{"id":1301,"date":"2023-04-18T22:47:13","date_gmt":"2023-04-18T14:47:13","guid":{"rendered":"http:\/\/www.wayln.com\/?p=1301"},"modified":"2023-04-18T22:47:13","modified_gmt":"2023-04-18T14:47:13","slug":"%e4%b8%bb%e6%88%90%e5%88%86%e5%88%86%e6%9e%90","status":"publish","type":"post","link":"https:\/\/www.wayln.com\/?p=1301","title":{"rendered":"\u4e3b\u6210\u5206\u5206\u6790"},"content":{"rendered":"<p>\u9996\u5148\uff0cPCA\u662f\u4e00\u79cd\u5e38\u7528\u7684\u6570\u636e\u964d\u7ef4\u6280\u672f\uff0c\u5176\u76ee\u7684\u662f\u4ece\u539f\u59cb\u6570\u636e\u4e2d\u63d0\u53d6\u51fa\u6700\u91cd\u8981\u7684\u4fe1\u606f\uff0c\u540c\u65f6\u51cf\u5c11\u6570\u636e\u7684\u7ef4\u5ea6\u3002PCA\u5c06\u539f\u59cb\u6570\u636e\u901a\u8fc7\u7ebf\u6027\u53d8\u6362\u8f6c\u6362\u4e3a\u65b0\u7684\u53d8\u91cf\uff0c\u4f7f\u5f97\u65b0\u7684\u53d8\u91cf\uff08\u4e5f\u79f0\u4e3a\u4e3b\u6210\u5206\uff09\u5177\u6709\u6700\u5927\u7684\u65b9\u5dee\u3002\u4e3b\u6210\u5206\u7684\u65b9\u5dee\u53cd\u6620\u4e86\u6570\u636e\u7684\u91cd\u8981\u6027\uff0c\u56e0\u6b64\u6211\u4eec\u53ef\u4ee5\u901a\u8fc7\u9009\u62e9\u524d\u51e0\u4e2a\u4e3b\u6210\u5206\u6765\u8868\u793a\u6570\u636e\u7684\u4e3b\u8981\u7279\u5f81\u3002<\/p>\n<p>\u4e0b\u9762\uff0c\u6211\u5c06\u4ecb\u7ecdPCA\u7684\u63a8\u5bfc\u8fc7\u7a0b\u3002<\/p>\n<p>\u5047\u8bbe\u6211\u4eec\u6709\u4e00\u4e2a <span class=\"katex math inline\">m<\/span> \u7ef4\u7684\u6570\u636e\u96c6 <span class=\"katex math inline\">X={x_1, x_2, &#8230;, x_n}<\/span>\uff0c\u5176\u4e2d\u6bcf\u4e2a <span class=\"katex math inline\">x_i<\/span> \u662f\u4e00\u4e2a <span class=\"katex math inline\">m<\/span> \u7ef4\u5411\u91cf\u3002\u6211\u4eec\u7684\u76ee\u6807\u662f\u901a\u8fc7\u7ebf\u6027\u53d8\u6362\u5c06 <span class=\"katex math inline\">X<\/span> \u8f6c\u6362\u4e3a\u4e00\u4e2a <span class=\"katex math inline\">k<\/span> \u7ef4\u7684\u65b0\u6570\u636e\u96c6 <span class=\"katex math inline\">Y={y_1, y_2, &#8230;, y_n}<\/span>\uff0c\u5176\u4e2d\u6bcf\u4e2a <span class=\"katex math inline\">y_i<\/span> \u662f\u4e00\u4e2a <span class=\"katex math inline\">k<\/span> \u7ef4\u5411\u91cf\uff0c<span class=\"katex math inline\">k&lt;m<\/span>\u3002\u8fd9\u4e2a\u7ebf\u6027\u53d8\u6362\u53ef\u4ee5\u8868\u793a\u4e3a\uff1a<\/p>\n<p>Y=PX<\/p>\n<p>\u5176\u4e2d <span class=\"katex math inline\">P<\/span> \u662f\u4e00\u4e2a <span class=\"katex math inline\">m \\times k<\/span> \u7684\u77e9\u9635\uff0c\u5b83\u5c06\u539f\u59cb\u6570\u636e <span class=\"katex math inline\">X<\/span> \u8f6c\u6362\u4e3a\u65b0\u7684\u6570\u636e <span class=\"katex math inline\">Y<\/span>\u3002<\/p>\n<p>\u6211\u4eec\u5e0c\u671b\u9009\u62e9\u4e00\u4e2a <span class=\"katex math inline\">P<\/span>\uff0c\u4f7f\u5f97\u65b0\u7684\u6570\u636e\u96c6 <span class=\"katex math inline\">Y<\/span> \u5177\u6709\u6700\u5927\u7684\u65b9\u5dee\uff0c\u56e0\u4e3a\u65b9\u5dee\u53cd\u6620\u4e86\u6570\u636e\u7684\u91cd\u8981\u6027\u3002\u56e0\u6b64\uff0c\u6211\u4eec\u9700\u8981\u627e\u5230\u4e00\u4e2a <span class=\"katex math inline\">P<\/span>\uff0c\u4f7f\u5f97 <span class=\"katex math inline\">Y<\/span> \u7684\u534f\u65b9\u5dee\u77e9\u9635 <span class=\"katex math