{"id":1653,"date":"2012-10-26T15:55:34","date_gmt":"2012-10-26T07:55:34","guid":{"rendered":"http:\/\/www.yewen.us\/blog\/?p=1653"},"modified":"2012-10-26T15:55:34","modified_gmt":"2012-10-26T07:55:34","slug":"machine-learning-3-linear-regression","status":"publish","type":"post","link":"https:\/\/www.yewen.us\/blog\/2012\/10\/machine-learning-3-linear-regression\/","title":{"rendered":"\u673a\u5668\u5b66\u4e60\u624b\u8bb0\u7cfb\u5217 3: \u7ebf\u6027\u56de\u5f52\u548c\u6700\u5c0f\u4e8c\u4e58\u6cd5"},"content":{"rendered":"

\u597d\u51e0\u4e2a\u6708\u6ca1\u518d\u7ee7\u7eed, \u6316\u5751\u4e0d\u586b\u662f\u4e0d\u5bf9\u7684, \u8fd8\u662f\u7ee7\u7eed\u5199\u624b\u8bb0.<\/p>\n

\u7ebf\u6027\u56de\u5f52<\/strong><\/p>\n

\u7ebf\u6027\u56de\u5f52\u4e00\u822c\u7528\u6765\u5b66\u4e60\u4e00\u7ef4\u81ea\u53d8\u91cf\u548c\u4e00\u7ef4\u56e0\u53d8\u91cf\u4e4b\u95f4\u7684\u7ebf\u6027\u5173\u7cfb. \u5982\u679c\u5b58\u5728\u4e00\u7ef4\u81ea\u53d8\u91cf x, \u540c\u65f6\u8fd8\u6709\u4e00\u7ef4\u56e0\u53d8\u91cf y = f(x), \u5982\u679c\u6709\u4e00\u5806\u5bf9\u4e0d\u540c x \u4e0b y \u7684\u89c2\u6d4b\u503c, \u5373 \u7684\u89c2\u6d4b\u5bf9, \u4e14\u5982\u679c x, y \u4e4b\u95f4\u5b58\u5728\u8f83\u660e\u663e\u7684\u7ebf\u6027\u5173\u7cfb, \u53ef\u4ee5\u7528 f(x) = a*x + b<\/code> \u8fd9\u6837\u7684\u65b9\u7a0b\u8868\u793a, \u5219\u53ef\u4ee5\u7528\u7ebf\u6027\u56de\u5f52\u7684\u65b9\u6cd5\u5b66\u4e60\u51fa a \u548c b \u7684\u503c, \u540c\u65f6\u4f30\u8ba1\u8fd9\u4e2a\u62df\u5408\u65b9\u6cd5\u7684\u8bef\u5dee r.<\/p>\n

\u6269\u5c55\u4e00\u4e0b, \u7ebf\u6027\u56de\u5f52\u4e5f\u53ef\u4ee5\u6307 y = a1<\/sub>*x1<\/sub> + a2<\/sub>*x2<\/sub> + ... + ak<\/sub>*xk<\/sub> + b<\/code> \u8fd9\u6837\u591a\u7ef4\u81ea\u53d8\u91cf\u548c\u4e00\u7ef4\u56e0\u53d8\u91cf\u4e4b\u95f4\u7684\u7ebf\u6027\u5173\u7cfb (\u591a\u9879\u5f0f\u91cc\u81ea\u53d8\u91cf\u7684\u6700\u9ad8\u5e42\u6b21\u90fd\u662f 1), \u540c\u6837\u4e5f\u53ef\u4ee5\u7528\u56de\u5f52\u7684\u65b9\u5f0f\u5b66\u51fa\u6765\u91cc\u9762\u4e0d\u540c\u7684\u7cfb\u6570\u548c\u5e38\u6570\u9879\u7684\u503c. \u53ea\u6709\u4e00\u7ef4\u81ea\u53d8\u91cf\u7684\u79f0\u4e4b\u4e3a\u4e00\u5143\u7ebf\u6027\u56de\u5f52, \u5426\u5219\u662f\u591a\u5143\u7ebf\u6027\u56de\u5f52.<\/p>\n

