03_回帰その1_基礎的事項.nextjournal.ipynb

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\u57fa\u790e\u7684\u4e8b\u9805\n\n\u3044\u3064\u3082\u306e\u3088\u3046\u306b\u53f3\u4e0a\u306e\u300cRemix\u300d\u30dc\u30bf\u30f3\u3092\u62bc\u3057\u3066\u81ea\u5206\u7528\u306e\u8907\u88fd\u3092\u4f5c\u6210\u3059\u308c\u3070\u3001\u5404\u30bb\u30eb\u3092\u300cShift+Enter\u300d\u3067\u5b9f\u884c\u3057\u306a\u304c\u3089\u8aad\u3080\u3053\u3068\u304c\u3067\u304d\u307e\u3059\n\n\u5fc5\u8981\u306a\u30d1\u30c3\u30b1\u30fc\u30b8\u3092\u8aad\u307f\u8fbc\u307f\u307e\u3059"]},{"cell_type":"code","metadata":{"nextjournal":{"id":"d0e5f6e2-8f99-40ab-903c-c380f8902046","kind":"code","language":"julia"}},"source":["using Plots, CSV, Statistics, LinearAlgebra"],"outputs":[],"execution_count":1},{"cell_type":"markdown","metadata":{},"source":["# 1\u5909\u6570\u306e\u56de\u5e30 (\u5358\u56de\u5e30)\n\n\u5207\u72471\u3067\u50be\u304d2\u306b\u30ce\u30a4\u30ba\u3092\u52a0\u3048\u305f\u30e9\u30d9\u30eb\u3042\u308a\u30c7\u30fc\u30bf$\\{y_n, x_n\\}_{n=1}^{N}$\u3092$N=100$\u70b9\u3064\u304f\u3063\u3066\u30d7\u30ed\u30c3\u30c8\u3057\u307e\u3059"]},{"cell_type":"code","metadata":{"nextjournal":{"id":"2ec348e6-8235-4306-a1a0-74c6be51fce0","kind":"code","language":"julia"}},"source":["N = 100\nx = rand(N) # rand\u95a2\u6570\u30920\u304b\u30891\u306e\u4e00\u69d8\u4e71\u6570\u306b\u5f93\u3063\u305f\u5165\u529bx\u3092N\u70b9\u3064\u304f\u308b\n\n#\u914d\u5217\u306e\u8981\u7d20\u305d\u308c\u305e\u308c\u306b\u3064\u3044\u3066\u6f14\u7b97\u3059\u308b\u5834\u5408\u306f\u30c9\u30c3\u30c8.\u3092\u4ed8\u3051\u308b\n#randn\u306f\u5e73\u5747\u30bc\u30ed\u3001\u5206\u65631\u306e\u6b63\u898f\u5206\u5e03\u306b\u5f93\u3063\u305f\u4e71\u6570\u3092\u4f5c\u308b\u95a2\u6570\ny = 1.0 .+ 2.0 .* x .+ 0.3 .* randn(N)\n\nscatter(x, y, xlabel=\"x\", ylabel=\"y\", legend=false)"],"outputs":[{"output_type":"execute_result","data":{"image\/svg+xml":"<?xml version=\"1.0\" encoding=\"utf-8\"?>\n<svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" 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fill-rule=\"evenodd\" fill-opacity=\"1\" stroke=\"#000000\" stroke-opacity=\"1\" stroke-width=\"3.2\"\/>\n<circle clip-path=\"url(#clip272)\" cx=\"286.543\" cy=\"1125.48\" r=\"14\" fill=\"#009af9\" fill-rule=\"evenodd\" fill-opacity=\"1\" stroke=\"#000000\" stroke-opacity=\"1\" stroke-width=\"3.2\"\/>\n<\/svg>\n"},"execution_count":1,"metadata":{}}],"execution_count":1},{"cell_type":"markdown","metadata":{},"source":["\u5358\u56de\u5e30\u3092\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u89e3\u304f\u305f\u3081\u306e\u95a2\u6570\u3092\u5b9a\u7fa9\u3057\u307e\u3057\u3087\u3046\n\n$$\n\\hat{\\bm{w}} = \\arg \\min\\ L(\\bm{y},\\bm{x} | \\bm{w}) = \\frac{1}{2} \\sum_{n=1}^{N} (y_n - w_0 - w_1 x_n)^2\n$$\n$$\n\\hat{w}_1 = \\frac{\\sum_{n=1}^{N}(x_n - \\bar{x})(y_n - \\bar{y})}{\\sum_{n=1}^{N}(x_n - \\bar{x})^2}\n$$\n$$\n\\hat{w}_0 = \\bar{y} - \\hat{w}_1 \\bar{x}\n$$\n$$\n\\bar{y} = \\frac{1}{N}\\sum_{n=1}^{N} y_n\n$$\n$$\n\\bar{x} = \\frac{1}{N}\\sum_{n=1}^{N} x_n\n$$\n"]},{"cell_type":"code","metadata":{"nextjournal":{"id":"b4a1ecef-39bb-49b8-8e5c-5b4c1c7a9712","kind":"code","language":"julia"}},"source":["function solve_linear_regression(x, y)\n  x_mean = mean(x) # mean\u306f\u7b97\u8853\u5e73\u5747\u3092\u6c42\u3081\u308b\u95a2\u6570\n  y_mean = mean(y)\n  w_1 = sum((x .- x_mean) .* (y .- y_mean)) \/ sum((x .