared Exponential × Periodic") { values.ell1 = Math.exp(arr[1]); values.ell2 = Math.exp(arr[2]); values.p = Math.exp(arr[3]); } const kernel = get_kernel(kernel_choice, sigma_value, values); const mean = [...Array(points.length)].map(() => 0); const x = points.map((d) => xscale.invert(d.x)); const y = points.map((d) => yscale.invert(d.y)); const Sigma = mathjs.add(apply_kernel(x, x, kernel), mathjs.multiply(mathjs.identity(x.length), delta))._data; const dens = multivariate_normal_log_density(mean, Sigma); return -dens(y); }32:T4fa,inputs = { const rows = []; if(kernel_choice == "Squared Exponential" || kernel_choice == "Rational Quadratic" || kernel_choice == "Periodic" || kernel_choice == "Matérn") rows.push([slider({ min: 0.01, max: 3, title: tex`\ell\text{: length-scale}`, description: "
" })]); if(kernel_choice == "Rational Quadratic") rows.push([slider({ min: 0.01, max: 5, title: tex`\alpha\text{: length-scale weighting}`, description: "
" })]); if(kernel_choice == "Squared Exponential × Periodic") { rows.push([slider({ min: 0.01, max: 3, title: tex`\ell_1\text{: squared exponential length-scale}`, description: "
" })]) rows.push([slider({ min: 0.01, max: 3, title: tex`\ell_2\text{: periodic length-scale}`, description: "
" })]) } if(kernel_choice == "Periodic" || kernel_choice == "Squared Exponential × Periodic") rows.push([slider({ min: 0.5, max: 10, title: tex`p\text{: period}`, description: "
" })]); if(kernel_choice == "AR(1)") rows.push([slider({ min: 0, max: 0.99, title: tex`\rho\text{: autocorrelation}`, description: "
" })]); if(kernel_choice == "Linear" || kernel_choice == "Linear × Linear") rows.push([slider({ min: 0, max: 10, title: tex`c\text{: offset}`, description: "
" })]); return rows; }33:[] 2c:{"id":"6e5496e311baac16","slug":"gaussian-processes","title":"Gaussian Process Playground","thumbnail":"56c756824bc6ab67c74eb9fa803fd34832c26fca1c26d4433b0126951e09c75c","access_level":"public","create_time":"$D2020-07-23T01:53:55.228Z","update_time":"$D2022-02-08T18:57:54.396Z","view_time":null,"first_public_time":"$D2020-07-23T18:36:50.917Z","has_unpublished_edits":null,"fork":false,"stars":22,"starred":false,"owner":{"type":"team","id":"a848bf10c248efb2","login":"herbps10","name":"Herb Susmann","avatar_url":"https://avatars-next.observableusercontent.com/3d8000aa2ff6515179603ce3451b0104272bd8b602dd87bb09f033573d6fa6a5","tier":"starter_2024"},"version":1216,"publish_version":null,"role":null,"authored":false,"authors":[{"type":"individual","id":"069af68ad7bd402a","login":"herbps10","name":"Herb Susmann","avatar_url":"https://avatars-next.observableusercontent.com/3d8000aa2ff6515179603ce3451b0104272bd8b602dd87bb09f033573d6fa6a5","tier":null}],"members":[{"type":"individual","id":"069af68ad7bd402a","login":"herbps10","name":"Herb Susmann","avatar_url":"https://avatars-next.observableusercontent.com/3d8000aa2ff6515179603ce3451b0104272bd8b602dd87bb09f033573d6fa6a5","tier":null,"role":"editor"}],"comments":[],"custom_thumbnail":false,"trashed":false,"lapsed":null,"subscribed":null,"forks":14,"collection":{"id":"9894182032aa1e5c","slug":"probability-distributions","title":"Statistics","owner":{"id":"a848bf10c248efb2","login":"herbps10","name":"Herb Susmann"}},"type":"notebook","body":{"title":"Gaussian Process Playground","cells":[{"id":0,"value":"$2e","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":457,"value":"md`\n# Playground\n_Click anywhere on the plot to add an observed data point. Click on a data point to remove it. Move the sliders to change the parameters of the covariance kernel function._\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":325,"value":"viewof kernel_choice = select({ title: \"Kernel\", options: [\"Squared Exponential\", \"Rational Quadratic\", \"Periodic\", \"AR(1)\", \"Linear\", \"Squared Exponential × Periodic\", \"Linear × Linear\", \"Constant\"] })","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":266,"value":"viewof delta = slider({ min: 0.0001, max: 0.1001, step: 0.001, value: 0.005, precision: 3, title: tex`\\delta\\text{: measurement error}` })","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":98,"value":"viewof sigma = slider({ min: 0.01, max: 2, value: 0.25, precision: 4, title: tex`\\sigma\\text{: covariance scale}` })","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":650,"value":"viewof values = inputsGroup(inputs, input_names)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":26,"value":"$2f","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":933,"value":"ml = {\n const button = html``;\n button.onclick = () => {\n optimize_and_update_inputs();\n }\n return button;\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1164,"value":"md`\nClick the button