\u5224\u65ad\u7ebf\u6027\u56de\u5f52\u597d\u574f, \u4e00\u822c\u5c31\u7528\u5e73\u65b9\u8bef\u5dee\u548c\u6765\u63cf\u8ff0, \u5176\u8868\u8fbe\u4e3a (f(x1<\/sub>)-y1<\/sub>)2<\/sup> + (f(x2<\/sub>)-y2<\/sub>)2<\/sup> + ... + (f(xk<\/sub>)-yk<\/sub>)2<\/sup>)<\/code>, \u6b64\u503c\u5982\u679c\u4e3a 0 \u5219\u8bf4\u660e\u81ea\u53d8\u91cf\u548c\u56e0\u53d8\u91cf\u5b58\u5728\u5b8c\u5168\u7684\u7ebf\u6027\u5173\u7cfb, \u5426\u5219\u662f\u8fd1\u4f3c\u7ebf\u6027, \u8d8a\u5c0f\u8fd1\u4f3c\u7684\u8d8a\u597d. \u8fd9\u4e2a\u4e1c\u897f\u770b\u7740\u6709\u6ca1\u6709\u4e00\u70b9\u9762\u719f? \u5176\u5b9e\u5c31\u662f\u673a\u5668\u5b66\u4e60\u624b\u8bb0\u7cfb\u5217 2: \u79bb\u7ebf\u6548\u679c\u8bc4\u4f30<\/a>\u91cc\u6700\u540e\u63d0\u5230\u7684 MSE \u7684\u975e\u5e73\u5747\u7248\u672c.<\/p>\n

\u89e3\u65b9\u7a0b\u6cd5<\/strong><\/p>\n

\u5047\u5982\u8f93\u5165\u6837\u672c\u5b58\u5728\u7edd\u5bf9\u7684\u7ebf\u6027\u5173\u7cfb, \u5373\u6700\u540e\u7684\u8bef\u5dee\u4e3a 0, \u5219\u95ee\u9898\u53d8\u4e3a\u89e3\u4e8c\u5143\u4e00\u6b21\u65b9\u7a0b (\u4e00\u5143\u7ebf\u6027\u56de\u5f52\u91cc\u7684\u7cfb\u6570\u52a0\u5e38\u6570) \u6216 N+1 \u5143\u4e00\u6b21\u65b9\u7a0b (\u591a\u5143\u7ebf\u6027\u56de\u5f52\u91cc N \u4e2a\u81ea\u53d8\u91cf\u7684\u7cfb\u6570\u52a0\u5e38\u6570). \u8fd9\u6ca1\u4ec0\u4e48\u597d\u8bf4\u7684, \u76f4\u63a5\u5bf9\u539f\u8f93\u5165\u76f4\u63a5\u6c42\u89e3\u5c31\u884c\u4e86, \u7c7b\u4f3c\u8ba1\u7b97\u65b9\u6cd5\u8fd9\u6837\u7684\u8bfe\u672c\u4e0a\u6709\u7684\u662f\u89e3\u6cd5, \u505a\u68af\u5ea6\u4e0b\u964d\u6216\u725b\u987f\u6cd5\u4e43\u81f3\u77e9\u9635\u5206\u89e3\u90fd\u53ef\u4ee5\u89e3.<\/p>\n

\u68af\u5ea6\u4e0b\u964d\u6cd5<\/strong><\/p>\n

\u8003\u8651\u5230\u7edd\u5927\u90e8\u5206\u60c5\u51b5\u4e0b\u4e0d\u5b58\u5728\u7edd\u5bf9\u7684\u7ebf\u6027\u5173\u7cfb, \u5219\u95ee\u9898\u53ef\u4ee5\u53d8\u6210\u600e\u4e48\u6c42\u5e73\u65b9\u8bef\u5dee\u548c\u7684\u6700\u5c0f\u503c\u70b9. \u5982\u679c\u662f\u4e00\u5143\u7ebf\u6027\u56de\u5f52, \u76ee\u6807\u51fd\u6570\u53d8\u4e3a g(a, b) = (f(x1<\/sub>)-y1<\/sub>)2<\/sup> + (f(x2<\/sub>)-y2<\/sub>)2<\/sup> + ... + (f(xk<\/sub>)-yk<\/sub>)2<\/sup><\/code><\/p>\n

