{"id":748,"date":"2013-06-22T05:46:12","date_gmt":"2013-06-21T21:46:12","guid":{"rendered":"http:\/\/www.shuizilong.com\/house\/?p=748"},"modified":"2013-09-03T14:59:35","modified_gmt":"2013-09-03T06:59:35","slug":"kth-farthest-point-in-unit-sphere","status":"publish","type":"post","link":"https:\/\/www.shuizilong.com\/house\/archives\/kth-farthest-point-in-unit-sphere\/","title":{"rendered":"\u5706\u5185 k \u8fdc\u70b9"},"content":{"rendered":"

Brief description: <\/h3>\n

…. \u7ed9\u5b9a\u4e00\u4e2a $$p$$ \u7ef4\u7684\u5355\u4f4d\u5706\uff0c\u968f\u673a\u5728\u91cc\u9762\u70b9 $$n$$ \u4e2a\u70b9\u3002\u3002\u3002
\n\u3002\u3002\u6c42\u7b2c $$k$$ \u8fdc\u70b9\u7684\u8ddd\u79bb\u7684\u671f\u671b\u3002
\n<\/p>\n

Background: <\/h3>\n

\u3002\u3002\u3002\u6574\u7406\u4e86\u4e00\u4e0b\u804a\u5929\u8bb0\u5f55\u3002\u3002\u53d1\u73b0\u8fd9\u4e2a\u95ee\u9898\u771f\u662f Nice !\u3002\u3002\u3002\u771f\u662f\u6709\u52a9\u4e8e\u63d0\u9ad8\u667a\u5546\u3002\u3002\u3002\u3002\uff09\u3002
\n\u6700\u65e9\u662f Diy \u7fa4\u91cc\u7684 __Sisyphus<\/a> \u795e\u725b\u63d0\u51fa\u7684\u95ee\u9898\u3002\u3002\u3002\u6211\u611f\u89c9\u6709\u70b9\u610f\u601d\u3002\u3002
\n\u5c31\u8ffd\u95ee\u4e86\u4e00\u4e0b\u6765\u6e90\u3002\u3002\u3002\u662f The Elements of Statistical Learning<\/em><\/a> \u4e66\u7b2c 23 \u9875\u7684\u4e00\u4e2a\u6817\u5b50\u91cc\u8f7b\u63cf\u6de1\u5199\u63d0\u51fa\u7684\u4e00\u4e2a\u7ed3\u8bba\u3002\u3002\u3002\u3002\u3002<\/p>\n

\u3002\u539f\u95ee\u9898\u662f\u3002\u3002
\n“The median distance from the origin to the closest data point … ” \uff08\u95ee\u6700\u8fd1\u70b9\u7684\u4e2d\u4f4d\u6570\u3002\u3002\u3002\uff08\u8fd9\u4e2a\u5e26\u5230\u5206\u5e03\u51fd\u6570\u91cc =1\/2 \u53cd\u89e3\u51fa\u6765\u5c31\u884c\u3002\u3002\uff09
\n\u3002\u3002\u4e0d\u8fc7\u8fdb\u4e00\u6b65\u8003\u8651\u8fd9\u4e2a\u4e1c\u897f\u7684\u671f\u671b\u7684\u8bdd\u786e\u5b9e\u6709\u610f\u601d\u591a\u4e86\u3002\u3002\u3002<\/p>\n

Analysis: <\/h3>\n

.\u6211\u4eec\u5bf9\u70b9\u6309\u7167\u5230\u539f\u70b9\u7684\u8ddd\u79bb\u4ece\u5927\u5230\u5c0f\u6392\u5e8f\u3002\u3002\u3002\u8bbe $$d_k$$ \u8868\u793a $$k$$ \u8fdc\u70b9\u7684\u8ddd\u79bb\u3002\u3002( \u4ece 1 \u6807\u53f7\u3002\u3002
\n\u3002\u3002\u9996\u5148\u6700\u8fdc\u70b9\u7684\u671f\u671b\u975e\u5e38\u597d\u6c42\u3002\u3002\u3002<\/p>\n

\u3002\u3002.\u8bbe $$V^n(r)$$ \u4e3a\u534a\u5f84\u4e3a $$r$$ \u7684 $$n$$ \u7ef4\u8d85\u7403\u7684\u4f53\u79ef\u3002\u3002.\u8003\u8651 $$d_1$$ \u7684\u5206\u5e03\u51fd\u6570\u3002\u3002\u3002\u3002\u663e\u7136\u5c31\u662f\u3002<\/p>\n

$$!P(d_1 \\leq x) = \\frac{V^n(x)}{V^n(1)} = x^{np} $$ <\/p>\n

\uff08\u8fd9\u4e00\u6b65\u9700\u8981\u7528\u4e24\u4e2a $$n$$ \u7ef4\u7403\u7684\u4f53\u79ef\u505a\u6bd4\u3002\u3002\u867d\u7136\u6211\u4eec\u4e0d\u77e5\u9053 $$n$$ \u7ef4\u7403\u4f53\u79ef\u516c\u5f0f<\/a>\u5177\u4f53\u662f\u5565\u3002\u3002\u4f46\u662f\u6bd4\u503c\u663e\u7136\u5c31\u662f\u8fd9\u4e2a blabla \u3002\u3002<\/p>\n

