2c:null 2d:["$","title",null,{"children":"Embed Me | Adrian's Workspace | Observable"}] 32:T47d,// Step 3: Compute the Auto Mutual Information (AMI) function computeAMI(data, maxTau, bins = 30) { const ami = []; const n = data.length; // Bin the data const [min, max] = extent(data); const binWidth = (max - min) / bins; const bin = value => Math.floor((value - min) / binWidth); for (let tau = 1; tau <= maxTau; tau++) { const jointFreq = new Map(); const freqX = new Map(); const freqY = new Map(); for (let i = 0; i < n - tau; i++) { const x = bin(data[i]); const y = bin(data[i + tau]); const xyKey = `${x},${y}`; jointFreq.set(xyKey, (jointFreq.get(xyKey) || 0) + 1); freqX.set(x, (freqX.get(x) || 0) + 1); freqY.set(y, (freqY.get(y) || 0) + 1); } let hX = 0, hY = 0, hXY = 0; jointFreq.forEach((count, key) => { const pXY = count / (n - tau); hXY -= pXY * Math.log(pXY); }); freqX.forEach(count => { const pX = count / (n - tau); hX -= pX * Math.log(pX); }); freqY.forEach(count => { const pY = count / (n - tau); hY -= pY * Math.log(pY); }); ami.push(hX + hY - hXY); } return ami; }33:T44a,// Step 7: Compute and visualize the AMI function visualizeAMI() { const maxTau = tauSlider; const showMinima = showMinimaCheckbox; const ami = computeAMI(timeSeries, maxTau); const localMinima = findLocalMinima(ami); const marks = [ Plot.line(ami.map((value, index) => ({ x: index + 1, y: value })), { x: "x", y: "y" , // stroke:'lightgrey' },) ]; if (showMinima) { marks.push(Plot.dot(localMinima, { x: "x", y: "y", fill: controls.color , r:20, stroke:'lightgrey', strokeWidth:3})); } return Plot.plot({ marks, x: { label: "τ", type: "linear", domain: [1, maxTau], }, y: { label: "AMI", }, style:{ //color: "lightgrey", background:'none', }, height: viewport.height, width: viewport.width, }); }34:T40c,function computeFNN(data, maxDim, tau = tauSelector, Rtol = 10, Atol = 2) { const fnn = []; const N = data.length; for (let d = 1; d <= maxDim; d++) { let falseNeighbors = 0; let totalNeighbors = 0; for (let i = 0; i < N - (d + 1) * tau; i++) { // Create the embedded vectors const X1 = []; const X2 = []; for (let k = 0; k < d; k++) { X1.push(data[i + k * tau]); X2.push(data[i + k * tau + 1]); } const distance = Math.sqrt(X1.reduce((sum, val, index) => sum + Math.pow(val - data[i + (index + 1) * tau], 2), 0)); const R = Math.abs(data[i + d * tau] - data[i + (d + 1) * tau]); if (distance > 0) { if ((R / distance > Rtol) || (R / Math.abs(data[i]) > Atol)) { falseNeighbors++; } totalNeighbors++; } } console.log(`Dimension: ${d}, False Neighbors: ${falseNeighbors}, Total Neighbors: ${totalNeighbors}`); fnn.push(totalNeighbors > 0 ? falseNeighbors / totalNeighbors : 0); } return fnn; } 35:T548,function create3DPhaseSpacePlot(data) { const trace = { x: data.map(d => d[0]), y: data.map(d => d[1]), z: data.map(d => d[2]), mode: "lines", type: "scatter3d", line: { width: 6, color:controls.color } }; const layout = { //title: "Phase Space", autosize: true, backgroundColor:'black', width: viewport.width, height:viewport.height, margin: { l: 50, r: 50, b: 100, t: 100, pad: 4}, paper_bgcolor: 'none', // Background color of the paper (around the plot) plot_bgcolor: 'none', style:{background:'none'}, // scene: { // xaxis: { title: "X" ,backgroundcolor: "rgb(0, 0, 0)",gridcolor: "rgb(0, 0, 0)", // showbackground: false, // zerolinecolor: "rgb(0, 0, 0)",}, // yaxis: { title: "Y" ,backgroundcolor: "rgb(0, 0, 0)",gridcolor: "rgb(0, 0, 0)", // showbackground: false, // zerolinecolor: "rgb(0, 0, 0)",}, // zaxis: { title: "Z" ,backgroundcolor: "rgb(0, 0, 0)",gridcolor: "rgb(0, 0, 0)", // showbackground: false, // zerolinecolor: "rgb(0, 0, 0)",}, // }, backgroundColor: "none", color: 'black' }; const element = html`

`; Plotly.newPlot(element, [trace], layout,); return element; }36:T43e,function createTimeSeriesPlot(data, valueIndex, color, label) { const times = data.map((_, i) => i * controls.dt); const values = data.map(d => d[valueIndex]); const trace = { x: times, y: values, mode: "lines", line: { color: color }, name: label }; const layout = { title: `${label} Time Series`, xaxis: { title: "Time", autorange: true, showgrid: false, zeroline: false, showline: false, autotick: true, ticks: '', showticklabels: false }, yaxis: { title: label , autorange: true, showgrid: false, zeroline: false, showline: false, autotick: true, ticks: '', showticklabels: false }, autosize: true, //paper_bgcolor: '#171717', // Background color of the paper (around the plot) //plot_bgcolor: '#171717', width: viewport.width, height: 300, style:{ //background:'#171717', } }; const container = html`

