{"id":3324,"date":"2022-02-27T15:00:05","date_gmt":"2022-02-27T07:00:05","guid":{"rendered":"https:\/\/egonlin.com\/?p=3324"},"modified":"2022-02-27T15:06:13","modified_gmt":"2022-02-27T07:06:13","slug":"%e7%ac%ac%e4%b8%80%e7%af%87%ef%bc%9a%e6%9c%80%e5%b0%8f%e4%ba%8c%e4%b9%98%e6%b3%95","status":"publish","type":"post","link":"https:\/\/egonlin.com\/?p=3324","title":{"rendered":"\u7b2c\u4e00\u7bc7\uff1a\u6700\u5c0f\u4e8c\u4e58\u6cd5"},"content":{"rendered":"

\u6700\u5c0f\u4e8c\u4e58\u6cd5<\/h1>\n

  \u6700\u5c0f\u4e8c\u4e58\u6cd5\uff0c\u53ef\u4ee5\u7406\u89e3\u4e3a\u6700\u5c0f\u5e73\u65b9\u548c\uff0c\u5373\u8bef\u5dee\u7684\u6700\u5c0f\u5e73\u65b9\u548c\uff0c\u5728\u7ebf\u6027\u56de\u5f52\u4e2d\uff0c$\u8bef\u5dee=\u771f\u5b9e\u503c-\u9884\u6d4b\u503c$\u3002\u6700\u5c0f\u4e8c\u4e58\u6cd5\u7684\u6838\u5fc3\u601d\u60f3\u5c31\u662f\u2014\u2014\u901a\u8fc7\u6700\u5c0f\u5316\u8bef\u5dee\u7684\u5e73\u65b9\u548c\uff0c\u4f7f\u5f97\u62df\u5408\u5bf9\u8c61\u65e0\u9650\u63a5\u8fd1\u76ee\u6807\u5bf9\u8c61\uff0c\u6700\u5c0f\u4e8c\u4e58\u6cd5\u4e00\u822c\u89e3\u51b3\u7ebf\u6027\u95ee\u9898\u3002<\/p>\n

\u6700\u5c0f\u4e8c\u4e58\u6cd5\u2014\u2014\u4ee3\u6570\u6cd5<\/h1>\n

  \u5047\u8bbe\u7ebf\u6027\u56de\u5f52\u7684\u5047\u8bbe\u51fd\u6570\u4e3a
\n$$
\n\\begin{align}
\nh_\\omega(x_0,x_1,\\cdots,x_n) & = \\omega_0x_0+\\omega_1x_1+\\cdots+\\omega_nxn \\
\n& = \\sum<\/em>{i=0}^n \\omega_ix_i
\n\\end{align}
\n$$
\n\u5176\u4e2d$n-1$\u662f\u7279\u5f81\u6570\u3002\u5982\u679c\u9488\u5bf9\u6240\u6709\u7684$\\omega_i\\quad(i=1,2,\\cdots,n)$\u800c\u8a00\uff0c\u5047\u8bbe\u51fd\u6570\u662f\u975e\u7ebf\u6027\u7684\uff0c\u4f46\u662f\u9488\u5bf9\u67d0\u4e00\u4e2a$\\omega_i$\u7684\u8bdd\uff0c\u7531\u4e8e\u53d8\u91cf\u53ea\u5269\u4e0b\u4e00\u4e2a$\\omega_i$\uff0c\u5047\u8bbe\u51fd\u6570\u5c31\u662f\u7ebf\u6027\u7684\uff0c\u65e2\u53ef\u4ee5\u4f7f\u7528\u6700\u5c0f\u4e8c\u4e58\u6cd5\u6c42\u89e3\u3002<\/p>\n

  \u901a\u8fc7\u7ebf\u6027\u56de\u5f52\u7684\u5047\u8bbe\u51fd\u6570\u65e2\u53ef\u4ee5\u5f97\u5230\u76ee\u6807\u51fd\u6570\u4e3a
\n$$
\n\\begin{align}
\nJ(\\omega_0,\\omega_1,\\cdots,\\omegan) & = \\sum<\/em>{j=1}^m (h\\omega(x^{(j)})-y^{(j)})^2 \\
\n& = \\sum<\/em>{j=1}^m(\\sum_{i=0}^n \\omega_ix_i^{(j)} – y^{(j)})^2
\n\\end{align}
\n$$
\n\u5176\u4e2d$m$\u4e3a\u6837\u672c\u6570\u3002<\/p>\n

