{"id":643,"date":"2026-05-10T14:22:04","date_gmt":"2026-05-10T06:22:04","guid":{"rendered":"https:\/\/blog.nanoturtle.cn\/?post_type=thread&#038;p=643"},"modified":"2026-05-23T14:04:35","modified_gmt":"2026-05-23T06:04:35","slug":"variance-minimization-principle","status":"publish","type":"thread","link":"https:\/\/blog.nanoturtle.cn\/index.php\/thread\/variance-minimization-principle\/","title":{"rendered":"\u79bb\u5dee\u5e73\u65b9\u548c\u6700\u5c0f\u5316\u7684\u6570\u5b66\u539f\u7406"},"content":{"rendered":"\n<p class=\"wp-block-paragraph\">\u79bb\u5dee\u5e73\u65b9\u548c\u6700\u5c0f\u5316\uff0c\u8fd9\u4e2a\u542c\u8d77\u6765\u6709\u70b9\u62d7\u53e3\u7684\u6982\u5ff5\uff0c\u5176\u5b9e\u662f\u7edf\u8ba1\u5b66\u91cc\u6700\u6734\u7d20\u7684\u76f4\u89c9\u4e4b\u4e00\u3002\u8bf4\u767d\u4e86\uff0c\u5c31\u662f\u5f53\u6211\u4eec\u60f3\u7528\u4e00\u7ec4\u6570\u91cc\u7684\u67d0\u4e2a\u201c\u4ee3\u8868\u503c\u201d\u6765\u6982\u62ec\u6574\u4f53\u65f6\uff0c\u600e\u4e48\u9009\u624d\u80fd\u8ba9\u6bcf\u4e2a\u6570\u636e\u70b9\u79bb\u8fd9\u4e2a\u4ee3\u8868\u503c\u7684\u201c\u8ddd\u79bb\u201d\u603b\u548c\u6700\u5c0f\u3002\u8fd9\u91cc\u7684\u201c\u8ddd\u79bb\u201d\u4e0d\u662f\u76f4\u7ebf\u8ddd\u79bb\uff0c\u800c\u662f\u5e73\u65b9\u540e\u7684\u8ddd\u79bb\u2014\u2014\u4e4b\u6240\u4ee5\u7528\u5e73\u65b9\uff0c\u662f\u56e0\u4e3a\u5b83\u65e2\u80fd\u907f\u514d\u6b63\u8d1f\u62b5\u6d88\uff0c\u53c8\u80fd\u653e\u5927\u504f\u79bb\u8f83\u5927\u7684\u70b9\uff0c\u8ba9\u4ee3\u8868\u503c\u66f4\u654f\u611f\u5730\u53cd\u6620\u6570\u636e\u7684\u96c6\u4e2d\u8d8b\u52bf\u3002<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">\u4e3a\u4ec0\u4e48\u504f\u504f\u662f\u5e73\u65b9\uff1f<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">\u4f60\u53ef\u80fd\u60f3\u8fc7\uff0c\u4e3a\u4ec0\u4e48\u4e0d\u7528\u7edd\u5bf9\u503c\uff1f\u7edd\u5bf9\u503c\u8ddd\u79bb\uff08\u5e73\u5747\u7edd\u5bf9\u8bef\u5dee\uff09\u4e5f\u80fd\u8861\u91cf\u504f\u79bb\uff0c\u4f46\u5b83\u5728\u6570\u5b66\u4e0a\u6709\u4e2a\u81f4\u547d\u7f3a\u9677\uff1a\u4e0d\u53ef\u5bfc\u3002\u5e73\u65b9\u548c\u51fd\u6570\u5904\u5904\u5149\u6ed1\uff0c\u6c42\u5bfc\u65b9\u4fbf\uff0c\u80fd\u76f4\u63a5\u901a\u8fc7\u5bfc\u6570\u7b49\u4e8e\u96f6\u627e\u5230\u6781\u503c\u70b9\u3002\u5bf9\u4e00\u7ec4\u6570\u636e (x_1, x_2, dots, x_n)\uff0c\u8bbe\u4ee3\u8868\u503c\u4e3a (c)\uff0c\u79bb\u5dee\u5e73\u65b9\u548c (S = sum (x_i &#8211; c)^2)\u3002\u5bf9 (c) \u6c42\u5bfc\u5e76\u4ee4\u5176\u4e3a\u96f6\uff0c\u5f97\u5230 (2sum (x_i &#8211; c) = 0)\uff0c\u89e3\u5f97 (c = frac{1}{n}sum