above to perform maximum likelihood parameter estimation. The data error ${tex`\\delta`} is treated as a fixed value, and all of the kernel parameters are estimated. The estimation is done via a Powell zero-order method from the [optimization-js](https://github.com/optimization-js/optimization-js) library. The method works best for simpler kernels like the Squared Exponential, and tends to bog down for other ones like the periodic kernel. The method finds only a local maximum, so in some situations pressing the button multiple times will yield different parameter estimates. The parameter values are limited to the range of the input sliders, so in some cases the maximum likelihood estimate for a parameter will be cut off. This tends to happen with the length-scale parameter, where sometimes the estimate will be very large when there is not enough information in the data to properly identify the parameter.\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":721,"value":"md`\n## Resources\n- David Duvenaud, _The Kernel Cookbook: Advice on Covariance Functions by David Duvenaud_ https://www.cs.toronto.edu/~duvenaud/cookbook/\n- Stan User's Guide, _Gaussian Process Regression_ https://mc-stan.org/docs/2_23/stan-users-guide/gaussian-process-regression.html\n- Scikit-learn Documentation, _Gaussian Processes_ https://scikit-learn.org/stable/modules/gaussian_process.html#kernels-for-gaussian-processes\n- Williams, C. K., & Rasmussen, C. E. (2006). _Gaussian Processes for Machine Learning_. Cambridge, MA: MIT press. http://www.gaussianprocess.org/gpml/\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":242,"value":"mutable points = [{ x: xscale(1), y: yscale(-0.6) }, { x: xscale(5), y: yscale(0.2) }, { x: xscale(8), y: yscale(-0.2) }]","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1104,"value":"$30","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":907,"value":"$31","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":998,"value":"maximum_likelihood = (kernel_choice, delta) => {\n let init = [];\n if(kernel_choice == \"Squared Exponential\") {\n init = [1, 1];\n }\n else if(kernel_choice == \"Rational Quadratic\") {\n init = [1, 1, 1];\n }\n else if(kernel_choice == \"Periodic\") {\n init = [1, 1, 1];\n }\n else if(kernel_choice == \"AR(1)\") {\n init = [1, 1];\n }\n else if(kernel_choice == \"Linear\") {\n init = [1, 1];\n }\n else if(kernel_choice == \"Linear × Linear\") {\n init = [1, 1];\n }\n else if(kernel_choice == \"Squared Exponential × Periodic\") {\n init = [1, 1, 1, 1];\n }\n else if(kernel_choice == \"Constant\") {\n init = [1];\n }\n \n const result = optim.minimize_Powell((arr) => neg_log_dens_optim(arr, kernel_choice, delta), init);\n \n return result;\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":27,"value":"radius = 7","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":129,"value":"xtilde = [...Array(201).keys()].map((i) => i / 200 * 10)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":644,"value":"$32","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":659,"value":"input_names = {\n const rows = [];\n if(kernel_choice == \"Squared Exponential\" || kernel_choice == \"Rational Quadratic\" || kernel_choice == \"Periodic\" || kernel_choice == \"Matérn\") rows.push([\"ell\"]);\n if(kernel_choice == \"Squared Exponential × Periodic\") {\n rows.push([\"ell1\"]);\n rows.push([\"ell2\"]);\n }\n if(kernel_choice == \"Rational Quadratic\") rows.push([\"alpha\"]);\n if(kernel_choice == \"Periodic\" || kernel_choice == \"Squared Exponential × Periodic\") rows.push([\"p\"]);\n if(kernel_choice == \"AR(1)\") rows.push([\"rho\"]);\n if(kernel_choice == \"Linear\" || kernel_choice == \"Linear × Linear\") rows.push([\"c\"]);\n\n return rows;\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":949,"value":"kernel = get_kernel(kernel_choice, sigma, values)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":142,"value":"get_kernel = (kernel_choice, sigma, values) => {\n if(kernel_choice == \"Rational Quadratic\") return rational_quadratic_kernel(sigma, values.ell, values.alpha);\n else if(kernel_choice == \"Squared Exponential\") return squared_exponential_kernel(sigma, values.ell);\n else if(kernel_choice == \"Periodic\") return periodic_kernel(sigma, values.ell, values.p);\n else if(kernel_choice == \"AR(1)\") return ar_kernel(values.rho, sigma);\n else if(kernel_choice == \"Linear\") return linear_kernel(sigma, values.c);\n else if(kernel_choice == \"Squared Exponential × Periodic\") return multiply_kernels(squared_exponential_kernel(sigma, values.ell1), periodic_kernel(1, values.ell2, values.p));\n else if(kernel_choice == \"Linear × Linear\") return multiply_kernels(linear_kernel(sigma, values.c), linear_kernel(1, values.c));\n