2a:null 2b:["$","title",null,{"children":"Time-varying regression model | Mattias Villani | Observable"}] 31:T67b,scatterplot = { const plt = Plot.plot({ title: md`Darker are closer to *t = *${obsnumber} (red dot)`, color: { //domain: [0,0.01/Math.log(n)], scheme: "blues", label: "Closeness in time to selected time period", legend: false, //domain: ["time series, xₜ", "filter mean", "95 % intervals"], //range: [colors[0], colors[2], "#F0F0F0"], //transform: (f) => Math.pow(f,1) }, marks: [ Plot.axisX({ label: null, fontSize: 10 }), Plot.axisY({ label: null, fontSize: 10 }), Plot.text(["x"], { frameAnchor: "bottom-right", fontSize: 16, dy: 30 }), Plot.text(["y"], { frameAnchor: "left", fontSize: 16, rotate: -0, dy: -185, dx: -20 }), Plot.ruleX([0]), Plot.ruleY([d3.min(simulatedDataLocal.map(d => d.y))]), Plot.line( [ {x: 0, y: simulatedDataLocal[obsnumber-1].alpha + 0*simulatedDataLocal[obsnumber-1].beta}, {x: 2, y: simulatedDataLocal[obsnumber-1].alpha + 2*simulatedDataLocal[obsnumber-1].beta} ], {x: "x", y: "y", stroke: colors[2], strokeWidth: 2} ), Plot.dot(simulatedDataLocal, {x: "x", y: "y", fill: "weight", strokeWidth: 2}), Plot.dot(simulatedDataLocal, {x: "x", y: "y", stroke: "gray", strokeWidth: 0.3, r: 3}), Plot.dot([simulatedDataLocal[obsnumber-1]], {x: "x", y: "y", fill: colors[2], stroke: colors[2], strokeWidth: 5}), Plot.tip(simulatedDataLocal, Plot.pointer({ x: "x", y: "y", title: (d) => ["time = ", d.time].join("") })) ] }) return plt }32:T443,paramevolplot = Plot.plot({ title: md`Evolution: β₀ = ${simulatedData[obsnumber-1].alpha.toFixed(3)} and β₁ = ${simulatedData[obsnumber-1].beta.toFixed(3)}`, y: { label: "coefficients" }, marks: [ Plot.ruleX([0]), Plot.ruleY([d3.min(simulatedData.map(d => d3.min([d.alpha,d.beta])))]), Plot.lineY(simulatedData, {x: "time", y: "alpha", stroke: colors[0]}), Plot.lineY(simulatedData, {x: "time", y: "beta", stroke: colors[1]}), Plot.dot([simulatedData[obsnumber-1]], {x: "time", y: "alpha", fill: colors[0], stroke: colors[0], r: 4}), Plot.dot([simulatedData[obsnumber-1]], {x: "time", y: "beta", fill: colors[1], stroke: colors[1], r: 4}), Plot.axisX({ label: null, fontSize: 10 }), Plot.axisY({ label: null, fontSize: 10 }), Plot.text(["time"], { frameAnchor: "bottom-right", fontSize: 16, dy: 30 }), Plot.text(["coeffs"], { frameAnchor: "left", fontSize: 16, rotate: -0, dy: -185, dx: -20 }), ] })33:[] 2e:{"id":"2eac471324f40b89","slug":"tvreg-simple","title":"Time-varying regression model","thumbnail":"8300e4311ffb0a800bdf755e1e5f22aa8b69cb7cc70c145039db42a3e0a4bc34","access_level":"public","create_time":"$D2026-05-28T19:26:49.537Z","update_time":"$D2026-05-29T12:39:38.103Z","view_time":null,"first_public_time":"$D2026-05-28T19:26:49.537Z","fork":{"id":"7e4661befb678e27","slug":"time-varying-regression-model","title":"Time-varying regression model","owner":{"id":"7592a7ab121f74a0","login":"mattiasvillani","name":"Mattias 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Villani","avatar_url":"https://avatars-next.observableusercontent.com/a440aaf39a3abefb86cbcc758fd7a5ab42555ae68d6825b52ec006f385a94d1d","tier":"pro_2024","role":"editor"}],"comments":[],"custom_thumbnail":false,"trashed":false,"subscribed":null,"forks":0,"collection":{"id":"738365a949244e81","slug":"talks","title":"Talks","owner":{"id":"7592a7ab121f74a0","login":"mattiasvillani","name":"Mattias Villani"}},"type":"notebook","body":{"title":"Time-varying regression model","cells":[{"id":0,"value":"# Time-varying regression model","pinned":false,"hidden":false,"mode":"md"},{"id":162,"value":"viewof input = Inputs.form({\n n: Inputs.range([10, 1000], {value: 100, step: 1, label: \"time series length\"}),\n sigmaeps: Inputs.range([0.1, 5], {value: 1, step: 0.1, label: tex`\\sigma_\\varepsilon`}),\n sigmaalpha: Inputs.range([Number.EPSILON, 2], {value: 1, step: 0.1, label: tex`\\sigma_{\\beta_0}`}),\n sigmabeta: Inputs.range([Number.EPSILON, 2], {value: 1, step: 0.1, label: tex`\\sigma_{\\beta_1}`})\n})","pinned":false,"hidden":false,"mode":"ojs"},{"id":1257,"value":"viewof obsnumber = Inputs.range([1, n], {label: \"regr line at time \", step: 1})","pinned":false,"hidden":false,"mode":"ojs"},{"id":781,"value":"viewof simulatebutton = Inputs.button(\"Simulate new data\")","pinned":false,"hidden":false,"mode":"ojs"},{"id":1581,"value":"bothplots = html`\n \n \n \n

