2c:null 2d:["$","title",null,{"children":"Enkel linjär regression - skattningar | Mattias Villani | Observable"}] 32:[] 30:{"id":"f1018d2b349b0408","slug":null,"title":"Enkel linjär regression - skattningar","thumbnail":"d5c31ec5a4ddd33c4445c7cac43326f65fa79e9b55d063ffc6138d39487b6bdb","access_level":"public","create_time":"$D2023-01-22T22:27:19.541Z","update_time":"$D2023-01-22T22:43:30.074Z","view_time":null,"first_public_time":"$D2023-01-22T22:27:19.541Z","has_unpublished_edits":null,"fork":{"id":"4ab3c3f7ade79d7c","slug":"samplingfordelningen-linear-regression","title":"Samplingfördelningen - linear regression","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":1147,"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":null,"type":"notebook","body":{"title":"Enkel linjär regression - skattningar","cells":[{"id":0,"value":"# Enkel linjär regression - skattningar","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":623,"value":"### Skattad regressionsmodell\n\nDen enkla linära regressionsmodellen är\n\n> ${tex`y = b_0 + b_1 x`} ","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":712,"value":"Modellens parametrar kan skattas med minsta-kvadrat-skattningarna\n\n> ${tex`b_1 = \\frac{s_{xy}}{s_x^2}`} där ${tex`s_x^2 = \\frac{\\sum (x_i-\\bar x)^2}{n-1}`} och ${tex`s_{xy} = \\frac{\\sum (x_i-\\bar x)(y_i-\\bar y)}{n-1}`}\n\noch\n\n> ${tex`b_0 = \\bar y -b_1 \\bar x`}.","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":743,"value":"### Regressionslinjen kommer variera från stickprov till stickprov\n\nI figuren nedan simulerar vi nya datamaterial (stickprov) från en underliggande populationsmodell med ett intercept på 1 och en lutning på 1. I varje datamaterial beräknas skattningarna ${tex`b_0`} och ${tex`b_1`}, vilket alltså ger en ny regressionslinje för varje simulerat datamaterial. ","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":33,"value":"#### Populationsmodell som vi simulerar data från","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":67,"value":"viewof n = Inputs.range([1, 1000], {value: 20, step: 1, label: tex`n`});","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":27,"value":"viewof beta0 = Inputs.range([-2, 2], {value: 1, step: 0.1, label: tex`\\beta_0`})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":29,"value":"viewof beta1 = Inputs.range([-2, 2], {value: 1, step: 0.1, label: tex`\\beta_1`})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":70,"value":"viewof sigmaepsilon = Inputs.range([0, 2], {value: 0.5, step: 0.1, label: tex`\\sigma_\\varepsilon`})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1113,"value":"viewof ciprob = Inputs.range([0, 0.99], {value: 0.0, step: 0.05, label: \"Confidence level\"})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":100,"value":"viewof data_estimate = Inputs.button(\"Simulera nytt datamaterial\", {value: null, reduce: () => sampleDataAndEstimate(n, beta0, beta1, sigmaepsilon)})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1142,"value":"viewof b_est = Inputs.button(\"Update\", {value: [], reduce: () => sampleOneDataAndEstimate(n, beta0, beta1, sigmaepsilon)}) ","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":241,"value":"estimates = tex`\\text{Skattad intercept: } b_0 = ${data_estimate[2].coefficients[0].toPrecision(4)} \\\\ \\text{Skattad lutning: } b_1 =${data_estimate[2].coefficients[1].toPrecision(4)}`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":80,"value":"plt = Plot.plot({\n color: {legend: true},\n x: {domain: [-1, 1]},\n y: {domain: [-3,3]},\n //y: {domain: [beta0 + beta1*d3.min(data, d => d.x)*1.2 -3*sigmaepsilon, beta0 + beta1*d3.max(data, d => d.x)*1.2 + 3*sigmaepsilon]},\n marks: [\n Plot.dot(data_estimate[0], {x: \"x\", y: \"y\", strokeOpacity: 0.8, r: 1.5, stroke: colors[1]}),\n Plot.linearRegressionY(data_estimate[0], {x: \"x\", y: \"y\", stroke: colors[6], strokeWidth: 2, ci: 0})\n ],\n})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":837,"value":"
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\n\n---\n## 🙈 Appendix 🙈\n\nHär följer bara kod och detaljer som du antagligen inte vill läsa.","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":848,"value":"#### Imports","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":185,"value":"import {lm} from \"@chrispahm/linear-models-in-observable-notebooks\"","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":198,"value":"import {summary} from \"@chrispahm/linear-models-in-observable-notebooks\"","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":23,"value":"jstat = require('jstat')","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":854,"value":"#### Functions","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":111,"value":"sampleDataAndEstimate = function sampleDataAndEstimate(n, beta0, beta1, sigmaepsilon){\n const x = d3.range(-1, 1, 2/n);\n const epsilon = jstat.randn(1,n)[0];\n const data = x.map(x => ({x: x, y: beta0 + beta1*x + jstat.randn(1,n)[0][0]*sigmaepsilon}));\n const linear_model = lm('y ~ x', data);\n b_est.push({sample_id: b_est.length/2 + 1, coef:linear_model.coefficients[0].toPrecision(4),type:\"intercept\"});\n b_est.push({sample_id: (b_est.length+1)/2,coef:linear_model.coefficients[1].toPrecision(4),type:\"lutning\"});\n return [data, b_est, linear_model]\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1084,"value":"sampleOneDataAndEstimate = function sampleOneDataAndEstimate(n, beta0, beta1, sigmaepsilon){\n const x = d3.range(-1, 1, 2/n);\n const epsilon = jstat.randn(1,n)[0];\n const data = x.map(x => ({x: x, y: beta0 + beta1*x + jstat.randn(1,n)[0][0]*sigmaepsilon}));\n const linear_model = lm('y ~ x', data);\n const b_est = [{sample_id: 1, coef:linear_model.coefficients[0].toPrecision(4),type:\"intercept\"},\n {sample_id: 1, coef:linear_model.coefficients[1].toPrecision(4),type:\"lutning\"}\n ];\n return [data, b_est, linear_model]\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":850,"value":"#### Data","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":843,"value":"#### Style","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":667,"value":"viewof md_colour = Inputs.color({label: \"Highlight color cells\", value: \"#F5F5DC\"})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":361,"value":"colors = d3.schemePaired","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"}],"files":"$W32","stdlib":"1","titling":"auto"}} 31:"$30:body"