- x_mean).^2) #sum\u306f\u548c\u3092\u3068\u308b\u95a2\u6570\n  w_0 = y_mean - w_1 * x_mean\n  return w_0, w_1\nend"],"outputs":[{"output_type":"execute_result","data":{"text\/plain":"solve_linear_regression"},"execution_count":1,"metadata":{}}],"execution_count":1},{"cell_type":"markdown","metadata":{},"source":["\u305d\u308c\u3067\u306f\u3001\u5148\u307b\u3069\u306e\u30c7\u30fc\u30bf\u3092\u4f7f\u3063\u3066\u5358\u56de\u5e30\u3092\u884c\u3063\u3066\u307f\u307e\u3057\u3087\u3046\u3002\u60aa\u304f\u306a\u3044\u7d50\u679c\u3067\u3059\u306d"]},{"cell_type":"code","metadata":{"nextjournal":{"id":"8b3739c7-9854-4260-8032-51d510c0caa4","kind":"code","language":"julia"}},"source":["w_0, w_1 = solve_linear_regression(x, y)"],"outputs":[{"output_type":"execute_result","data":{"text\/plain":"(0.981166, 2.04889)"},"execution_count":1,"metadata":{}}],"execution_count":1},{"cell_type":"markdown","metadata":{},"source":["\u56de\u5e30\u306e\u7d50\u679c\u3092\u30c7\u30fc\u30bf\u306b\u91cd\u306d\u3066\u30d7\u30ed\u30c3\u30c8\u3057\u3066\u307f\u3066\u3001\u76ee\u3067\u78ba\u8a8d\u3057\u307e\u3057\u3087\u3046"]},{"cell_type":"code","metadata":{"nextjournal":{"id":"1582aa79-d6a4-4d4b-8fad-aebdc1e59cf5","kind":"code","language":"julia"}},"source":["x_grid = range(0.0, stop=1.0, length=100)\ny_hat = w_0 .+ w_1 .* x_grid\n\nscatter(x, y, xlabel=\"x\", ylabel=\"y\", label=\"data\")\nplot!(x_grid, y_hat, lw=3, label=\"regression\", legend=:topleft)  #regression\u306f\u56de\u5e30\u306e\u610f\u5473"],"outputs":[{"output_type":"execute_result","data":{"image\/svg+xml":"<?xml version=\"1.0\" encoding=\"utf-8\"?>\n<svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"600\" height=\"400\" viewBox=\"0 0 2400 1600\">\n<defs>\n  <clipPath id=\"clip310\">\n    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(\u91cd\u56de\u5e30)\n\n$x_1$\u3068$x_2$\u306e2\u3064\u306e\u5165\u529b\u304c\u3042\u308a\u3001\u5207\u72471\u3067\u305d\u308c\u305e\u308c\u50be\u304d2\u3068-2\u306b\u30ce\u30a4\u30ba\u3092\u52a0\u3048\u305f\u30e9\u30d9\u30eb\u3042\u308a\u30c7\u30fc\u30bf$\\{y_n, \\bm{x}_n\\}_{n=1}^{N}$\u3092$N=100$\u70b9\u3064\u304f\u3063\u3066\u30d7\u30ed\u30c3\u30c8\u3057\u307e\u3059"]},{"cell_type":"code","metadata":{"nextjournal":{"id":"72e0efef-7817-409c-bbeb-cab1f1e1588f","kind":"code","language":"julia"}},"source":["N = 100\nx_1 = rand(N)\nx_2 = rand(N)\ny = 1.0 .+ 2.0 .* x_1 - 2.0 .* x_2 .+ 0.3 .* randn(N)\n\np1 = scatter(x_1, y, xlabel=\"x_1\", ylabel=\"y\", legend=false)\np2 = scatter(x_2, y, xlabel=\"x_2\", ylabel=\"y\", legend=false)\nplot(p1, p2, layout=(2, 1), legend=false)"],"outputs":[{"output_type":"execute_result","data":{"image\/svg+xml":"<?xml version=\"1.0\" encoding=\"utf-8\"?>\n<svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" 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fill=\"#009af9\" fill-rule=\"evenodd\" fill-opacity=\"1\" stroke=\"#000000\" stroke-opacity=\"1\" stroke-width=\"3.2\"\/>\n<circle clip-path=\"url(#clip353)\" cx=\"1585.5\" cy=\"1274.04\" r=\"14\" fill=\"#009af9\" fill-rule=\"evenodd\" fill-opacity=\"1\" stroke=\"#000000\" stroke-opacity=\"1\" stroke-width=\"3.2\"\/>\n<circle clip-path=\"url(#clip353)\" cx=\"564.36\" cy=\"1159.81\" r=\"14\" fill=\"#009af9\" fill-rule=\"evenodd\" fill-opacity=\"1\" stroke=\"#000000\" stroke-opacity=\"1\" stroke-width=\"3.2\"\/>\n<circle