\u30c7\u30fc\u30bf\u304b\u3089\u56de\u5e30\u76f4\u7dda \\( y=\\hat{\\alpha} + \\hat{\\beta} x \\) \u3092\u6700\u5c0f\u4e8c\u4e57\u6cd5\u306b\u3088\u3063\u3066\u6c42\u3081\u305f\u5834\u5408\u306b\u3001\u56de\u5e30\u4fc2\u6570 \\(\\hat{\\alpha}, \\hat{\\beta}\\)\u306f\u6b21\u306e\u5f0f\u3067\u8868\u3055\u308c\u307e\u3059\u3002<\/p>\n\n\n\n
$$ \\hat{\\beta} = r_{xy}\\frac{s_{y}}{s_{x}} $$<\/p>\n\n\n\n
$$ \\hat{\\alpha} = \\bar{y} – \\hat{\\beta}\\bar{x_i} $$<\/p>\n\n\n\n
\\(r_{xy} : \u76f8\u95a2\u4fc2\u6570\\)<\/p>\n\n\n\n
\\(s_{x} : x\u306e\u6a19\u6e96\u504f\u5dee, s_{y} : y\u306e\u6a19\u6e96\u504f\u5dee,\\)<\/p>\n\n\n\n
\u4eca\u56de\u306f\u6b8b\u5dee\u5e73\u65b9\u548c\u306e\u5f0f\u304b\u3089\u3053\u306e\u56de\u5e30\u4fc2\u6570\u306e\u5f0f\u3092\u5c0e\u51fa\u3059\u308b\u904e\u7a0b\u3092\u30e1\u30e2\u3057\u3066\u3044\u304d\u307e\u3059\u3002<\/p>\n\n\n\n
\u307e\u305a\u3001\u6700\u5c0f\u4e8c\u4e57\u6cd5\u306b\u3088\u308a\u56de\u5e30\u4fc2\u6570\u3092\u6c42\u3081\u308b\u306b\u3042\u305f\u308a\u3001\u6b8b\u5dee\u5e73\u65b9\u548c\u306e\u5f0f\u3092\u7528\u610f\u3057\u307e\u3059\u3002<\/p>\n\n\n\n
$$ S(\\hat{\\alpha}, \\hat{\\beta}) = \\sum^{n}_{i=1}(y_i – \\hat{y})^2 = \\sum^{n}_{i=1}(y_i – (\\hat{\\alpha} + \\hat{\\beta} x_i ))^2 $$<\/p>\n\n\n\n
\u3053\u306e\\( S(\\hat{\\alpha}, \\hat{\\beta}) \\)\u304c\u6700\u5c0f\u306b\u306a\u308b\\(\\hat{\\alpha},\\hat{\\beta}\\)\u3092\u6c42\u3081\u308c\u3070\u3088\u3044\u306e\u3067\u3001\\( S(\\hat{\\alpha}, \\hat{\\beta}) \\)\u3092\u305d\u308c\u305e\u308c\\(\\hat{\\alpha}, \\hat{\\beta}\\)\u3067\u504f\u5fae\u5206\u3057\u305f\u5f0f\u30920\u3068\u7f6e\u3044\u305f\u5f0f\u3092\u6c42\u3081\u307e\u3059\u3002<\/p>\n\n\n\n
$$ \\frac{\\partial S}{\\partial \\hat{\\alpha}} = 2\\times(-1)\\times\\sum^{n}_{i=1}(y_i – \\hat{\\alpha} – \\hat{\\beta} x_i ) =0 \\\\ \u3000 \\sum^{n}_{i=1}y_i = \\sum^{n}_{i=1}\\hat{\\alpha} + \\sum^{n}_{i=1}\\hat{\\beta}x_i \\\\ = n\\hat{\\alpha} + \\hat{\\beta}\\sum^{n}_{i=1}x_i$$<\/p>\n\n\n\n
\u4e21\u8fba\u3092\\(n\\)\u3067\u5272\u308b\u3068\u3001<\/p>\n\n\n\n
$$ \\frac{\\sum^{n}_{i=1}y_i}{n} = \\hat{\\alpha} + \\frac{\\sum^{n}_{i=1}x_i}{n} \\\\ \\bar{y} = \\hat{\\alpha} + \\hat{\\beta}\\bar{x} \\\\ \\hat{\\alpha} = \\bar{y} – \\hat{\\beta}\\bar{x}$$<\/p>\n\n\n\n
\u3053\u308c\u3067\u3001\\(\\hat{\\alpha}\\)\u304c\u6c42\u307e\u308a\u307e\u3057\u305f\u3002<\/p>\n\n\n\n
