29:null 2a:["$","title",null,{"children":"ARTFIMA - Tempered fractional differencing | Mattias Villani | Observable"}] 2f:[] 2d:{"id":"f53e7faa8779708f","slug":"artfima-tempered-fractional-differencing","title":"ARTFIMA - Tempered fractional differencing","thumbnail":"15fffb22b64735a0dc9833b0c075a2a4c8e49d8beb54d1b276a575fd00995440","access_level":"public","create_time":"$D2023-06-13T16:52:08.204Z","update_time":"$D2023-12-07T09:36:25.267Z","view_time":null,"first_public_time":"$D2023-06-13T16:52:08.204Z","has_unpublished_edits":null,"fork":false,"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":99,"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":0,"collection":{"id":"b4f68ca0e73a1ae6","slug":"time-series","title":"Time Series","owner":{"id":"7592a7ab121f74a0","login":"mattiasvillani","name":"Mattias Villani"}},"type":"notebook","body":{"title":"ARTFIMA - Tempered fractional differencing","cells":[{"id":0,"value":"# ARTFIMA - Tempered fractional differencing","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":86,"value":"The ${tex`\\mathrm{ARTFIMA}(p,d,q)`} process is of the form\n\n${tex`\\phi_{p}(L)(1-L)^{d}x_{t}=\\theta_{q}(L)\\varepsilon_{t}`}\n\nwhere ${tex`\\phi_{p}(L) = 1 - \\phi_1 L - \\ldots - \\phi_p L^p`} and ${tex`\\theta_{p}(L) = 1 + \\theta_1 L + \\ldots + \\theta_q L^q`} is the AR and MA polynomials in the lag operator *L* such that ${tex`L^k x_t = x_{t-k}`}.","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":53,"value":"The tempered fractional differencing operator with differencing order ${tex`d \\in \\mathbb{R}`} and tempering ${tex`\\lambda>0`} is of the form:\n\n${tex`(1-e^{-\\lambda})^{d}x_{t}\\overset{\\mathrm{def}}{=}\\sum_{j=0}^{\\infty}\\omega_j x_{t-j}`}\n\nwhere the weights are given by\n\n${tex`\\omega_j = (-1)^{j}\\frac{\\Gamma(1+d)}{\\Gamma(1+d-j)j!}e^{-\\lambda j}`}\n","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":17,"value":"viewof params = Inputs.form([\n Inputs.range([-5, 5], {\n value: 0,\n step: 0.1,\n label: tex`\\text{differencing, }d`\n }),\n Inputs.range([0, 2], {\n value: 0,\n step: 0.01,\n label: tex`\\text{tempering, }\\lambda`\n }),\n Inputs.range([1, 30], {\n value: 10,\n step: 1,\n label: \"number of lags:\"\n })\n])","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":45,"value":"Plot.plot({\n y: {label: \"weight, ω\"},\n x: {label: \"lag\"},\n marks: [\n Plot.ruleY([0]),\n Plot.lineY(data, {x: \"lag\", y: \"weight\", stroke: \"orange\", strokeWidth: 3}),\n Plot.dot(data, {x: \"lag\", y: \"weight\", fill: \"steelblue\", strokeWidth: 3})\n ]\n})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":9,"value":"
\n
\n
\n
\n
\n
\n
\n
\n
\n
\n
\n
\n
\n
\n
\n
\n\n---\n## 🙈 Appendix 🙈\n\nHere is some code and data details that you probably do not want to read.","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":20,"value":"#### Imports","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":7,"value":"mathjs = require('mathjs')","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":22,"value":"#### Functions","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":5,"value":"function ω(lag) {\n return (-1) ** lag * (mathjs.gamma(1 + d) / (mathjs.factorial(lag) * mathjs.gamma(1 + d - lag))) * Math.exp(-lag * lambda);\n}\n","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":24,"value":"#### Data","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":29,"value":"d = params[0]","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":27,"value":"lambda = params[1]","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":31,"value":"nlags = params[2]","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":15,"value":"data = Array.from({ length: nlags }, (_, index) => index).map(d => ({lag: d, weight: ω(d)}))","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"}],"files":"$W2f","stdlib":"1","titling":"auto"}} 2e:"$2d:body"