clip-path=\"url(#clip353)\" cx=\"2149.35\" cy=\"1240.1\" r=\"14\" fill=\"#009af9\" fill-rule=\"evenodd\" fill-opacity=\"1\" stroke=\"#000000\" stroke-opacity=\"1\" stroke-width=\"3.2\"\/>\n<circle clip-path=\"url(#clip353)\" cx=\"1537.13\" cy=\"1140.66\" r=\"14\" fill=\"#009af9\" fill-rule=\"evenodd\" fill-opacity=\"1\" stroke=\"#000000\" stroke-opacity=\"1\" stroke-width=\"3.2\"\/>\n<circle clip-path=\"url(#clip353)\" cx=\"1966.05\" cy=\"1105.15\" r=\"14\" fill=\"#009af9\" fill-rule=\"evenodd\" fill-opacity=\"1\" stroke=\"#000000\" stroke-opacity=\"1\" stroke-width=\"3.2\"\/>\n<circle clip-path=\"url(#clip353)\" cx=\"1908.36\" cy=\"1198.8\" r=\"14\" fill=\"#009af9\" fill-rule=\"evenodd\" fill-opacity=\"1\" stroke=\"#000000\" stroke-opacity=\"1\" stroke-width=\"3.2\"\/>\n<circle clip-path=\"url(#clip353)\" cx=\"954.787\" cy=\"1055.13\" r=\"14\" fill=\"#009af9\" fill-rule=\"evenodd\" fill-opacity=\"1\" stroke=\"#000000\" stroke-opacity=\"1\" stroke-width=\"3.2\"\/>\n<circle clip-path=\"url(#clip353)\" cx=\"1000.17\" cy=\"1038\" r=\"14\" fill=\"#009af9\" fill-rule=\"evenodd\" fill-opacity=\"1\" stroke=\"#000000\" stroke-opacity=\"1\" stroke-width=\"3.2\"\/>\n<circle clip-path=\"url(#clip353)\" cx=\"294.589\" cy=\"863.544\" r=\"14\" fill=\"#009af9\" fill-rule=\"evenodd\" fill-opacity=\"1\" stroke=\"#000000\" stroke-opacity=\"1\" stroke-width=\"3.2\"\/>\n<circle clip-path=\"url(#clip353)\" cx=\"796.25\" cy=\"987.072\" r=\"14\" fill=\"#009af9\" fill-rule=\"evenodd\" fill-opacity=\"1\" stroke=\"#000000\" stroke-opacity=\"1\" stroke-width=\"3.2\"\/>\n<circle clip-path=\"url(#clip353)\" cx=\"409.366\" cy=\"968.04\" r=\"14\" fill=\"#009af9\" fill-rule=\"evenodd\" fill-opacity=\"1\" stroke=\"#000000\" stroke-opacity=\"1\" stroke-width=\"3.2\"\/>\n<circle clip-path=\"url(#clip353)\" cx=\"1628.28\" cy=\"1185.31\" r=\"14\" fill=\"#009af9\" fill-rule=\"evenodd\" fill-opacity=\"1\" stroke=\"#000000\" stroke-opacity=\"1\" stroke-width=\"3.2\"\/>\n<circle clip-path=\"url(#clip353)\" cx=\"1846.51\" cy=\"1174.53\" r=\"14\" fill=\"#009af9\" fill-rule=\"evenodd\" fill-opacity=\"1\" stroke=\"#000000\" stroke-opacity=\"1\" stroke-width=\"3.2\"\/>\n<\/svg>\n"},"execution_count":1,"metadata":{}}],"execution_count":1},{"cell_type":"markdown","metadata":{},"source":["\u91cd\u56de\u5e30\u3092\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u89e3\u304f\u305f\u3081\u306e\u95a2\u6570\u3092\u5b9a\u7fa9\u3057\u307e\u3057\u3087\u3046\n\n$$\n\\hat{\\bm{w}}=\\arg \\min \\ L(\\bm{w}) = \\frac{1}{2} \\sum_{n=1}^{N} (y_n - w_0 - \\sum_{i=1}^{p} w_i x_{n,i})^2 = \\| \\bm{y} - X \\bm{w} \\ \\|\n$$\n$$\n\\hat{\\bm{w}} = (X^T X)^{-1} X^T y\n$$\n"]},{"cell_type":"code","metadata":{"nextjournal":{"id":"03e71815-526a-4c1f-87d1-191930c4c86a","kind":"code","language":"julia"}},"source":["function solve_multiple_regression(X, y)\n  # X'\u306fX\u306e\u8ee2\u7f6e\u3092\u3068\u308a\u307e\u3059\u3002inv\u95a2\u6570\u306f\u9006\u884c\u5217\u3092\u8a08\u7b97\u3057\u307e\u3059\n  # \u672c\u5f53\u306e\u5b9f\u88c5\u3067\u306f\u3001pinv\u95a2\u6570\u3084\u7279\u7570\u5024\u5206\u89e3(SVD)\u3092\u7528\u3044\u308b\u307b\u3046\u304c\u5b89\u5b9a\u306b\u8a08\u7b97\u3067\u304d\u307e\u3059\n  w = inv(X' * X) * X' * y\n  return w\nend"],"outputs":[{"output_type":"execute_result","data":{"text\/plain":"solve_multiple_regression"},"execution_count":1,"metadata":{}}],"execution_count":1},{"cell_type":"markdown","metadata":{},"source":["\u305d\u308c\u3067\u306f\u3001\u5148\u307b\u3069\u306e\u30c7\u30fc\u30bf\u3092\u4f7f\u3063\u3066\u91cd\u56de\u5e30\u3092\u884c\u3063\u3066\u307f\u307e\u3057\u3087\u3046\u3002\u5148\u307b\u3069\u3068\u540c\u69d8\u306b\u60aa\u304f\u306a\u3044\u7d50\u679c\u3067\u3059\u306d"]},{"cell_type":"code","metadata":{"nextjournal":{"id":"0fd791c7-84e6-40d1-8dfb-fa2446ab9745","kind":"code","language":"julia"}},"source":["X = ones(N, 3)\nX[:, 2] .