31:T413,viewof data = html``32:T822,// Modified fminsearch function with bounds: Please note that this implementation is not optimized. // Adapted from: https://github.com/jonasalmeida/fminsearch // See also: https://observablehq.com/@christophe-yamahata/nonlinear-regression-in-javascript fminsearch = function(fun, Parm0, x, y, Opt) { if (!Opt) Opt = {}; if (!Opt.maxIter) Opt.maxIter = 1000; if (!Opt.step) { Opt.step = Parm0.map(p => p !== 0 ? p / 100 : 1); // Initial step 1/100 of initial value or 1 if zero } if (typeof Opt.display === 'undefined') Opt.display = true; if (!Opt.objFun) { Opt.objFun = function(y, yp) { return y.map((yi, i) => Math.pow((yi - yp[i]), 2)).reduce((a, b) => a + b); }; } if (!Opt.bounds) { Opt.bounds = Parm0.map(() => [-Infinity, Infinity]); // Default no bounds } const cloneVector = V => V.map(v => v); const applyBounds = (P, bounds) => P.map((p, i) => Math.max(bounds[i][0], Math.min(bounds[i][1], p))); // Initialize variables let P0 = cloneVector(Parm0); let P1 = cloneVector(Parm0); const n = P0.length; let step = cloneVector(Opt.step); // Ensure step is reinitialized const funParm = P => Opt.objFun(y, fun(x, P)); for (let i = 0; i < Opt.maxIter; i++) { let improved = false; for (let j = 0; j < n; j++) { P1 = cloneVector(P0); P1[j] += step[j]; P1 = applyBounds(P1, Opt.bounds); if (funParm(P1) < funParm(P0)) { step[j] *= 1.2; // Move faster in the same direction P0 = cloneVector(P1); improved = true; } else { step[j] *= -0.5; // Reverse and reduce step size } } // Add a convergence check to terminate early if no improvement if (!improved) { break; } if (Opt.display && i > (Opt.maxIter - 10)) { console.log(`Iteration ${i + 1}:`, `Objective Function Value: ${funParm(P0)}`, `Parameters:`, P0); } } return P0; }; 33:[] 2f:{"title":"Gaussian regression in JavaScript","cells":[{"id":0,"value":"# Gaussian regression in JavaScript","pinned":false,"hidden":false,"mode":"md"},{"id":543,"value":"

This notebook calculates Gaussian regression using user-introduced data.

","pinned":false,"hidden":false,"mode":"html"},{"id":506,"value":"

Given a dataset of x, y pairs, a Gaussian function ${tex`y = f(x)`} is evaluated:

","pinned":false,"hidden":false,"mode":"html"},{"id":511,"value":"tex.block`\nf(x) = A \\times \\exp\\left(-\\frac{1}{2} (\\frac{x - \\mu}{\\sigma})^2\\right)\n`","pinned":false,"hidden":false,"mode":"ojs"},{"id":524,"value":"where ${tex`A`}, ${tex`\\mu`}, and ${tex`\\sigma`} are the parameters to be determined.","pinned":false,"hidden":false,"mode":"html"},{"id":366,"value":"

① Provide x, y data

\nEnter or paste free-form x, y paired data in the following window:^(*)

* Enter data: Use a single space between x and y data, press enter to introduce a new point.\n
Paste data: Copy x,y data from a spreadsheet.\n

","pinned":false,"hidden":false,"mode":"html"},{"id":216,"value":"$31","pinned":false,"hidden":false,"mode":"ojs"},{"id":561,"value":"
\n

② Define the guess parameters

Before conducting Gaussian regression, define the initial guess parameters:

","pinned":false,"hidden":false,"mode":"html"},{"id":1128,"value":"viewof A_init = Inputs.number([-Infinity, Infinity], {step: 0.01, label: htl.html`Amplitude (${tex`\\textbf{A}`})`, value: 0.19})","pinned":false,"hidden":false,"mode":"ojs"},{"id":1139,"value":"viewof mu_init = Inputs.number([-Infinity, Infinity], {step: 0.5, label: htl.html`Mean (${tex`\\mu`})`, value: 1304.5})","pinned":false,"hidden":false,"mode":"ojs"},{"id":1150,"value":"viewof stdev_init = Inputs.number([0, Infinity], {step: 0.05, label: html`Standard
deviation (${tex`\\sigma`})`, value:12.4})","pinned":false,"hidden":false,"mode":"ojs"},{"id":576,"value":"
\n