\u6b21\u306b\\(\\hat{\\beta}\\)\u3092\u6c42\u3081\u308b\u305f\u3081\u3001\\(S\\)\u3092\\(\\hat{\\beta}\\)\u3067\u504f\u5fae\u5206\u30570\u3068\u304a\u304d\u307e\u3059\u3002<\/p>\n\n\n\n
$$ \\frac{\\partial S}{\\partial \\hat{\\beta}} = 2\\times(-1)\\times\\sum^{n}_{i=1}(y_i – \\hat{\\alpha} – \\hat{\\beta} x_i ) x_i =0 \\\\ \\sum^{n}_{i=1}x_i y_i = \\sum^{n}_{i=1}\\hat{\\alpha} x_i + \\sum^{n}_{i=1}\\hat{\\beta} x_i^2 \\\\ \\sum^{n}_{i=1}x_i y_i = \\hat{\\alpha}\\sum^{n}_{i=1}x_i + \\hat{\\beta}\\sum^{n}_{i=1}x_i^2$$<\/p>\n\n\n\n
\u4e21\u8fba\u3092\\(n\\)\u3067\u5272\u308b\u3068\u3001<\/p>\n\n\n\n
$$ \\frac{\\sum^{n}_{i=1}x_iy_i}{n} = \\hat{\\alpha}\\frac{\\sum^{n}_{i=1}x_i}{n} + \\hat{\\beta}{\\frac{\\sum^{n}_{i=1}x_i^2}{n}} \\\\ \\frac{\\sum^{n}_{i=1}x_iy_i}{n} = \\hat{\\alpha}\\bar{x} + \\hat{\\beta}{\\frac{\\sum^{n}_{i=1}x_i^2}{n}} $$<\/p>\n\n\n\n
\u3053\u3053\u3067\u3001\u5148\u306b\u6c42\u3081\u305f\\(\\hat{\\alpha} = \\bar{y} – \\hat{\\beta}\\bar{x}\\)\u3092\u4ee3\u5165\u3057\u3066\u3001<\/p>\n\n\n\n
$$ \\frac{\\sum^{n}_{i=1}x_iy_i}{n} = \\bar{x}\\bar{y} – \\hat{\\beta}\\bar{x}^2 + \\hat{\\beta}{\\frac{\\sum^{n}_{i=1}x_i^2}{n}} \\\\ = \\bar{x}\\bar{y} + \\hat{\\beta}(\\frac{\\sum^{n}_{i=1}x_i^2}{n}-\\bar{x}^2)$$<\/p>\n\n\n\n
\u3053\u3053\u3067\u3001\\( \\frac{\\sum^{n}_{i=1}x_i^2}{n}-\\bar{x}^2\\)\u306f\\(x\\)\u306e\u5206\u6563\u3092\u8868\u3059\u305f\u3081\u3001\u3053\u308c\u3092\\(s_{xx}\\)\u3068\u304a\u304f\u3068\u3001<\/p>\n\n\n\n
$$\\frac{\\sum^{n}_{i=1}x_iy_i}{n} = \\bar{x}\\bar{y} + \\hat{\\beta}s_{xx} \\\\ \\frac{\\sum^{n}_{i=1}x_iy_i }{n}- \\bar{x}\\bar{y}= \\hat{\\beta}s_{xx}$$<\/p>\n\n\n\n
\u3053\u3053\u3067\\(\\frac{\\sum^{n}_{i=1}x_iy_i}{n} – \\bar{x}\\bar{y}\\) \u306f\u6b21\u306e\u3088\u3046\u306b\u5909\u5f62\u3067\u304d\u308b\u3002<\/p>\n\n\n\n
$$ \\frac{\\sum^{n}_{i=1}x_iy_i}{n} + \\bar{x}\\bar{y} – 2\\bar{x}\\bar{y} \\\\ = \\frac{\\sum^{n}_{i=1}x_iy_i}{n} + \\frac{\\sum^{n}_{i=1}\\bar{x}\\bar{y}}{n} – \\frac{\\sum^{n}_{i=1}x_i}{n}\\bar{y} – \\frac{\\sum^{n}_{i=1}y_i}{n}\\bar{x} = \\frac{\\sum^{n}_{i=1}(x_iy_i – x_i\\bar{y} – \\bar{x}y_i + \\bar{x}\\bar{y})}{n}$$<\/p>\n\n\n\n
\u3088\u3063\u3066\\(\\frac{\\sum^{n}_{i=1}x_iy_i}{n}- \\bar{x}\\bar{y}\\) \u306f\\(x, y\\)\u306e\u5171\u5206\u6563\u3092\u8868\u3059\u305f\u3081\u3001\u3053\u308c\u3092\\(s_{xy}\\)\u3068\u304a\u304f\u3068\u3001<\/p>\n\n\n\n