= x_1\nX[:, 3] .= x_2\n\nw = solve_multiple_regression(X, y)"],"outputs":[{"output_type":"execute_result","data":{"text\/plain":"3-element Array{Float64,1}:\n  1.09163\n  1.86706\n -2.05144"},"execution_count":1,"metadata":{}}],"execution_count":1},{"cell_type":"markdown","metadata":{},"source":["\u56de\u5e30\u306e\u7d50\u679c\u3092\u30c7\u30fc\u30bf\u306b\u91cd\u306d\u3066\u30d7\u30ed\u30c3\u30c8\u3057\u3066\u307f\u3066\u3001\u76ee\u3067\u78ba\u8a8d\u3057\u307e\u3057\u3087\u3046"]},{"cell_type":"code","metadata":{"nextjournal":{"id":"3413d65a-4d1a-4477-b3de-fb1e7f95c83b","kind":"code","language":"julia"}},"source":["y_hat = w[1] .+ w[2] .* x_1 .+ w[3] .* x_2\n\np1 = scatter(x_1, y, xlabel=\"x_1\", ylabel=\"y\", label=\"data\")\nscatter!(x_1, y_hat, label=\"regression\", legend=:topleft) #regression\u306f\u56de\u5e30\u306e\u610f\u5473\np2 = scatter(x_2, y, xlabel=\"x_2\", ylabel=\"y\", label=\"data\")\nscatter!(x_2, y_hat, label=\"regression\", legend=:topright)\n  plot(p1, p2, layout=(2, 1))"],"outputs":[{"output_type":"execute_result","data":{"image\/svg+xml":"<?xml version=\"1.0\" encoding=\"utf-8\"?>\n<svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"600\" height=\"400\" viewBox=\"0 0 2400 1600\">\n<defs>\n  <clipPath id=\"clip390\">\n    <rect x=\"0\" y=\"0\" width=\"2400\" height=\"1600\"\/>\n  <\/clipPath>\n<\/defs>\n<path clip-path=\"url(#clip390)\" d=\"\nM0 1600 L2400 1600 L2400 0 L0 0  Z\n  \" fill=\"#ffffff\" fill-rule=\"evenodd\" fill-opacity=\"1\"\/>\n<defs>\n  <clipPath id=\"clip391\">\n    <rect x=\"480\" y=\"0\" width=\"1681\" height=\"1600\"\/>\n  <\/clipPath>\n<\/defs>\n<path clip-path=\"url(#clip390)\" d=\"\nM205.723 623.18 L2352.76 623.18 L2352.76 47.2441 L205.723 47.2441  Z\n  \" fill=\"#ffffff\" fill-rule=\"evenodd\" fill-opacity=\"1\"\/>\n<defs>\n  <clipPath id=\"clip392\">\n    <rect x=\"205\" y=\"47\" width=\"2148\" height=\"577\"\/>\n  <\/clipPath>\n<\/defs>\n<polyline 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,100.0,1.5539,24,666,20.2,28.79,34.37,17.9\n28.6558,0.0,18.1,0,0.597,5.155,100.0,1.5894,24,666,20.2,210.97,20.08,16.3\n45.7461,0.0,18.1,0,0.693,4.519,100.0,1.6582,24,666,20.2,88.27,36.98,7.0\n18.0846,0.0,18.1,0,0.679,6.434,100.0,1.8347,24,666,20.2,27.25,29.05,7.2\n10.8342,0.0,18.1,0,0.679,6.782,90.8,1.8195,24,666,20.2,21.57,25.79,7.5\n25.9406,0.0,18.1,0,0.679,5.304,89.1,1.6475,24,666,20.2,127.36,26.64,10.4\n73.5341,0.0,18.1,0,0.679,5.957,100.0,1.8026,24,666,20.2,16.45,20.62,8.8\n11.8123,0.0,18.1,0,0.718,6.824,76.5,1.794,24,666,20.2,48.45,22.74,8.4\n11.0874,0.0,18.1,0,0.718,6.411,100.0,1.8589,24,666,20.2,318.75,15.02,16.7\n7.02259,0.0,18.1,0,0.718,6.006,95.3,1.8746,24,666,20.2,319.98,15.7,14.2\n12.0482,0.0,18.1,0,0.614,5.648,87.6,1.9512,24,666,20.2,291.55,14.1,20.8\n7.05042,0.0,18.1,0,0.614,6.103,85.1,2.0218,24,666,20.2,2.52,23.29,13.4\n8.79212,0.0,18.1,0,0.584,5.565,70.6,2.0635,24,666,20.2,3.65,17.16,11.7\n15.8603,0.0,18.1,0,0.679,5.896,95.4,1.9096,24,666,20.2,7.68,24.39,8.3\n12.2472,0.0,18.1,0,0