③ Compare data with the fitting curve

After specifying the guess parameters, compare the input data with the fitting curve generated by the Gaussian regression algorithm. Adjust the initialization parameters as necessary to improve the fit. The cacluated Gaussian regression function is:

","pinned":false,"hidden":false,"mode":"html"},{"id":1499,"value":"{\nreturn tex`\n f(x) = ${p_fitting[0].toFixed(3)} \\times \\exp\\left(-\\frac{1}{2} (\\frac{x - ${p_fitting[1].toFixed(3)}}{${p_fitting[2].toFixed(3)}})^2\\right)\n`\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":1491,"value":"

For calculation purposes, copy the following line:

","pinned":false,"hidden":false,"mode":"html"},{"id":1476,"value":"{\n var gaussian_function = \"y = \" + p_fitting[0].toFixed(6) + \"*exp(-0.5*((x-\" +p_fitting[1].toFixed(6)+\")/\"+p_fitting[2].toFixed(6)+\")^2)\";\n \n return html`${gaussian_function}`;\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":115,"value":"Plot.plot({\n inset: 0,\n grid: true,\n height: 500,\n width: 700,\n x: {label: \"x → \",\n tickFormat: \"f\",\n },\n y: {label: \"↑ y \"},\n style: { \n fontSize: \"14px\",\n fontWeight: \"bold\",\n },\n marks: [\n Plot.ruleX([Math.min(...x)*0.999,Math.max(...x)*1.001]),\n Plot.ruleY([Math.min(...y,0),Math.max(...y)]),\n\n Math.max(...init_curve.map(d => d.y)) > Math.max(...y)?\n Plot.ruleY([Math.max(...init_curve.map(d => d.y))],{stroke: \"red\"}):null,\n\n Math.max(...fitting_curve.map(d => d.y)) > Math.max(...y)?\n Plot.ruleY([Math.max(...fitting_curve.map(d => d.y))],{stroke: \"black\"}):null,\n \n Plot.line(init_curve, {x: \"x\", y: \"y\", curve: \"catmull-rom\", stroke: color_guess, strokeDasharray: \"7,4\"}),\n Plot.line(fitting_curve, {x: \"x\", y: \"y\", curve: \"catmull-rom\", stroke: color_fitting}),\n Plot.dot(data_points, {x: \"x\", y: \"y\", curve: \"catmull-rom\", stroke: color_dot, fill: 'white', r: 4}), \n ]\n})","pinned":false,"hidden":false,"mode":"ojs"},{"id":990,"value":"viewof color_dot = Inputs.color({ label: \"Data points\", value: \"#4472C4\", datalist: [\"#4472C4\", \"#22BF8A\", \"#FF8000\", \"#C50707\", \"#000000\"] })","pinned":false,"hidden":false,"mode":"ojs"},{"id":144,"value":"viewof color_guess = Inputs.color({ label: \"Initial guess\", value: \"#FF8000\", datalist: [\"#FF8000\", \"#C50707\", \"#DDB4FF\", \"#9C9C9C\", \"#000000\"] })","pinned":false,"hidden":false,"mode":"ojs"},{"id":1298,"value":"viewof color_fitting = Inputs.color({ label: \"Best-fit\", value: \"#22BF8A\", datalist: [\"#176A00\", \"#00B4FF\", \"#AACC07\", \"#9C9C9C\", \"#000000\"] })","pinned":false,"hidden":false,"mode":"ojs"},{"id":65,"value":"---\n## Appendix","pinned":false,"hidden":false,"mode":"md"},{"id":212,"value":"### Data and parameters","pinned":false,"hidden":false,"mode":"md"},{"id":274,"value":"// x = [ 0, 10, 20,30, 40, 50, 60,70, 80, 90, 100, 110, 120, 130, 140, 150, 160, 170, 180, 190, 200];\nx = {\n var data_x = [];\n var data_ = data.replaceAll(\" \",\"\") // data manipulation to make this notebook compatible\n // with data copied from https://observablehq.com/@christophe-yamahata/on-line-tool-calque\n data_ = data_.replaceAll(\" \",\"\\t\").split(\"\\n\");\n const n = data_.length;\n for(var i=0; i < n; i++){\n if(!