\u6211\u4eec\u7684\u76ee\u6807\u5c31\u662f\u8ba9\u8fd9\u4e2a\u76ee\u6807\u51fd\u6570\u503c\u6700\u5c0f, \u9009\u5b9a\u4e00\u7ec4 \u7684\u521d\u59cb\u503c, \u7136\u540e\u6c42\u5176\u68af\u5ea6\u65b9\u5411, \u6bcf\u6b21\u524d\u8fdb\u4e00\u4e2a\u5c0f\u6b65\u957f, \u518d\u6c42\u68af\u5ea6\u524d\u8fdb, \u76f4\u5230\u76ee\u6807\u51fd\u6570\u503c\u4e0d\u518d\u4e0b\u964d, \u8bf4\u660e\u6211\u4eec\u5df2\u7ecf\u8d70\u5230\u4e86\u4e00\u4e2a\u6781\u503c\u70b9\u9644\u8fd1, \u7ec8\u6b62\u8fed\u4ee3. \u5bf9\u4e00\u5143\u7ebf\u6027\u56de\u5f52, \u68af\u5ea6\u65b9\u5411\u662f\u5bf9\u51fd\u6570\u6c42\u504f\u5bfc\u5f97\u5230\u7684\u5411\u91cf\u65b9\u5411.<\/p>\n

\u53e6\u5916\u9700\u8981\u6ce8\u610f\u7684\u662f, \u68af\u5ea6\u4e0b\u964d\u4e0d\u4e00\u5b9a\u80fd\u627e\u5230\u6700\u4f18\u89e3, \u53ef\u80fd\u4f1a\u5728\u67d0\u4e2a\u5c40\u90e8\u6700\u4f18\u89e3\u90a3\u9677\u8fdb\u53bb\u5c31\u51fa\u4e0d\u6765\u4e86.<\/p>\n

\u8fd9\u90e8\u5206\u66f4\u8be6\u7ec6\u7684\u63a8\u5bfc\u8bf7\u89c1\u53c2\u8003\u8d44\u6599\u91cc “\u673a\u5668\u5b66\u4e60\u4e2d\u7684\u6570\u5b66(1)” \u4e00\u7bc7, \u91cc\u9762\u7684\u516c\u5f0f\u548c\u56fe\u505a\u7684\u5f88\u8d5e, \u601d\u8def\u4e5f\u6bd4\u6211\u6e05\u6670.<\/p>\n

\u725b\u987f\u6cd5<\/strong><\/p>\n

\u68af\u5ea6\u4e0b\u964d\u4e00\u822c\u9047\u5230\u7684\u95ee\u9898\u8fed\u4ee3\u6b65\u957f\u4e0d\u597d\u9009, \u9009\u592a\u5c0f\u5230\u6781\u503c\u70b9\u592a\u6162, \u641e\u592a\u5927\u53c8\u4f1a\u5728\u6781\u503c\u70b9\u9644\u8fd1\u65f6\u56e0\u4e3a\u6b65\u957f\u592a\u5927\u8df3\u8fc7\u53bb\u4e86.<\/p>\n

\u725b\u987f\u6cd5\u6700\u5927\u7684\u8d21\u732e\u5c31\u662f\u540c\u65f6\u7ed9\u51fa\u4e86\u68af\u5ea6\u65b9\u5411\u548c\u8fed\u4ee3\u6b65\u957f, \u51e0\u4e4e\u662f\u4e00\u6b65\u5230\u4f4d\u7684\u6c42\u89e3. \u65b9\u6cd5\u540c\u89e3\u65b9\u7a0b\u4e00\u6837, \u5bf9\u65b0\u7684\u635f\u5931\u76ee\u6807\u51fd\u6570\u6c42\u89e3, \u53ea\u662f\u4e00\u6b21\u89e3\u53ef\u80fd\u8fd8\u4e0d\u591f\u597d, \u9700\u8981\u591a\u505a\u51e0\u6b21\u8fed\u4ee3. \u4e00\u822c\u68af\u5ea6\u4e0b\u964d\u53ef\u80fd\u9700\u8981\u4e0a\u5343\u8f6e\u7684\u8fed\u4ee3, \u800c\u725b\u987f\u6cd5\u51e0\u6b21\u8fed\u4ee3\u5c31\u80fd\u5230\u6781\u503c\u70b9\u4e86.<\/p>\n