.584,5.837,59.7,1.9976,24,666,20.2,24.65,15.69,10.2\n37.6619,0.0,18.1,0,0.679,6.202,78.7,1.8629,24,666,20.2,18.82,14.52,10.9\n7.36711,0.0,18.1,0,0.679,6.193,78.1,1.9356,24,666,20.2,96.73,21.52,11.0\n9.33889,0.0,18.1,0,0.679,6.38,95.6,1.9682,24,666,20.2,60.72,24.08,9.5\n8.49213,0.0,18.1,0,0.584,6.348,86.1,2.0527,24,666,20.2,83.45,17.64,14.5\n10.0623,0.0,18.1,0,0.584,6.833,94.3,2.0882,24,666,20.2,81.33,19.69,14.1\n6.44405,0.0,18.1,0,0.584,6.425,74.8,2.2004,24,666,20.2,97.95,12.03,16.1\n5.58107,0.0,18.1,0,0.713,6.436,87.9,2.3158,24,666,20.2,100.19,16.22,14.3\n13.9134,0.0,18.1,0,0.713,6.208,95.0,2.2222,24,666,20.2,100.63,15.17,11.7\n11.1604,0.0,18.1,0,0.74,6.629,94.6,2.1247,24,666,20.2,109.85,23.27,13.4\n14.4208,0.0,18.1,0,0.74,6.461,93.3,2.0026,24,666,20.2,27.49,18.05,9.6\n15.1772,0.0,18.1,0,0.74,6.152,100.0,1.9142,24,666,20.2,9.32,26.45,8.7\n13.6781,0.0,18.1,0,0.74,5.935,87.9,1.8206,24,666,20.2,68.95,34.02,8.4\n9.39063,0.0,18.1,0,0.74,5.627,93.9,1.8172,24,666,20.2,396.9,22.88,12.8\n22.0511,0.0,18.1,0,0.74,5.818,92.4,1.8662,24,666,20.2,391.45,22.11,10.5\n9.72418,0.0,18.1,0,0.74,6.406,97.2,2.0651,24,666,20.2,385.96,19.52,17.1\n5.66637,0.0,18.1,0,0.74,6.219,100.0,2.0048,24,666,20.2,395.69,16.59,18.4\n9.96654,0.0,18.1,0,0.74,6.485,100.0,1.9784,24,666,20.2,386.73,18.85,15.4\n12.8023,0.0,18.1,0,0.74,5.854,96.6,1.8956,24,666,20.2,240.52,23.79,10.8\n10.6718,0.0,18.1,0,0.74,6.459,94.8,1.9879,24,666,20.2,43.06,23.98,11.8\n6.28807,0.0,18.1,0,0.74,6.341,96.4,2.072,24,666,20.2,318.01,17.79,14.9\n9.92485,0.0,18.1,0,0.74,6.251,96.6,2.198,24,666,20.2,388.52,16.44,12.6\n9.32909,0.0,18.1,0,0.713,6.185,98.7,2.2616,24,666,20.2,396.9,18.13,14.1\n7.52601,0.0,18.1,0,0.713,6.417,98.3,2.185,24,666,20.2,304.21,19.31,13.0\n6.71772,0.0,18.1,0,0.713,6.749,92.6,2.3236,24,666,20.2,0.32,17.44,13.4\n5.44114,0.0,18.1,0,0.713,6.655,98.2,2.3552,24,666,20.2,355.29,17.73,15.2\n5.09017,0.0,18.1,0,0.713,6.297,91.8,2.3682,24,666,20.2,385.09,17.27,16.1\n8.24809,0.0,18.1,0,0.713,7.393,99.3,2.4527,24,666,20.2,375.87,16.74,17.8\n9.51363,0.0,18.1,0,0.713,6.728,94.1,2.4961,24,666,20.2,6.68,18.71,14.9\n4.75237,0.0,18.1,0,0.713,6.525,86.5,2.4358,24,666,20.2,50.92,18.13,14.1\n4.66883,0.0,18.1,0,0.713,5.976,87.9,2.5806,24,666,20.2,10.48,19.01,12.7\n8.20058,0.0,18.1,0,0.713,5.936,80.3,2.7792,24,666,20.2,3.5,16.94,13.5\n7.75223,0.0,18.1,0,0.713,6.301,83.7,2.7831,24,666,20.2,272.21,16.23,14.9\n6.80117,0.0,18.1,0,0.713,6.081,84.4,2.7175,24,666,20.2,396.9,14.7,20.0\n4.81213,0.0,18.1,0,0.713,6.701,90.0,2.5975,24,666,20.2,255.23,16.42,16.4\n3.69311,0.0,18.1,0,0.713,6.376,88.4,2.5671,24,666,20.2,391.43,14.65,17.7\n6.65492,0.0,18.1,0,0.713,6.317,83.0,2.7344,24,666,20.2,396.9,13.99,19.5\n5.82115,0.0,18.1,0,0.713,6.513,89.9,2.8016,24,666,20.2,393.82,10.29,20.2\n7.83932,0.0,18.1,0,0.655,6.209,65.4,2.9634,24,666,20.2,396.9,13.22,21.4\n3.1636,0.0,18.1,0,0.655,5.759,48.2,3.0665,24,666,20.2,334.4,14.13,19.9\n3.77498,0.0,18.1,0,0.655,5.952,84.7,2.8715,24,666,20.2,22.01,17.15,19.0\n4.42228,0.0,18.1,0,0.584,6.003,94.5,2.5403,24,666,20.2,331.29,21.32,19.1\n15.5757,0.0,18.1,0,0.58,5.926,71.0,2.9084,24,666,20.2,368.74,18.13,19.1\n13.0751,0.0,18.1,0,0.58,5.713,56.7,2.8237,24,666,20.2,396.9,14.76,20.1\n4.34879,0.0,18.1,0,0.58,6.167,84.0,3.0334,24,666,20.2,396.9,16.29,19.9\n4.03841,0.0,18.1,0,0.532,6.229,90.7,3.0993,24,666,20.2,395.33,12.87,19.6\n3.56868,0.0,18.1,0,0.58,6.437,75.0,2.8965,24,666,20.2,393.37,14.36,23.2\n4.