2a:null 2b:["$","title",null,{"children":"Piecewise constant model | Mattias Villani | Observable"}] 31:T584,viewof mu_vect = { switch (input.K) { case 1: return Inputs.form([Inputs.range([0, 12], {value: 2, step: 0.1, label: tex`\mu_1`})]) case 2: return Inputs.form([ Inputs.range([0, 10], {value: 2, step: 0.1, label: tex`\mu_1`}), Inputs.range([0, 10], {value: 4, step: 0.1, label: tex`\mu_2`}) ]) break; case 3: return Inputs.form([ Inputs.range([0, 10], {value: 2, step: 0.1, label: tex`\mu_1`}), Inputs.range([0, 10], {value: 4, step: 0.1, label: tex`\mu_2`}), Inputs.range([0, 10], {value: 6, step: 0.1, label: tex`\mu_3`}) ]) break; case 4: return Inputs.form([ Inputs.range([0, 10], {value: 2, step: 0.1, label: tex`\mu_1`}), Inputs.range([0, 10], {value: 4, step: 0.1, label: tex`\mu_2`}), Inputs.range([0, 10], {value: 6, step: 0.1, label: tex`\mu_3`}), Inputs.range([0, 10], {value: 8, step: 0.1, label: tex`\mu_4`}) ]) break; case 5: return Inputs.form([ Inputs.range([0, 10], {value: 2, step: 0.1, label: tex`\mu_1`}), Inputs.range([0, 10], {value: 4, step: 0.1, label: tex`\mu_2`}), Inputs.range([0, 10], {value: 6, step: 0.1, label: tex`\mu_3`}), Inputs.range([0, 10], {value: 8, step: 0.1, label: tex`\mu_4`}), Inputs.range([0, 10], {value: 10, step: 0.1, label: tex`\mu_5`}) ]) break; } }32:T6be,viewof thresholds2 = { if (!changethresholds) { return Inputs.form([Inputs.text({value: "Automatically selected thresholds", label: "", disabled: true})]) }else{ switch (input.K) { case 1: return Inputs.form([Inputs.text({value: "Automatically selected thresholds", label: "", disabled: true})]) case 2: return Inputs.form([ Inputs.range([1, n], {value: Math.floor(n/2), step: 1, label: tex`t_1`}), ]) break; case 3: return Inputs.form([ Inputs.range([1, Math.floor(n/2)], {value: Math.floor(n/3), step: 1, label: tex`t_1`}), Inputs.range([Math.floor(n/2) + 1, n], {value: Math.floor(2*n/3), step: 1, label: tex`t_2`}) ]) break; case 4: return Inputs.form([ Inputs.range([1, Math.floor(n/3)], {value: Math.floor(n/4), step: 1, label: tex`t_1`}), Inputs.range([Math.floor(n/3) + 1, Math.floor(2*n/3)], {value: Math.floor(2n/4), step: 1, label: tex`t_2`}), Inputs.range([Math.floor(2*n/3) + 1, n], {value: Math.floor(3*n/4), step: 1, label: tex`t_3`}) ]) break; case 5: return Inputs.form([ Inputs.range([1, Math.floor(n/4)], {value: Math.floor(n/5), step: 1, label: tex`t_1`}), Inputs.range([Math.floor(n/4) + 1, Math.floor(2*n/4)], {value: Math.floor(2*n/5), step: 1, label: tex`t_2`}), Inputs.range([Math.floor(2*n/4) + 1, Math.floor(3*n/4)], {value: Math.floor(3*n/5), step: 1, label: tex`t_3`}), Inputs.range([Math.floor(3*n/4) + 1, n], {value: Math.floor(4*n/5), step: 1, label: tex`t_4`}) ]) break; } } } 33:[] 2e:{"id":"94ede687704d9b03","slug":"piecewise-constant-model","title":"Piecewise constant model","thumbnail":"1b8dbbb12abd432080ca052430257ae3d636de24c228d4a8869854a2e9edf9a3","access_level":"public","create_time":"$D2023-12-16T10:56:27.509Z","update_time":"$D2025-01-08T07:33:21.092Z","view_time":null,"first_public_time":"$D2023-12-16T12:02:24.781Z","fork":{"id":"77fe0ce480074335","slug":"spectral-density-seasonal-arma-p-q-p-q-process","title":"Spectral density of a seasonal ARMA(p,q)×(P,Q) process","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":1104,"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":1,"collection":{"id":"b4f68ca0e73a1ae6","slug":"time-series","title":"Time