inline\">C_Y<\/span> \u7684\u4e3b\u5bf9\u89d2\u7ebf\u4e0a\u7684\u5143\u7d20\u6700\u5927\u3002<\/p>\n<p>\u4e3a\u4e86\u6c42\u89e3 <span class=\"katex math inline\">P<\/span>\uff0c\u6211\u4eec\u53ef\u4ee5\u4f7f\u7528\u7279\u5f81\u503c\u5206\u89e3\uff08Eigenvalue Decomposition\uff09\u6765\u6c42\u89e3\u534f\u65b9\u5dee\u77e9\u9635 <span class=\"katex math inline\">C_X<\/span> \u7684\u7279\u5f81\u5411\u91cf\u548c\u7279\u5f81\u503c\u3002\u5177\u4f53\u6765\u8bf4\uff0c\u6211\u4eec\u53ef\u4ee5\u6309\u7167\u4ee5\u4e0b\u6b65\u9aa4\u6c42\u89e3 <span class=\"katex math inline\">P<\/span>\uff1a<\/p>\n<p>\u5c06\u539f\u59cb\u6570\u636e <span class=\"katex math inline\">X<\/span> \u8fdb\u884c\u4e2d\u5fc3\u5316\u5904\u7406\uff0c\u5373\u5bf9\u6bcf\u4e2a\u7279\u5f81\u6c42\u5747\u503c\u5e76\u5c06\u6bcf\u4e2a\u7279\u5f81\u7684\u503c\u51cf\u53bb\u5747\u503c\uff0c\u5f97\u5230\u4e2d\u5fc3\u5316\u540e\u7684\u6570\u636e <span class=\"katex math inline\">X_c<\/span>\u3002<br \/>\n\u8ba1\u7b97\u4e2d\u5fc3\u5316\u540e\u7684\u6570\u636e\u7684\u534f\u65b9\u5dee\u77e9\u9635 <span class=\"katex math inline\">C_X=\\frac{1}{n-1}X_c^TX_c<\/span>\u3002<br \/>\n\u5bf9\u534f\u65b9\u5dee\u77e9\u9635 <span class=\"katex math inline\">C_X<\/span> \u8fdb\u884c\u7279\u5f81\u503c\u5206\u89e3\uff0c\u5f97\u5230\u7279\u5f81\u5411\u91cf <span class=\"katex math inline\">V<\/span> \u548c\u7279\u5f81\u503c <span class=\"katex math inline\">\\lambda<\/span>\u3002<br \/>\n\u9009\u62e9\u524d <span class=\"katex math inline\">k<\/span> \u4e2a\u7279\u5f81\u503c\u5bf9\u5e94\u7684\u7279\u5f81\u5411\u91cf\u7ec4\u6210\u77e9\u9635 <span class=\"katex math inline\">P<\/span>\u3002<br \/>\n\u9700\u8981\u6ce8\u610f\u7684\u662f\uff0c\u7279\u5f81\u5411\u91cf\u662f\u6309\u7167\u5176\u5bf9\u5e94\u7684\u7279\u5f81\u503c\u5927\u5c0f\u4ece\u5927\u5230\u5c0f\u6392\u5e8f\u7684\uff0c\u56e0\u6b64\u524d <span class=\"katex math inline\">k<\/span> \u4e2a\u7279\u5f81\u5411\u91cf\u5bf9\u5e94\u7684\u7279\u5f81\u503c\u662f\u534f\u65b9\u5dee\u77e9\u9635 <span class=\"katex math inline\">C_X<\/span> \u7684\u524d <span class=\"katex math inline\">k<\/span> \u5927\u7279\u5f81\u503c\u3002<\/p>\n<p>\u901a\u8fc7\u4e0a\u8ff0\u6b65\u9aa4\uff0c\u6211\u4eec\u5c31\u53ef\u4ee5\u5f97\u5230\u4e00\u4e2a <span class=\"katex math inline\">m \\times k<\/span> \u7684\u77e9\u9635 <span class=\"katex math inline\">P<\/span>\uff0c\u5b83\u5c06\u539f\u59cb\u6570\u636e <span class=\"katex math inline\">X<\/span> \u8f6c\u6362\u4e3a\u4e00\u4e2a <span class=\"katex math inline\">k<\/span> \u7ef4\u7684\u65b0\u6570\u636e\u96c6 <span class=\"katex math inline\">Y<\/span>\u3002\u5176\u4e2d\uff0c<span class=\"katex math inline\">P<\/span> \u7684\u5217\u5411\u91cf\u662f\u534f\u65b9\u5dee\u77e9\u9635 <span class=\"katex math inline\">C_X<\/span> \u7684\u7279\u5f81\u5411\u91cf\uff0c<span class=\"katex math inline\">k<\/span> \u662f\u6211\u4eec\u9009\u62e9\u7684\u65b0\u6570\u636e\u96c6\u7684\u7ef4\u5ea6\u3002\u8fd9\u4e9b\u7279\u5f81\u5411\u91cf\u662f\u7ecf\u8fc7\u5f52\u4e00\u5316\u5904\u7406\u7684\uff0c\u4f7f\u5f97\u5b83\u4eec\u7684\u957f\u5ea6\u4e3a <span class=\"katex math inline\">1<\/span>\u3002<\/p>\n<p>\u5f53\u6211\u4eec\u5c06\u539f\u59cb\u6570\u636e <span