64689,0.0,18.1,0,0.614,6.98,67.6,2.5329,24,666,20.2,374.68,11.66,29.8\n8.05579,0.0,18.1,0,0.584,5.427,95.4,2.4298,24,666,20.2,352.58,18.14,13.8\n6.39312,0.0,18.1,0,0.584,6.162,97.4,2.206,24,666,20.2,302.76,24.1,13.3\n4.87141,0.0,18.1,0,0.614,6.484,93.6,2.3053,24,666,20.2,396.21,18.68,16.7\n15.0234,0.0,18.1,0,0.614,5.304,97.3,2.1007,24,666,20.2,349.48,24.91,12.0\n10.233,0.0,18.1,0,0.614,6.185,96.7,2.1705,24,666,20.2,379.7,18.03,14.6\n14.3337,0.0,18.1,0,0.614,6.229,88.0,1.9512,24,666,20.2,383.32,13.11,21.4\n5.82401,0.0,18.1,0,0.532,6.242,64.7,3.4242,24,666,20.2,396.9,10.74,23.0\n5.70818,0.0,18.1,0,0.532,6.75,74.9,3.3317,24,666,20.2,393.07,7.74,23.7\n5.73116,0.0,18.1,0,0.532,7.061,77.0,3.4106,24,666,20.2,395.28,7.01,25.0\n2.81838,0.0,18.1,0,0.532,5.762,40.3,4.0983,24,666,20.2,392.92,10.42,21.8\n2.37857,0.0,18.1,0,0.583,5.871,41.9,3.724,24,666,20.2,370.73,13.34,20.6\n3.67367,0.0,18.1,0,0.583,6.312,51.9,3.9917,24,666,20.2,388.62,10.58,21.2\n5.69175,0.0,18.1,0,0.583,6.114,79.8,3.5459,24,666,20.2,392.68,14.98,19.1\n4.83567,0.0,18.1,0,0.583,5.905,53.2,3.1523,24,666,20.2,388.22,11.45,20.6\n0.15086,0.0,27.74,0,0.609,5.454,92.7,1.8209,4,711,20.1,395.09,18.06,15.2\n0.18337,0.0,27.74,0,0.609,5.414,98.3,1.7554,4,711,20.1,344.05,23.97,7.0\n0.20746,0.0,27.74,0,0.609,5.093,98.0,1.8226,4,711,20.1,318.43,29.68,8.1\n0.10574,0.0,27.74,0,0.609,5.983,98.8,1.8681,4,711,20.1,390.11,18.07,13.6\n0.11132,0.0,27.74,0,0.609,5.983,83.5,2.1099,4,711,20.1,396.9,13.35,20.1\n0.17331,0.0,9.69,0,0.585,5.707,54.0,2.3817,6,391,19.2,396.9,12.01,21.8\n0.27957,0.0,9.69,0,0.585,5.926,42.6,2.3817,6,391,19.2,396.9,13.59,24.5\n0.17899,0.0,9.69,0,0.585,5.67,28.8,2.7986,6,391,19.2,393.29,17.6,23.1\n0.2896,0.0,9.69,0,0.585,5.39,72.9,2.7986,6,391,19.2,396.9,21.14,19.7\n0.26838,0.0,9.69,0,0.585,5.794,70.6,2.8927,6,391,19.2,396.9,14.1,18.3\n0.23912,0.0,9.69,0,0.585,6.019,65.3,2.4091,6,391,19.2,396.9,12.92,21.2\n0.17783,0.0,9.69,0,0.585,5.569,73.5,2.3999,6,391,19.2,395.77,15.1,17.5\n0.22438,0.0,9.69,0,0.585,6.027,79.7,2.4982,6,391,19.2,396.9,14.33,16.8\n0.06263,0.0,11.93,0,0.573,6.593,69.1,2.4786,1,273,21.0,391.99,9.67,22.4\n0.04527,0.0,11.93,0,0.573,6.12,76.7,2.2875,1,273,21.0,396.9,9.08,20.6\n0.06076,0.0,11.93,0,0.573,6.976,91.0,2.1675,1,273,21.0,396.9,5.64,23.9\n0.10959,0.0,11.93,0,0.573,6.794,89.3,2.3889,1,273,21.0,393.45,6.48,22.0\n0.04741,0.0,11.93,0,0.573,6.03,80.8,2.505,1,273,21.0,396.9,7.88,11.9\n"},"execution_count":1,"metadata":{}}],"execution_count":1},{"cell_type":"markdown","metadata":{},"source":["\u3082\u3046\u4e00\u5ea6\u5404\u5217\u306e\u610f\u5473\u3092\u78ba\u8a8d\u3057\u307e\u3059\n\n 1. CRIM \u5730\u57df\u3054\u3068\u306e\u4e00\u4eba\u5f53\u305f\u308a\u306e\u72af\u7f6a\u7387 (\u753a\u3054\u3068\u306e\u4e00\u4eba\u3042\u305f\u308a) (\u8aac\u660e\u5909\u6570)\n 2. ZN 25,000 \u5e73\u65b9\u30d5\u30a3\u30fc\u30c8\u4ee5\u4e0a\u306e\u4f4f\u5b85\u533a\u753b\u306e\u5272\u5408 (\u8aac\u660e\u5909\u6570)\n 3. INDUS\t\u5c0f\u58f2\u696d\u4ee5\u5916\u306e\u5546\u696d\u304c\u5360\u3081\u308b\u571f\u5730\u9762\u7a4d\u306e\u5272\u5408 (\u8aac\u660e\u5909\u6570)\n 4. CHAS \u30c1\u30e3\u30fc\u30eb\u30ba\u5ddd\u6cbf\u3044\u304b\u3069\u3046\u304b\u306e\u5909\u6570\uff08\u5ddd\u6cbf\u30441\u3001\u305d\u308c\u4ee5\u59160\u3001\u8aac\u660e\u5909\u6570\uff09\n 5. NOX \u4e00\u9178\u5316\u7a92\u7d20\u6fc3\u5ea6\uff08ppgm\u5358\u4f4d\u3001\u8aac\u660e\u5909\u6570\uff09\n 6. RM 