`; Plotly.newPlot(container, [trace], layout); return container; }37:[] 30:{"id":"6b6c7e64607f7ab1","slug":null,"title":"Embed Me","thumbnail":"f74381595ebcd7a67f9a02543b7472d55ce52f24e2fbe98197205ec54baf5a22","access_level":"public","create_time":"$D2024-06-17T16:24:47.320Z","update_time":"$D2025-10-02T08:42:27.482Z","view_time":null,"first_public_time":"$D2024-06-18T14:50:21.115Z","has_unpublished_edits":null,"fork":false,"stars":0,"starred":false,"owner":{"type":"team","id":"4c66c6a854db2a43","login":"adrian7-ws","name":"Adrian's Workspace","avatar_url":"https://avatars-next.observableusercontent.com/aa397fb8648715d791246e7fc0033e2269f7b9ed21184cb2505babc79ad99dcd","tier":"starter_2024","bio":""},"version":766,"publish_version":null,"role":null,"authored":false,"authors":[{"type":"individual","id":"39835b898099e8aa","login":"adrianp","name":"Adrian Pelcaru","avatar_url":"https://avatars-next.observableusercontent.com/d39dfcd622b59da8f369378f3d1ca34e415aa3214d6603e139ed5ad071142b4c","tier":null,"bio":"I am a theoretical physics PhD student at the Technical University Dresden with a background in Biology and theoretical Biophysics. My main research focus is on the advancement of causal inference in nonlinear dynamical systems and its application."}],"members":[{"type":"individual","id":"39835b898099e8aa","login":"adrianp","name":"Adrian Pelcaru","avatar_url":"https://avatars-next.observableusercontent.com/d39dfcd622b59da8f369378f3d1ca34e415aa3214d6603e139ed5ad071142b4c","tier":null,"role":"editor"},{"type":"team","id":"4c66c6a854db2a43","login":"adrian7-ws","name":"Adrian's Workspace","avatar_url":"https://avatars-next.observableusercontent.com/aa397fb8648715d791246e7fc0033e2269f7b9ed21184cb2505babc79ad99dcd","tier":"starter_2024","role":"viewer"}],"comments":[],"custom_thumbnail":false,"trashed":false,"lapsed":null,"subscribed":null,"forks":0,"collection":null,"type":"notebook","body":{"titling":"auto","title":"Embed Me","cells":[{"id":0,"value":"# Embed Me","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":677,"value":"viewof controls = multiplexInputs(\n {\n sigma: Inputs.range([9, 11], {value:10, label: \"σ\", step: 0.1}),\n beta : Inputs.range([2, 4], {value:8/3, label: \"β\", step: 0.1}),\n rho : Inputs.range([20, 30], {value:28, label: \"ρ\", step: 0.1}),\n dt : Inputs.range([0.001, 0.01], {value: 0.01, label:\"dt\", step:0.0001}),\n numSteps : Inputs.range([100, 50000], {value: 1000, label:\"time\", step:1}),\n color : Inputs.color({label: \"Favorite color\", value: \"#ed719e\"})\n },\n { label: 'Controls',open:false, inline: false }\n)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":495,"value":"viewof plotX =createTimeSeriesPlot(lorenzData, 0, controls.color, \"X(t)\");","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":510,"value":"viewof plotY = createTimeSeriesPlot(lorenzData, 1, controls.color, \"Y(t)\");\n","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":514,"value":"viewof plotZ = createTimeSeriesPlot(lorenzData, 2,controls.color, \"Z(t)\");","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":432,"value":"html`

`;","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":371,"value":"//create3DPhaseSpacePlot(lorenzData);\nviewof plot3D = create3DPhaseSpacePlot(lorenzData);","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":5,"value":"// Step 1: Generate Lorenz attractor data\nfunction lorenzAttractor() {\n let x = 1, y = 1, z = 1; // Initial conditions\n const data = [];\n\n for (let i = 0; i < controls.numSteps; i++) {\n const dx = controls.sigma * (y - x);\n const dy = x * (controls.rho - z) - y;\n const dz = x * y - controls.beta * z;\n\n x += dx * controls.dt;\n y += dy * controls.dt;\n z += dz * controls.dt;\n\n data.push([x, y, z]);\n }\n\n return data;\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":19,"value":"// Step 2: Create the time series data\nlorenzData = lorenzAttractor();","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":23,"value":"viewof dimension = select({\n title: \"Select time series\",\n options: [0,1,2],\n value: 1 // Default to τ = 1\n});","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":29,"value":"timeSeries = lorenzData.map(d => d[dimension]);","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":26,"value":"$32","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":108,"value":"// Step 4: Function to find local minima in AMI\nfunction findLocalMinima(ami) {\n const localMinima = [];\n for (let i = 1; i < ami.length - 1; i++) {\n if (ami[i] < ami[i - 1] && ami[i] < ami[i + 1]) {\n localMinima.push({ x: i + 1, y: ami[i] });\n }\n }\n return localMinima;\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":571,"value":"## The Auto Mutual information\n\nThe Auto Mutual Information (AMI) is