2d:null 2e:["$","title",null,{"children":"Poisson time series model | Mattias Villani | Observable"}] 33:[] 31:{"id":"86d4a1a50929b18f","slug":"poisson-time-series-model","title":"Poisson time series model","thumbnail":"fbc55b967f56a968b956e168e9c3bce1865797cad030b6fe3c045cbe9bff3e1c","access_level":"public","create_time":"$D2023-12-17T17:22:34.737Z","update_time":"$D2026-02-04T04:28:17.381Z","view_time":null,"first_public_time":"$D2023-12-17T18:38:46.392Z","has_unpublished_edits":null,"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":1242,"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"}],"comments":[],"custom_thumbnail":false,"trashed":false,"lapsed":null,"subscribed":null,"forks":1,"collection":{"id":"b4f68ca0e73a1ae6","slug":"time-series","title":"Time Series","owner":{"id":"7592a7ab121f74a0","login":"mattiasvillani","name":"Mattias Villani"}},"type":"notebook","body":{"title":"Poisson time series model","cells":[{"id":0,"value":"# Poisson time series model","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":194,"value":"A useful model for time series of counts ${tex`Y \\in \\{0,1,2,\\ldots \\}`} is a Poisson distribution with time-varying intensity ${tex`\\lambda_t = \\exp(z_t)`}, where ${tex`z_t`} is a continuous process with autocorrelation, most commonly a random walk or an autoregressive process of order 1:\n\n${tex`\n\\begin{aligned} \ny_t \\vert z_t &\\overset{\\mathrm{indep}}{\\sim} \\mathrm{Pois}(\\exp(z_t)) \\\\ \nz_t &= \\mu + \\phi(z_{t-1} - \\mu) + \\eta_t, \\qquad \\eta_t \\sim N(0, \\sigma^2_\\eta)\n\\end{aligned}\n`}\nNote that ${tex`\\lambda_t = \\exp(z_t)`} is guaranteed to be positive for all *t*, as required for the Poisson distribution.\n","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":463,"value":"#### Settings","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":162,"value":"viewof input = Inputs.form({\n n: Inputs.range([10, 1000], {value: 100, step: 1, label: \"time series length\"}),\n mu: Inputs.range([-5, 5], {value: 0, step: 0.1, label: tex`\\text{mean log intensity }\\mu`}),\n phi: Inputs.range([-0.99, 0.99], {value: 0.8, step: 0.01, label: tex`\\text{AR(1) parameter }\\phi`}),\n sigmaeta: Inputs.range([0.01, 2], {value: 1, step: 0.1, label: tex`\\text{std innovation, }\\sigma_\\eta`})\n})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1118,"value":"viewof plotselector = Inputs.checkbox([\"time series, yₜ\", \"intensity, λₜ\"], {label: \"plot:\", value: [\"time series, yₜ\", \"intensity, λₜ\"]})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":781,"value":"viewof simulatebutton = Inputs.button(\"Simulate new data\")","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1041,"value":"timeplot = Plot.plot({\n color: {\n legend: true,\n domain: [\"time series, yₜ\", \"intensity, λₜ\"],\n range: [colors[0], colors[1]]\n },\n marks: [\n Plot.ruleX([0]),\n Plot.ruleY([d3.min(simulatedData.map(d => d.y))]),\n\n Plot.dot(simulatedData, {\n filter: d => d.type == \"time series, yₜ\" && plotselector.includes(\"time series, yₜ\"), \n x: \"time\", y: \"y\", fill: \"type\", strokeWidth: 2}),\n Plot.lineY(simulatedData, {\n filter: d => d.type == \"intensity, λₜ\" && plotselector.includes(\"intensity, λₜ\"), \n x: \"time\", y: \"y\", stroke: \"type\", strokeWidth: 2})\n ]\n})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":13,"value":"
\n
\n
\n
\n
\n
\n
\n
\n
\n
\n
\n
\n
\n
\n
\n
\n\n---\n## 🙈 Appendix 🙈\nHere follow code and details that you probably do not want to read.","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":15,"value":"#### Imports","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":11,"value":"math = require(\"mathjs\");","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":816,"value":"jStat = require(\"jstat\");","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1230,"value":"import {colors} from \"@mattiasvillani/statistics-tools\"","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":96,"value":"#### Data","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":1013,"value":"n = input.n","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":94,"value":"#### Functions","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":1004,"value":"function simPoisTimeSeries(n, mu, phi, sigmaeta) {\n\n simulatebutton\n\n const data = [];\n let z = mu;\n for (let t = 0; t < n; t++) {\n z = mu + phi*(z - mu) + sigmaeta*d3.randomNormal()();\n data.push({time: t + 1, y: Math.exp(z), type: \"intensity, λₜ\"});\n data.push({time: t + 1, y: jStat.poisson.sample(Math.exp(z)), type: \"time series, yₜ\"});\n }\n\n return data;\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":519,"value":"simulatedData = simPoisTimeSeries(n, input.mu, input.phi, input.sigmaeta)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"}],"files":"$W33","stdlib":"1","titling":"auto"}} 32:"$31:body"