\n ${paramevolplot}\n

\n ${scatterplot}\n

\n \n\n`","pinned":false,"hidden":false,"mode":"ojs"},{"id":1041,"value":"$31","pinned":false,"hidden":false,"mode":"ojs"},{"id":1367,"value":"$32","pinned":false,"hidden":false,"mode":"ojs"},{"id":1480,"value":"

","pinned":false,"hidden":false,"mode":"html"},{"id":1482,"value":"You can change how fast the color shading decays with the time distance to the chosen time period using this slider:","pinned":false,"hidden":false,"mode":"md"},{"id":1474,"value":"viewof sigma_bw = Inputs.range([.001, 0.3], {value: 0.075, label: \"Color sensitivity\", step: 0.001})","pinned":false,"hidden":false,"mode":"ojs"},{"id":13,"value":"
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\n\n---\n## 🙈 Appendix 🙈\nHere follow code and details that you probably do not want to read.","pinned":false,"hidden":false,"mode":"md"},{"id":1447,"value":"#### Plots","pinned":false,"hidden":false,"mode":"md"},{"id":15,"value":"#### Imports","pinned":false,"hidden":false,"mode":"md"},{"id":11,"value":"math = require(\"mathjs\");","pinned":false,"hidden":false,"mode":"ojs"},{"id":816,"value":"jStat = require(\"jstat\");","pinned":false,"hidden":false,"mode":"ojs"},{"id":1549,"value":"import {colors} from \"@mattiasvillani/statistics-tools\"","pinned":true,"hidden":false,"mode":"ojs"},{"id":96,"value":"#### Data","pinned":false,"hidden":false,"mode":"md"},{"id":1013,"value":"n = input.n","pinned":false,"hidden":false,"mode":"ojs"},{"id":94,"value":"#### Functions","pinned":false,"hidden":false,"mode":"md"},{"id":1004,"value":"function simLocalRegression(n, sigmaeps, sigmaalpha, sigmabeta) {\n\n simulatebutton\n\n const data = [];\n let alpha = 0;\n let beta = 0;\n const x = d3.range(n).map(d => d3.randomUniform(0, 2)())\n\n for (let t = 0; t < n; t++) {\n alpha = alpha + sigmaalpha*d3.randomNormal()();\n beta = beta + sigmabeta*d3.randomNormal()();\n data.push({time: t + 1, y: alpha + beta*x[t] + sigmaeps*d3.randomNormal()(), x: x[t], alpha: alpha, beta: beta});\n }\n\n return data;\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":519,"value":"simulatedData = simLocalRegression(n, input.sigmaeps, input.sigmaalpha, input.sigmabeta)","pinned":false,"hidden":false,"mode":"ojs"},{"id":1216,"value":"function weightedData(obsnumber, sigma){\n\n const localData = [];\n let denseval = d3.range(n).map(t => jStat.normal.pdf(t, obsnumber-1, n*sigma))\n const sumdenseval = d3.sum(denseval)\n const weights = denseval.map(d => d/sumdenseval)\n for (let t = 0; t < n; ++t) {\n localData[t] = simulatedData[t]\n localData[t].weight = weights[t]\n }\n return localData\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":1245,"value":"simulatedDataLocal = weightedData(obsnumber, sigma_bw)","pinned":false,"hidden":false,"mode":"ojs"},{"id":1537,"value":"simulatedDataLocal[80].weight","pinned":true,"hidden":false,"mode":"ojs"}],"files":"$W33","stdlib":"1","titling":"auto"}} 2f:"$2e:body"