diff --git a/M2/TP2-linear-regression-etu.ipynb b/M2/TP2-linear-regression-etu.ipynb new file mode 100644 index 0000000000000000000000000000000000000000..f02a1d8853ef99baeef300e1791128c60df8ada4 --- /dev/null +++ b/M2/TP2-linear-regression-etu.ipynb @@ -0,0 +1,20658 @@ +{ + "cells": [ + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# TP2 : Linear regression\n", + "## Parameters estimation by least square\n", + "\n", + "### Datas : Cholesterol $(y_i)_{i=1,\\ldots,n}$ level versus ages variable $(x)_{i=1,\\ldots,n}$\n", + "\n", + "### Assumptions\n", + "We suppose that for $i=1,\\ldots,n$\n", + "$$Y_i=\\beta_0+\\beta_1 x+\\varepsilon_i$$\n", + "\n", + "with\n", + "- $E(\\varepsilon_i)=0$\n", + "- The $(\\varepsilon_i)_{i=1,\\dots,n}$ are i.i.d.\n", + "- The distributions of $(\\varepsilon_i)_{i=1,\\dots,n}$ are $\\mathcal{N}(0,\\sigma^2)$\n", + "\n", + "### Mean square estimations\n", + "\n", + "We use the mean square estimation of the parameters $\\beta_0$ and $\\beta_1$: \n", + "\n", + "\\begin{array}{ll}\n", + "Min\\quad\n", + "f(\\beta)=\\frac{1}{2}\\sum_{i=1}^{n}r_{i}^{2}(\\beta)& =\\frac{1}{2}\\|r(\\beta)\\|^2\\\\\n", + "\\beta \\in \\mathbf{R}^{2}&\n", + "\\end{array}\n", + "\n", + "where $r$ is the residual function\n", + "$$r:\\mathbf{R}^2\\rightarrow \\mathbf{R}^n$$\n", + "$$r_i(\\beta) = y_i - \\beta_{0}+\\beta_{1}x_{i}$$\n", + "\n", + "So we can write\n", + "$$r(\\beta) = y - X\\beta,$$\n", + "where\n", + "\n", + "$$y = \\begin{pmatrix} y_1\\\\ \\vdots \\\\ y_n\\end{pmatrix}\\;\\,\\texttt{and}\\;\\;\n", + " X = \\begin{pmatrix} 1 & x_1\\\\ \\vdots & \\vdots \\\\ 1 & x_n\\end{pmatrix}$$\n", + "\n" + ] + }, + { + "cell_type": "code", + "execution_count": 12, + "metadata": {}, + "outputs": [ + { + "data": { + "image/png": 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LaTeXStrings\n", + "using Statistics\n", + "using DataFrames\n", + "using GLM\n", + "\n", + "y=[1.8, 2.3, 2, 2.4, 2, 2.5, 2.6, 2.6, 2.9, 2.3, 2.4, 2.1, 2.5, 2.7, 2.7, 3, 3.1, 2.3, 2.5, 3, 3.3, 2.7, 3.1, 2.9, 3.4, 2.4, 3.4, 3.7, 2.8, 3.3, 3.5, 3.3, 2.6]\n", + "x=[25, 25, 25, 25, 25, 25, 25, 35, 35, 35, 35, 35, 35, 35, 45, 45, 45, 45, 45, 45, 45, 45, 55, 55, 55, 55, 55, 65, 65, 65, 65, 65, 65]\n", + "n = length(x)\n", + "\n", + "p = plot(x,y, seriestype = :scatter, xlabel = (L\"x\"), ylabel = (L\"y\"))\n", + "a , b = .5 , 0.04\n", + "plot!(p,x,a .+ b*x)\n", + "\n", + "r(a,b) = y - (a .+ b*x)\n", + "\n", + "r(a,b)\n", + "\n", + "i = 11\n", + "plot!(p, [x[i],x[i]], [y[i],a+b*x[i]],linewidth=3)\n", + "annotate!