2a:null 2b:["$","title",null,{"children":"Local level model with dynamic shrinkage process prior | Mattias Villani | Observable"}] 31:T950,A problem with the above model is that it cannot simultaneously achieve the following three desirable properties: 1. periods of (nearly) no change in the level (${tex`\nu_t \approx 0`} for a longer time stretch) 2. abrupt jump in the level at time *t* (very large ${tex`\nu_t`}) 3. extended periods of rapid change in the level The **dynamic shrinkage process** tries to satisfy all of the three properties, by replacing the global variance ${tex`\sigma_\nu^2`} with a global variance ${tex`\tau^2`} and local variances ${tex`\lambda_t^2`} at every *t*: ${tex.block` \begin{aligned} y_t &= \mu_t + \varepsilon_t, \hspace{2.1cm} \varepsilon_t \overset{iid}{\sim} N(0, \sigma_\varepsilon^2) \\ \mu_t &= \mu_{t-1} + \nu_t, \hspace{1.75cm} \nu_t \overset{indep}{\sim} N(0, \tau^2 \lambda_t^2) \\ \log \lambda_t^2 &= \phi \log \lambda_{t-1}^2 + \eta_t, \hspace{1cm} \eta_t \overset{iid}{\sim}Z(\alpha, \beta, 0, \sigma_\eta) \end{aligned} `} where ${tex`Z(\alpha, \beta, \mu, \sigma)`} is the heavy tailed [Z-distribution](https://observablehq.com/@mattiasvillani/z-distribution). The dynamic shrinkage process is a time series extension of the horseshoe prior from regression. The connection can be seen by setting the persistence of the local volatility ${tex`\phi`} to zero. It can be shown that ${tex.block` \eta_t \sim Z(1/2, 1/2, 0, 1) \Longleftrightarrow \lambda_t = \exp(\eta_t/2) \sim C⁺(0,1), `} so that the local standard deviation follows the half Cauchy in the original horseshoe prior. The global standard deviation ${tex`\tau`} can also be assigned the ${tex`C⁺(0,1)`} prior as in the original horseshoe, or fixed to some suitable value. The global-local extension of the usual iid normal innovations gives properties 1 and 2 above. By introducing persistence in the log-volatility ${tex`\log \lambda_t^2`} by the ${tex`\phi`} parameter we also obtain the third property with persistent periods of rapidly changing parameters. For example with ${tex`\phi`} close to one we get that - a large positive ${tex`\eta_t`} will remain in the system for a while and keep ${tex`\lambda_t`} at a high level. Parameters will move a lot during this period. - a small (negative) ${tex`\eta_t`} reduces the time variation in ${tex`\mu_t`} and will help to keep this low time variation for an extended period. Parameters will stand more or less still during this period.32:T787,timeplot = { let xt = simulatedData.filter(d => d.type == "time series, xₜ").map((d) => d.x); let mut = simulatedData.filter(d => d.type == "local level, μₜ").map((d) => d.x); let taulambdat = simulatedData.filter(d => d.type == "total stdev, τ⋅λₜ").map((d) => d.x); const y2 = d3.scaleLinear(d3.extent(taulambdat), d3.extent(xt)); return Plot.plot({ //title: md`Local level model with error stdev ${tex`\sigma_\varepsilon = ${input.sigmaeps}`} and // global stdev ${tex`\tau = ${input.tau.toFixed(5)}`}`, y: {axis: "left", label: "xₜ, μₜ", domain: d3.extent(xt.concat(mut))}, color: { legend: true, domain: ["time series, xₜ", "local level, μₜ", "total standard deviation, τ⋅λₜ"], range: [colors[0], colors[1], colors[2]] //scheme: d3.Paired }, marks: [ Plot.axisY(y2.ticks(), {filter: plotselector.includes("total stdev, τ⋅λₜ"), color: colors[2], anchor: "right", label: labelrightaxis, y: y2, tickFormat: y2.tickFormat()}), Plot.ruleX([0]), Plot.ruleX([input.n], {filter: plotselector.includes("total stdev, τ⋅λₜ")}), Plot.ruleY([d3.min(simulatedData.map(d => d.x))]), Plot.lineY(simulatedData, {filter: d => d.type == "time series, xₜ" && plotselector.includes("time series, xₜ"), x: "time", y: "x", stroke: "type", strokeWidth: 2}), Plot.lineY(simulatedData, {filter: d => d.type == "local level, μₜ" && plotselector.includes("local level, μₜ"), x: "time", y: "x", stroke: "type", strokeWidth: 2}), Plot.lineY(simulatedData, Plot.mapY((D) => D.map(y2), { filter: d => d.type == "total stdev, τ⋅λₜ" && plotselector.includes("total stdev, τ⋅λₜ"), x: "time", y: "x", stroke: colors[2]})) ] }) }33:[] 