2b:null 2c:["$","title",null,{"children":"Autoregressive processes - simulation and autocorrelation | Mattias Villani | Observable"}] 31:T421,viewof phi = { switch (siminputs[0]) { case 1: return Inputs.form([ Inputs.range([-2, 2], {value: 0.7, step: 0.01, label: tex`\phi_1`}) ]) break; case 2: return Inputs.form([ Inputs.range([-2, 2], {value: 0.7, step: 0.01, label: tex`\phi_1`}), Inputs.range([-2, 2], {value: 0.0, step: 0.01, label: tex`\phi_2`}) ]) break; case 3: return Inputs.form([ Inputs.range([-2, 2], {value: 0.7, step: 0.01, label: tex`\phi_1`}), Inputs.range([-2, 2], {value: 0.0, step: 0.01, label: tex`\phi_2`}), Inputs.range([-2, 2], {value: 0.0, step: 0.01, label: tex`\phi_3`}) ]) break; case 4: return Inputs.form([ Inputs.range([-2, 2], {value: 0.7, step: 0.01, label: tex`\phi_1`}), Inputs.range([-2, 2], {value: 0.0, step: 0.01, label: tex`\phi_2`}), Inputs.range([-2, 2], {value: 0.0, step: 0.01, label: tex`\phi_3`}), Inputs.range([-2, 2], {value: 0.0, step: 0.01, label: tex`\phi_4`}) ]) break; } } 32:[] 2f:{"id":"47feedd1709f5ceb","slug":"ar-simulation-and-autocorrelation","title":"Autoregressive processes - simulation and autocorrelation","thumbnail":"1fca22379e5d1f134eac0ed418321372330bab573a98d5a15df6dcdd4515726c","access_level":"public","create_time":"$D2024-05-21T11:49:26.482Z","update_time":"$D2025-05-13T19:26:31.824Z","view_time":null,"first_public_time":"$D2024-05-21T11:49:26.482Z","fork":{"id":"112b4245ebf148d9","slug":"ar-processes","title":"Autoregressive processes - simulation and priors","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":850,"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,"subscribed":null,"forks":0,"collection":{"id":"b4f68ca0e73a1ae6","slug":"time-series","title":"Time Series","owner":{"id":"7592a7ab121f74a0","login":"mattiasvillani","name":"Mattias Villani"}},"type":"notebook","body":{"title":"Autoregressive processes - simulation and autocorrelation","cells":[{"id":38,"value":"# Autoregressive processes - simulation and autocorrelation","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":406,"value":"### Autoregressive process of order *p*\nLet the model be an autoregressive process of order *p* in so called steady-state form where ${tex`\\mu`} is the unconditional mean of the process:","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":408,"value":"model_text = tex`\n\\begin{aligned}\nX_t = \\mu + \\phi_1(X_{t-1}-\\mu)+ \\ldots + \\phi_p(X_{t-p}-\\mu) + \\varepsilon_t, \\, \\text{ where } \\, \\varepsilon_t \\overset{\\mathrm{iid}}{\\sim}N(0,\\sigma^2)\n\\end{aligned}\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":668,"value":"Here is a simulator for the AR(p) process:","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":723,"value":"viewof T = Inputs.range([5, 1000], {value: 50, step: 1, label: tex`\\text{length series, }T`})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":666,"value":"viewof siminputs = Inputs.form([\n Inputs.range([1, 4], {value: 2, step: 1, label: tex`\\text{number of lags, }p`}),\n Inputs.range([-3, 3], {value: 0.0, step: 0.1, label: tex`\\text{mean, }\\mu`}),\n Inputs.range([0, 2], {value: 0.5, step: 0.1, label: tex`\\text{error stdev, }\\sigma`})\n ])","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":672,"value":"$31","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":846,"value":"viewof simulate = Inputs.button(\"Simulate!\")","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":840,"value":"\n \n \n \n

\n ${plt_timeseries}\n

\n \n

\n ${pltacf}\n

\n \n","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":109,"value":"
\n
\n
\n
\n
\n
\n
\n
\n
\n
\n
\n
\n
\n
\n
\n
\n\n---\n## 🙈 Appendix 🙈\n\nHere is only code and details that you probably don't want to read.","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":792,"value":"#### Plots","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":829,"value":"plt_timeseries = Plot.plot({\n title: \"Time series\",\n x: {\n label: \"time, t\",\n axis: true, \n domain: [0,T]\n },\n y: {\n label: \"X(t)\",\n axis: true,\n },\n marks: [\n Plot.ruleY([d3.min(timeseries.map(d => d.x))]),\n Plot.ruleX([0]),\n Plot.lineY(timeseries, {filter: d => d.time < T, x: \"time\", y: \"x\", stroke: \"steelblue\", strokeWidth: 2})\n ]\n})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":835,"value":"pltacf = Plot.plot({\n title: \"Estimated autocorrelation function\",\n //title: md`Estimated ACF of AR(${p}) process`,\n marks: [\n //Plot.axisX({ticks: d3.range(n_acf_lags)}),\n Plot.ruleY([0]),\n Plot.ruleX([0]),\n //Plot.lineY(acf.map((d,i) => ({lag: i, acf: d})), {x: \"lag\", y: \"acf\", stroke: \"green\"}),\n Plot.dot(acf.map((d,i) => ({lag: i, acf: d})), {filter: d => d.lag > 0, x: \"lag\", y: \"acf\", fill: \"green\", stroke: \"green\"}),\n Plot.ruleX(acf.map((d,i) => ({lag: i, acf: d})), {filter: d => d.lag > 0, x: \"lag\", y: \"acf\", stroke: \"green\", strokeWidth: 2}),\n ]\n})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":221,"value":"#### Imports","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":26,"value":"jstat = require('jstat')","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":747,"value":"mathjs = require('mathjs')","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":778,"value":"import {computeACF, simulateAR} from \"@mattiasvillani/statistics-tools\"","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":620,"value":"### Functions","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":223,"value":"#### Data","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":547,"value":"p = siminputs[0]","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":549,"value":"mu = siminputs[1]","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":684,"value":"sigma = siminputs[2]","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":662,"value":"timeseries = {\n simulate;\n return simulateAR(1000, mu, phi, sigma)\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":775,"value":"n_acf_lags = Math.max(5*p,10)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":773,"value":"acf = computeACF(timeseries.map(d => d.x), n_acf_lags)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":229,"value":"#### Style","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":202,"value":"","pinned":false,"hidden":false,"mode":"html","author":"$undefined","summary":"$undefined"},{"id":58,"value":"viewof md_colour = Inputs.color({label: \"Highlight color cells\", value: \"#F5F5DC\"})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"}],"files":"$W32","stdlib":"1","titling":"auto"}} 30:"$2f:body"