Series","owner":{"id":"7592a7ab121f74a0","login":"mattiasvillani","name":"Mattias Villani"}},"type":"notebook","body":{"title":"Piecewise constant model","cells":[{"id":0,"value":"# Piecewise constant model","pinned":false,"hidden":false,"mode":"md"},{"id":194,"value":"The piece-wise constant model with ${tex`K`} segments is of the form\n\n${tex`\n\\begin{aligned} y_t &= \\mu_t + \\varepsilon_t, \\quad \\varepsilon_t \\sim N(0, \\sigma_\\varepsilon^2) \\\\ \\mu_t &= \\begin{cases} \\mu_1 & \\text{if $1 \\leq t \\leq t_1$} \\\\ \\mu_2 & \\text{if $t_1 < t \\leq t_2$} \\\\ \\vdots & \\vdots \\\\ \\mu_K & \\text{if $t_{K-1} < t \\leq T$}, \\\\ \\end{cases}\\end{aligned}\n`}\n\n","pinned":false,"hidden":false,"mode":"md"},{"id":463,"value":"#### Model settings","pinned":false,"hidden":false,"mode":"md"},{"id":162,"value":"viewof input = Inputs.form({\n K: Inputs.range([1, 5], {value: 3, step: 1, label: \"number of segments\"}),\n n: Inputs.range([10, 1000], {value: 200, step: 1, label: \"time series length\"})\n})","pinned":false,"hidden":false,"mode":"ojs"},{"id":465,"value":"#### Parameter settings","pinned":false,"hidden":false,"mode":"md"},{"id":282,"value":"$31","pinned":false,"hidden":false,"mode":"ojs"},{"id":1067,"value":"viewof sigmaeps = Inputs.range([0.1, 4], {value: 1, step: 0.1, label: tex`\\text{error stdev, }\\sigma_\\varepsilon`})","pinned":false,"hidden":false,"mode":"ojs"},{"id":1073,"value":"viewof changethresholds = Inputs.toggle({label: \"change thresholds\", value: false})","pinned":false,"hidden":false,"mode":"ojs"},{"id":1071,"value":"$32","pinned":false,"hidden":false,"mode":"ojs"},{"id":781,"value":"viewof simulatebutton = Inputs.button(\"Simulate new data\")","pinned":false,"hidden":false,"mode":"ojs"},{"id":1041,"value":"timeplot = Plot.plot({\n title: md`Piecewise constant model with *K*=${input.K} segments`,\n marks: [\n Plot.ruleX([0]),\n Plot.ruleY([d3.min(simulatedData.map(d => d.x))]),\n Plot.lineY(simulatedData, {x: \"time\", y: \"mu\", stroke: \"orange\", strokeWidth: 2}),\n Plot.lineY(simulatedData, {x: \"time\", y: \"x\", stroke: \"steelblue\", strokeWidth: 1, tip: false})\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":816,"value":"jStat = require(\"jstat\");","pinned":false,"hidden":false,"mode":"ojs"},{"id":96,"value":"#### Data","pinned":false,"hidden":false,"mode":"md"},{"id":644,"value":"K = input.K","pinned":false,"hidden":false,"mode":"ojs"},{"id":1013,"value":"n = input.n","pinned":false,"hidden":false,"mode":"ojs"},{"id":1010,"value":"thresholds = {\n if (!changethresholds) {\n return d3.range(n/K, n, n/K)\n }else{\n return thresholds2\n }\n \n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":94,"value":"#### Functions","pinned":false,"hidden":false,"mode":"md"},{"id":1004,"value":"function simPiecewiseConstant(n, sigmaeps, K) {\n\n simulatebutton\n \n const timeSeries = [];\n \n // Generate white noise sequence\n const ε = d3.range(n).map(d => sigmaeps*d3.randomNormal()())\n let mu;\n for (let t = 0; t < n; t++) {\n for (let k = 0; k < K; k++) {\n if (t <= thresholds[k]){\n mu = mu_vect[k];\n break;\n }else{\n mu = mu_vect[k];\n }\n }\n timeSeries[t] = {time: t + 1, x: mu + ε[t], mu: mu} ;\n }\n\n return timeSeries;\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":519,"value":"simulatedData = simPiecewiseConstant(n, sigmaeps, K)","pinned":false,"hidden":false,"mode":"ojs"}],"files":"$W33","stdlib":"1","titling":"auto"}} 2f:"$2e:body"