(isNaN(parseFloat(data_[i].split('\\t')[0])) || isNaN(parseFloat(data_[i].split('\\t')[1])))) {\n data_x.push(parseFloat(data_[i].split('\\t')[0])); \n }\n }\n return data_x;\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":52,"value":"// y = [ 0, 0.0003, 0.0015, 0.0053, 0.017, 0.045, 0.1, 0.21, 0.37, 0.6, 0.9, 1.2, 1.7, 2.1, 2.6, 3.2, 3.7, 4.3, 4.9, 5.5, 6.1 ]\ny = {\n var data_y = [];\n var data_ = data.replaceAll(\" \",\"\") // data manipulation to make this notebook compatible\n // with data copied from https://observablehq.com/@christophe-yamahata/on-line-tool-calque\n data_ = data_.replaceAll(\" \",\"\\t\").split(\"\\n\");\n const n = data_.length;\n for(var i=0; i < n; i++){\n if(!(isNaN(parseFloat(data_[i].split('\\t')[0])) || isNaN(parseFloat(data_[i].split('\\t')[1])))) {\n data_y.push(parseFloat(data_[i].split('\\t')[1])); \n }\n }\n return data_y;\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":153,"value":"data_points = {\n var data = [];\n const n = x.length;\n for(var i = 0; i < n; i++) \n {\n data.push({\"x\":x[i],\"y\":y[i]}); \n }\n return data;\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":1281,"value":"init_curve = create_fitting_curve(p_init)","pinned":false,"hidden":false,"mode":"ojs"},{"id":137,"value":"fitting_curve = create_fitting_curve(p_fitting)","pinned":false,"hidden":false,"mode":"ojs"},{"id":1180,"value":"mutable p_init = [A_init,mu_init,stdev_init]","pinned":false,"hidden":false,"mode":"ojs"},{"id":1257,"value":"// Calculate p_fitting with reactivity to all dependencies\np_fitting = {\n console.log(\"p_init:\", p_init);\n console.log(\"iterations:\", iterations);\n console.log(\"Opt:\", Opt);\n return fminsearch(model, p_init, x, y, Opt);\n};","pinned":false,"hidden":false,"mode":"ojs"},{"id":1253,"value":"// Since this notebook focuses on Gaussian regression, adjusting the number of iterations is not a needed feature. \nviewof iterations = Inputs.range([100, 2000], {value: 1000, step: 100, label: \"Iterations\"})","pinned":true,"hidden":false,"mode":"ojs"},{"id":1250,"value":"// Define bounds for the parameters\nOpt = {\n return {\n maxIter: iterations, // Optional\n bounds: [\n [-Infinity, Infinity], // Bounds for p[0]\n [-Infinity, Infinity], // Bounds for p[1]\n [0, Infinity], // Bounds for p[2]\n ],\n };\n};","pinned":true,"hidden":false,"mode":"ojs"},{"id":1122,"value":"// Gaussian model\nmodel = (x, p) => x.map(x_i => p[0] * Math.exp(-0.5 * Math.pow((x_i - p[1]) / p[2], 2)));","pinned":true,"hidden":false,"mode":"ojs"},{"id":72,"value":"### Javascript functions","pinned":false,"hidden":false,"mode":"md"},{"id":1274,"value":"create_fitting_curve = function(parameters){\n const data = [];\n const n = 1000;\n const x_min = Math.min(...x);\n const x_max = Math.max(...x);\n\n const x_ = d3.range(x_min, x_max, (x_max - x_min) / n).map(d => d.toFixed(4));\n const y_ = model(x_,parameters);\n\n for (let i = 0; i < x_.length; i++) {\n data.push({\n x: parseFloat(x_[i]),\n y: y_[i]\n });\n }\n return data;\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":1259,"value":"$32","pinned":false,"hidden":false,"mode":"ojs"}],"files":"$W33","stdlib":"1","titling":"auto"}