2e:{"id":"ddd78e9a44772ec2","slug":"local-level-model-dsp-prior","title":"Local level model with dynamic shrinkage process 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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"},{"type":"team","id":"7592a7ab121f74a0","login":"mattiasvillani","name":"Mattias Villani","avatar_url":"https://avatars-next.observableusercontent.com/a440aaf39a3abefb86cbcc758fd7a5ab42555ae68d6825b52ec006f385a94d1d","tier":"pro_2024","role":"viewer"}],"comments":[],"custom_thumbnail":false,"trashed":false,"subscribed":null,"forks":2,"collection":{"id":"b4f68ca0e73a1ae6","slug":"time-series","title":"Time Series","owner":{"id":"7592a7ab121f74a0","login":"mattiasvillani","name":"Mattias Villani"}},"type":"notebook","body":{"title":"Local level model with dynamic shrinkage process prior","cells":[{"id":0,"value":"# Local level model with dynamic shrinkage process prior","pinned":false,"hidden":false,"mode":"md"},{"id":194,"value":"The traditional [local level model](https://observablehq.com/@mattiasvillani/local-level-model) has parameter innovations that are iid Gaussian with constant global (time-invariant) variance ${tex`\\sigma_\\nu^2`}:\n\n${tex.block`\n\\begin{aligned}\n y_t &= \\mu_t + \\varepsilon_t, \\hspace{0.75cm} \\varepsilon_t \\overset{iid}{\\sim} N(0, \\sigma_\\varepsilon^2) \\\\\n \\mu_t &= \\mu_{t-1} + \\nu_t, \\quad \\nu_t \\overset{iid}{\\sim} N(0, \\sigma_\\nu^2)\n\\end{aligned}\n`}","pinned":false,"hidden":false,"mode":"md"},{"id":1195,"value":"$31","pinned":false,"hidden":false,"mode":"md"},{"id":463,"value":"#### Settings","pinned":false,"hidden":false,"mode":"md"},{"id":162,"value":"viewof input = Inputs.form({\n n: Inputs.range([10, 1000], {value: 100, step: 1, label: \"time series length\"}),\n sigmaeps: Inputs.range([0.01, 5], {value: 0.2, step: 0.1, label: tex`\\text{std error, }\\sigma_\\varepsilon`}),\n tau: Inputs.range([0, 0.1], {value: 0.001, step: 0.0001, label: tex`\\text{global std, }\\tau`}),\n phi: Inputs.range([-0.99, 0.99], {value: 0.7, step: 0.01, label: tex`\\text{persistence, }\\phi`}),\n alpha: Inputs.range([Number.EPSILON, 3], {value: 0.5, step: 0.1, label: tex`\\alpha`}),\n beta: Inputs.range([Number.EPSILON, 3], {value: 0.5, step: 0.1, label: tex`\\beta`}),\n sigma: Inputs.range([Number.EPSILON, 3], {value: 1, step: 0.1, label: tex`\\sigma_\\eta`})\n})","pinned":false,"hidden":false,"mode":"ojs"},{"id":1282,"value":"viewof plotselector = Inputs.checkbox([\"time series, xₜ\", \"local level, μₜ\", \"total stdev, τ⋅λₜ\"], {label: \"plot:\", value: [\"time series, xₜ\", \"local level, μₜ\", \"total stdev, τ⋅λₜ\"]})","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":"$32","pinned":false,"hidden":false,"mode":"ojs"},{"id":13,"value":"
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\n\n---\n## 🙈 Appendix 🙈\nHere follow code and details that you probably do not want to read.","pinned":false,"hidden":false,"mode":"md"},{"id":15,"value":"#### Imports","pinned":false,"hidden":false,"mode":"md"},{"id":11,"value":"math = require(\"mathjs\");","pinned":false,"hidden":false,"mode":"ojs"},{"id":1185,"value":"import {colors, simLocalLevelDSP} from \"@mattiasvillani/statistics-tools\"","pinned":false,"hidden":false,"mode":"ojs"},{"id":816,"value":"jStat = require(\"jstat\");","pinned":false,"hidden":false,"mode":"ojs"},{"id":96,"value":"#### Data","pinned":false,"hidden":false,"mode":"md"},{"id":1399,"value":"showdist = plotselector","pinned":false,"hidden":false,"mode":"ojs"},{"id":1013,"value":"n = input.n","pinned":false,"hidden":false,"mode":"ojs"},{"id":1394,"value":"labelrightaxis = {\n if (plotselector.includes(\"total stdev, τ⋅λₜ\")){ return \"τ⋅λₜ\"}\n else{\n return \"\"\n }\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":94,"value":"#### Functions","pinned":false,"hidden":false,"mode":"md"},{"id":519,"value":"simulatedData = {\n simulatebutton\n return simLocalLevelDSP(n, input.sigmaeps, input.tau, input.phi, input.alpha, input.beta, input.sigma)\n}","pinned":false,"hidden":false,"mode":"ojs"}],"files":"$W33","stdlib":"1","titling":"auto"}} 2f:"$2e:body"