class=\"katex math inline\">X<\/span> \u6620\u5c04\u5230\u65b0\u7684\u6570\u636e\u96c6 <span class=\"katex math inline\">Y<\/span> \u4e2d\u65f6\uff0c\u53ef\u4ee5\u5c06 <span class=\"katex math inline\">X<\/span> \u4e2d\u6bcf\u4e2a\u6570\u636e\u70b9 <span class=\"katex math inline\">x_i<\/span> \u5206\u522b\u8f6c\u6362\u4e3a <span class=\"katex math inline\">Y<\/span> \u4e2d\u7684\u5411\u91cf <span class=\"katex math inline\">y_i<\/span>\u3002\u53ef\u4ee5\u5c06 <span class=\"katex math inline\">y_i<\/span> \u770b\u4f5c\u662f <span class=\"katex math inline\">x_i<\/span> \u5728\u65b0\u7684\u5750\u6807\u7cfb\u4e0b\u7684\u6295\u5f71\uff0c\u5176\u4e2d\u5750\u6807\u7cfb\u7684\u57fa\u5411\u91cf\u662f <span class=\"katex math inline\">P<\/span> \u7684\u5217\u5411\u91cf\u3002\u56e0\u6b64\uff0c\u6211\u4eec\u53ef\u4ee5\u4f7f\u7528 <span class=\"katex math inline\">Y<\/span> \u4e2d\u7684\u6570\u636e\u70b9 <span class=\"katex math inline\">y_i<\/span> \u6765\u4ee3\u66ff\u539f\u59cb\u6570\u636e\u70b9 <span class=\"katex math inline\">x_i<\/span>\uff0c\u4ece\u800c\u8fbe\u5230\u964d\u7ef4\u7684\u76ee\u7684\u3002<\/p>\n<p>\u603b\u7684\u6765\u8bf4\uff0c\u4e3b\u6210\u5206\u5206\u6790\u53ef\u4ee5\u5c06\u9ad8\u7ef4\u6570\u636e\u964d\u7ef4\u5230\u4f4e\u7ef4\u7a7a\u95f4\uff0c\u540c\u65f6\u4fdd\u7559\u4e86\u539f\u59cb\u6570\u636e\u7684\u4e3b\u8981\u7279\u5f81\u3002\u8fd9\u4f7f\u5f97\u6211\u4eec\u53ef\u4ee5\u66f4\u597d\u5730\u7406\u89e3\u548c\u53ef\u89c6\u5316\u6570\u636e\uff0c\u5e76\u66f4\u5bb9\u6613\u5730\u8fdb\u884c\u540e\u7eed\u5206\u6790\u3002<\/p>\n","protected":false},"excerpt":{"rendered":"<p>\u9996\u5148\uff0cPCA\u662f\u4e00\u79cd\u5e38\u7528\u7684\u6570\u636e\u964d\u7ef4\u6280\u672f\uff0c\u5176\u76ee\u7684\u662f\u4ece\u539f\u59cb\u6570\u636e\u4e2d\u63d0\u53d6\u51fa\u6700\u91cd\u8981\u7684\u4fe1\u606f\uff0c\u540c\u65f6\u51cf\u5c11\u6570\u636e\u7684\u7ef4\u5ea6\u3002PCA\u5c06\u539f\u59cb [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[21,53,17],"tags":[],"class_list":["post-1301","post","type-post","status-publish","format-standard","hentry","category-21","category-53","category-17"],"_links":{"self":[{"href":"https:\/\/www.wayln.com\/index.php?rest_route=\/wp\/v2\/posts\/1301","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.wayln.com\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.wayln.com\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.wayln.com\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.wayln.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=1301"}],"version-history":[{"count":2,"href":"https:\/\/www.wayln.com\/index.php?rest_route=\/wp\/v2\/posts\/1301\/revisions"}],"predecessor-version":[{"id":1303,"href":"https:\/\/www.wayln.com\/index.php?rest_route=\/wp\/v2\/posts\/1301\/revisions\/1303"}],"wp:attachment":[{"href":"https:\/\/www.wayln.com\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=1301"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.wayln.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=1301"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.wayln.com\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=1301"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}