\u6700\u5c0f\u4e8c\u4e58\u6cd5<\/strong><\/p>\n

\u4f1f\u5927\u7684\u9ad8\u65af\u540c\u5b66\u63d0\u51fa\u5e76\u8bc1\u660e\u4e86\u6700\u5c0f\u4e8c\u4e58\u6cd5\u662f\u6700\u597d\u89e3\u7b54, \u8bc1\u660e\u8fc7\u7a0b\u7565… \u76f4\u63a5\u770b\u7ef4\u57fa\u6216\u767e\u5ea6\u767e\u79d1\u4e0a\u7684\u539f\u6587\u5427 (\u6570\u5b66\u4e0d\u597d\u4f24\u4e0d\u8d77).<\/p>\n

\u5e94\u7528<\/strong><\/p>\n

\u867d\u7136\u8fd9\u4e2a\u65b9\u6cd5\u770b\u8d77\u6765\u5f88\u7b80\u5355\u7c97\u66b4, \u4f46\u662f\u5f88\u591a\u65f6\u5019\u53d8\u5316\u786e\u5b9e\u5c31\u662f\u7ebf\u6027\u7684. \u6bd4\u5982\u5728\u5f88\u591a\u8bba\u6587\u548c\u5de5\u4e1a\u5b9e\u8df5\u4e2d, \u5927\u5bb6\u8ba4\u4e3a\u540c\u7b49\u8d28\u91cf\u7684\u5e7f\u544a\u6216\u641c\u7d22\u7ed3\u679c, \u653e\u5728\u4ece\u4e0a\u5230\u4e0b\u4e0d\u540c\u7684\u4f4d\u7f6e\u4e0a, \u5176\u70b9\u51fb\u7387\u548c\u4f4d\u7f6e\u7684\u5173\u7cfb\u7b26\u5408\u7ebf\u6027\u5173\u7cfb, \u5373 ctr(rank) = a*rank + b.<\/p>\n

\u5728\u516d\u6708\u7684\u4e00\u6b21\u968f\u7b14\u6742\u8bb0<\/a>\u91cc, \u63d0\u5230\u4e86\u8fd9\u6837\u7684\u95ee\u9898:<\/p>\n

\u5982\u4e0b\u5f0f\u5b50\u91cc\u4e0d\u540c\u7684\u963f\u62c9\u4f2f\u6570\u5b57\u53ea\u662f\u4e00\u4e2a\u7b26\u53f7, \u5b9e\u9645\u8868\u793a\u7684\u53ef\u80fd\u662f\u5176\u4ed6\u6570\u5b57
\n967621 = 3
\n797321 = 1
\n378581 = 4
\n422151 = 0
\n535951 = 1
\n335771 = 0<\/p>\n

\u6839\u636e\u4e0a\u8ff0\u5f0f\u5b50, \u5224\u65ad\u4e0b\u5f0f\u7b49\u4e8e?
\n565441 = ?<\/p><\/blockquote>\n

\u5047\u8bbe\u6bcf\u4e2a\u5f0f\u5b50\u6700\u540e\u505a\u7684\u90fd\u662f\u52a0\u6cd5, \u5e76\u628a\u5b57\u7b26 0~9 \u6620\u5c04\u5230 x0<\/sub>~x9<\/sub>, \u5219\u7edf\u8ba1\u4e0d\u540c\u5b57\u6bcd\u51fa\u73b0\u7684\u6b21\u6570\u5c31\u53ef\u4ee5\u5217\u7ebf\u6027\u8fd4\u7a0b, \u53ef\u4ee5\u5c06\u7b2c\u4e00\u4e2a\u5f0f\u5b50\u8868\u793a\u4e3a<\/p>\n

x0<\/sub>*0 + x1<\/sub>*1 + x2<\/sub>*1 + x3<\/sub>*0 + x4<\/sub>*0 + x5<\/sub>*0 + x6<\/sub>*2 + x7<\/sub>*1 + x8<\/sub>*0 + x9<\/sub>*1 = 3<\/p><\/blockquote>\n