  \u5229\u7528\u76ee\u6807\u51fd\u6570\u5206\u522b\u5bf9$\\omegai$\u6c42\u504f\u5bfc\uff0c\u5e76\u4e14\u4ee4\u5bfc\u6570\u4e3a0\uff0c\u5373
\n$$
\n\\sum<\/em>{j=1}^m \\sum_{i=0}^n (\\omega_ix_i^{(j)} – y^{(j)})x_i^{(j)} = 0
\n$$
\n  \u901a\u8fc7\u6c42\u89e3\u4e0a\u5f0f\uff0c\u53ef\u4ee5\u5f97\u5230$n+1$\u5143\u4e00\u6b21\u65b9\u7a0b\u7ec4\uff0c\u901a\u8fc7\u6c42\u89e3\u8fd9\u4e2a\u65b9\u7a0b\u7ec4\u5c31\u53ef\u4ee5\u7684\u5f97\u5230\u6240\u6709\u7684$\\omega_i$\u3002<\/p>\n

\"\"<\/div><\/p>\n

\u6700\u5c0f\u4e8c\u4e58\u6cd5\u2014\u2014\u77e9\u9635\u6cd5<\/h1>\n

[rml_readmore]\uff1a
\n  \u6700\u5c0f\u4e8c\u4e58\u6cd5\u77e9\u9635\u6cd5\u6bd4\u4ee3\u6570\u6cd5\u7b80\u5355\u4e0d\u5c11\u3002\u6211\u4eec\u628a\u4ee3\u6570\u6cd5\u4e2d\u7ebf\u6027\u56de\u5f52\u7684\u5047\u8bbe\u51fd\u6570\u53ef\u4ee5\u5199\u6210
\n$$
\nh<\/em>\\omega(X) = X\\omega
\n$$
\n\u5176\u4e2d$h_\\omega(X)$\u662f$m1$\u7ef4\u7684\u5411\u91cf\uff0c$X$\u662f$m<\/em>n$\u7ef4\u7684\u77e9\u9635\uff0c$\\omega$\u662f$n*1$\u7ef4\u7684\u5411\u91cf\uff0c$m$\u4e3a\u6837\u672c\u6570\uff0c$n$\u4e3a\u7279\u5f81\u6570\u3002<\/p>\n

  \u901a\u8fc7\u4e0a\u8ff0\u77e9\u9635\u5f62\u5f0f\u7684\u5047\u8bbe\u51fd\u6570\u53ef\u4ee5\u5f97\u5230\u77e9\u9635\u5f62\u5f0f\u7684\u76ee\u6807\u51fd\u6570\u4e3a
\n$$
\nJ(\\omega)={\\frac{1}{2}}(X\\omega-Y)^T(X\\omega-Y)
\n$$
\n\u5176\u4e2d${\\frac{1}{2}}$\u53ea\u662f\u4e3a\u4e86\u65b9\u4fbf\u8ba1\u7b97\u3002<\/p>\n

  \u76ee\u6807\u51fd\u6570\u5bf9$\\omega$\u6c42\u5bfc\u53d60\uff0c\u53ef\u4ee5\u5f97
\n$$
\n\\nabla_\\omega{J(\\omega)} = X^T(X\\omega-Y) =0
\n$$
\n  \u4e0a\u8ff0\u6c42\u504f\u5bfc\u4f7f\u7528\u4e86\u77e9\u9635\u6c42\u5bfc\u94fe\u5f0f\u6cd5\u5219\u548c\u4e24\u4e2a\u77e9\u9635\u6c42\u5bfc\u7684\u516c\u5f0f
\n$$
\n\\begin{align}
\n& \\nabla_X(X^TX) = 2X \\
\n& \\nablaXf(AX+B) = A^T\\nabla<\/em>{AX+B}f
\n\\end{align}
\n$$
\n  \u901a\u8fc7\u5bf9\u4e0a\u8ff0\u5f0f\u5b50\u6574\u7406\u53ef\u5f97
\n$$
\n\\begin{align}
\n& X^TX\\omega=X^TX\\quad{\u4e24\u8fb9\u540c\u65f6\u4e58}(X^TX)^{-1} \\
\n& \\omega = (X^TX)^{-1}X^TY
\n\\end{align}
\n$$
\n  \u901a\u8fc7\u4e0a\u8ff0\u7684\u5316\u7b80\u53ef\u4ee5\u76f4\u63a5\u5bf9\u5411\u91cf$\\omega$\u6c42\u5bfc\uff0c\u800c\u4e0d\u9700\u8981\u5bf9$\\omega$\u4e2d\u7684\u6bcf\u4e00\u4e2a\u5143\u7d20\u6c42\u504f\u5bfc\u3002<\/p>\n