$$s_{xy} = \\hat{\\beta}s_{xx} \\\\ \\hat{\\beta} = \\frac{s_{xy}}{s_{xx}}=\\frac{s_{xy}}{s_xs_y}\\cdot \\frac{s_y}{s_x} $$<\/p>\n\n\n\n
\u3053\u3053\u3067\u76f8\u95a2\u4fc2\u6570\\(r_{xy} = \\frac{s_{xy}}{s_{x}s_{y}}\\)\u3068\u304a\u304f\u3068\u3001<\/p>\n\n\n\n
$$ \\hat{\\beta} = r_{xy}\\frac{s_y}{s_x}$$<\/p>\n\n\n\n
<\/p>\n\n\n\n
\u3088\u3063\u3066\u3001\u56de\u5e30\u4fc2\u6570\u306f\u4ee5\u4e0b\u306e\u5f0f\u3067\u6c42\u307e\u308b\u3053\u3068\u304c\u5206\u304b\u308a\u307e\u3057\u305f\u3002<\/p>\n\n\n\n
$$ \\hat{\\beta} = r_{xy}\\frac{s_{y}}{s_{x}} $$<\/p>\n\n\n\n
$$ \\hat{\\alpha} = \\bar{y} – \\hat{\\beta}\\bar{x_i} $$<\/p>\n","protected":false},"excerpt":{"rendered":"
\u30c7\u30fc\u30bf\u304b\u3089\u56de\u5e30\u76f4\u7dda \\( y=\\hat{\\alpha} + \\hat...<\/p>\n","protected":false},"author":1,"featured_media":4910,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_locale":"ja","_original_post":"https:\/\/obenkyolab.com\/?p=4862","footnotes":""},"categories":[6],"tags":[],"class_list":["post-4862","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-6","ja"],"_links":{"self":[{"href":"https:\/\/obenkyolab.com\/index.php?rest_route=\/wp\/v2\/posts\/4862","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/obenkyolab.com\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/obenkyolab.com\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/obenkyolab.com\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/obenkyolab.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=4862"}],"version-history":[{"count":51,"href":"https:\/\/obenkyolab.com\/index.php?rest_route=\/wp\/v2\/posts\/4862\/revisions"}],"predecessor-version":[{"id":4975,"href":"https:\/\/obenkyolab.com\/index.php?rest_route=\/wp\/v2\/posts\/4862\/revisions\/4975"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/obenkyolab.com\/index.php?rest_route=\/wp\/v2\/media\/4910"}],"wp:attachment":[{"href":"https:\/\/obenkyolab.com\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=4862"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/obenkyolab.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=4862"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/obenkyolab.com\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=4862"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}