x_i)\u2014\u2014\u6b63\u662f\u7b97\u672f\u5e73\u5747\u6570\u3002\u8fd9\u4e2a\u7ed3\u679c\u7b80\u6d01\u5f97\u4ee4\u4eba\u6109\u60a6\uff1a\u6700\u5c0f\u5316\u79bb\u5dee\u5e73\u65b9\u548c\u7684\u89e3\uff0c\u5c31\u662f\u5747\u503c\u3002<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">\u5206\u7ec4\u573a\u666f\u4e0b\u7684\u6570\u5b66\u53d8\u5f62<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">\u5f53\u95ee\u9898\u53d8\u6210\u201c\u628a\u6570\u636e\u5206\u6210\u4e24\u7ec4\uff0c\u4f7f\u4e24\u7ec4\u5404\u81ea\u7684\u7ec4\u5185\u79bb\u5dee\u5e73\u65b9\u548c\u4e4b\u548c\u6700\u5c0f\u201d\u65f6\uff0c\u539f\u7406\u4f9d\u7136\u6210\u7acb\uff0c\u4f46\u591a\u4e86\u4e00\u5c42\u7ec4\u5408\u4f18\u5316\u3002\u5047\u8bbe\u6570\u636e\u5df2\u6392\u5e8f\uff08\u6392\u5e8f\u662f\u524d\u63d0\uff0c\u56e0\u4e3a\u6700\u4f18\u5206\u7ec4\u5728\u6709\u5e8f\u6570\u636e\u4e0a\u4e00\u5b9a\u662f\u8fde\u7eed\u5206\u5272\uff09\uff0c\u603b\u79bb\u5dee\u5e73\u65b9\u548c (D^2 = sum (x_i &#8211; bar{x})^2) \u53ef\u4ee5\u5206\u89e3\u4e3a\u7ec4\u5185\u5e73\u65b9\u548c\u4e0e\u7ec4\u95f4\u5e73\u65b9\u548c\u4e4b\u548c\uff1a<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">[ D^2 = sum_{i=1}^{k} (x_i &#8211; bar{x}_1)^2 + sum_{i=k+1}^{n} (x_i &#8211; bar{x}_2)^2 + k(bar{x}_1 &#8211; bar{x})^2 + (n-k)(bar{x}_2 &#8211; bar{x})^2 ]<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u5176\u4e2d (bar{x}_1, bar{x}_2) \u5206\u522b\u662f\u4e24\u7ec4\u5747\u503c\u3002\u6ce8\u610f\uff0c\u603b\u5e73\u65b9\u548c (D^2) \u662f\u56fa\u5b9a\u7684\uff0c\u6240\u4ee5\u6700\u5c0f\u5316\u7ec4\u5185\u5e73\u65b9\u548c\u7b49\u4ef7\u4e8e\u6700\u5927\u5316\u7ec4\u95f4\u5e73\u65b9\u548c\u3002\u800c\u7ec4\u95f4\u5e73\u65b9\u548c\u53ea\u4f9d\u8d56\u4e8e\u4e24\u7ec4\u5747\u503c\u4e0e\u603b\u5747\u503c\u7684\u504f\u79bb\uff0c\u4ee5\u53ca\u5404\u7ec4\u6837\u672c\u91cf\u3002\u8fd9\u4e2a\u5173\u7cfb\u8ba9\u95ee\u9898\u4ece\u201c\u679a\u4e3e\u6240\u6709\u5206\u7ec4\u201d\u964d\u7ef4\u6210\u201c\u679a\u4e3e\u5206\u5272\u70b9\u201d\u2014\u2014\u56e0\u4e3a\u4e00\u65e6\u5206\u5272\u70b9\u786e\u5b9a\uff0c\u4e24\u7ec4\u5747\u503c\u5c31\u7531\u524d\u7f00\u548c\u76f4\u63a5\u7b97\u51fa\uff0c\u7ec4\u5185\u5e73\u65b9\u548c\u4e5f\u80fd\u901a\u8fc7\u603b\u5e73\u65b9\u548c\u51cf\u53bb\u7ec4\u95f4\u5e73\u65b9\u548c\u5feb\u901f\u5f97\u5230\uff0c\u65e0\u9700\u91cd\u590d\u8ba1\u7b97\u6bcf\u4e2a\u70b9\u5230\u5747\u503c\u7684\u5dee\u3002<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">\u4e00\u4e2a\u53cd\u76f4\u89c9\u7684\u7ec6\u8282<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">\u5f88\u591a\u4eba\u4ee5\u4e3a\uff0c\u6700\u4f18\u5206\u7ec4\u4e00\u5b9a\u662f\u8ba9\u4e24\u7ec4\u6570\u636e\u201c\u5c3d\u91cf\u5e73\u8861\u201d\uff0c\u6bd4\u5982\u6837\u672c\u91cf\u63a5\u8fd1\u3002\u4f46\u6570\u5b66\u4e0a\u5e76\u975e\u5982\u6b64\u3002\u7ec4\u95f4\u5e73\u65b9\u548c (k(bar{x}_1 &#8211; bar{x})^2 + (n-k)(bar{x}_2 &#8211; bar{x})^2) \u4e2d\uff0c\u6743\u91cd (k) \u548c (n-k) \u4f1a\u653e\u5927\u5747\u503c\u5dee\u5f02\u8f83\u5927\u7684\u90a3\u4e00\u7ec4\u7684\u5f71\u54cd\u3002\u5982\u679c\u4e00\u7ec4\u6570\u636e\u6781\u7aef\u504f\u79bb\u603b\u5747\u503c\uff0c\u5373\u4f7f\u6837\u672c\u91cf\u5f88\u5c0f\uff0c\u4e5f\u53ef\u80fd\u4ea7\u751f\u5f88\u5927\u7684\u7ec4\u95f4\u5e73\u65b9\u548c\uff0c\u4ece\u800c\u8ba9\u7ec4\u5185\u5e73\u65b9\u548c\u66f4\u5c0f\u3002\u4e3e\u4e2a\u4f8b\u5b50\uff1a\u6570\u636e [1, 2, 100]\uff0c\u603b\u5747\u503c\u7ea634.33\u3002\u82e5\u5206\u6210 [1,2] \u548c [100]\uff0c\u7ec4\u5185\u5e73\u65b9\u548c\u4ec5\u4e3a (1-1.5)^2+(2-1.5)^2=0.5\uff0c\u52a0\u4e0a\u53e6\u4e00\u7ec40\uff0c\u603b\u548c0.5\uff1b\u82e5\u5206\u6210 [1] \u548c [2,100]\uff0c\u7ec4\u5185\u5e73\u65b9\u548c\u4e3a0 + (2-51)^2+(100-51)^2=4802\uff0c\u5929\u5dee\u5730\u522b\u3002\u6240\u4ee5\u6700\u4f18\u5206\u7ec4\u5f80\u5f80\u4f1a\u628a\u79bb\u7fa4\u503c\u5355\u72ec\u62ce\u51fa\u6765\u3002<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">\u4ece\u6570\u5b66\u5230\u7b97\u6cd5<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">\u4e00\u7ef4\u6570\u636e\u7684\u6700\u4f18\u5206\u7ec4\uff0c\u53ea\u9700\u904d\u5386 (n-1) \u4e2a\u5206\u5272\u70b9\uff0c\u6bcf\u6b21\u7528\u524d\u7f00\u548c\u8ba1\u7b97\u5747\u503c\uff0c\u590d\u6742\u5ea6 (O(n))\u3002\u4f46\u9ad8\u7ef4\u60c5\u51b5\uff08\u6bd4\u5982K-means\u805a\u7c7b\uff09\u5c31\u6ca1\u8fd9\u4e48\u5e78\u8fd0\u4e86\uff0c\u56e0\u4e3a\u65e0\u6cd5\u6392\u5e8f\uff0c\u5fc5\u987b\u8fed\u4ee3\u903c\u8fd1\u3002\u4e0d\u8fc7\u6838\u5fc3\u601d\u60f3\u4e00\u8109\u76f8\u627f\uff1a\u79bb\u5dee\u5e73\u65b9\u548c\u6700\u5c0f\u5316\uff0c\u672c\u8d28\u4e0a\u662f\u5728\u5bfb\u627e\u6570\u636e\u5185\u5728\u7684\u201c\u91cd\u5fc3\u201d\uff0c\u65e0\u8bba\u8fd9\u4e2a\u91cd\u5fc3\u662f\u4e00\u4e2a\u70b9\uff0c\u8fd8\u662f\u591a\u4e2a\u7c07\u7684\u4e2d\u5fc3\u3002<\/p>\n","protected":false},"featured_media":0,"comment_status":"open","ping_status":"closed","template":"","class_list":["post-643","thread","type-thread","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/blog.nanoturtle.cn\/index.php\/wp-json\/wp\/v2\/thread\/643","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/blog.nanoturtle.cn\/index.php\/wp-json\/wp\/v2\/thread"}],"about":[{"href":"https:\/\/blog.nanoturtle.cn\/index.php\/wp-json\/wp\/v2\/types\/thread"}],"replies":[{"embeddable":true,"href":"https:\/\/blog.nanoturtle.cn\/index.php\/wp-json\/wp\/v2\/comments?post=643"}],"wp:attachment":[{"href":"https:\/\/blog.nanoturtle.cn\/index.php\/wp-json\/wp\/v2\/media?parent=643"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}