(p,x[i],mean([y[i],a+b*x[i]]),text(L\"r_{11}(a,b)\", :left, :green))\n", + "plot(p,legend=false)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "### Exercise\n", + "\n", + "For solving the optimization problem we have to sole the linear system $X^TX\\beta = X^Ty$\n", + "\n", + "Use the \\ operator for solving linear equation" + ] + }, + { + "cell_type": "code", + "execution_count": 15, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "[1.639575381679391, 0.024909351145038124]\n", + "StatsModels.TableRegressionModel{LinearModel{GLM.LmResp{Vector{Float64}}, GLM.DensePredChol{Float64, LinearAlgebra.CholeskyPivoted{Float64, Matrix{Float64}, Vector{Int64}}}}, Matrix{Float64}}\n", + "\n", + "chol ~ 1 + age\n", + "\n", + "Coefficients:\n", + "────────────────────────────────────────────────────────────────────────\n", + " Coef. Std. Error t Pr(>|t|) Lower 95% Upper 95%\n", + "────────────────────────────────────────────────────────────────────────\n", + "(Intercept) 1.63958 0.189994 8.63 <1e-09 1.25208 2.02707\n", + "age 0.0249094 0.00413632 6.02 <1e-05 0.0164733 0.0333454\n", + "────────────────────────────────────────────────────────────────────────\n", + "\u001b[91m\u001b[1mTest Failed\u001b[22m\u001b[39m at \u001b[39m\u001b[1m/Users/gergaud/git-ENS/Julia-TSE/etudiants/M2/jl_notebook_cell_df34fa98e69747e1a8f8a730347b8e2f_X15sZmlsZQ==.jl:18\u001b[22m\n", + " Expression: βhat ≈ coef(lm1)\n", + " Evaluated: [0.0, 0.0] ≈ [1.639575381679391, 0.024909351145038124]\n", + "\n" + ] + }, + { + "ename": "Test.FallbackTestSetException", + "evalue": "Test.FallbackTestSetException(\"There was an error during testing\")", + "output_type": "error", + "traceback": [ + "Test.FallbackTestSetException(\"There was an error during testing\")\n", + "\n", + "Stacktrace:\n", + " [1] record(ts::Test.FallbackTestSet, t::Union{Test.Error, Test.Fail})\n", + " @ Test /Applications/Julia-1.10.app/Contents/Resources/julia/share/julia/stdlib/v1.10/Test/src/Test.jl:1000\n", + " [2] do_test(result::Test.ExecutionResult, orig_expr::Any)\n", + " @ Test /Applications/Julia-1.10.app/Contents/Resources/julia/share/julia/stdlib/v1.10/Test/src/Test.jl:705\n", + " [3] top-level scope\n", + " @ /Applications/Julia-1.10.app/Contents/Resources/julia/share/julia/stdlib/v1.10/Test/src/Test.jl:672" + ] + } + ], + "source": [ + "# \n", + "# Use of the GLM Package\n", + "datas = DataFrame(age=x, chol = y)\n", + "lm1 = lm(@formula(chol ~ age), datas)\n", + "println(coef(lm1))\n", + "println(lm1)\n", + "\n", + "# ... To complete and modify\n", + "\n", + "X = [ones(n) x]\n", + "\n", + "βhat = [0., 0.]\n", + "βhat₀ = βhat[1]\n", + "βhat₁ = βhat[2]\n", + "p = plot(x,y, seriestype = :scatter, xlabel = (L\"x\"), ylabel = (L\"y\"))\n", + "plot!(p,x,βhat₀ .+ βhat₁*x)\n", + "\n", + "println(@test βhat ≈ coef(lm1))" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "### Confidence intervals of the parameters\n", + "\n", + "- We note $r_i = y_i - (\\hat{\\beta}_0 + \\hat{\\beta}_1x_i)$, then the unbiased estimation of the variance $\\sigma^2$ is\n", + "\n", + "$$\\hat{\\sigma}^2\\frac{\\sum_ir_i^2}{n-2}$$\n", + "\n", + "- As $\\hat{\\beta} = (X^TX)^{-1}X^Ty$ the statistic $B=(X^TX)^{-1}X^TY = (X^TX)^{-1}X^T(X\\beta+\\varepsilon)=\\beta +(X^TX)^{-1}X^T\\varepsilon$ as the $\\mathcal{N}(\\beta,\\sigma^2(X^TX)^{-1})$ distribution. \n", + "\n", + "Si les postulats sont vérifiés alors les\n", + "intervalles de confiances de $\\beta _{0},\\beta _{1}$ et $\\sigma ^{2}$ au\n", + "seuil $(1-\\alpha )$ sont:\n", + "\n", + "\n", + "- \n", + "$$\n", + "\\sigma ^{2}\\in \\left[ \\frac{\\sum_{i=1}^{n}r_{i}^{2}}{\\chi _{1-\\alpha /2}};%\n", + "\\frac{\\sum_{i=1}^{n}r_{i}^{2}}{\\chi _{\\alpha /2}}\\right] \n", + "$$\n", + "\n", + "where $\\chi _{1-\\alpha /2}$ and $\\chi _{\\alpha /2}$ sont lues\n", + "dans le table du Khi-2 à $\\nu =n-2$ ddl.\n", + "\n", + "\\item \n", + "$$\n", + "\\begin{array}{ccc}\n", + "\\beta _{1} & \\in & [\\hat{\\beta}_{1}-t_{1-\\alpha /2}\\hat{\\sigma}%\n", + "_{B_{1}}^{{}};\\hat{\\beta}_{1}+t_{1-\\alpha /2}\\hat{\\sigma}_{B_{1}}^{{}}] \\\\ \n", + "\\beta _{0} & \\in & [\\hat{\\beta}_{0}-t_{1-\\alpha /2}\\hat{\\sigma}%\n", + "_{B_{0}}^{{}};\\hat{\\beta}_{0}+t_{1-\\alpha /2}\\hat{\\sigma}_{B_{0}}^{{}}]\n", + "\\end{array}\n", + "$$\n", + "\n", + "où \n", + "- $\\hat{\\sigma}_{B_{1}}^{2} = \\dfrac{\\hat{\\sigma}^{2}}{SCE_{x}} $\n", + "- $\\hat{\\sigma}_{B_{0}}^{2} = \\hat{\\sigma}^{2}(\\dfrac{1}{n}+\\dfrac{\\bar{x}^{2}}{SCE_{x}})$\n", + "- $t_{1-\\alpha /2}$ from the Student law with $n-2$ degrees of freedom\n", + "\n", + "\n" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "### Exercise\n", + "1. Compute the $n$ residuals in the vector res\n", + "2. Compute the estimation of residual variance $\\sigma^2$ and of the standard error $\\sigma$\n", + "3. Compute the estimation of the variance, covariance matrix $\\Sigma$.\n", + "4. Compute the estimation of the standard error of $\\hat{\\beta}_0$ and $\\hat{\\beta}_1$.\n", + "5. Compute the confidence intervalle of $\\beta_0$ and $\\beta_1$\n" + ] + }, + { + "cell_type": "code", + "execution_count": 17, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "\u001b[91m\u001b[1mTest Failed\u001b[22m\u001b[39m at \u001b[39m\u001b[1m/Users/gergaud/git-ENS/Julia-TSE/etudiants/M2/jl_notebook_cell_df34fa98e69747e1a8f8a730347b8e2f_X21sZmlsZQ==.jl:4\u001b[22m\n", + " Expression: res ≈ residuals(lm1)\n", + " Evaluated: [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0 … 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0] ≈ [-0.4623091603053442, 0.03769083969465559, -0.26230916030534424, 0.13769083969465568, -0.26230916030534424, 0.23769083969465576, 0.33769083969465585, 0.08859732824427491, 0.38859732824427473, -0.21140267175572536 … -0.10958969465648805, 0.39041030534351195, -0.609589694656488, 0.39041030534351195, 0.4413167938931313, -0.4586832061068691, 0.04131679389313092, 0.2413167938931311, 0.04131679389313092, -0.6586832061068688]\n", + "\n" + ] + }, + { + "ename": "Test.FallbackTestSetException", + "evalue": "Test.FallbackTestSetException(\"There was an error