{"id":782,"date":"2024-12-03T17:16:23","date_gmt":"2024-12-03T09:16:23","guid":{"rendered":"https:\/\/blog.sssn.tech\/?p=782"},"modified":"2024-12-03T17:17:04","modified_gmt":"2024-12-03T09:17:04","slug":"svr-%e6%94%af%e6%8c%81%e5%90%91%e9%87%8f%e6%9c%ba%e5%9c%a8%e5%9b%9e%e5%bd%92%e9%97%ae%e9%a2%98%e4%b8%ad%e7%9a%84%e5%ba%94%e7%94%a8","status":"publish","type":"post","link":"https:\/\/blog.sssn.tech\/?p=782","title":{"rendered":"SVR: \u652f\u6301\u5411\u91cf\u673a\u5728\u56de\u5f52\u95ee\u9898\u4e2d\u7684\u5e94\u7528"},"content":{"rendered":"

SVR: \u652f\u6301\u5411\u91cf\u673a\u5728\u56de\u5f52\u95ee\u9898\u4e2d\u7684\u5e94\u7528<\/h1>\n

\u5728\u9605\u8bfb\u8fd9\u7bc7\u535a\u5ba2\u524d, \u8bf7\u786e\u4fdd\u81ea\u5df1\u5bf9SVM\u6709\u521d\u6b65\u7684\u4e86\u89e3, \u5982\u679c\u8fd8\u4e0d\u6e05\u695aSVM\u662f\u4ec0\u4e48, \u53ef\u4ee5\u770b\u4e0b\u6211\u4e0a\u7bc7\u4ecb\u7ecdSVM\u7684\u535a\u5ba2<\/p>\n