1\u6238\u3042\u305f\u308a\u306e\u90e8\u5c4b\u6570 (\u8aac\u660e\u5909\u6570)\n 7. AGE 1940\u5e74\u4ee5\u524d\u306b\u5efa\u8a2d\u3055\u308c\u305f\u7269\u4ef6\u306e\u5272\u5408 (\u8aac\u660e\u5909\u6570)\n 8. DIS \u5730\u57df\u304b\u30895\u3064\u306e\u96c7\u7528\u65bd\u8a2d\u307e\u3067\u306e\u91cd\u307f\u4ed8\u304d\u8ddd\u96e2 (\u8aac\u660e\u5909\u6570)\n 9. RAD \u9ad8\u901f\u9053\u8def\u3078\u306e\u30a2\u30af\u30bb\u30b9\u3057\u3084\u3059\u3055 (\u8aac\u660e\u5909\u6570)\n10. TAX 10,000\u30c9\u30eb\u3042\u305f\u308a\u306e\u56fa\u5b9a\u8cc7\u7523\u7a0e\u306e\u5272\u5408 (\u8aac\u660e\u5909\u6570)\n11. PTRATIO \u751f\u5f92\u3068\u6559\u5e2b\u306e\u6bd4\u7387 (\u8aac\u660e\u5909\u6570)\n12. B\t\u5730\u57df\u306b\u304a\u3051\u308b\u9ed2\u4eba\u306e\u5272\u5408 (\u8aac\u660e\u5909\u6570)\n13. LSTAT\t\u5730\u57df\u306b\u304a\u3051\u308b\u4f4e\u6240\u5f97\u8005\u306e\u5272\u5408 (\u8aac\u660e\u5909\u6570) \n14. MEDV \u4f4f\u5b85\u4fa1\u683c\u306e\u4e2d\u592e\u5730 (1000\u30c9\u30eb\u5358\u4f4d) **(\u76ee\u7684\u5909\u6570)**"]},{"cell_type":"code","metadata":{"nextjournal":{"id":"cf275f19-76c5-42da-9b59-e482dae61783","kind":"code","language":"julia"}},"source":["#\u4f5c\u6210\u3057\u305f\u95a2\u6570\u3067\u8a08\u7b97\u3067\u304d\u308b\u3088\u3046\u306b64\u30d3\u30c3\u30c8\u6d6e\u52d5\u5c0f\u6570\u70b9Float64\u306eVector\u578b(=1\u6b21\u5143Array\u578b)\u3078\u5909\u63db\u3059\u308b\ny = Vector{Float64}(boston.MEDV)\n#\u4f5c\u6210\u3057\u305f\u95a2\u6570\u3067\u8a08\u7b97\u3067\u304d\u308b\u3088\u3046\u306b64\u30d3\u30c3\u30c8\u6d6e\u52d5\u5c0f\u6570\u70b9Float64\u306eMatrfix\u578b(=2\u6b21\u5143Array\u578b)\u3078\u5909\u63db\u3059\u308b\nx = Matrix{Float64}(boston[:, 1:13])"],"outputs":[{"output_type":"execute_result","data":{"text\/plain":"506\u00d713 Array{Float64,2}:\n 0.00632  18.0   2.31  0.0  0.538  6.575  \u2026  1.0  296.0  15.3  396.9    4.98\n 0.02731   0.0   7.07  0.0  0.469  6.421     2.0  242.0  17.8  396.9    9.14\n 0.02729   0.0   7.07  0.0  0.469  7.185     2.0  242.0  17.8  392.83   4.03\n 0.03237   0.0   2.18  0.0  0.458  6.998     3.0  222.0  18.7  394.63   2.94\n 0.06905   0.0   2.18  0.0  0.458  7.147     3.0  222.0  18.7  396.9    5.33\n 0.02985   0.0   2.18  0.0  0.458  6.43   \u2026  3.0  222.0  18.7  394.12   5.21\n 0.08829  12.5   7.87  0.0  0.524  6.012     5.0  311.0  15.2  395.6   12.43\n 0.14455  12.5   7.87  0.0  0.524  6.172     5.0  311.0  15.2  396.9   19.15\n 0.21124  12.5   7.87  0.0  0.524  5.631     5.0  311.0  15.2  386.63  29.93\n 0.17004  12.5   7.87  0.0  0.524  6.004     5.0  311.0  15.2  386.71  17.1\n \u22ee                                 \u22ee      \u22f1               \u22ee            \n 0.26838   0.0   9.69  0.0  0.585  5.794     6.0  391.0  19.2  396.9   14.1\n 0.23912   0.0   9.69  0.0  0.585  6.019     6.0  391.0  19.2  396.9   12.92\n 0.17783   0.0   9.69  0.0  0.585  5.569     6.0  391.0  19.2  395.77  15.1\n 0.22438   0.0   9.69  0.0  0.585  6.027  \u2026  6.0  391.0  19.2  396.9   14.33\n 0.06263   0.0  11.93  0.0  0.573  6.593     1.0  273.0  21.0  391.99   9.67\n 0.04527   0.0  11.93  0.0  0.573  6.12      1.0  273.0  21.0  396.9    9.08\n 0.06076   0.0  11.93  0.0  0.573  6.976     1.0  273.0  21.0  396.9    5.64\n 0.10959   0.0  11.93  0.0  0.573  6.794     1.0  273.0  21.0  393.45   6.48\n 0.04741   0.0  11.93  0.0  0.573  6.03   \u2026  1.0  273.0  21.0  396.9    7.88"},"execution_count":1,"metadata":{}}],"execution_count":1},{"cell_type":"code","metadata":{"nextjournal":{"id":"9cd7a290-724e-4a49-b9b3-597d73920333","kind":"code","language":"julia"}},"source":["N = length(y)\nX = ones(N, 14)\nX[:, 2:14] .