29:null 2a:["$","title",null,{"children":"Spectral density - ARTFIMA(p,q,d,λ) | Mattias Villani | Observable"}] 2f:T491,viewof ϕvect = { switch (modelsettings.p) { case 0: return Inputs.form([Inputs.text({value: "No AR component", label: "", disabled: true})]) case 1: return Inputs.form([ Inputs.range([-1, 1], {value: 0.5, step: 0.01, label: tex`\phi_1`}) ]) break; case 2: return Inputs.form([ Inputs.range([-2, 2], {value: 0.5, step: 0.01, label: tex`\phi_1`}), Inputs.range([-1, 1], {value: 0.0, step: 0.01, label: tex`\phi_2`}) ]) break; case 3: return Inputs.form([ Inputs.range([-1, 1], {value: 0.5, step: 0.01, label: tex`\phi_1`}), Inputs.range([-1, 1], {value: 0.0, step: 0.01, label: tex`\phi_2`}), Inputs.range([-1, 1], {value: 0.0, step: 0.01, label: tex`\phi_3`}) ]) break; case 4: return Inputs.form([ Inputs.range([-1, 1], {value: 0.5, step: 0.01, label: tex`\phi_1`}), Inputs.range([-1, 1], {value: 0.0, step: 0.01, label: tex`\phi_2`}), Inputs.range([-1, 1], {value: 0.0, step: 0.01, label: tex`\phi_3`}), Inputs.range([-1, 1], {value: 0.0, step: 0.01, label: tex`\phi_4`}) ]) break; } }30:T4a5,viewof θvect = { switch (modelsettings.q) { case 0: return Inputs.form([Inputs.text({value: "No MA component", label: "", disabled: true})]) case 1: return Inputs.form([ Inputs.range([-1, 1], {value: 0.5, step: 0.01, label: tex`\theta_1`}) ]) break; case 2: return Inputs.form([ Inputs.range([-2, 2], {value: 0.5, step: 0.01, label: tex`\theta_1`}), Inputs.range([-1, 1], {value: 0.0, step: 0.01, label: tex`\theta_2`}) ]) break; case 3: return Inputs.form([ Inputs.range([-1, 1], {value: 0.5, step: 0.01, label: tex`\theta_1`}), Inputs.range([-1, 1], {value: 0.0, step: 0.01, label: tex`\theta_2`}), Inputs.range([-1, 1], {value: 0.0, step: 0.01, label: tex`\theta_3`}) ]) break; case 4: return Inputs.form([ Inputs.range([-1, 1], {value: 0.5, step: 0.01, label: tex`\theta_1`}), Inputs.range([-1, 1], {value: 0.0, step: 0.01, label: tex`\theta_2`}), Inputs.range([-1, 1], {value: 0.0, step: 0.01, label: tex`\theta_3`}), Inputs.range([-1, 1], {value: 0.0, step: 0.01, label: tex`\theta_4`}) ]) break; } }31:[] 2d:{"id":"dd93272ddaed9a2a","slug":"spectral-density-artfima-p-q-process","title":"Spectral density - ARTFIMA(p,q,d,λ)","thumbnail":"21110835d27043fdea2428fe51daa0def19d44d8e87242ab34416dfea2add7d9","access_level":"public","create_time":"$D2023-12-01T10:14:45.854Z","update_time":"$D2023-12-07T20:04:23.290Z","view_time":null,"first_public_time":"$D2023-12-01T10:38:06.337Z","has_unpublished_edits":null,"fork":{"id":"be5efbee215d5826","slug":"spectral-density-of-arma","title":"Spectral density - ARMA(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":"education_2026"},"version":348,"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":"Spectral density - ARTFIMA(p,q,d,λ)","cells":[{"id":0,"value":"# Spectral density - ARTFIMA(*p*,*q*,*d*,λ) ","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":194,"value":"The ARTFIMA(*p*,*q*,*d*,λ) process is of the form\n\n${tex`\n\\phi_p(L)\\Delta^{d,\\lambda}(Y_t-\\mu) = \\theta_q(L)\\varepsilon_t,\n`}\n\nwhere\n\n${tex`\\phi_p(L) = 1 - \\phi_1L - \\cdots - \\phi_pL^p`} is the AR lag polynomial,
\n${tex`\\theta_q(L) = 1 + \\theta_1L + \\cdots + \\theta_qL^q`}is the MA \nlag polynomial,
\n${tex`\\{ \\varepsilon_t \\}_{t=1}`} is an iid sequence from ${tex`N(0,\\sigma_\\varepsilon^2)`} and
\n${tex`\\Delta^{d,\\lambda}`} is the tempered fractional differenting operator defined by

\n\n${tex`\\Delta^{d,\\lambda} = (1- e^{-\\lambda}L)^d Y_t = \\sum_{j=0}^\\infty (-1)^j \\frac{\\Gamma(1+d)}{\\Gamma(1+d -j)}e^{-\\lambda j} Y_{t-j}`}\n\nThe spectral density of an ARTFIMA(*p*,*q*,*d*,λ) process is