during testing\")", + "output_type": "error", + "traceback": [ + "Test.FallbackTestSetException(\"There was an error during testing\")\n", + "\n", + "Stacktrace:\n", + " [1] record(ts::Test.FallbackTestSet, t::Union{Test.Error, Test.Fail})\n", + " @ Test /Applications/Julia-1.10.app/Contents/Resources/julia/share/julia/stdlib/v1.10/Test/src/Test.jl:1000\n", + " [2] do_test(result::Test.ExecutionResult, orig_expr::Any)\n", + " @ Test /Applications/Julia-1.10.app/Contents/Resources/julia/share/julia/stdlib/v1.10/Test/src/Test.jl:705\n", + " [3] top-level scope\n", + " @ /Applications/Julia-1.10.app/Contents/Resources/julia/share/julia/stdlib/v1.10/Test/src/Test.jl:672" + ] + } + ], + "source": [ + " # Residue\n", + " # To modify\n", + "res = zeros(size(x)) \n", + "println(\"Test residuals : \",@test res ≈ residuals(lm1))\n", + "\n", + "# estimation of σ²\n", + "# To modify\n", + "σ²hat = 0. \n", + "σhat = 0.\n", + "\n", + "println(\"σ²hat = \", σ²hat)\n", + "\n", + "# Variances, covariances matrix of B\n", + "# To modify\n", + "Σ = zeros(2,2) # Variances, covariances matrix of B\n", + "println(\"Σ = \", Σ)\n", + "covB = vcov(lm1) # GLM Package\n", + "println(\"Test Variance, covariance matrix : \", @test Σ ≈ covB)\n", + "\n", + "using Distributions\n", + "using HypothesisTests\n", + "\n", + "# Confidence interval of β₀ and β₁\n", + "# To modify\n", + "α = 0.05\n", + "\n", + "t_crit = quantile(TDist(n-2),1-α/2)\n", + "\n", + "σhat₀ = 0.\n", + "σhat₁ = 0.\n", + "println(\"test of the variance of σhat₀ : \", @test σhat₀ ≈ stderror(lm1)[1])\n", + "println(\"test of the variance of σhat₁ : \", @test σhat₁ ≈ stderror(lm1)[2])\n", + "\n", + "# at the end uncommment the 2 following lines\n", + "#println(\"β₀ ∈ [\", βhat₀ - t_crit*σhat₀, \",\",βhat₀ + t_crit*σhat₀,\"]\")\n", + "#println(\"β₁ ∈ [\", βhat₁ - t_crit*σhat₁, \",\",βhat₁ + t_crit*σhat₁,\"]\")" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "### Confidence Bands and Prediction Bands\n", + "- The condidence band for the mean is\n", + "$$\\hat{\\beta}_0 +\\hat{\\beta}_1x \\pm t_{1-\\alpha/2}\\hat{\\sigma}\\sqrt{\\frac{1}{n}+\\frac{(x-\\bar{x})^2}{SCE_x}}$$\n", + "- The Prediction band is \n", + "$$\\hat{\\beta}_0 +\\hat{\\beta}_1x \\pm t_{1-\\alpha/2}\\hat{\\sigma}\\sqrt{1+\\frac{1}{n}+\\frac{(x-\\bar{x})^2}{SCE_x}}$$\n", + "\n", + "### Exercice\n", + "Plot these confidence and prevision bands on the graph p" + ] + }, + { + "cell_type": "code", + "execution_count": 18, + "metadata": {}, + "outputs": [], + "source": [ + "# To complete\n", + "\n", + "\n", + "# Confidence Bands\n", + "# To complete\n", + "\n", + "\n", + "# Prevision Bands\n", + "# To complete\n" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Simulation of the distribution the estimates of $\\beta_0$ and $\\beta_1$\n", + "\n", + "We consider the model\n", + "$$Y_i = \\beta_0 + \\beta_1 x_i + \\varepsilon_i$$\n", + "with\n", + "- $(\\varepsilon_i)_{i=1,\\ldots,n}$ i.i.d.