\u56de\u5f52\u95ee\u9898\u6982\u8ff0<\/h2>\n

\u73b0\u5728\u6709\u4e00\u7ed9\u5b9a\u7684\u6570\u636e\u96c6<\/p>\n

T = \\{(x_1, y_1), (x_2, y_2), ..., (x_l, y_l)\\}<\/code><\/pre>\n
\u5176\u4e2d<\/p>\n
x_i \\in \\mathbb{R}^n, y_i \\in \\mathbb{R}<\/code><\/pre>\n
\u6211\u4eec\u8981\u6839\u636e\u6570\u636e\u96c6, \u786e\u5b9a\u4e00\u4e2a\u5728\\mathbb{R}^n<\/code>\u4e0a\u7684\u5b9e\u503c\u51fd\u6570y=f(x)<\/code>, \u6765\u63a8\u65ad\u5c06\u6765\u7ed9\u51fa\u7684\u4efb\u610f\u6a21\u5f0fx<\/code>\u5bf9\u5e94\u7684y<\/code>\u503c<\/p>\n
\"image-20241203153038309\"<\/p>\n
\u56de\u987eSVM\u5206\u7c7b<\/h2>\n
\u5bf9\u4e8e\u4e00\u7ed9\u5b9a\u7684\u4e8c\u5206\u7c7b\u6570\u636e\u96c6, \u6211\u4eec\u8981\u786e\u5b9a\u4e00\u4e2a\u8d85\u5e73\u9762<\/p>\n
y = w \\cdot x + b<\/code><\/pre>\n
\u6765\u505a\u51fa\u5206\u7c7b, \u6211\u4eec\u5e0c\u671b\u79bb\u8d85\u5e73\u9762\u6700\u8fd1\u7684\u6837\u672c\u70b9\u4e4b\u95f4\u8ddd\u79bb\u6700\u5927, \u8fd9\u4e2a\u95ee\u9898\u53ef\u4ee5\u8868\u793a\u4e3a<\/p>\n
\\underset{w, b}{min} \\frac{1}{2} ||w||^2 \\\\\ns.t. \\, y_i(w \\cdot x_i + b) \\ge 1, \\forall i<\/code><\/pre>\n
\u4e0d\u719f\u6089\u8fd9\u4e2a\u8868\u793a\u7684\u540c\u5b66, \u8bf7\u53bb\u770b\u770b\u5f00\u5934\u7ed9\u51fa\u7684\u4ecb\u7ecdSVM\u7684\u535a\u5ba2<\/p>\n
\"image-20241203153530197\"<\/p>\n
\u56de\u5f52\u95ee\u9898: SVR<\/h2>\n
\u5728SVR(Support Vector Regression)\u4e2d, \u540c\u6837\u662f\u8ba1\u7b97\u95f4\u9694, \u4e0d\u540c\u7684\u662f, \u6211\u4eec\u5e0c\u671b\u79bb\u5e73\u9762\u6700\u8fdc\u7684\u6837\u672c\u70b9\u4e4b\u95f4\u95f4\u9694\u6700\u5c0f<\/p>\n
    \n
  • SVM: \u79bb\u5e73\u9762\u6700\u8fd1\u7684\u70b9\u95f4\u9694\u6700\u5927<\/li>\n
  • SVR: \u79bb\u5e73\u9762\u6700\u8fdc\u7684\u70b9\u95f4\u9694\u6700\u5c0f<\/li>\n<\/ul>\n
    \"image-20241203153759903\"<\/p>\n
    \u5728\u4e0a\u56fe\u7684\u56de\u5f52\u95ee\u9898\u4e2d, \u663e\u7136\u53f3\u56fe\u7684\u56de\u5f52\u66f4\u597d\u4e00\u4e9b<\/p>\n
    \u76f8\u6bd4\u4e8eSVM\u7684\u4f18\u5316\u76ee\u6807<\/p>\n
    \\underset{w, b}{min} \\frac{1}{2} ||w||^2 \\\\\ns.t. \\, y_i(w \\cdot x_i + b) \\ge 1, \\forall i<\/code><\/pre>\n
    SVR\u7684\u4f18\u5316\u76ee\u6807\u88ab\u4fee\u6539\u4e3a<\/p>\n
    \\underset{w, b}{min} \\frac{1}{2} ||w||^2 \\\\\ns.t. \\, |y_i - (w \\cdot x_i + b)| \\le \\varepsilon, \\forall i<\/code><\/pre>\n
    \u5176\u4e2d<\/p>\n
      \n
    • \\varepsilon<\/code>\u662f\u4eba\u4e3a\u8bbe\u5b9a\u7684\u53c2\u6570, \u4ee3\u8868\u5141\u8bb8\u7684\u6700\u5927\u94c5\u9524\u8bef\u5dee<\/li>\n<\/ul>\n
      \u5728SVM\u4e2d, \u6211\u4eec\u5e0c\u671b\u79bb\u5e73\u9762\u6700\u8fd1\u7684\u70b9\u95f4\u9694\u6700\u5927, \u5373, \u4e0a\u4e0b\u4e34\u754c\u5e73\u9762\u95f4\u9694\u6700\u5927, \u8fd9\u4e2a\u95f4\u9694\u662f<\/p>\n
      d_{svm} = \\frac{2}{||w||}<\/code><\/pre>\n
      \u800c\u5728SVR\u4e2d, \u6211\u4eec\u5e0c\u671b\u79bb\u5e73\u9762\u6700\u8fdc\u7684\u70b9\u95f4\u9694\u6700\u5c0f, \u5373, \u4e0a\u4e0b\u4e34\u754c\u5e73\u9762\u95f4\u9694\u6700\u5927, \u8fd9\u4e2a\u95f4\u9694\u662f<\/p>\n
      d_{svm} = \\frac{2 \\varepsilon}{\\sqrt{1+||w||^2}}<\/code><\/pre>\n
      \u843d\u5728\u4e34\u754c\u5e73\u9762\u4e0a\u7684\u70b9, \u5c31\u88ab\u53eb\u505a\u652f\u6301\u5411\u91cf<\/p>\n