the nonlinear generalization of the autocorrelation function. It takes one time series, lets say ${tex`X`} shifts it by amount ${tex`\\tau`} and computes the mutual information. AMI is defined as:\n\n${tex.block`AMI(X, \\tau) = \\sum_{x_{t} \\in X} \\sum_{ x_{t+\\tau} \\in X} p(x_{t}, x_{t + \\tau}) \\log\\frac{p(x_{t}, x_{t+\\tau})}{p(x_{t}) p(x_{t+\\tau})},`}\n\n","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":32,"value":"// Step 4: Create a slider for selecting the maximum τ\nviewof tauSlider = slider({\n min: 1,\n max: 300,\n step: 1,\n value: 20,\n title: \"Maximum τ\"\n});","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":104,"value":"// Step 6: Create a checkbox for showing local minima\nviewof showMinimaCheckbox = checkbox({\n options: [{ label: \"Show Local Minima\", value: true }],\n value: [true]\n});","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":34,"value":"$33","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":82,"value":"// Custom extent function\nfunction extent(data) {\n let min = Infinity, max = -Infinity;\n for (const value of data) {\n if (value < min) min = value;\n if (value > max) max = value;\n }\n return [min, max];\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":69,"value":"vizAmi = visualizeAMI();","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":163,"value":"## The False Nearest Neighbor Algorithm","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":177,"value":"ami = computeAMI(timeSeries, tauSlider);","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":185,"value":"localMinima = findLocalMinima(ami);","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":189,"value":"// Step 4: Create a dropdown for selecting the tau value\nviewof tauSelector = select({\n title: \"Select τ\",\n options: localMinima.map(tau => tau.x),\n value: 1 // Default to τ = 1\n});","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":168,"value":"// Step 4: Create a slider for selecting the embedding dimension\nviewof dimSlider = slider({\n min: 1,\n max: 20,\n step: 1,\n value: 5,\n title: \"Maximum Embedding Dimension\"\n});\n","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":172,"value":"visualizeFNN();","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":211,"value":"$34","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":275,"value":"$35","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":346,"value":"// Step 5: Compute and visualize the FNN results\nfunction visualizeFNN() {\n const maxDim = dimSlider;\n const showMinima = showMinimaCheckbox;\n const fnn = computeFNN(timeSeries, maxDim);\n const localMinima = findLocalMinima(fnn);\n\n const marks = [\n Plot.line(fnn.map((value, index) => ({ x: index + 1, y: value })), { x: \"x\", y: \"y\" }, {width:5})\n ];\n \n if (showMinima) {\n marks.push(Plot.dot(localMinima, { x: \"x\", y: \"y\", fill: controls.color ,r:10, stroke:'darkgrey', strokeWidth:2,}));\n }\n\n return Plot.plot({\n marks,\n x: {\n label: \"E\",\n type: \"linear\",\n domain: [1, maxDim],\n },\n y: {\n label: \"FNN\",\n },\n height: viewport.height,\n width: viewport.width\n });\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":373,"value":"$36","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":689,"value":"viewof viewportSizing = Inputs.radio(['Display', 'Export'], {\n value: 'Display',\n label: 'Size viewport for:'\n})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":691,"value":"viewport = ({\n width: viewportSizing === 'Export' ? 640 : width,\n height: viewportSizing === 'Export' ? 480 : Math.max(400, width * 0.7),\n pixelRatio: viewportSizing === 'Export' ? 1 : devicePixelRatio\n})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":8,"value":"## Imports","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":10,"value":"import { inputsGroup } from '@bumbeishvili/input-groups'","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":53,"value":"import {ternarySlider} from \"@yurivish/ternary-slider\"","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":56,"value":"import {slider, checkbox, select} from \"@jashkenas/inputs\"","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":15,"value":"d3 = require(\"d3@6\")","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":39,"value":"import { Plot } from \"@observablehq/plot\";","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":263,"value":"Plotly = require('https://cdn.plot.ly/plotly-latest.min.js')","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":674,"value":"import { multiplexInputs } from '@rreusser/multiplex-inputs'","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"}],"files":"$W37","stdlib":"1"}} 31:"$30:body"