\n", + "- The $\\varepsilon_i$ have the $\\mathcal{N}(0,\\sigma^2)$ distribution\n", + "- $(x_1,\\ldots,x_n) = (1,2,\\ldots,10)$\n", + "- $\\beta_0 = 2.0, \\beta_1 = 1.5$ and $\\sigma = 1.5$\n", + "\n", + "We recall that the statistic $B$ which estimate the parameter $\\beta$ follows the $\\mathcal{N}(\\beta,\\Sigma)$ distribution, with $\\Sigma=\\sigma^2(X^TX)^{-1}$.\n" + ] + }, + { + "cell_type": "code", + "execution_count": 21, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Percentage of points inside the ellipse : 0.9534\n" + ] + }, + { + "data": { + "image/png": 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1670.42,1066.07 1676.21,1068.33 1681.97,1070.55 1687.7,1072.74 1693.39,1074.89 1699.05,1077.01 1704.67,1079.09 1710.26,1081.14 1715.81,1083.15 1721.33,1085.12 1726.81,1087.06 1732.25,1088.96 1737.65,1090.82 1743.01,1092.65 1748.33,1094.44 1753.62,1096.19 1758.86,1097.91 1764.06,1099.59 1769.22,1101.22 1774.33,1102.83 1779.41,1104.39 1784.44,1105.91 1789.43,1107.4 1794.37,1108.85 1799.27,1110.26 1804.12,1111.62 1808.93,1112.96 1813.69,1114.25 1818.41,1115.5 1823.08,1116.71 1827.7,1117.88 1832.27,1119.02 1836.79,1120.11 1841.27,1121.16 1845.69,1122.18 1850.07,1123.15 1854.4,1124.08 1858.67,1124.97 1862.89,1125.83 1867.07,1126.64 1871.19,1127.41 1875.26,1128.14 1879.27,1128.83 1883.23,1129.48 1887.14,1130.08 1891,1130.65 1894.8,1131.18 1898.54,1131.66 1902.23,1132.1 1905.87,1132.51 1909.45,1132.87 1912.97,1133.19 1916.44,1133.47 1919.85,1133.7 1923.2,1133.9 1926.49,1134.05 1929.73,1134.17 1932.91,1134.24 1936.03,1134.27 1939.09,1134.26 1942.09,1134.21 1945.03,1134.11 1947.92,1133.98 1950.74,1133.8 1953.5,1133.58 1956.21,1133.32 1958.85,1133.02 1961.43,1132.68 1963.95,1132.3 1966.4,1131.88 1968.8,1131.41 1971.13,1130.91 1973.4,1130.36 1975.61,1129.77 1977.76,1129.14 1979.84,1128.47 1981.86,1127.76 1983.81,1127.01 1985.7,1126.22 1987.53,1125.39 1989.3,1124.51 1991,1123.6 1992.63,1122.65 1994.2,1121.65 1995.71,1120.62 1997.15,1119.54 1998.53,1118.43 1999.84,1117.28 2001.09,1116.08 2002.27,1114.85 2003.38,1113.58 2004.43,1112.27 2005.42,1110.91 2006.34,1109.52 2007.19,1108.1 2007.98,1106.63 2008.7,1105.12 2009.35,1103.58 2009.94,1101.99 2010.46,1100.37 2010.92,1098.71 2011.31,1097.02 2011.63,1095.28 2011.89,1093.51 2012.08,1091.7 2012.2,1089.86 2012.26,1087.97 2012.25,1086.05 2012.18,1084.1 2012.04,1082.1 2011.83,1080.07 2011.55,1078.01 2011.21,1075.91 2010.8,1073.77 2010.33,1071.6 2009.79,1069.39 2009.18,1067.15 2008.51,1064.87 2007.77,1062.56 2006.97,1060.22 2006.1,1057.84 2005.16,1055.43 2004.16,1052.98 2003.09,1050.5 2001.96,1047.99 2000.76,1045.44 1999.5,1042.87 1998.17,1040.26 1996.77,1037.62 1995.31,1034.94 1993.79,1032.24 1992.2,1029.5 1990.55,1026.73 1988.83,1023.94 1987.05,1021.11 1985.21,1018.25 1983.3,1015.36 1981.32,1012.44 