= x\n\n# \u3053\u3053\u3067solve_multiple_regression()\u3092\u547c\u3073\u51fa\u3057\u3066w\u3092\u6c42\u3081\u308b\nw = solve_multiple_regression(X, y)\n\n# w\u3092\u6574\u7406\u3057\u3066\u30d7\u30ea\u30f3\u30c8\nfor p = 1:14\n  println(\"w_$(p-1): $(w[p])\")\nend"],"outputs":[],"execution_count":1},{"cell_type":"code","metadata":{"nextjournal":{"id":"8c3d8794-5224-48eb-8b54-82a6874f2dd7","kind":"code","language":"julia"}},"source":["# \u5c0f\u30ec\u30dd\u30fc\u30c8\u8ab2\u984c \u8a2d\u554f1\n\n# \u6700\u3082\u5927\u304d\u306a\u50be\u304d(w_i)\u3092\u3082\u3064\u8aac\u660e\u5909\u6570\u306f\u4f55\u3067\u3057\u305f\u304b\uff1f\u307e\u305f\u305d\u306e\u7406\u7531\u3092\u8003\u5bdf\u3057\u3066\u8ff0\u3079\u3066\u304f\u3060\u3055\u3044\u3002\n# \u89e3\u7b54\u4f8b: \n# CHAS \u30c1\u30e3\u30fc\u30eb\u30ba\u5ddd\u6cbf\u3044\u304b\u3069\u3046\u304b\u306e\u5909\u6570\u3002\n# \u30c1\u30e3\u30fc\u30eb\u30ba\u5ddd\u6cbf\u3044\u306f\u4ed6\u306b\u6bd4\u3079\u3066\u767a\u5c55\u3057\u3066\u304a\u308a\u304c\u826f\u304f\u5730\u4fa1\u304c\u9ad8\u3044\u304b\u3089\u3002\n\n# \u30d7\u30ed\u30c3\u30c8\u3057\u3066\u307f\u3066\u81ea\u5206\u306e\u76ee\u3067\u78ba\u304b\u3081\u3066\u304f\u3060\u3055\u3044\n# \u4f8b\u3048\u3070CRIM\u306e\u5834\u5408\u3092\u30d7\u30ed\u30c3\u30c8\u3057\u3066\u3044\u307e\u3059\u30022\u30f6\u6240\u306eCRIM\u3092\u66f8\u304d\u63db\u3048\u3066\u304f\u3060\u3055\u3044\u3002\n\nscatter(boston.RM, boston.MEDV, xlabel=\"RM\", ylabel=\"MEDV\")"],"outputs":[{"output_type":"execute_result","data":{"image\/svg+xml":"<?xml version=\"1.0\" encoding=\"utf-8\"?>\n<svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"600\" height=\"400\" viewBox=\"0 0 2400 1600\">\n<defs>\n  <clipPath id=\"clip510\">\n    <rect x=\"0\" y=\"0\" width=\"2400\" height=\"1600\"\/>\n  <\/clipPath>\n<\/defs>\n<path clip-path=\"url(#clip510)\" d=\"\nM0 1600 L2400 1600 L2400 0 L0 0  Z\n  \" fill=\"#ffffff\" fill-rule=\"evenodd\" fill-opacity=\"1\"\/>\n<defs>\n  <clipPath id=\"clip511\">\n    <rect x=\"480\" y=\"0\" width=\"1681\" height=\"1600\"\/>\n  <\/clipPath>\n<\/defs>\n<path clip-path=\"url(#clip510)\" d=\"\nM202.459 1423.18 L2352.76 1423.18 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\u30d7\u30ed\u30c3\u30c8\u3057\u3066\u307f\u3066\u81ea\u5206\u306e\u76ee\u3067\u78ba\u304b\u3081\u3066\u304f\u3060\u3055\u3044\n# \u4f8b\u3048\u3070CRIM\u306e\u5834\u5408\u3092\u30d7\u30ed\u30c3\u30c8\u3057\u3066\u3044\u307e\u3059\u30022\u30f6\u6240\u306eCRIM\u3092\u66f8\u304d\u63db\u3048\u3066\u304f\u3060\u3055\u3044\u3002\n\nscatter(boston.NOX, boston.MEDV, xlabel=\"NOX\", ylabel=\"MEDV\")"],"outputs":[{"output_type":"execute_result","data":{"image\/svg+xml":"<?xml version=\"1.0\" encoding=\"utf-8\"?>\n<svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"600\" height=\"400\" viewBox=\"0 0 2400 1600\">\n<defs>\n  <clipPath id=\"clip550\">\n    <rect x=\"0\" y=\"0\" width=\"2400\" height=\"1600\"\/>\n  <\/clipPath>\n<\/defs>\n<path clip-path=\"url(#clip550)\" d=\"\nM0 1600 L2400 1600 L2400 0 L0 0  Z\n  \" fill=\"#ffffff\" fill-rule=\"evenodd\" fill-opacity=\"1\"\/>\n<defs>\n  <clipPath id=\"clip551\">\n    <rect x=\"480\" y=\"0\" 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