\n\n${tex`\nf(\\omega)=\\frac{\\sigma^{2}}{2\\pi}\\left|\\frac{\\theta_q(e^{-i\\omega})}{\\phi_{p}(e^{-i\\omega})}\\right|^{2} \\text{ for } -\\pi < \\omega \\leq \\pi \n`}","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":247,"value":"#### Model settings","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":162,"value":"viewof modelsettings = Inputs.form({\n p: Inputs.range([0, 4], {value: 1, step: 1, label: \"number of AR lags\"}),\n q: Inputs.range([0, 4], {value: 1, step: 1, label: \"number of MA lags\"})\n})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":249,"value":"#### Parameter settings","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":244,"value":"viewof σ = Inputs.range([0.01, 5], {label: tex`\\text{error stdev, }\\sigma`, step: 0.01})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":116,"value":"$2f","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":121,"value":"$30","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":340,"value":"viewof diffsettings = Inputs.form({\n d: Inputs.range([-5, 5], {value: 0, step: 0.1, label: tex`\\text{differencing, }d`}),\n lambda: Inputs.range([0, 5], {value: 0, step: 0.01, label: tex`\\text{tempering, }\\lambda`}),\n})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":152,"value":"Plot.plot({\n y: {label: \"Spectral density f(ω)\"},\n x: {label: \"Angular frequency ω\"},\n title: md`Spectral density of an ARTFIMA(p=${modelsettings.p},q=${modelsettings.q}, d = ${d}, λ = ${lambda}) process`,\n marks: [\n Plot.ruleY([0]),\n Plot.ruleX([0]),\n Plot.lineY(specdensdata, {x: \"ω\", y: \"specdens\", stroke: \"steelblue\", strokeWidth: 2})\n ]\n})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":13,"value":"
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\n\n---\n## 🙈 Appendix 🙈\nHere follow code and details that you probably do not want to read.","pinned":false,"hidden":false,"mode":"md","author":"$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":"$undefined"},{"id":96,"value":"#### Data","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":289,"value":"lambda = diffsettings.lambda","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":292,"value":"d = diffsettings.d","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":254,"value":"ϕvect_value = {\n if (ϕvect[0] == \"No AR component\"){\n return [0]\n }else{\n return ϕvect\n }\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":256,"value":"θvect_value = {\n if (θvect[0] == \"No MA component\"){\n return [0]\n }else{\n return θvect\n }\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":99,"value":"specdensdata = {\n const ωgrid = d3.range(0,Math.PI, 0.01);\n return ωgrid.map(ω => ({ω: ω, specdens: SpecDens(ω, ϕvect_value, θvect_value, d, lambda, σ)}))\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":"$undefined"},{"id":5,"value":"function SpecDens(ω, ϕvect, θvect, d, lambda, σ){\n const eiw = math.exp(math.complex(0,-ω))\n const eiw_fracdiff = math.exp(math.complex(-lambda,-ω))\n var ARpoly = math.complex(1,0);\n var lag = 0;\n for (let ϕ of ϕvect){\n lag = lag + 1;\n ARpoly = math.add(ARpoly, math.multiply(math.complex(-ϕ,0),math.pow(eiw,lag)))\n } \n var MApoly = math.complex(1,0);\n var lag = 0;\n for (let θ of θvect){\n lag = lag + 1;\n MApoly = math.add(MApoly, math.multiply(math.complex(θ,0),math.pow(eiw,lag)))\n }\n return (σ**2/Math.PI)*(math.abs(MApoly)**2 / math.abs(ARpoly)**2)*math.pow(math.abs(math.subtract(math.complex(1,0), eiw_fracdiff)),-2*d)\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"}],"files":"$W31","stdlib":"1","titling":"auto"}} 2e:"$2d:body"