\u6700\u5c0f\u4e8c\u4e58\u6cd5\u4f18\u7f3a\u70b9<\/h1>\n

\u4f18\u70b9<\/h2>\n
    \n
  1. \u7b80\u6d01\u9ad8\u6548\uff0c\u6bd4\u68af\u5ea6\u4e0b\u964d\u6cd5\u65b9\u4fbf<\/li>\n<\/ol>\n

    \u7f3a\u70b9<\/h2>\n
      \n
    1. \u6700\u5c0f\u4e8c\u4e58\u6cd5\u9700\u8981\u8ba1\u7b97$X^TX$\u7684\u9006\u77e9\u9635\uff0c\u53ef\u80fd$X^TX$\u6ca1\u6709\u9006\u77e9\u9635(\u4e00\u822c\u9700\u8981\u8003\u8651\u4f7f\u7528\u5176\u4ed6\u7684\u4f18\u5316\u7b97\u6cd5\uff0c\u6216\u8005\u91cd\u65b0\u5904\u7406\u6570\u636e\u8ba9$X^TX$\u6709\u9006\u77e9\u9635)<\/li>\n
    2. \u5f53\u7279\u5f81\u6570$n$\u975e\u5e38\u5927\u7684\u65f6\u5019\uff0c$X^TX$\u7684\u8ba1\u7b97\u91cf\u975e\u5e38\u5927(\u4f7f\u7528\u968f\u673a\u68af\u5ea6\u4e0b\u964d\u6cd5\u6216\u4f7f\u7528\u964d\u7ef4\u7b97\u6cd5\u964d\u4f4e\u7279\u5f81\u7ef4\u5ea6)<\/li>\n
    3. \u6700\u5c0f\u4e8c\u4e58\u6cd5\u53ea\u6709\u62df\u5408\u51fd\u6570\u4e3a\u7ebf\u6027\u7684\u65f6\u5019\u624d\u53ef\u4ee5\u4f7f\u7528(\u60f3\u529e\u6cd5\u901a\u8fc7\u67d0\u4e9b\u673a\u5de7\u8ba9\u62df\u5408\u51fd\u6570\u8f6c\u5316\u4e3a\u7ebf\u6027\u7684)<\/li>\n<\/ol>\n","protected":false},"excerpt":{"rendered":"

      \u6700\u5c0f\u4e8c\u4e58\u6cd5   \u6700\u5c0f\u4e8c\u4e58\u6cd5\uff0c\u53ef\u4ee5\u7406\u89e3\u4e3a\u6700\u5c0f\u5e73\u65b9\u548c\uff0c\u5373\u8bef\u5dee\u7684\u6700\u5c0f\u5e73\u65b9\u548c\uff0c\u5728\u7ebf\u6027\u56de\u5f52\u4e2d\uff0c$\u8bef\u5dee […]<\/p>\n","protected":false},"author":3,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":[],"categories":[276,303,304],"tags":[],"_links":{"self":[{"href":"https:\/\/egonlin.com\/index.php?rest_route=\/wp\/v2\/posts\/3324"}],"collection":[{"href":"https:\/\/egonlin.com\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/egonlin.com\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/egonlin.com\/index.php?rest_route=\/wp\/v2\/users\/3"}],"replies":[{"embeddable":true,"href":"https:\/\/egonlin.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=3324"}],"version-history":[{"count":0,"href":"https:\/\/egonlin.com\/index.php?rest_route=\/wp\/v2\/posts\/3324\/revisions"}],"wp:attachment":[{"href":"https:\/\/egonlin.com\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=3324"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/egonlin.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=3324"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/egonlin.com\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=3324"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}