28:null 29:["$","title",null,{"children":"Local level model | Mattias Villani | Observable"}] 2f:[] 2c:{"id":"b8266f4d5273a4d2","slug":null,"title":"Local level model","thumbnail":"72a8d7c952d4eee979139b0abd3cb28cb2a7f0b2bca66b5efc4e6f8356b24e89","access_level":"public","create_time":"$D2026-05-28T20:07:43.858Z","update_time":"$D2026-05-28T20:07:43.858Z","view_time":null,"first_public_time":"$D2026-05-28T20:07:43.858Z","fork":{"id":"b4b92d0305b26b8a","slug":"local-level-model","title":"Local level model","owner":{"id":"7592a7ab121f74a0","login":"mattiasvillani","name":"Mattias Villani"}},"stars":0,"starred":false,"owner":{"type":"team","id":"7592a7ab121f74a0","login":"mattiasvillani","name":"Mattias Villani","avatar_url":"https://avatars-next.observableusercontent.com/a440aaf39a3abefb86cbcc758fd7a5ab42555ae68d6825b52ec006f385a94d1d","tier":"pro_2024"},"version":1199,"publish_version":null,"role":null,"authored":false,"authors":[{"type":"individual","id":"56499be8cf43f0ef","login":"mattiasvillani","name":"Mattias Villani","avatar_url":"https://avatars-next.observableusercontent.com/a440aaf39a3abefb86cbcc758fd7a5ab42555ae68d6825b52ec006f385a94d1d","tier":null}],"members":[{"type":"individual","id":"56499be8cf43f0ef","login":"mattiasvillani","name":"Mattias Villani","avatar_url":"https://avatars-next.observableusercontent.com/a440aaf39a3abefb86cbcc758fd7a5ab42555ae68d6825b52ec006f385a94d1d","tier":null,"role":"editor"},{"type":"team","id":"7592a7ab121f74a0","login":"mattiasvillani","name":"Mattias 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":null,"type":"notebook","body":{"title":"Local level model","cells":[{"id":0,"value":"# Local level model","pinned":false,"hidden":false,"mode":"md"},{"id":194,"value":"The local level model is\n\n${tex`\n\\begin{aligned}\n y_t &= \\mu_t + \\varepsilon_t, \\hspace{0.75cm} \\varepsilon_t \\sim N(0, \\sigma_\\varepsilon^2) \\\\\n \\mu_t &= \\mu_{t-1} + \\eta_t, \\quad \\eta_t \\sim N(0, \\sigma_\\eta^2)\n\\end{aligned}\n`}\n\n","pinned":false,"hidden":false,"mode":"md"},{"id":463,"value":"#### Settings","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([Number.EPSILON, 5], {value: 1, step: 0.1, label: tex`\\text{std error, }\\sigma_\\varepsilon`}),\n sigmaeta: Inputs.range([Number.EPSILON, 2], {value: 1, step: 0.1, label: tex`\\text{std innovation, }\\sigma_\\eta`})\n})","pinned":false,"hidden":false,"mode":"ojs"},{"id":1118,"value":"viewof plotselector = Inputs.checkbox([\"time series, xₜ\", \"local level, μₜ\"], {label: \"plot:\", value: [\"time series, xₜ\", \"local level, μₜ\"]})","pinned":false,"hidden":false,"mode":"ojs"},{"id":781,"value":"viewof simulatebutton = Inputs.button(\"Simulate new data\")","pinned":false,"hidden":false,"mode":"ojs"},{"id":1041,"value":"timeplot = Plot.plot({\n color: {\n legend: true,\n domain: [\"time series, xₜ\", \"local level, μₜ\"],\n range: [colors[0], colors[1]]\n },\n marks: [\n Plot.ruleX([0]),\n Plot.ruleY([d3.min(simulatedData.map(d => d.x))]),\n Plot.lineY(simulatedData, {\n filter: d => d.type == \"time series, xₜ\" && plotselector.includes(\"time series, xₜ\"), \n x: \"time\", y: \"x\", stroke: \"type\", strokeWidth: 2}),\n Plot.lineY(simulatedData, {\n filter: d => d.type == \"local level, μₜ\" && plotselector.includes(\"local level, μₜ\"), \n x: \"time\", y: \"x\", stroke: \"type\", strokeWidth: 2})\n ]\n})","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":15,"value":"#### Imports","pinned":false,"hidden":false,"mode":"md"},{"id":11,"value":"math = require(\"mathjs\");","pinned":false,"hidden":false,"mode":"ojs"},{"id":1185,"value":"import {simLocalLevel} from \"@mattiasvillani/statistics-tools\"","pinned":false,"hidden":false,"mode":"ojs"},{"id":816,"value":"jStat = require(\"jstat\");","pinned":false,"hidden":false,"mode":"ojs"},{"id":1196,"value":"import {colors} from \"@mattiasvillani/statistics-tools\"","pinned":false,"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":519,"value":"simulatedData = {\n simulatebutton\n return simLocalLevel(n, input.sigmaeps, input.sigmaeta)\n}","pinned":false,"hidden":false,"mode":"ojs"}],"files":"$W2f","stdlib":"1","titling":"auto"}} 2d:"$2c:body"