1979.29,1009.5 1977.19,1006.52 1975.03,1003.52 1972.8,1000.49 1970.51,997.426 1968.16,994.338 1965.75,991.223 1963.28,988.081 1960.74,984.911 1958.15,981.716 1955.49,978.494 1952.77,975.247 1949.99,971.974 1947.15,968.675 1944.25,965.353 1941.3,962.005 1938.28,958.634 1935.2,955.238 1932.07,951.819 1928.87,948.377 1925.62,944.913 1922.31,941.425 1918.94,937.916 1915.52,934.385 1912.03,930.833 1908.5,927.259 1904.9,923.665 1901.25,920.051 1897.55,916.417 1893.79,912.763 1889.97,909.089 1886.1,905.397 1882.18,901.687 1878.2,897.958 1874.17,894.212 1870.09,890.448 1865.96,886.667 1861.77,882.87 1857.53,879.057 1853.24,875.227 1848.91,871.382 1844.52,867.522 1840.08,863.648 1835.59,859.759 1831.05,855.856 1826.47,851.939 1821.83,848.01 1817.15,844.068 1812.43,840.113 1807.65,836.146 1802.83,832.168 1797.97,828.179 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DataFrame([xVals,yVals], [:x, :y]) # create a DataFrame from the datas\n", + " model = lm(@formula(y ~ x), data) # linear regresseion\n", + " return (coef(model)) # return the estimation of β₀ and β₁\n", + "end\n", + "\n", + "ests = [coefEst() for _ in 1:N] # generating N sample of the parameters (in an array)\n", + "\n", + "# The statistic B which estimate β follows the N(β,Σ) distribution with, Σ = σ²(XᵀX)⁻¹\n", + "Σ⁻¹ = (1/σ^2)*X'*X # Compute the inverse of the variance, covariance matrix \n", + "A = cholesky(Σ⁻¹).U # Z = A(B-β) follows the N(0,I) distribution\n", + "A⁻¹ = inv(A)\n", + "# The radius of a standard bidimensional normal distribution follows a Rayleigh law\n", + "radius = quantile(Rayleigh(),1-α) # Z₁² + Z₂² follows a χ^2 with 2 dof distribution\n", + "isInEllipse(b) = norm(A*(b-β)) <= radius # function which indicate if b is in the ellipse or not.\n", + "estIn = isInEllipse.(ests) # Vectorize the function\n", + "\n", + "println(\"Percentage of points inside the ellipse : \", sum(estIn)/N)\n", + "scatter(first.(ests[estIn]), last.(ests[estIn]), color = :green, ms=2, msw=0) # display points inside the ellipse \n", + "scatter!(first.(ests[.!estIn]), last.(ests[.!estIn]), color = :red, ms=2, msw=0) # display points outside the ellipse\n", + "# Calculus of the ellipse\n", + "ellipsePts = [radius*A⁻¹*[cos(t),sin(t)] + β for t in 0:0.01:2*pi] # compute the ellipse\n", + "scatter!([β₀], [β₁], c=:red, ms=5, msw=2) # display the center of the ellipse\n", + "plot!(first.(ellipsePts), last.(ellipsePts), c=:blue, lw=2, legend=false,xlabel=L\"\\hat{\\beta}_0\", ylabel=L\"\\hat{\\beta}_1\")\n" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [] + } + ], + "metadata": { + "kernelspec": { + "display_name": "Julia 1.10.5", + "language": "julia", + "name": "julia-1.10" + }, + "language_info": { + "file_extension": ".jl", + "mimetype": "application/julia", + "name": "julia", + "version": "1.10.5" + } + }, + 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