\u5fae\u5206\u5f97\u5230\u5bc6\u5ea6\u51fd\u6570\u3002
\n$$!P(d_1 = x) = npx^{np-1} $$<\/p>\n

\u4e58\u4ee5 $$x$$ \u518d\u79ef\u5206\u5f97\u5230\u671f\u671b\u3002\u3002\u3002
\n$$!E(d_1) = \\int_{0}^{1} xP(d_1 = x) \\mathrm{d}x = \\int_{0}^{1} npx^{np} \\mathrm{d}x = \\frac{np}{np+1} $$<\/p>\n

\u7136\u540e\u7b2c $$k$$ \u8fdc\u7684\u671f\u671b\u5c31\u662f\u3002\u3002
\n$$!E(d_k) = \\prod_{i=n-k}^n \\frac{ip}{ip+1}$$<\/p>\n

<\/br>
\n<\/br>
\n<\/br>
\n<\/br>
\n<\/br>
\n<\/br>
\n<\/br>
\n<\/br><\/p>\n

So why ?\u3002\u3002\u3002<\/p>\n

\u3002\u3002\u3002\u6211\u4eec\u8003\u8651\u7b2c $$k$$ \u8fdc\u70b9\u8ddd\u79bb\u7684\u671f\u671b\u5df2\u7ecf\u5f97\u5230\u4e86\u3002\u3002\u73b0\u5728\u6765\u6c42 $$k+1$$ \u8fdc\u7684\u3002\u3002
\n\u90a3\u4e48\u5c31\u76f8\u5f53\u4e8e\u5728\u4e00\u4e2a\u534a\u5f84\u4e3a\u7b2c $$k$$ \u8fdc\u671f\u671b\u7684\u8d85\u7403\u5185\u90e8\u968f\u673a\u53d6 $$n-k$$ \u4e2a\u70b9\u3002\u65f6\u7684\u6700\u8fdc\u70b9\u8ddd\u79bb\u7684\u671f\u671b\u3002\u3002\u3002\u3002
\n\u3002\u3002\u6240\u4ee5\u6700\u540e\u7ed3\u679c\u5c31\u662f\u4e00\u5806\u6700\u8fdc\u70b9\u7684\u671f\u671b\u4e58\u4ee5\u6765\u3002\u3002\u3002\u3002\u3002\u662f\u4e0d\u662f\u5f88\u597d\u4e3d\u6d01\uff1f\u3002\u3002\u3002\u3002<\/p>\n

External link: <\/h3>\n

http:\/\/roosephu.github.io\/2013\/06\/21\/kth-point\/<\/a>
\nhttp:\/\/acm.hdu.edu.cn\/showproblem.php?pid=4653<\/a><\/p>\n","protected":false},"excerpt":{"rendered":"

Brief description: …. \u7ed9\u5b9a\u4e00\u4e2a $$p$$ \u7ef4\u7684\u5355\u4f4d\u5706\uff0c\u968f\u673a\u5728\u91cc\u9762\u70b9 $$n$$ \u4e2a\u70b9\u3002\u3002\u3002 \u3002\u3002\u6c42\u7b2c $$k$$ \u8fdc\u70b9\u7684\u8ddd\u79bb\u7684\u671f\u671b\u3002<\/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":"","jetpack_publicize_message":"","jetpack_publicize_feature_enabled":true,"jetpack_social_post_already_shared":true,"jetpack_social_options":{"image_generator_settings":{"template":"highway","enabled":false}}},"categories":[122],"tags":[117],"class_list":["post-748","post","type-post","status-publish","format-standard","hentry","category-122","tag-117"],"jetpack_publicize_connections":[],"jetpack_featured_media_url":"","jetpack_shortlink":"https:\/\/wp.me\/p2tdP7-c4","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/www.shuizilong.com\/house\/wp-json\/wp\/v2\/posts\/748","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.shuizilong.com\/house\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.shuizilong.com\/house\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.shuizilong.com\/house\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.shuizilong.com\/house\/wp-json\/wp\/v2\/comments?post=748"}],"version-history":[{"count":1,"href":"https:\/\/www.shuizilong.com\/house\/wp-json\/wp\/v2\/posts\/748\/revisions"}],"predecessor-version":[{"id":749,"href":"https:\/\/www.shuizilong.com\/house\/wp-json\/wp\/v2\/posts\/748\/revisions\/749"}],"wp:attachment":[{"href":"https:\/\/www.shuizilong.com\/house\/wp-json\/wp\/v2\/media?parent=748"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.shuizilong.com\/house\/wp-json\/wp\/v2\/categories?post=748"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.shuizilong.com\/house\/wp-json\/wp\/v2\/tags?post=748"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}