\u5176\u4ed6\u7c7b\u63a8. \u5bf9\u8fd9\u5806\u5f0f\u5b50\u6c42\u89e3\u5c31\u53ef\u4ee5\u5f97\u5230\u4e0d\u540c\u6570\u5b57\u5bf9\u5e94\u7684\u771f\u5b9e\u6570\u503c, \u53ef\u4ee5\u5f97\u5230 565441 = 1. \/\/ \u5177\u4f53\u4ee3\u7801\u548c\u65b9\u6cd5\u4e0b\u6b21\u7ed9\u51fa<\/p>\n

\u53c2\u8003\u8d44\u6599<\/strong><\/p>\n

* Wiki Least squares: http:\/\/en.wikipedia.org\/wiki\/Least_squares<\/a>
\n* Wiki Mean Squared Error: http:\/\/en.wikipedia.org\/wiki\/Mean_squared_error<\/a>
\n* \u4e2d\u6587\u7ef4\u57fa \u6700\u5c0f\u4e8c\u4e58\u6cd5: http:\/\/zh.wikipedia.org\/wiki\/%E6%9C%80%E5%B0%8F%E4%BA%8C%E4%B9%98%E6%B3%95<\/a>
\n* \u767e\u5ea6\u767e\u79d1 \u7ebf\u6027\u56de\u5f52: http:\/\/baike.baidu.com\/view\/449540.htm<\/a>
\n* \u767e\u5ea6\u767e\u79d1 \u6700\u5c0f\u4e8c\u4e58\u6cd5: http:\/\/baike.baidu.com\/view\/139822.htm<\/a>
\n* \u4eba\u4eba\u4e0a\u7684\u65e5\u5fd7 \u5e7c\u513f\u56ed\u7684\u9898\u76ee\u548c\u673a\u5668\u5b66\u4e60\u7684\u5173\u7cfb: http:\/\/blog.renren.com\/share\/30314\/13432269197<\/a>
\n* \u673a\u5668\u5b66\u4e60\u4e2d\u7684\u6570\u5b66(1): http:\/\/www.cnblogs.com\/LeftNotEasy\/archive\/2010\/12\/05\/mathmatic_in_machine_learning_1_regression_and_gradient_descent.html<\/a><\/p>\n

\/\/ \u672c\u6587\u5f00\u5199\u4e8e 9.24, \u62d6\u5230 10.26 \u624d\u9a6c\u9a6c\u864e\u864e\u5b8c\u6210, \u6700\u540e\u90a3\u4e2a\u9898\u7684\u89e3\u4e5f\u6ca1\u5199, \u5404\u79cd\u9519\u4e71\u540e\u7eed\u518d\u4fee\u6539\u6216\u8865\u5145\u5427<\/p>\n","protected":false},"excerpt":{"rendered":"

\u597d\u51e0\u4e2a\u6708\u6ca1\u518d\u7ee7\u7eed, \u6316\u5751\u4e0d\u586b\u662f\u4e0d\u5bf9\u7684, \u8fd8\u662f\u7ee7\u7eed\u5199\u624b\u8bb0. \u7ebf\u6027\u56de\u5f52 \u7ebf\u6027\u56de\u5f52\u4e00\u822c\u7528\u6765\u5b66\u4e60\u4e00\u7ef4\u81ea\u53d8\u91cf\u548c\u4e00\u7ef4\u56e0\u53d8\u91cf […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"jetpack_post_was_ever_published":false,"_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":false,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[5,7,9],"tags":[306,310,323,386],"class_list":["post-1653","post","type-post","status-publish","format-standard","hentry","category-work-life","category-tech-notes","category-machine-learning","tag-306","tag-310","tag-323","tag-386"],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"jetpack_shortlink":"https:\/\/wp.me\/p4aR5e-qF","_links":{"self":[{"href":"https:\/\/www.yewen.us\/blog\/wp-json\/wp\/v2\/posts\/1653","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.yewen.us\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.yewen.us\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.yewen.us\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.yewen.us\/blog\/wp-json\/wp\/v2\/comments?post=1653"}],"version-history":[{"count":0,"href":"https:\/\/www.yewen.us\/blog\/wp-json\/wp\/v2\/posts\/1653\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.yewen.us\/blog\/wp-json\/wp\/v2\/media?parent=1653"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.yewen.us\/blog\/wp-json\/wp\/v2\/categories?post=1653"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.yewen.us\/blog\/wp-json\/wp\/v2\/tags?post=1653"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}