      \u5728\u53e6\u4e00\u65b9\u9762, ||w||^2<\/code>\u4f5c\u4e3a\u5e73\u9762\u6cd5\u5411\u91cf\u7684\u8303\u6570, \u5176\u503c\u4e5f\u4ee3\u8868\u4e86\u5e73\u9762\u53c2\u6570\u7684\u590d\u6742\u7a0b\u5ea6, \u6700\u5c0f\u5316\u4ed6\u53ef\u4ee5\u63a7\u5236\u590d\u6742\u5ea6, \u907f\u514d\u8fc7\u62df\u5408, \u8fd9\u4e5f\u53ef\u4ee5\u88ab\u770b\u4f5c\u662f\u4e00\u79cd\u6b63\u5219\u5316<\/p>\n
      \u8fdb\u4e00\u6b65\u63a8\u5e7f: \u975e\u7ebf\u5f62\u62df\u5408\u4e0e\u677e\u5f1b\u56e0\u5b50<\/h2>\n
      \u975e\u7ebf\u5f62\u62df\u5408<\/strong><\/p>\n
      \u4e3a\u4e86\u8ba9\u6211\u4eec\u53ef\u4ee5\u62df\u5408\u51fa\u66f2\u7ebf(\u9762), \u6211\u4eec\u628a\u76ee\u6807\u51fd\u6570\u4ece<\/p>\n
      y = w \\cdot x + b<\/code><\/pre>\n
      \u63a8\u5e7f\u5230\u975e\u7ebf\u5f62\u60c5\u51b5\u5219\u662f<\/p>\n
      y = w \\cdot \\varphi(x) + b<\/code><\/pre>\n
      \u4e3a\u4e86\u53bb\u9664\u5f02\u5e38\u6570\u636e\u5bf9\u7ed3\u679c\u7684\u5f71\u54cd, \u540c\u6837\u7684, SVR\u4e5f\u53ef\u4ee5\u5f15\u5165\u677e\u5f1b\u56e0\u5b50<\/p>\n
      \\underset{w, b}{min} \\frac{1}{2} ||w||^2 + C \\sum_{i=1}^{l} (\\xi_i + \\xi_i^*)\n\\\\\ns.t. \\, w \\cdot \\varphi(x_i) + b - y_i \\le \\varepsilon + \\xi_i, \\forall i \\\\\ns.t. \\, y_i - w \\cdot \\varphi(x_i) - b \\le \\varepsilon + \\xi_i^*, \\forall i \\\\\n\\xi_i, \\xi_i^* \\ge 0<\/code><\/pre>\n
      \u5176\u4e2d<\/p>\n
        \n
      • \\xi_i, \\xi_i^*<\/code>\u4e3a\u6837\u672c\u70b9\u677e\u5f1b\u53d8\u91cf\n
          \n
        • \\xi_i<\/code>\u662f\u6b63\u5411\u8d85\u51fa\u91cf<\/li>\n
        • \\xi_i^*<\/code>\u662f\u8d1f\u5411\u8d85\u51fa\u91cf<\/li>\n<\/ul>\n<\/li>\n
        • C<\/code>\u7528\u6765\u5e73\u8861\u5bbd\u5bb9\u5ea6, \u8d8a\u5927\u8d8a\u4e25\u683c, \u8d8a\u5c0f\u8d8a\u5bbd\u5bb9<\/li>\n<\/ul>\n
          SVR\u53ea\u5bf9\u95f4\u9694\u5916\u7684\u6837\u672c\u8fdb\u884c\u60e9\u7f5a, \u5f53\u6837\u672c\u70b9\u5728\u95f4\u9694\u5185\u65f6, \u4e0d\u8ba1\u7b97\u635f\u5931<\/p>\n","protected":false},"excerpt":{"rendered":"
          SVR: \u652f\u6301\u5411\u91cf\u673a\u5728\u56de\u5f52\u95ee\u9898\u4e2d\u7684\u5e94\u7528 \u5728\u9605\u8bfb\u8fd9\u7bc7\u535a\u5ba2\u524d, \u8bf7\u786e\u4fdd\u81ea\u5df1\u5bf9SVM\u6709 […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[16,30,18],"tags":[],"class_list":["post-782","post","type-post","status-publish","format-standard","hentry","category-16","category-30","category-18"],"_links":{"self":[{"href":"https:\/\/blog.sssn.tech\/index.php?rest_route=\/wp\/v2\/posts\/782","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/blog.sssn.tech\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/blog.sssn.tech\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/blog.sssn.tech\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/blog.sssn.tech\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=782"}],"version-history":[{"count":2,"href":"https:\/\/blog.sssn.tech\/index.php?rest_route=\/wp\/v2\/posts\/782\/revisions"}],"predecessor-version":[{"id":784,"href":"https:\/\/blog.sssn.tech\/index.php?rest_route=\/wp\/v2\/posts\/782\/revisions\/784"}],"wp:attachment":[{"href":"https:\/\/blog.sssn.tech\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=782"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blog.sssn.tech\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=782"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blog.sssn.tech\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=782"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}