else if(kernel_choice == \"Constant\") return constant_kernel(sigma);\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":200,"value":"conditional_distribution = (x, y, xtilde, kernel) => {\n if(x.length == 0) return ({ mean: [{ x: 0, y: 0 }] });\n const Sigma = mathjs.add(apply_kernel(x, x, kernel), mathjs.multiply(mathjs.identity(x.length), delta));\n const Sigmainv = mathjs.inv(Sigma);\n const Omega = apply_kernel(xtilde, xtilde, kernel);\n const K = apply_kernel(x, xtilde, kernel);\n \n const mean = mathjs.multiply(mathjs.multiply(mathjs.transpose(K), Sigmainv), y)\n ._data;\n \n const variance = mathjs.diag(mathjs.subtract(Omega, mathjs.multiply(mathjs.multiply(mathjs.transpose(K), Sigmainv), K)))\n ._data;\n \n const sd = mathjs.sqrt(variance);\n \n return mean.map((d, i) => ({ \n x: xtilde[i], \n mean: d, \n var: variance[i], \n sd: sd[i], \n lower: d - 1.96 * sd[i], \n upper: d + 1.96 * sd[i],\n lower2: d - 1.28 * sd[i],\n upper2: d + 1.28 * sd[i]\n }));\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":124,"value":"apply_kernel = (x1s, x2s, kernel) => {\n const covariance = [];\n for(let i = 0; i < x1s.length; i++) {\n covariance.push([]);\n for(let j = 0; j < x2s.length; j++) {\n covariance[i][j] = kernel(x1s[i], x2s[j]);\n }\n }\n return mathjs.matrix(covariance);\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1183,"value":"constant_kernel = (sigma) => {\n const sigmasq = Math.pow(sigma, 2);\n return (x1, x2) => sigmasq\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":114,"value":"squared_exponential_kernel = (sigma, ell) => {\n const sigmasq = Math.pow(sigma, 2);\n const ellsq = 2 * Math.pow(ell, 2);\n return (x1, x2) => sigmasq * Math.exp(-Math.pow(x1 - x2, 2) / ellsq)\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":315,"value":"rational_quadratic_kernel = (sigma, ell, alpha) => {\n const sigmasq = Math.pow(sigma, 2);\n const denom = 2 * alpha * Math.pow(ell, 2);\n return (x1, x2) => sigmasq * Math.pow(1 + Math.pow(x1 - x2, 2) / denom, -alpha);\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":473,"value":"periodic_kernel = (sigma, ell, p) => {\n const sigmasq = Math.pow(sigma, 2);\n const ellsq = Math.pow(ell, 2);\n return (x1, x2) => sigmasq * Math.exp(-2*Math.pow(Math.sin(Math.PI * Math.abs(x1 - x2) / p), 2) / ellsq);\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":797,"value":"multiply_kernels = (kernel1, kernel2) => (x1, x2) => kernel1(x1, x2) * kernel2(x1, x2)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":771,"value":"linear_kernel = (sigma, c) => {\n const sigmasq = Math.pow(sigma, 2);\n return (x1, x2) => sigmasq + (x1 - c) * (x2 - c);\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":358,"value":"ar_kernel = (rho, sigma) => {\n const sigmasq = Math.pow(sigma, 2);\n return (x1, x2) => sigmasq * Math.pow(rho, Math.abs(x1 - x2))\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":69,"value":"xscale = d3.scaleLinear()\n .domain([0, 10])\n .range([margin.left, width - margin.right])","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":71,"value":"yscale = d3.scaleLinear()\n .domain([-1, 1])\n .range([height - margin.bottom, margin.top])","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":73,"value":"xAxis = g => g\n .attr(\"transform\", `translate(0,${height - margin.bottom})`)\n .attr(\"pointer-events\", \"none\")\n .call(d3.axisBottom(xscale))","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":75,"value":"yAxis = g => g\n .attr(\"transform\", `translate(${margin.left},0)`)\n .attr(\"pointer-events\", \"none\")\n .call(d3.axisLeft(yscale))","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":30,"value":"height = width * 1/2","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":63,"value":"margin = ({top: 20, right: 30, bottom: 30, left: 40})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":4,"value":"import {select, slider, button, checkbox} from \"@jashkenas/inputs\"","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":641,"value":"import { inputsGroup } from \"@bumbeishvili/input-groups\"","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":6,"value":"d3 = require('d3@5')","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":8,"value":"mathjs = require('mathjs@7.6.0')","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":896,"value":"import { multivariate_normal_log_density } from \"@herbps10/distributions-and-their-gradients\"","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":925,"value":"optim = require(\"https://unpkg.com/optimization-js@latest/dist/optimization.js\")","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"}],"files":"$W33","stdlib":"1","titling":"auto"}} 2d:"$2c:body"